BV Solutions to Hyperbolic Systems

by Vanishing Viscosity

Alberto Bressan

S.I.S.S.A., Trieste 34014, Italy.

1 - Introduction

Aim of these notes is to provide a self-contained presentation of recents results on hyperbolic

systems of conservation laws, based on the vanishing viscosity approach.

A system of conservation laws in one space dimension takes the form

ut + f(u)x = 0 . (1.1)

Here u = (u1, . . . , un) is the vector of conserved quantities while the components of f =

(f1, . . . , fn) are called the fluxes. Integrating (1.1) over a fixed interval [a, b] we find

d

dt

∫ b

a

u(t, x) dx =∫ b

a

ut(t, x) dx = −∫ b

a

f(u(t, x)

)x

dx

= f(u(t, a)

)− f(u(t, b)

)= [inflow at a]− [outflow at b] .

Each component of the vector u thus represents a quantity which is neither created nor destroyed:

its total amount inside any given interval [a, b] can change only because of the flow across boundary

points.

Systems of the form (1.1) are commonly used to express the fundamental balance laws of

continuum physics, when small viscosity or dissipation effects are neglected. For a comprehensive

1

discussion of conservation laws and their derivation from basic principles of physics we refer to the

book of Dafermos [D1].

Smooth solutions of (1.1) satisfy the equivalent quasilinear system

ut + A(u)ux = 0 , (1.2)

where A(u) .= Df(u) is the Jacobian matrix of first order partial derivatives of f . We notice,

however, that if u has a jump at some point x0, then the left hand side of (1.2) contains a product

of the discontinuous function x 7→ A(u(x)

)with the distributional derivative ux, which in this

case contains a Dirac mass at the point x0. In general, such a product is not well defined. The

quasilinear system (1.2) is thus meaningful only within a class of continuous functions. On the

other hand, working with the equation in divergence form (1.1) allows us to consider discontinuous

solutions as well, interpreted in distributional sense. We say that a locally integrable function

u = u(t, x) is a weak solution of (1.1) if t 7→ u(t, ·) is continuous as a map with values in L1loc

and moreover ∫ ∫ {uφt + f(u)φx

}dxdt = 0 (1.3)

for every differentiable function with compact support φ ∈ C1c .

The above system is called strictly hyperbolic if each matrix A(u) .= Df(u) has n real,

distinct eigenvalues λ1(u) < · · · < λn(u). One can then find dual bases of left and right eigenvectors

of A(u), denoted by l1(u), . . . , ln(u) and r1(u), . . . , rn(u), normalized according to

|ri| = 1 , li · rj ={

1 if i = j,0 if i 6= j. (1.4)

To appreciate the effect of the non-linearity, consider first the case of a linear system with

constant coefficients

ut + Aux = 0. (1.5)

Call λ1 < · · · < λn the eigenvalues of the matrix A, and let li, ri be the corresponding left and

right eigenvectors as in (1.4). The general solution of (1.5) can be written as a superposition of

independent linear waves:

u(t, x) =∑

i

φi(x− λit)ri , φi(y) .= li · u(0, y) .

Notice that here the solution is completely decoupled along the eigenspaces of A, and each com-

ponent travels with constant speed, given by the corresponding eigenvalue of A.

In the nonlinear case (1.2) where the matrix A depends on the state u, new features will

appear in the solutions.

2

x

u(t)u(0)

figure 1

(i) Since the eigenvalues λi now depend on u, the shape of the various components in the solution

will vary in time (fig. 1). Rarefaction waves will decay, and compression waves will become

steeper, possibly leading to shock formation in finite time.

(ii) Since the eigenvectors ri also depend on u, nontrivial interactions between different waves will

occur (fig. 2). The strength of the interacting waves may change, and new waves of different

families can be created, as a result of the interaction.

x

tnonlinearlinear

figure 2

The strong nonlinearity of the equations and the lack of regularity of solutions, also due to

the absence of second order terms that could provide a smoothing effect, account for most of the

difficulties encountered in a rigorous mathematical analysis of the system (1.1). It is well known

that the main techniques of abstract functional analysis do not apply in this context. Solutions

cannot be represented as fixed points of continuous transformations, or in variational form, as

critical points of suitable functionals. Dealing with vector valued functions, comparison principles

based on upper or lower solutions cannot be used. Moreover, the theory of accretive operators and

contractive nonlinear semigroups works well in the scalar case [C], but does not apply to systems.

3

For the above reasons, the theory of hyperbolic conservation laws has largely developed by ad hoc

methods, along two main lines.

1. The BV setting, considered by Glimm [G]. Solutions are here constructed within a space of

functions with bounded variation, controlling the BV norm by a wave interaction functional.

2. The L∞ setting, considered by Tartar and DiPerna [DP2], based on weak convergence and a

compensated compactness argument.

Both approaches yield results on the global existence of weak solutions. However, the method

of compensated compactness appears to be suitable only for 2× 2 systems. Moreover, it is only in

the BV setting that the well-posedness of the Cauchy problem could recently be proved, as well as

the stability and convergence of vanishing viscosity approximations. Throughout the following we

thus restrict ourselves to the study of BV solutions, referring to [DP2] or [Se] for the alternative

approach based on compensated compactness.

Since the pioneering work of Glimm, the basic building block toward the construction and the

analysis of more general solutions has been provided by the Riemann problem, i.e. the initial

value problem with piecewise constant data

u(0, x) = u(x) ={

u− if x < 0,u+ if x > 0.

(1.6)

This was first introduced by B.Riemann (1860) in the context of isentropic gas dynamics. A century

later, P.Lax [Lx] and T.P.Liu [L1] solved the Riemann problem for more general n × n systems.

The new approach of Bianchini [Bi] now applies to all strictly hyperbolic systems, not necessarily

in conservation form. Solutions are always found in the self-similar form u(t, x) = U(x/t). The

central position taken by the Riemann problem is related to a symmetry of the equations (1.1). If

u = u(t, x) is a solution of (1.1), then for any θ > 0 the function

uθ(t, x) .= u(θt, θx)

provides another solution. The solutions which are invariant under these rescalings of the indepen-

dent variables are precisely those which correspond to some Riemann data (1.6).

For a general Cauchy problem, both the Glimm scheme [G] and the method of front tracking

[D1], [DP1], [B1], [BaJ], [HR] yield approximate solutions of a general Cauchy problem by piecing

together a large number of Riemann solutions. For initial data with small total variation, this

approach is successful because one can provide a uniform a priori bound on the amount of new

waves produced by nonlinear interactions, and hence on the total variation of the solution. It is

safe to say that, in the context of weak solutions with small total variation, nearly all results on

4

the existence, uniqueness, continuous dependence and qualitative behavior have relied on a careful

analysis of the Riemann problem.

In [BiB] a substantially different perspective has emerged from the study of vanishing viscosity

approximations. Solutions of (1.1) are here obtained as limits for ε → 0 of solutions to the parabolic

problems

uεt + A(uε)uε

x = εuεxx (1.7)

with A(u) .= Df(u). This approach is very natural and has been considered since the 1950’s.

However, complete results had been obtained only in the scalar case [O], [K]. For general n × n

systems, the main difficulty lies in establishing the compactness of the approximating sequence.

We observe that uε(t, x) solves (1.7) if and only if uε(t, x) = u(t/ε, x/ε) for some function u which

satisfies

ut + A(u)ux = uxx . (1.8)

In the analysis of vanishing viscosity approximations, the key step is to derive a priori estimates

on the total variation and on the stability of solutions of (1.8). For this parabolic system, the

rescaling (t, x) 7→ (θt, θx) no longer determines a symmetry. Hence the Riemann data no longer

hold a privileged position. The role of basic building block is now taken by the viscous travelling

profiles, i.e. solutions of the form

u(t, x) = U(x− λt).

Of course, the function U must then satisfy the second order O.D.E.

U ′′ =(A(U)− λ

)U ′.

In this new approach, the profile u(·) of a viscous solution is viewed locally as a superposition of

viscous travelling waves. More precisely, let a smooth function u : IR 7→ IRn be given. At each

point x, looking at the third order jet (u, ux, uxx) we seek travelling profiles U1, . . . , Un such that

U ′′i =

(A(Ui)− σi

)U ′

i (1.9)

for some speed σi close to the characteristic speed λi, and moreover

Ui(x) = u(x) i = 1, . . . , n , (1.10)

∑

i

U ′i(x) = ux(x) ,

∑

i

U ′′i (x) = uxx(x) . (1.11)

It turns out that this decomposition is unique provided that the travelling profiles are chosen

within suitable center manifolds. We let ri be the unit vector parallel to U ′i , so that U ′

i = viri for

5

some scalar vi. One can show that ri remains close to the eigenvector ri(u) of the Jacobian matrix

A(u) .= Df(u), but ri 6= ri(u) in general. The first equation in (1.11) now yields the decomposition

ux =∑

i

viri . (1.12)

If u = u(t, x) is a solution of (1.8), we can think of vi as the density of i-waves in u. The remarkable

fact is that these components satisfy a system of evolution equations

vi,t + (λivi)x − vi,xx = φi i = 1, . . . , n , (1.13)

where the source terms φi on the right hand side are INTEGRABLE over the whole domain {x ∈IR , t > 0}. Indeed, we can think of the sources φi as new waves produced by interactions between

viscous waves. Their total strength is controlled by means of viscous interaction functionals,

somewhat similar to the one introduced by Glimm in [G] to study the hyperbolic case. Since the

left hand side of (1.13) is in conservation form and the vectors ri have unit length, for an arbitrarily

large time t we obtain the bound

‖ux(t)∥∥L1 ≤

∑

i

∥∥vi(t)∥∥L1 ≤

∑

i

(∥∥vi(t0)∥∥ +

∫ t

t0

∫ ∣∣φi(s, x)∣∣ dxds

). (1.14)

This argument yields global BV bounds and stability estimates for viscous solutions. In turn,

letting ε → 0 in (1.7), a standard compactness argument yields the convergence of uε to a weak

solution u of (1.1).

The plan of these notes is as follows. In Section 2 we briefly review the basic theory of hyper-

bolic systems of conservation laws: shock and rarefaction waves, entropies, the Liu admissibility

conditions and the Riemann problem. For initial data with small total variation, we also recall the

main results concerning the existence, uniqueness and stability of solutions to the general Cauchy

problem. Section 3 contains the statement of the main new results on vanishing viscosity limits and

an outline of the proof. In Section 4 we derive some preliminary estimates which can be obtained

by standard parabolic techniques, representing a viscous solution in terms of convolutions with a

heat kernel. Section 5 discusses in detail the local decomposition of a solution as superposition of

viscous travelling waves. The evolution equations (1.13) for the components vi and the strength

of the source terms φi are then studied in Section 6. This will provide the crucial estimate on

the total variation of viscous solutions, uniformly in time. In Section 7 we briefly examine the

stability and some other properties of viscous solutions. The existence, uniqueness and stability of

vanishing viscosity limits are then discussed in Section 8.

6

2 - Review of hyperbolic conservation laws

In most of this section, we shall consider a strictly hyperbolic system of conservation laws

satisfying the additional hypothesis

(H) For each i = 1, . . . , n, the i-th field is either genuinely nonlinear, so that Dλi(u) · ri(u) > 0

for all u, or linearly degenerate, with Dλi(u) · ri(u) = 0 for all u.

We observe that the i-th characteristic field is genuinely nonlinear iff the eigenvalue λi is

strictly increasing along each integral curve of the corresponding field of eigenvectors ri. It is

linearly degenerate when λi is constant along each such curve.

2.1. Centered Rarefaction Waves.

We begin by studying a special type of solutions, in the form of centered rarefaction waves.

Fix a state u0 ∈ IRn and an index i ∈ {1, . . . , n}. Let ri(u) be a field of i-eigenvectors of the

Jacobian matrix A(u) = Df(u). The integral curve of the vector field ri through the point u0 is

called the i-rarefaction curve through u0. It is obtained by solving the initial value problem for

the O.D.E. in state space:du

ds= ri(u), u(0) = u0. (2.1)

We shall denote this curve as

s 7→ Ri(s)(u0).

Clearly, the parametrization depends on the choice of the eigenvectors ri. In particular, if we impose

the normalization∣∣ri(u)

∣∣ ≡ 1, then the rarefaction curve Ri will be parametrized by arc-length.

Let the i-th field be genuinely nonlinear. Given two states u−, u+, assume that u+ lies on the

positive i-rarefaction curve through u−, i.e. u+ = Ri(σ)(u−) for some σ > 0. For each s ∈ [0, σ],

define

λi(s) = λi

(Ri(s)(u−)

).

By genuine nonlinearity, the map s 7→ λi(s) is strictly increasing. Hence, for every λ ∈[λi(u−), λi(u+)

], there is a unique value s ∈ [0, σ] such that λ = λi(s). We claim that, for

t > 0, the function

u(t, x) =

u− if x/t < λi(u−),Ri(s)(u−) if x/t = λi(s) ∈

[λi(u−), λi(u+)

],

u+ if x/t > λi(u+),(2.2)

is a continuous solution of the Riemann problem (1.1), (1.6). Indeed, by construction it follows

limt→0+

∥∥u(t, ·)− u∥∥L1 = 0.

7

Moreover, the equation (1.2) is trivially satisfied in the sectors where x/t < λi(u−) or x/t > λi(u+),

since here ut = ux = 0. Next, consider what happens in the intermediate sector{λi(u−) < x/t <

λi(u+)}. Since u is constant along each ray through the origin {x/t = c}, we have

ut(t, x) +x

tux(t, x) = 0.

Our construction at (2.2) now yields x/t = λi

(u(t, x)

). Moreover, ux is parallel to ri(u), hence it is

an eigenvector of the Jacobian matrix A(u) with eigenvalue λi(u). This implies ut + A(u)ux = 0 ,

proving our claim. Notice that the assumption σ > 0 is essential for the validity of this construction.

In the opposite case σ < 0, the definition (2.2) would yield a triple-valued function in the region

where x/t ∈ ]λi(u+), λi(u−)

[.

2.2. Shocks and Admissibility Conditions.

The simplest type of discontinuous solution of (1.1) is given by a shock :

U(t, x) ={

u+ if x > λt,u− if x < λt,

(2.3)

for some left and right states u−, u+ ∈ IRn and a speed λ ∈ IR. Using the divergence theorem,

one checks that the identity (1.2) is satisfied for all test functions φ if and only if the following

Rankine-Hugoniot conditions hold:

λ (u+ − u−) = f(u+)− f(u−) . (2.4)

As usual, we denote by A(u) = Df(u) the n × n Jacobian matrix of f at u. For any u, v,

consider the averaged matrix

A(u, v) .=∫ 1

0

A(θu + (1− θ)v

)dθ

and call λi(u, v), i = 1, . . . , n its eigenvalues. We can then write (2.4) in the equivalent form

λ (u+ − u−) =∫ 1

0

Df(θu+ + (1− θ)u−

) · (u+ − u−) dθ = A(u+, u−) · (u+ − u−). (2.5)

In other words, the Rankine-Hugoniot conditions hold if and only if the jump u+ − u− is an

eigenvector of the averaged matrix A(u+, u−) and the speed λ coincides with the corresponding

eigenvalue.

The representation (2.5) is very useful for studying the set of pairs (u−, u+) that can be

connected by a shock. Given a left state u− = u0, for each i = 1, . . . , n one can prove that there

8

u0

R i

iS

r (u )0i

figure 3

exists a one-parameter curve of right states u+ = Si(s)(u0) which satisfy the Rankine-Hugoniot

conditions (2.4) for a suitable speed λ = λi(s). This curve can also be parametrized by arc-length:

s 7→ Si(s)(u0).

It has a second order tangency with the corresponding rarefaction curve Ri at the point u0 (see

fig. 3).

To study the behavior of a general weak solution at a point of discontinuity, we first recall

some definitions.

A function u = u(t, x) has an approximate jump discontinuity at the point (τ, ξ) if there

exists vectors u+ 6= u− and a speed λ such that

limr→0+

1r2

∫ r

−r

∫ r

−r

∣∣u(τ + t, ξ + x)− U(t, x)∣∣ dxdt = 0 , (2.6)

with U as in (2.3). We say that u is approximately continuous at the point (τ, ξ) if the above

limit hold with u+ = u− (and λ arbitrary).

If u is now a solution of the system of conservation laws (1.1) having an approximate jump

at a point (τ, ξ), one can prove that the states u−, u+ and the speed λ again satisfy the Rankine-

Hugoniot conditions (see Theorem 4.1 in [B3]).

In order to achieve uniqueness of solutions to initial value problems, one needs to supplement

the conservation equations (1.1) with additional admissibility conditions, to be satisfied at points

of jump. We recall here two basic approaches.

A continuously differentiable function η : IRn 7→ IR is called an entropy for the system of

conservation laws (1.1), with entropy flux q : IRn 7→ IR, if it satisfies the identity

Dη(u) ·Df(u) = Dq(u). (2.7)

9

An immediate consequence of (2.7) is that, if u = u(t, x) is a C1 solution of (2.1), then

η(u)t + q(u)x = 0. (2.8)

For a smooth solution u, not only the quantities u1, . . . , un are conserved, but the additional

conservation law (2.8) holds as well. However one should be aware that, for a discontinuous

solution u, the quantity η(u) may not be conserved. A standard admissibility condition for weak

solutions can now be formulated as follows.

Let η be a convex entropy for the system (1.1), with entropy flux q. A weak solution u is

entropy-admissible if

η(u)t + q(u)x ≤ 0 (2.9)

in distribution sense, i.e. ∫∫ {η(u)ϕt + q(u)ϕx

}dxdt ≥ 0 (2.10)

for every function ϕ ≥ 0, continuously differentiable with compact support.

In analogy with (2.4), if u is an entropy admissible solution, at every point of approximate

jump one can show that (2.9) implies

λ[η(u+)− η(u−)

] ≥ q(u+)− q(u−). (2.11)

The above admissibility condition can be useful only if some nontrivial convex entropy exists.

This is not always the case. Indeed, according to (2.7), to construct an entropy we must solve a

system of n equations for the two scalar functions η, q. This system is overdetermined whenever

n ≥ 3. In continuum physics, however, one finds various systems of conservation laws endowed

with significant entropies.

An alternative admissibility condition, due to Liu [L2], is particularly useful because it can

be applied to any system, even in the absence of nontrivial entropies., To understand its meaning,

consider first a scalar conservation law. In this case, any two states u−, u+ satisfy the equations

(2.4), provided that we choose the speed

λ =f(u+)− f(u−)

u+ − u−. (2.12)

To test the stability of the shock (2.3) w.r.t. a small perturbation, fix any intermediate state

u∗ ∈ [u−, u+]. We can slightly perturb the initial data by splitting the initial jump in two smaller

jumps (fig. 4), say

u(0, x) =

{u− if x < 0,u∗ if 0 < x < ε,u+ if x > ε.

10

uu u

PP+

+

*

*

+u

*u

λ

λ+

−

−u

f(u)

x0 ε

f

−u−P

figure 4 figure 5

Clearly, the original shock can be stable only if the speed λ− of the jump behind is faster than

the speed λ+ of the jump ahead, i.e.

λ− .=f(u∗)− f(u−)

u∗ − u−≥ f(u+)− f(u∗)

u+ − u∗= λ+ . (2.13)

In fig. 5, the first speed corresponds to the slope of the segment P−P ∗ while the second speed is

the slope of P ∗P+. On the other hand, by (2.12) the speed λ of the original shock is the slope of

the segment P−P+. Therefore the inequality (2.13) holds if and only if

f(u∗)− f(u−)u∗ − u−

≥ f(u+)− f(u−)u+ − u−

. (2.14)

The condition (2.13) cannot be extended to the vector valued case, because in general one

cannot find intermediate states u∗ such that both couples (u−, u∗) and (u∗, u+) are connected by a

shock. However, the condition (2.14) has a straightforward generalization to n×n systems. Denote

by s 7→ Si(s)(u−) the i-shock curve through the point u−, and let λi(s) be the speed of the shock

connecting u− with Si(s)(u−). Now consider any right state

u+ = Si(σ)(u−)

on this shock curve. Following [L2], we say that the shock (2.3) satisfies the Liu admissibility

condition if

λi(s) ≥ λi(σ) for all s ∈ [0, σ] (2.15)

(or for all s ∈ [σ, 0], in case where σ < 0). In other words, if u∗ = Si(s)(u−) is any intermediate

state along the shock curve through u−, the speed of the shock (u−, u∗) should be larger or equal

to the speed of the original shock (u−, u+).

11

2.3. Solution of the Riemann Problem.

Let the assumptions (H) hold, so that each characteristic field is either genuinely nonlinear or

linearly degenerate. Relying on the previous analysis, the solution of the general Riemann problem

ut + f(u)x = 0 , u(0, x) ={

u− if x < 0,u+ if x > 0.

(2.16)

can be obtained by finding intermediate states ω0 = u−, ω1, . . . , ωn = u+ such that each pair of

adiacent states ωi−1, ωi can be connected by an i-shock, or by a centered rarefaction i-wave. By

the implicit function theorem, this can always be done provided that the two states u−, u+ are

sufficiently close. Following [Lx], the complete solution is now obtained by piecing together the

solutions of the n Riemann problems

ut + f(u)x = 0, u(0, x) ={

ωi−1 if x < 0,ωi if x > 0, (2.17)

on different sectors of the t-x plane. A typical situation is illustrated in fig. 6.

If the assumption (H) is removed, one can still solve the Riemann problem in terms of n + 1

constant states ω0, . . . , ωn. However, in this case each couple of states ωi−1, ωi can be connected

by a large number of adiacent shocks and rarefaction waves (fig. 7). The analysis of the Riemann

problem thus becomes far more complicated [L1].

ωu = u =

t

ω3

x

ω2ω1

0− +

shock

shockrarefaction

ω u = ω3+u =

0

2ω1ω

−

x

t

figure 6 figure 7

2.4. Glimm and front tracking approximations.

Approximate solutions for the general Cauchy problem can be constructed by patching to-

gether solutions of several Riemann problems. In the Glimm scheme (fig. 8), one works with a

fixed grid in the t-x plane, with mesh sizes ∆t, ∆x. At time t = 0 the initial data is approximated

12

by a piecewise constant function, with jumps at grid points. Solving the corresponding Riemann

problems, a solution is constructed up to a time ∆t sufficiently small so that waves emerging from

different nodes do not interact. At time t1.= ∆t, we replace the solution u(∆t, ·) by a piecewise

constant function having jumps exactly at grid points. Solving the new Riemann problems at

every one of these points, one can prolong the solution to the next time interval [∆t, 2∆t]. At

time t2.= 2∆t, the solution is again approximated by a piecewise constant functions with jumps

exactly at grid points, etc. . . A key ingredient of the Glimm scheme is the restarting procedure.

At each time tj.= j ∆t, a natural way to approximate a BV function with a piecewise constant

one is by taking its average value on each subinterval Ji.= [xi−1, xi]. However, this procedure may

generate an arbitrarily large amount of oscillations. Instead, the Glimm scheme is based on random

sampling: a point yi is selected at random inside each interval Ji and the old value u(tj−, yi) at

this particular point is taken as the new value of u(tj , x) for all x ∈ Ji. An excellent introduction

to the Glimm scheme can be found in the book by J. Smoller [Sm].

t x

t∆

∆

2

∆

x*

* **

* *

0 2 4

k

ky

k−1

∆xx

figure 8

An alternative technique for contructing approximate solutions is by wave-front tracking

(fig. 9). This method was introduced by Dafermos [D1] in the scalar case and later developed

by various authors [DP1], [B1], [BaJ]. It now provides an efficient tool in the study of general n×n

systems of conservation laws, both for theoretical and numerical purposes [B3], [HR].

The initial data is here approximated with a piecewise constant function, and each Riemann

problem is solved approximately, within the class of piecewise constant functions. In particular, if

the exact solution contains a centered rarefaction, this must be approximated by a rarefaction fan,

containing several small jumps. At the first time t1 where two fronts interact, the new Riemann

problem is again approximately solved by a piecewise constant function. The solution is then

prolonged up to the second interaction time t2, where the new Riemann problem is solved, etc. . .

13

Comparing this method with the previous one of Glimm, we see that in the Glimm scheme

one specifies a priori the nodal points where the the Riemann problems are to be solved. On the

other hand, in a solution constructed by wave-front tracking the locations of the jumps and of the

interaction points depend on the solution itself. No restarting procedure is needed and the map

t 7→ u(t, ·) is thus continuous with values in L1loc.

x

t

0

t1

t3

t4

t2

σ’

xα

xβ

σ

figure 9

In the end, both algorithms produce a sequence of approximate solutions, whose convergence

relies on a compactness argument based on uniform bounds on the total variation. We sketch

the main idea involved in these a priori BV bounds. Consider a piecewise constant function

u : IR 7→ IRn, say with jumps at points x1 < x2 < · · · < xN . Call σα the amplitude of the jump at

xα. The total strength of waves is then defined as

V (u) .=∑α

|σα| . (2.18)

Clearly, this is an equivalent way to measure the total variation. Along a solution u = u(t, x)

constructed by front tracking, the quantity V (t) = V(u(t, ·)) may well increase at interaction

times. To provide global a priori bounds, following [G] one introduces a wave interaction potential,

defined as

Q(u) =∑

(α,β)∈A|σα σβ |, (2.19)

where the summation runs over the set A of all couples of approaching waves. Roughly speaking,

we say that two wave-fronts located at xα < xβ are approaching if the one at xα has a faster speed

than the one at xβ (hence the two fronts are expected to collide at a future time). Now consider a

time τ where two incoming wave-fronts interact, say with strengths σ, σ′ (for example, take τ = t1

14

in fig. 9). The difference between the outgoing waves emerging from the interaction and the two

incoming waves σ, σ′ is of magnitude O(1) · |σσ′|. On the other hand, after time τ the two incoming

waves are no longer approaching. This accounts for the decrease of the functional Q in (2.19) by

the amount |σσ′|. Observing that the new waves generated by the interaction could approach all

other fronts, the change in the functionals V,Q across the interaction time τ is estimated as

∆V (τ) = O(1) · |σσ′| , ∆Q(τ) = −|σσ′|+O(1) · |σσ′|V (τ−).

If the initial data has small total variation, for a suitable constant C0 the quantity

Υ(t) .= V(u(t, ·)) + C0 Q

(u(t, ·))

is monotone decreasing in time. This argument provides a uniform BV bound on all approximate

solutions. Using Helly’s compactness theorem, one obtains the strong convergence in L1loc of a

subsequence of approximate solutions, and hence the existence of a weak solution.

Theorem 1. Consider the Cauchy problem

ut + f(u)x = 0 , u(0, x) = u(x) , (2.20)

for a system of conservation laws with a smooth flux f defined in a neighborhood of the origin.

Assume that the system is strictly hyperbolic and satisfies the conditions (H). Then, for a sufficiently

small δ > 0 the following holds. For every initial condition u(0, x) = u(x) with

‖u‖L∞ < δ , Tot.Var.{u} < δ , (2.21)

the Cauchy problem has a weak solution, defined for all times t ≥ 0, obtained as limit of front

tracking approximations.

2.5. A semigroup of solutions.

For solutions obtained as limits of Glimm or front tracking approximations, the uniqueness

and the continuous dependence on the initial data were first established in [BC] for systems of two

equations and then in [BCP] for n × n systems. Relying on some major new ideas introduced by

Liu and Yang in [LY1], [LY2] a much simpler proof could then be worked out in [BLY]. Here the

key step is the construction of a Lyapunov functional Φ(u, v), equivalent to the L1 distance

1C‖u− v‖L1 ≤ Φ(u, v) ≤ C‖u− v‖L1 ,

and almost decreasing in time along each pair of ε-approximate solutions constructed by front

trackingd

dtΦ

(u(t), v(t)

) ≤ O(1) · ε .

15

A comprehensive treatment of the above existence-uniqueness theory for BV solutions can be found

in [B3], [HR]. The main result is as follows.

Theorem 2. With the same assumptions as Theorem 1, the weak solution of the Cauchy problem

(2.20) obtained as limit of Glimm or front tracking approximations is unique and satisfies the Liu

entropy conditions. Adopting a semigroup notation, this solution can be written as

u(t, ·) = Stu .

The semigroup S : D × [0,∞[ 7→ D can be defined on a closed domain D ⊂ L1 containing all

functions with sufficiently small total variation, It is uniformly Lipschitz continuous w.r.t. both

time and the initial data:

∥∥Stu− Ssv∥∥L1 ≤ L ‖u− v‖L1 + L′ |t− s| . (2.22)

We refer to the flow generated by a system of conservation laws as a Riemann semigroup,

because it is entirely determined by specifying how Riemann problems are solved. As proved in

[B2], if two semigroups S, S′ yield the same solutions to all Riemann problems, then they coincide,

up to the choice of their domains.

w(0)

w(t)

w(t+h)

w(T)

Shw(t)

ST

ST−t

w(0)

w(t)

ST−t−h

w(t+h)

figure 10

From (2.22) one can deduce the error bound (fig. 10)

∥∥w(T )− ST w(0)∥∥L1 ≤ L ·

∫ T

0

{lim infh→0+

∥∥w(t + h)− Shw(t)∥∥L1

h

}dt , (2.23)

16

valid for every Lipschitz continuous map w : [0, T ] 7→ D taking values inside the domain of the

semigroup. We can think of t 7→ w(t) as an approximate solution of (1.1), while t 7→ Stw(0) is the

exact solution having the same initial data. The distance∥∥w(T )− ST w(0)

∥∥L1 is clearly bounded

by the length of the curve γ : [0, T ] 7→ L1, with γ(t) .= ST−tw(t). According to (2.23), this length

is estimated by the integral of an instantaneous error rate, amplified by the Lipschitz constant L

of the semigroup.

Using (2.23), one can estimate the distance between a front tracking approximation and the

corresponding exact solution. For approximate solutions constructed by the Glimm scheme, a

direct application of this same formula is not possible because of the additional errors introduced

by the restarting procedure at each time tk.= k ∆t. However, relying on a careful analysis of Liu

[L3], one can construct a front tracking approximate solution having the same initial and terminal

values as the Glimm solution. By this technique, in [BM] the authors proved the estimate

lim∆x→0

∥∥uGlimm(τ, ·)− uexact(τ, ·)∥∥L1√

∆x · | ln∆x| = 0 (2.24)

for every τ > 0. In other words, letting the mesh sizes ∆x, ∆t → 0 while keeping their ratio

∆x/∆t constant, the L1 norm of the error in the Glimm approximate solution tends to zero at a

rate slightly slower than√

∆x.

2.6. Uniqueness and characterization of entropy weak solutions

The stability result stated in Theorem 1 refers to a special class of weak solutions: those

obtained as limits of Glimm or front tracking approximations. It is desirable to have a uniqueness

theorem valid for general weak solutions, without reference to any particular constructive proce-

dure. Results in this direction were proved in, [BLF], [BG], [BLw]. They are all based on the error

formula (2.23). In the proofs, one considers a weak solution u = u(t, x) of the Cauchy problem

ut + f(u)x = 0 , u(0, x) = u(x) . (2.25)

Assuming that u satisfies suitable entropy and regularity conditions, one shows that

lim infh→0+

∥∥u(t + h)− Shu(t)∥∥L1

h= 0 (2.26)

at almost every time t. By (2.23), u thus coincides with the semigroup trajectory t 7→ Stu(0) = Stu

obtained as limit of front tracking approximations. Of course, this implies uniqueness. We state

here the main result in [BG]. A function u = u(t, x) is said to have tame oscillation if there exist

constants C ′, β > 0 such that the following holds. For every τ > 0 and a < b, the oscillation of u

over the triangle

∆τa,b

.={(s, y) ; s ≥ t , a + β(s− τ) < y < b− β(s− τ)

}

17

satisfies

Osc.{u; ∆τa,b} ≤ C ′ · Tot.Var.

{u(τ, ·); ]a, b[

}. (2.27)

We recall that the oscillation of a u on a set ∆ is defined as

Osc.{u; ∆} .= sup(s,y), (s′,y′)∈∆

∣∣u(s, y)− u(s′, y′)∣∣ .

Theorem 3. In the same setting as Theorem 1, let u = u(t, x) be a weak solution of the Cauchy

problem (2.20) satisfying the tame oscillation condition (2.27) and the Liu admissibility condition

(2.15) at every point of approximate jump. Then u coincides with the corresponding semigroup

trajectory, i.e.

u(t, ·) = Stu (2.28)

for all t ≥ 0. In particular, the solution that satisfies the above conditions is unique.

An additional characterization of these admissible weak solutions, based on local integral

estimates, was given in [B2]. To state this result, we need to introduce some notations. Given a

function u = u(t, x) and a point (τ, ξ), we denote by U ](u;τ,ξ) the solution of the Riemann problem

with initial data

u− = limx→ξ−

u(τ, x), u+ = limx→ξ+

u(τ, x). (2.29)

In addition, we define U [(u;τ,ξ) as the solution of the linear hyperbolic Cauchy problem with constant

coefficients

wt + A[wx = 0, w(0, x) = u(τ, x). (2.30)

Here A[ .= A(u(τ, ξ)

). Observe that (2.30) is obtained from the quasilinear system (1.2) by

“freezing” the coefficients of the matrix A(u) at the point (τ, ξ) and choosing u(τ) as initial data.

A notion of “good solution” can now be introduced, by locally comparing a function u with the

self-similar solution of a Riemann problem and with the solution of a linear hyperbolic system with

constant coefficients. More precisely, we say that a function u = u(t, x) is a viscosity solution

of the system (1.1) if t 7→ u(t, ·) is continuous as a map with values into L1loc, and moreover the

following integral estimates hold.

(i) At every point (τ, ξ), for every β′ > 0 one has

limh→0+

1h

∫ β′h

−β′h

∣∣∣u(τ + h, ξ + x)− U ](u;τ,ξ)(h, x)

∣∣∣ dx = 0. (2.31)

(ii) There exist constants C, β > 0 such that, for every τ ≥ 0 and a < ξ < b, one has

lim suph→0+

1h

∫ b−βh

a+βh

∣∣∣u(τ + h, x)− U [(u;τ,ξ)(h, x)

∣∣∣ dx ≤ C ·(Tot.Var.

{u(τ); ]a, b[

})2

. (2.32)

18

The above definition was introduced in [B2], with the expectation that the “viscosity solu-

tions” in the above integral sense should precisely coincide with the limits of vanishing viscosity

approximations (1.7). This conjecture was proved only many years later in [BiB]. The main re-

sult in [B2] shows that the integral estimates (2.31)-(2.32) completely characterize the semigroup

trajectories.

Theorem 4. Let S : D× [0,∞[×D be the semigroup generated by the system of conservation laws

(1.1), whose trajectories are limits of front tracking approximations. A function u : [0, T ] 7→ D is

a viscosity solution of (1.1) if and only if u(t) = Stu(0) for all t ∈ [0, T ].

3 - The Vanishing Viscosity Approach

In view of the previous uniqueness and stability results, it is natural to expect that the entropy

weak solutions of the hyperbolic system (1.1) should coincide with the unique limits of solutions

to the parabolic system

uεt + f(uε)x = ε uε

xx (3.1)

letting the viscosity coefficient ε → 0. For smooth solutions, this convergence is easy to show.

However, one should keep in mind that a weak solution of the hyperbolic system (1.1) in general

is only a function with bounded variation, possibly with a countable number of discontinuities. In

this case, as the smooth functions uε approach the discontinuous solution u, near points of jump

their gradients uεx must tend to infinity (fig. 11), while their second derivatives uε

xx become even

more singular. Therefore, establishing the convergence uε → u is a highly nontrivial matter. In

earlier literature, results in this direction relied on three different approaches:

1 - Comparison principles for parabolic equations. For a scalar conservation law, the

existence, uniqueness and global stability of vanishing viscosity solutions was first established by

Oleinik [O] in one space dimension. The famous paper by Kruzhkov [K] covers the more general

class of L∞ solutions and is also valid in several space dimensions.

2 - Singular perturbations. This technique was developed by Goodman and Xin [GX], and

covers the case where the limit solution u is piecewise smooth, with a finite number of non-

interacting, entropy admissible shocks. See also [Yu] and [Ro], for further results in this direction.

3 - Compensated compactness. With this approach, introduced by Tartar and DiPerna [DP2],

one first considers a weakly convergent subsequence uε ⇀ u. For a class of 2 × 2 systems, one

can show that this weak limit u actually provides a distributional solution to the nonlinear system

(1.1). The proof relies on a compensated compactness argument, based on the representation of

19

the weak limit in terms of Young measures, which must reduce to a Dirac mass due to the presence

of a large family of entropies.

x

u

u

ε

figure 11

Since the hyperbolic Cauchy problem is known to be well posed within a space of functions with

small total variation, it seems natural to develop a theory of vanishing viscosity approximations

within the same space BV. This was indeed accomplished in [BiB], in the more general framework

of nonlinear hyperbolic systems not necessarily in conservation form. The only assumptions needed

here are the strict hyperbolicity of the system and the small total variation of the initial data.

Theorem 5. Consider the Cauchy problem for the hyperbolic system with viscosity

uεt + A(uε)uε

x = ε uεxx uε(0, x) = u(x) . (3.2)

Assume that the matrices A(u) are strictly hyperbolic, smoothly depending on u in a neighborhood

of the origin. Then there exist constants C, L,L′ and δ > 0 such that the following holds. If

Tot.Var.{u} < δ , ‖u‖L∞ < δ, (3.3)

then for each ε > 0 the Cauchy problem (3.2)ε has a unique solution uε, defined for all t ≥ 0.

Adopting a semigroup notation, this will be written as t 7→ uε(t, ·) .= Sεt u. In addition, one has

BV bounds : Tot.Var.{Sε

t u} ≤ C Tot.Var.{u} . (3.4)

L1 stability :∥∥Sε

t u− Sεt v

∥∥L1 ≤ L

∥∥u− v∥∥L1 , (3.5)

∥∥Sεt u− Sε

s u∥∥L1 ≤ L′

(|t− s|+ ∣∣√εt−√εs

∣∣)

. (3.6)

Convergence: As ε → 0+, the solutions uε converge to the trajectories of a semigroup S such

that ∥∥Stu− Ssv∥∥L1 ≤ L ‖u− v‖L1 + L′ |t− s| . (3.7)

These vanishing viscosity limits can be regarded as the unique vanishing viscosity solutions of

the hyperbolic Cauchy problem

ut + A(u)ux = 0, u(0, x) = u(x) . (3.8)

20

In the conservative case A(u) = Df(u), every vanishing viscosity solution is a weak solution

of

ut + f(u)x = 0, u(0, x) = u(x) , (3.9)

satisfying the Liu admissibility conditions.

Assuming, in addition, that each field is genuinely nonlinear or linearly degenerate, the vanish-

ing viscosity solutions coincide with the unique limits of Glimm and front tracking approximations.

In the genuinely nonlinear case, an estimate on the rate of convergence of these viscous ap-

proximations was provided in [BY]:

Theorem 6. For the strictly hyperbolic system of conservation laws (3.9), assume that every char-

acteristic field is genuinely nonlinear. At any time t > 0, the difference between the corresponding

solutions of (3.2) and (3.9) can be estimated as

∥∥uε(t, ·)− u(t, ·)∥∥L1 = O(1) · (1 + t)

√ε| ln ε| Tot.Var.{u} .

The following remarks relate Theorem 5 with previous literature.

(i) Concerning the global existence of weak solutions to the system of conservation laws (3.9),

Theorem 5 contains the famous result of Glimm [G]. It also slightly extends the existence theorem

of Liu [L4], which does notrequire the assumption (H). Its main novelty lies in the case where the

system (3.8) is non-conservative. However, one should be aware that for non-conservative systems

the limit solution might change if we choose a viscosity matrix different from the identity, say

ut + A(u)ux = ε(B(u)ux

)x

.

The analysis of viscous solutions, with a more general viscosity matrix B(u), is still a wide open

problem.

(ii) Concerning the uniform stability of entropy weak solutions, the results previously available

for n× n hyperbolic systems [BC1], [BCP], [BLY] always required the assumption (H). For 2× 2

systems, this condition was somewhat relaxed in [AM]. On the other hand, Theorem 5 established

this uniform stability for completely general n×n strictly hyperbolic systems, without any reference

to the assumption (H).

(iii) For the viscous system (1.10), previous results in [L5], [SX], [SZ], [Yu] have established the

stability of special types of solutions, such as travelling viscous shocks or viscous rarefactions,

w.r.t. suitably small perturbations. Taking ε = 1, our present theorem yields at once the uniform

21

Lipschitz stability of all viscous solutions with sufficiently small total variation, w.r.t. the L1

distance. On the other hand, our approach has not yet been used to study the stability and

convergence of viscous solutions with large total variation. For a result in this direction using

singular perturbations, see [R].

We give below a general outline of the proof of Theorem 5. The main steps will then be

discussed in more detail in the following sections.

1. As a preliminary, we observe that uε is a solution of (3.2) if and only if the rescaled function

u(t, x) .= uε(εt, εx) is a solution of the parabolic system with unit viscosity

ut + A(u)ux = uxx , (3.10)

with initial data u(0, x) = u(εx). Clearly, the stretching of the space variable has no effect on the

total variation. Notice however that the values of uε on a fixed time interval [0, T ] correspond to

the values of u on the much longer time interval [0, T/ε]. To obtain the desired BV bounds for the

viscous solutions uε, we can confine all our analysis to solutions of (3.10), but we need estimates

uniformly valid for all times t ≥ 0 , depending only on the total variation of the initial data u.

2. Consider a solution of (3.10), with initial data

u(0, x) = u(x) (3.11)

having small total variation, say

Tot.Var.{u} ≤ δ0 .

Using standard parabolic estimates, one can then prove that the Cauchy problem (3.10)-(3.11) has

a unique solution, defined on an initial time interval [0, t] with t ≈ δ−20 . For t ∈ [0, t] the total

variation remains small:

Tot.Var.{u(t)

}=

∥∥ux(t)∥∥L1 = O(1) · δ0 ,

while all higher derivatives decay quickly. In particular

∥∥uxx(t)∥∥L1 = O(1) · δ0√

t,

∥∥uxxx(t)∥∥L1 = O(1) · δ0

t.

As long as the total variation remains small, one can prolong the solution also for larger times

t > t. In this case, uniform bounds on higher derivatives remain valid.

3. To establish uniform bounds on the total variation, valid for arbitrarily large times, we decom-

pose the gradient along a basis of unit vectors ri = ri(u, ux, uxx), say

ux =∑

i

viri (3.12)

22

We then derive an evolution equation for these gradient components, of the form

vi,t + (λivi)x − vi,xx = φi i = 1, . . . , n , (3.13)

4. A careful examination of the source terms φi in (3.13) shows that they arise mainly because of

interactions among different viscous waves. Their total strength can thus be estimated in terms of

suitable interaction potentials. This will allow us to prove the implication

∥∥ux(t)∥∥L1 ≤ δ0 for all t ∈ [t, T ],

∑

i

∫ T

t

∫ ∣∣φi(t, x)∣∣ dxdt ≤ δ0

=⇒∫ T

t

∫ ∣∣φi(t, x)∣∣ dxdt = O(1) · δ2

0 i = 1, . . . , n .

(3.14)

Since the left hand side of (3.13) is in conservation form, one has

‖ux(t)∥∥L1 ≤

∑

i

∥∥vi(t)∥∥L1 ≤

∑

i

(∥∥vi(t)∥∥ +

∫ t

t

∫ ∣∣φi(s, x)∣∣ dxds

). (3.15)

By the implication (3.14), for δ0 sufficiently small we obtain uniform bounds on the total strength

of the source terms φi, and hence on the BV norm of u(t, ·).

5. Similar techniques can also be applied to a solution z = z(t, x) of the variational equation

zt +[DA(u) · z]

ux + A(u)zx = zxx , (3.16)

which describes the evolution of a first order perturbation to a solution u of (3.11). Assuming that

the total variation of u remains small, one proves an estimate of the form

∥∥z(t, ·)∥∥L1 ≤ L

∥∥z(0, ·)∥∥L1 for all t ≥ 0 , (3.17)

valid for every solution u of (3.10) having small total variation and every L1 solution of the

corresponding system (3.16).

6. Relying on (3.17), a standard homotopy argument yields the Lipschitz continuity of the flow

of (3.10) w.r.t. the initial data, uniformly in time. Indeed, let any two solutions u, v of (3.10) be

given (fig. 12). We can connect them by a smooth path of solutions uθ, whose initial data satisfy

uθ(0, x) .= θu(0, x) + (1− θ)v(0, x) , θ ∈ [0, 1] .

The distance∥∥u(t, ·)−v(t, ·)

∥∥L1 at any later time t > 0 is clearly bounded by the length of the path

θ 7→ uθ(t). In turn, this can be computed by integrating the norm of a tangent vector. Calling

23

u(0)

v(0)

z (0)θ

u (0)θ

u (t)θ

z (t)θ

u(t)

v(t)

figure 12

zθ .= duθ/dθ, each vector zθ is a solution of the corresponding equation (3.19), with u replaced by

uθ. Using (3.18) we thus obtain

∥∥u(t, ·)− v(t, ·)∥∥L1 ≤

∫ 1

0

∥∥∥∥d

dθuθ(t)

∥∥∥∥L1

dθ ≤∫ 1

0

∥∥zθ(t)∥∥L1 dθ

≤ L

∫ 1

0

∥∥zθ(0)∥∥L1 dθ = L

∥∥u(0, ·)− v(0, ·)∥∥L1 .

(3.18)

7. By the simple rescaling of coordinates t 7→ εt, x 7→ εx, all of the above estimates remain valid

for solutions uε of the system (3.2)ε. In particular this yields the bounds (3.4) and (3.5).

8. By a compactness argument, these BV bounds imply the existence of a strong limit uεm → u

in L1loc, at least for some subsequence εm → 0. In the conservative case where A = Df , it is now

easy to show that this limit u provides a weak solution to the Cauchy problem (3.9).

9. At this stage, it only remains to prove that the limit is unique, i.e. it does not depend on

the choice of the sequence εm → 0. For a system in conservative form, and with the standard

assumption (H) that each field is either genuinely nonlinear or linearly degenerate, we can apply

the uniqueness theorem in [BG] and conclude that the limit of vanishing viscosity approximations

coincides with the limit of Glimm and of front tracking approximations.

10. To handle the general non-conservative case, some additional work is required. We first

consider Riemann initial data and show that in this special case the vanishing viscosity solution is

unique and can be accurately described. Then we prove that any weak solution obtained as limit

vanishing viscosity approximations is also a “viscosity solution” in the sense that it satisfies the local

integral estimates (2.31)-(2.32), where U ] is now the unique non-conservative solution of a Riemann

problem. By an argument introduced in [B2], a Lipschitz semigroup is completely determined as

24

soon as one specifies its local behavior for piecewise constant initial data. Characterizing its

trajectories as “viscosity solutions” we thus obtain the uniqueness of the semigroup of vanishing

viscosity limits.

4 - Parabolic estimates

Following a standard approach, one can write the parabolic system (3.11) in the form

ut − uxx = −A(u)ux (4.1)

regarding the hyperbolic term A(u)ux as a first order perturbation of the heat equation. The

general solution of (4.1) can then be represented as

u(t) = G(t) ∗ u(0)−∫ t

0

G(t− s) ∗[A(u(s)

)ux(s)

]ds

in terms of convolutions with the Gauss kernel

G(t, x) =1

2√

πte−x2/4t. (4.2)

From the above integral formula, one can derive local existence, uniqueness and regularity estimates

for solutions of (4.1). Since we shall be dealing with a solution u = u(t, x) having small total

variation, a more effective representation is the following. Consider the state

u∗ .= limx→−∞

u(t, x) , (4.3)

which is independent of time. We then define the matrix A∗ .= A(u∗) and let λ∗i , r∗i , l∗i be

the corresponding eigenvalues and right and left eigenvectors, normalized as in (1.4). It will be

convenient to use “•” to denote a directional derivative, so that z •A(u) .= DA(u) · z indicates the

derivative of the matrix valued function u 7→ A(u) in the direction of the vector z. The systems

(3.10) and (3.16) can now be written respectively as

ut + A∗ux − uxx =(A∗ −A(u)

)ux , (4.4)

zt + A∗zx − zxx =(A∗ −A(u)

)zx −

(z •A(u)

)ux . (4.5)

Observe that, if u is a solution of (4.4), then z = ux is a particular solution of the variational

equation (4.5). Therefore, as soon as one proves an a priori bound on zx or zxx, the same estimate

will be valid also for the corresponding derivatives uxx, uxxx.

25

In both of the equations (4.4), (4.5), we regard the right hand side as a perturbation of the

linear parabolic system with constant coefficients

wt + A∗wx − wxx = 0 . (4.6)

We denote by G∗ the Green kernel for (4.6), so that any solution admits the integral representation

w(t, x) =∫

G∗(t, x− y)w(0, y) dy .

The matrix valued function G∗ can be explicitly computed. If w solves (4.6), then the i-th com-

ponent wi.= l∗i · w satisfies the scalar equation

wi,t + λ∗i wi,x − wi,xx = 0 .

Therefore wi(t) = G∗i (t) ∗ wi(0), where

G∗i (t, x) =1

2√

πtexp

{− (x− λ∗i t)

2

4t

}.

It is now clear that this Green kernel G∗ = G∗(t, x) satisfies the bounds

∥∥G∗(t)∥∥L1 ≤ κ ,

∥∥G∗x(t)∥∥L1 ≤

κ√t,

∥∥G∗xx(t)∥∥L1 ≤

κ

t, (4.7)

for some constant κ and all t > 0.

In the following, we consider an initial data u(0, ·) having small total variation, but possibly

discontinuous. We shall prove the local existence of solutions and some estimates on the decay of

higher order derivatives. To get a feeling on this rate of decay, let us first take a look at the most

elementary case.

Example 4.1. The solution to the Cauchy problem for the heat equation

ut − uxx = 0 , u(0, x) ={

0 if x < 0,δ0 if x > 0,

is computed explicitly as

u(t, x) = δ0

∫ ∞

0

G(t, x− y) dy .

Observing that the Gauss kernel (4.2) satisfies

G(t, x) = t−1/2 G(1, t−1/2x),

for every k ≥ 1 one obtains the estimates∥∥∥∥

∂k−1

∂xk−1G(t)

∥∥∥∥L∞

≤∥∥∥∥

∂k

∂xkG(t)

∥∥∥∥L1

= O(1) · t−k/2 .

26

In the present example we have ux(t, x) = δ0G(t, x). Therefore

∥∥ux(t)∥∥L∞ ≤ ∥∥uxx(t)

∥∥L1 = O(1) · δ0√

t, (4.8)

∥∥uxx(t)∥∥L∞ ≤ ∥∥uxxx(t)

∥∥L1 = O(1) · δ0

t, (4.9)

∥∥uxxx(t)∥∥L∞ = O(1) · δ0

t√

t. (4.10)

In the remainder of this chapter, our analysis will show that the same decay rates hold for

solutions of the perturbed equation (4.4), restricted to some initial interval [0, t]. More precisely,

let δ0 measure the order of magnitude of the total variation in the initial data, so that

Tot.Var.{ux(0, ·)} = O(1) · δ0 . (4.11)

Then:

• There exists an initial interval [0, t], with t = O(1) · δ−20 on which the solution of (4.4) is well

defined. Its derivatives decay according to the estimates (4.8)–(4.10).

• As long as the total variation remains small, say

∥∥ux(t)∥∥L1 ≤ δ0 , (4.12)

the solution can be prolonged in time. In this case, for t > t the higher derivatives satisfy the

bounds ∥∥ux(t)∥∥L∞ ,

∥∥uxx(t)∥∥L1 = O(1) · δ2

0 , (4.13)

∥∥uxx(t)∥∥L∞ ,

∥∥uxxx(t)∥∥L1 = O(1) · δ3

0 , (4.14)

∥∥uxxx(t)∥∥L∞ = O(1) · δ4

0 . (4.15)

The situation is summarized in fig. 13. All the estimates drawn here as solid lines will be

obtained by standard parabolic-type arguments. If any breakdown occurs in the solution, the first

estimate that fails must be the one in (4.12), concerning the total variation. Toward a proof of

global existence of solutions, the most difficult step is to show that (4.12) remains valid for all

times t ∈ [t, ∞[ (the broken line in fig. 13). This will require hyperbolic-type estimates, based on

the local decomposition of the gradient ux as a sum of travelling waves, and on a careful analysis

of all interaction terms. In the remainder of this section we work out a proof of local existence and

a priori estimates for solutions of (4.4) and (4.5).

27

0

δ0

uxxx

xxuxu

δ0−2~t

L

L

L

1

~

1

1

t

figure 13

Proposition 4.2 (Local existence). For δ0 > 0 suitably small, the following holds. Consider

initial data

u(0, x) = u(x) , z(0, x) = z(x) . (4.16)

such that

Tot.Var.{u} ≤ δ0

4κ, z ∈ L1 . (4.17)

Then the equations (4.4), (4.5) have unique solutions u = u(t, x), z = z(t, x) defined on the time

interval t ∈ [0, 1]. Moreover

∥∥ux(t)∥∥L1 ≤

δ0

2,

∥∥z(t)∥∥L1 ≤

δ0

2for all t ∈ [0, 1] . (4.18)

Proof. The couple (u, ux), consisting of the solution of (4.2) together with its derivative, will be

obtained as the unique fixed point of a contractive transformation. For notational convenience, we

assume here

u∗ .= limx→−∞

u(x) = 0.

28

Of course this is not restrictive, since it can always be achieved by a translation of coordinates.

Consider the space E of all mappings

(u, z) : [0, 1] 7→ L∞ × L1 ,

with norm ∥∥(u, z)∥∥

E

.= supt

max{∥∥u(t)

∥∥L∞ ,

∥∥z(t)∥∥L1

}.

On E we define the transformation T (u, z) = (u, z) by setting

u(t) .= G∗(t) ∗ u +∫ t

0

G∗(t− s) ∗ [A∗ −A(u(s))

]z(s) ds ,

z(t) .= G∗x(t) ∗ u +∫ t

0

G∗x(t− s) ∗ [A∗ −A(u(s))

]z(s) ds .

Of course, the above definition implies z = ux. The assumption Tot.Var.{u} ≤ δ0/4κ together

with the bounds (4.7) yields

∥∥G∗(t) ∗ u∥∥L∞ ≤

∥∥G∗x(t) ∗ u∥∥L1 ≤

δ0

4. (4.19)

Moreover, if∥∥u(s)

∥∥L∞ ≤ δ0

2,

∥∥z(s)∥∥L1 ≤

δ0

2for all s ∈ [0, 1] ,

then ∥∥∥∥∫ t

0

G∗(t− s) ∗ [A∗ −A(u(s))

]z(s) ds

∥∥∥∥L∞

≤∫ t

0

κ ‖DA‖∥∥u(s)∥∥L∞

∥∥z(s)∥∥L1 ds

≤ tκκAδ20

4and similarly

∥∥∥∥∫ t

0

G∗x(t− s) ∗ [A∗ −A(u(s))

]z(s) ds

∥∥∥∥L1

≤∫ t

0

κ√t− s

‖DA‖∥∥u(s)∥∥L∞

∥∥z(s)∥∥L1 ds

≤ 2√

t κκAδ20

4.

Combining the above estimates with (4.19), we see that the transformation T maps the domain

D .={

(u, z) : [0, 1] 7→ L∞ × L1 ;∥∥u(t)

∥∥L∞ ,

∥∥z(t)∥∥L1 ≤ δ0/2 for all t ∈ [0, 1]

}

into itself. To prove that T is a strict contraction, we compute the difference T (u, z) − T (u′, z′).

The norm of the first component is estimated as

∥∥u− u′∥∥L∞ =

∥∥∥∥∫ t

0

G∗(t− s) ∗{[

A∗ −A(u(s))](

z(s)− z′(s))

+[A(u′(s))−A(u(s))

]z′(s)

}ds

∥∥∥∥L∞

≤∫ t

0

∥∥G∗(t− s)∥∥L∞

{‖DA‖

∥∥u(s)∥∥L∞

∥∥z(s)− z′(s)∥∥L1 + ‖DA‖

∥∥u(s)− u′(s)∥∥L∞

∥∥z′(s)∥∥L1

}ds

≤ 2√

tκκAδ0

∥∥(u− u′, z − z′)∥∥

E.

29

An entirely similar computation yields

∥∥z(t)− z′(t)∥∥L1 ≤ 2

√tκκAδ0

∥∥(u− u′, z − z′)∥∥

E.

Therefore, for δ0 small enough, the map T is a strict contraction.

By the contraction mapping theorem, a unique fixed point exists, inside the domain D. Clearly,

this provides the solution of (4.4) with the prescribed initial data.

Having constructed a solution u of (4.4), we now prove the existence of a solution z of the

linearized variational system (4.5), with initial data z ∈ L1. Consider the space E′ of all mappings

z : [0, 1] 7→ W1,1 with norm

‖z‖E′.= sup

tmax

{∥∥z(t)∥∥L1 ,

√t∥∥zx(t)

∥∥L1

}.

On E′ we define the transformation T (z) = z by setting

z(t) .= G∗(t) ∗ z +∫ t

0

G∗(t− s) ∗{[

A∗ −A(u(s))]zx(s)− [

z(s) •A(u(s))]ux(s)

}ds .

The bounds (4.7) now yield

∥∥G∗(t) ∗ z∥∥L1 ≤ κ‖z‖L1 ,

∥∥G∗x(t) ∗ z∥∥L1 ≤

κ√t‖z‖L1 . (4.20)

Moreover, using the identity

∫ t

0

1√s(t− s)

ds =∫ 1

0

1√σ(1− σ)

dσ = π < 4 (4.21)

one obtains the bound

∥∥∥∥∫ t

0

G∗x(t− s) ∗{[

A∗ −A(u(s))]zx(s)− [

z(s) •A(u(s))]ux(s)

}ds

∥∥∥∥L1

≤∫ t

0

κ√t− s

{‖DA‖

∥∥u(s)∥∥L∞

∥∥zx(s)∥∥L1 + ‖DA‖

∥∥zx(s)∥∥L1

∥∥ux(s)∥∥L1

}

≤ 4κκAδ0 ‖z‖E′ .

Assuming that δ0 is small enough, it is now clear that the linear transformation T ′ is a strict

contraction in the space E′. Hence it has a unique fixed point, which provides the desired solution

of (4.5).

30

Having established the local existence of a solution, we now study the decay of its higher order

derivatives.

Proposition 4.3. Let u, z be solutions of the systems (4.4) and (4.5), satisfying the bounds

∥∥ux(t)∥∥L1 ≤ δ0,

∥∥z(t)∥∥L1 ≤ δ0, (4.22)

for some constant δ0 < 1 and all t ∈ [0, t ], where

t.=

(1

400κκA δ0

)2

, κA.= sup

u

(‖DA‖+ ‖D2A‖) (4.23)

and κ is the constant in (4.7). Then for t ∈ [0, t] the following estimates hold.

∥∥uxx(t)∥∥L1 ,

∥∥zx(t)∥∥L1 ≤

2κδ0√t

, (4.24)

∥∥uxxx(t)∥∥L1 ,

∥∥zxx(t)∥∥L1 ≤

5κ2δ0

t, (4.25)

∥∥uxxx(t)∥∥L∞ ,

∥∥zxx(t)∥∥L∞ ≤ 16κ3δ0

t√

t. (4.26)

Proof. We begin with (4.24). The function zx can be represented in terms of convolutions with

the Green kernel G∗, as

zx(t) = G∗x(t) ∗ z(0) +∫ t

0

G∗x(t− s) ∗{(

A∗ −A(u))zx(s)− (

z •A(u))ux(s)

}ds . (4.27)

Using (4.7) and (4.22) we obtain

∥∥∥∥∫ t

0

G∗x(t− s) ∗{(

A∗ −A(u))zx(s)− (

z •A(u))ux(s)

}ds

∥∥∥∥L1

≤∫ t

0

∥∥G∗x(t− s)∥∥L1 ·

{∥∥ux(s)∥∥L1

∥∥DA∥∥L∞

∥∥zx(s)∥∥

L1 +∥∥z(s)

∥∥L∞

∥∥DA∥∥L∞

∥∥ux(s)∥∥L1

}ds

≤ 2δ0κ∥∥DA

∥∥L∞ ·

∫ t

0

1√t− s

∥∥zx(s)∥∥L1 ds .

Consider first the case of smooth initial data. We shall argue by contradiction. If (4.24) fails for

some t ≤ t, by continuity will be a first time τ < t at which (4.24) is satisfied as an equality. In

this case, recalling the identity (4.21) we compute

∥∥zx(τ)∥∥L1 ≤

κ√τ

δ0 + 2κδ0

∥∥DA∥∥L∞ ·

∫ τ

0

1√τ − s

2δ0κ√s

ds

<κδ0√

τ+ 16κ2κAδ2

0

∥∥DA∥∥L∞ ≤ 2κδ0√

τ,

31

reaching a contradiction. Hence, (4.24) must be satisfied as a strict inequality for all t ∈ [0, t].

A similar technique is used to establish (4.25). Indeed, we can write

zxx(t) = G∗x(t/2) ∗ zx(t/2)−∫ t

t/2

G∗x(t− s) ∗{(

z •A(u))ux(s) +

(A(u)−A∗

)zx(s)

}xds . (4.28)

We prove (4.16) first in the case zxx = uxxx, then in the general case. If (4.25) is satisfied as an

equality at a first time τ < t, using (4.7), (4.24) and recalling the definitions of the constants t, κA

in (4.23), we compute

∥∥zxx(τ)∥∥L1 ≤

κ√τ/2

· 2κδ0√τ/2

+∫ τ

τ/2

κ√τ − s

·{∥∥zx •A(u)ux(s)

∥∥L1 +

∥∥z • (ux •A(u)

)ux(s)

∥∥L1

+∥∥z •A(u)uxx(s)

∥∥L1 +

∥∥ux •A(u)zx(s)∥∥L1 +

∥∥∥(A(u)−A∗

)zxx(s)

∥∥∥L1

}ds

≤ 2κ2δ0

τ/2+

∫ τ

τ/2

κ√τ − s

·{

δ0‖DA‖L∞∥∥zxx(s)

∥∥L1 + δ0‖D2A‖L∞

∥∥uxx(s)∥∥2

L1

+ δ0‖DA‖L∞∥∥uxxx(s)

∥∥L1 + δ0‖DA‖L∞

∥∥zxx(s)∥∥L1 + δ0‖DA‖L∞

∥∥zxx(s)∥∥L1

}ds

≤ 4κ2δ0

τ+ κδ0

(4κ2δ2

0‖D2A‖L∞ + 20κ2δ0‖DA‖L∞) ∫ τ

τ/2

1s√

τ − sds ,

<4κ2δ0

τ+ 20κ3κAδ2

0 ·4√τ/2

<5κ2δ0

τ.

Again, we conclude that at time τ the bound (4.16) is still satisfied as a strict inequality, thus

reaching a contradiction.

Finally, assume that τ < t is the first time where the bound (4.26) is satisfied as an equality.

Using (4.7) and the previous bounds (4.24)-(4.25), from the representation (4.28) we now obtain

∥∥zxx(τ)∥∥

L∞ ≤ κ√τ/2

· 5κ2δ0

τ/2+

∫ τ

τ/2

κ√τ − s

·{∥∥zx •A(u)ux(s)

∥∥L∞ +

∥∥z • (ux •A(u))ux(s)∥∥

L∞

+∥∥z •A(u)uxx(s)

∥∥L∞ +

∥∥ux •A(u)zx(s)∥∥

L∞ +∥∥(A(u)−A∗)zxx(s)

∥∥L∞

}ds

≤ 10√

2κ3δ0

τ√

τ+

(8κ4δ3

0‖D2A‖+ 46κ4δ20‖DA‖)

∫ τ

τ/2

1s3/2

√τ − s

ds

≤ 15κ3δ0

τ√

τ+ 46κ4κAδ2

0 ·4

τ/2<

16κ3δ0

τ√

τ.

Therefore, the bound (4.26) must be satisfied as a strict inequality for all t ≤ t.

Corollary 4.4. In the same setting as Proposition 4.2, assume that the bounds (4.12) hold on a

larger interval [0, T ]. Then for all t ∈ [t, T ] there holds∥∥uxx(t)

∥∥L1 ,

∥∥ux(t)∥∥L∞ ,

∥∥zx(t)∥∥L1 = O(1) · δ2

0 , (4.29)

32

∥∥uxxx(t)∥∥L1 ,

∥∥uxx(t)∥∥L∞ ,

∥∥zxx(t)∥∥L1 = O(1) · δ3

0 , (4.30)

∥∥uxxx(t)∥∥L∞ ,

∥∥zxx(t)∥∥L∞ = O(1) · δ4

0 . (4.31)

Proof. This follows by applying Proposition 4.3 on the interval [t− t, t].

The next result provides a first estimate on the time interval where the solution u of (4.4)

exists and remain small.

Proposition 4.5. For δ0 > 0 sufficiently small, the following holds. Consider initial data u, z as

in (4.16), satisfying the bounds

Tot.Var.{u} ≤ δ0

4κ, ‖z‖L1 ≤ δ0

4κ. (4.32)

Then the systems (4.4), (4.5) admit unique solutions u = u(t, x), z = z(t, x) defined on the whole

interval [0, t], with t defined at (4.23). Moreover, one has

∥∥ux(t)∥∥L1 ≤

δ0

2,

∥∥z(t)∥∥L1 ≤

δ0

2for all t ∈ ]0, t]. (4.33)

Proof. Because of the local existence result proved in Proposition 4.2, it suffices to check that the

bounds (4.32) remain valid for all t ≤ t. The solution of (4.5) can be written in the form

z(t) = G∗(t) ∗ z +∫ t

0

G∗(t− s) ∗{(

A∗ −A(u))zx(s)− (

z •A(u))ux(s)

}ds .

As before, we first establish the result for z = ux, then for a general solution z of (4.5). Assume

that there exists a first time τ < t where the bound in (4.17) is satisfied as an equality. Using (4.7)

and (4.24) and recalling the definition of t at (4.23), from the above identity we obtain

∥∥z(τ)∥∥L1 ≤ κ ‖z‖L1 +

∫ τ

0

κ ·{‖DA‖∥∥ux(s)

∥∥L1

∥∥zx(s)∥∥L1 + ‖DA‖∥∥zx(s)

∥∥L1

∥∥ux(s)∥∥L1

}ds

≤ κδ0

4κ+

∫ τ

0

2κδ20√

s‖DA‖ ds

≤ δ0

4+ 4κκAδ2

0

√τ <

δ0

2,

reaching a contradiction.

33

To simplify the proof, we assumed here the same bounds on the functions ux and z. However,

observing that z solves a linear homogeneous equation, similar estimates can be immediately

derived without any restriction on the initial size of z ∈ L1. In particular, from Proposition 4.5 it

follows

Corollary 4.6. Let u = u(t, x), z = z(t, x) be solutions of (4.4), (4.5) respectively, with∥∥ux(0)∥∥L1 ≤ δ0/4κ. Then u, z are well defined on the whole interval [0, t] in (4.13), and sat-

isfy

∥∥ux(t)∥∥L1 ≤ 2κ

∥∥ux(0)∥∥L1 ,

∥∥z(t)∥∥L1 ≤ 2κ

∥∥z(0)∥∥L1 , for all t ≤ t . (4.34)

5 - Decomposition by Travelling Wave Profiles

Given a smooth function u : IR 7→ IRn, at each point x we want to decompose the gradient ux

as a sum of gradients of viscous travelling waves, say

ux(x) =∑

i

viri =∑

i

U ′i(x) . (5.1)

We recall that, for the parabolic system

ut + A(u)ux = uxx , (5.2)

a travelling wave profile is a solution of the special form u(t, x) = U(x − σt). For the application

that we have in mind, we can assume that the function u has small total variation, and that all of

its derivatives are uniformly small. However, in order to come up with a unique decomposition of

the form (1.10)-(1.11), we still face two major obstacles.

1. The system (1.9)–(1.11) is highly underdetermined. Indeed, by (1.9)-(1.10) each travelling

profile Ui provides a solution to the Cauchy problem

U ′′i =

(A(Ui)− σi

)U ′

i , Ui(x) = u(x) . (5.3)

Since this is a second order equation, we can still choose the derivative U ′(x) ∈ IRn arbitrarily,

as well as the speed σi. The general solution of (5.3) thus depends on n + 1 scalar parameters.

Summing over all n families of waves, we obtain n(n + 1) free parameters. On the other hand, the

system ∑

i

U ′i(x) = ux(x) ,

∑

i

U ′′i (x) = uxx(x) (5.4)

34

consists of only n + n scalar equations. When n ≥ 2, these are not enough to uniquely determine

the travelling profiles U1, . . . , Un.

2. For particular values of ux, uxx, the equations (5.3)-(5.4) may not have any solution at all. This

can already be seen for a scalar conservation law

ut + λ(u)ux = uxx .

In this case, the above equations reduce to

U ′′ =(λ(U)− σ

)U ′ , U(x) = u(x) ,

U ′(x) = ux(x) , U ′′(x) = uxx(x) .

The solution is now unique. The only choice for the speed is

σ = λ(u(x)

)− uxx(x)ux(x)

.

The smallness assumption on ux, uxx is not of much use here. If ux = 0 while uxx 6= 0, no solution

is found. This typically occurs at points where u has a local maximum or minimum.

To overcome the first obstacle, we shall consider not the family of all travelling profiles, but

only certain subfamilies. More precisely, for each i = 1, . . . , n, our decomposition will involve only

those travelling profiles Ui which lie on a suitable center manifold.

The second difficulty will be removed by introducing a cutoff function. In a normal situation,

our decomposition will be uniquely determined by (5.3)-(5.4). In the exceptional cases where the

cutoff function is active, only the first equation in (5.4) will be satisfied, allowing for some error in

the second equation.

5.1. Construction of a Center Manifold

To carry out our program, we begin by selecting certain families of travelling waves, depending

on the correct number of parameters to fit the data. Observing that the system (5.4) consists of

n + n scalar equations, this number of parameters is easy to guess. To achieve a unique solution,

through any given state u we need to construct n families of travelling wave profiles Ui, each

depending on two scalar parameters. The construction will be achieved by an application of the

center manifold theorem.

Travelling waves for the parabolic system (5.2) correspond to (possibly unbounded) solutions

of (A(U)− σ

)U ′ = U ′′. (5.5)

35

We write (5.5) as a first order system on the space IRn × IRn × IR:

u = v ,

v =(A(u)− σ

)v ,

σ = 0 .

(5.6)

Let a state u∗ be given and fix an index i ∈ {1, . . . , n}. Linearizing (5.6) at the equilibrium point

P ∗ .=(u∗, 0, λi(u∗)

)we obtain the linear system

u = v ,

v =(A(u∗)− λi(u∗)

)v ,

σ = 0 .

(5.7)

Let {r∗1 , . . . , r∗n} and {l∗1, . . . , l∗n} be dual bases of right and left eigenvectors of A(u∗) normalized

as in (1.4). We call (V1, . . . , Vn) the coordinates of a vector v ∈ IRn w.r.t. this basis, so that

|r∗i | = 1, v =∑

j

Vjr∗j , Vj

.= l∗j · v .

The center subspace N for (5.7) (see fig. 14) consists of all vectors (u, v, σ) ∈ IRn × IRn × IR such

that

Vj = 0 for all j 6= i, (5.8)

and therefore has dimension n + 2. By the center manifold theorem (see the Appendix), there

exists a smooth manifold M⊂ IRn+n+1, tangent to N at the stationary point P ∗, which is locally

invariant under the flow of (5.6). This manifold has dimension n + 2 and can be locally defined by

the n− 1 equations

Vj = ϕj(u, Vi, σ) j 6= i. (5.9)

M

P*

N

figure 14

36

We seek a set of coordinates on this center manifold, involving n + 2 free parameters. As a

preliminary, observe that a generic point on the center subspace N can be described as

P = (u, v, σ) = (u, Vir∗i , σ) .

We can thus regard (u, vi, σ) ∈ IRn+2 as coordinates of the point P . Notice that vi = ±|v| is the

signed strength of the vector v, while σ is the wave speed. Similarly, we will show that a generic

point on the center manifold M can be written as

P = (u, v, σ) = (u, viri, σ) ,

where again vi is the signed strength of v and ri is a unit vector parallel to v. In general, this unit

vector is not constant as in the linear case, but depends on u, vi, σ. All the information about the

center manifold is thus encoded in the map (u, vi, σ) 7→ ri(u, vi, σ).

We can assume that the n− 1 smooth scalar functions ϕj in (5.9) are defined on the domain

D .={

(u, Vi, σi) ; |u− u∗| < ε, |Vi| < ε,∣∣σi − λi(u∗)| < ε

}⊂ IRn+2 , (5.10)

for some small ε > 0. The tangency condition implies

ϕj(u, Vi, σ) = O(1) ·(|u− u∗|2 + |Vi|2 +

∣∣σ − λi(u∗)∣∣2

). (5.11)

By construction, every trajectory

t 7→ P (t) .=(u(t), v(t), σ(t)

)

of (5.6), which remains within a small neighborhood of the point P ∗ .=(u∗, 0, λi(u∗)

)for all t ∈ IR,

must lie entirely on the manifold M. In particular, M contains all viscous i-shock profiles joining

a pair of states u−, u+ sufficiently close to u∗. Moreover, all equilibrium points (u, 0, σ) with

|u− u∗| < ε and∣∣σ − λi(u∗)

∣∣ < ε must lie on M. Hence

ϕj(u, 0, σ) = 0 for all j 6= i. (5.12)

By (5.12) and the smoothness of the functions ϕj , we can “factor out” the component Vi and write

ϕj(u, Vi, σ) = ψj(u, Vi, σ) · Vi,

for suitable smooth functions ψj . From (5.11) it follows

ψj → 0 as (u, Vi, σ) → (u∗, 0, λi(u∗)

). (5.13)

37

On the manifold M we thus have

v =∑

k

Vkr∗k = Vi ·r∗i +

∑

j 6=i

ψj(u, Vi, σ) r∗j

.= Vi r]

i (u, Vi, σ). (5.14)

By (5.13), the function r] defined by the last equality in (5.14) satisfies

r]i (u, Vi, σ) → r∗i as (u, Vi, σ) → (

u∗, 0, λi(u∗)). (5.15)

Notice that, for a travelling profile Ui of the i-th family, the derivative v = U ′i should be ”almost

parallel” to the eigenvector r∗i . This is indeed confirmed by (5.15).

We can now define the new variable

vi = vi(u, Vi, σ) .= Vi ·∣∣r]

i (u, Vi, σ)∣∣. (5.16)

As (u, Vi, σ) ranges in a small neighborhood of(u∗, 0, λi(u∗)

), by (5.15) the vector r]

i remains close

to the unit vector r∗i.= ri(u∗). In particular, its length |r]

i | remains uniformly positive. Hence the

transformation Vi ←→ vi is invertible and smooth. We can thus reparametrize the center manifold

M in terms of the variables (u, vi, σ) ∈ IRn × IR× IR. Moreover, we define the unit vector

ri(u, vi, σ) .=r]i

|r]i |

. (5.17)

Observe that ri is also a smooth function of its arguments. With the above definitions, in alternative

to (5.9) we can express the manifold M by means of the equation

v = viri(u, vi, σ) . (5.18)

The above construction of a center manifold can be repeated for every i = 1, . . . , n. We thus

obtain n center manifolds Mi ⊂ IRn+n+1 and vector functions ri = ri(u, vi, σi) such that

|ri| ≡ 1, (5.19)

Mi ={(u, v, σi) ; v = vi ri(u, vi, σi)

}, (5.20)

as (u, vi, σi) ∈ IRn × IR× IR ranges in a neighborhood of(u∗, 0, λi(u∗)

).

Next, we derive some identities for future use. The partial derivatives of ri = ri(u, vi, σi)

w.r.t. its arguments will be written as

ri,u.=

∂

∂uri , ri,v

.=∂

∂viri , ri,σ

.=∂

∂σiri .

38

Of course ri,u is an n×n matrix, while ri,v, ri,σ are n-vectors. Higher order derivatives are denoted

as ri,uσ, ri,σσ . . . We claim that

ri(u, 0, σi) = ri(u) for all u, σi . (5.21)

Indeed, consider again the equation for a viscous travelling i-wave:

uxx =(A(u)− σi

)ux . (5.22)

For a solution contained in the center manifold, taking the derivative w.r.t. x of

ux = v = viri(u, vi, σi) (5.23)

and using (5.22) one finds

vi,xri + viri,x =(A(u)− σi

)viri . (5.24)

Since |ri| ≡ 1, the vector ri is perpendicular to its derivative ri,x. Taking the inner product of

(5.24) with ri we thus obtain

vi,x = (λi − σi)vi , (5.25)

where we defined the ”generalized eigenvalue” λi = λi(u, vi, σi) as the inner product

λi.=

⟨ri , A(u)ri

⟩. (5.26)

Writing out the derivative

ri,x = ri,uux + ri,vvi,x + ri,σσx

and using (5.25), (5.23) and the identity σx = 0 clearly valid for a travelling wave, from (5.24) we

finally obtain

(λi − σi)viri + vi

(ri,urivi + ri,v(λi − σi)vi

)=

(A(u)− σi

)viri . (5.27)

This yields a fundamental identity satisfied by our ”generalized eigenvectors” ri, namely

(A(u)− λi

)ri = vi

(ri,uri + ri,v(λi − σi)

). (5.28)

Notice how (5.28) replaces the familiar identity

(A(u)− λi

)ri = 0 (5.29)

satisfied by the usual eigenvectors and eigenvalues of the matrix A(u). The presence of some small

terms on the right hand side of (5.28) is of great importance. Indeed, in the evolution equations

39

(3.13), these terms achieve a crucial cancellation with other source terms that would otherwise not

be integrable.

Comparing (5.28) with (5.29) we see that, as vi → 0, the unit vector ri(u, vi, σi) approaches

an eigenvector of the matrix A(u), while λi(u, vi, σi) approaches the corresponding eigenvalue. By

continuity, this establishes our claim (5.21).

By (5.21), when vi = 0 the vector ri does not depend on the speed σi, hence ri,σ(u, 0, σi) = 0.

In turn, by the smoothness of the vector field ri we also have

ri(u, vi, σi)− ri(u) = O(1) · vi,

ri,uσ = O(1) · vi,

ri,σ = O(1) · vi,

ri,σσ = O(1) · vi.(5.30)

Observing that the vectors ri,v and ri,σ are both perpendicular to ri, from (5.28) and the definition

(5.26) we deduce

λi(u, vi, σi)− λi(u) = O(1) · vi, λi,σ = O(1) · vi. (5.31)

We conclude our analysis of trajectories on the center manifold by proving the identity

−σivi,x + (λivi)x − vi,xx = 0, (5.32)

valid for a travelling wave profile. Toward this goal, we first differentiate (5.24) and obtain

vi,xxri + 2vi,xri,x + viri,xx =(A(u)viri

)x− σivi,xri − σiviri,x . (5.33)

From the identities ⟨ri, ri,x

⟩= 0 ,

⟨ri, ri,xx

⟩= −⟨

ri,x, ri,x

⟩,

taking the inner product of (5.24) with ri,x we find

⟨ri, ri,xx

⟩vi = −⟨

ri,x, A(u)ri

⟩vi . (5.34)

On the other hand, taking the inner product of (5.33) with ri we find

vi,xx +⟨ri, ri,xx

⟩vi =

⟨ri, (A(u)rivi)x

⟩− σivi,x .

By (5.34), this yields (5.32).

The significance of the identity (5.32) can be explained as follows. Consider a very special

solution of the parabolic system (5.2), say

u = u(t, x) = Ui(x− σit)

40

consisting of a travelling i-wave on the center manifold. The corresponding decomposition (5.1)

will then contain one single term:

ux(t, x) = vi(t, x)ri(t, x) = U ′i(x− σit) ,

while vj ≡ 0 for all other components j 6= i. We seek the evolution equation satisfied by the scalar

component vi. Since we are dealing with a travelling wave, we have

vi,t + σivi,x = 0 .

Inserting this in (5.32) we thus conclude

vi,t + (λivi)x − vi,xx = 0. (5.35)

In other words, in connection with a travelling wave profile, the source terms φi in (3.13) vanish

identically.

5.2. Wave decomposition

Let u : IR 7→ IRn be a smooth function with small total variation. At each point x, we seek a

decomposition of the gradient ux in the form (5.1), where ri = ri(u, vi, σi) are the vectors defining

the center manifold at (5.20). To uniquely determine the ri, we must first define the wave strengths

vi and speeds σi in terms of u, ux, uxx.

Consider first the special case where u is precisely the profile of a viscous travelling wave of

the j-th family (contained in the center manifold Mj). In this case, our decomposition should

clearly contain one single component:

ux = vj rj(u, vj , σj) . (5.36)

It is easy to guess what vj , σj should be. Indeed, since by construction |rj | = 1, the quantity

vj = ±|ux|

is the signed strength of the wave. Notice also that for a travelling wave the vectors ux and ut are

always parallel, because ut = −σjux where σj is the speed of the wave. We can thus write

ut = uxx −A(u)ux = ωj rj(u, vj , σj) (5.37)

for some scalar ωj . The speed of the wave is now obtained as σj = −ωj/vj .

Motivated by the previous analysis, as a first attempt we define

ut.= uxx −A(u)ux (5.38)

41

and try to find scalar quantities vi, ωi such that

ux =∑

i

vi ri(u, vi, σi),

ut =∑

i

ωi ri(u, vi, σi),σi = −ωi

vi. (5.39)

The trouble with (5.39) is that the vectors ri are defined only for speeds σi close to the i-th

characteristic speed λ∗i.= λi(u∗). However, when ux ≈ 0 one has vi ≈ 0 and the ratio ωi/vi may

become arbitrarily large.

1η

−δ1

δ1

s

(s)

θ(s)

figure 15

To overcome this problem, we introduce a cutoff function (fig. 15). Fix δ1 > 0 sufficiently

small. Define a smooth odd function θ : IR 7→ [−2δ1, 2δ1] such that

θ(s) ={

s if |s| ≤ δ1

0 if |s| ≥ 3δ1|θ′| ≤ 1, |θ′′| ≤ 4/δ1 . (5.40)

We now rewrite (5.39) in terms of the new variable wi, related to ωi by ωi.= wi−λ∗i vi. We require

that σi coincides with −ωi/vi only when this ratio is sufficiently close to λ∗i.= λi(u∗). Our basic

equations thus take the form

ux =∑

i

vi ri(u, vi, σi),

ut =∑

i

(wi − λ∗i vi) ri(u, vi, σi),(5.41)

where

ut = uxx −A(u)ux , σi = λ∗i − θ

(wi

vi

). (5.42)

42

Notice that σi is not well defined when vi = wi = 0. However, recalling (5.21), in this case we have

ri = ri(u) regardless of σi. Hence the two equations in (5.41) are still meaningful.

Remark 5.1. By construction, the vectors viri are the gradients of viscous travelling waves Ui

such that

Ui(x) = u(x), U ′i(x) = viri , U ′′

i =(A(u)− σi

)U ′

i .

From the first equation in (5.41) it follows

ux(x) =∑

i

U ′i(x) .

If σi = λ∗i − wi/vi for all i = 1, . . . , n, i.e. if none of the cutoff functions is active, then

uxx(x) = ut + A(u)ux =∑

i

(wi − λ∗i vi)ri + A(u)∑

i

viri

=∑

i

(A(u)− σi

)viri =

∑

i

U ′′i (x) .

In this case, both of the equalities in (5.4) hold. Notice however that the second equality in (5.4)

may fail if |wi/vi| > δ1 for some i.

We now show that the system of equations (5.41)-(5.42) uniquely defines the components

vi, wi. Moreover, we study the smoothness of these components, as functions of u, ux, uxx.

Lemma 5.2. For |u−u∗|, |ux| and |uxx| sufficiently small, the system of 2n equations (5.41) has a

unique solution (v, w) = (v1, . . . , vn, w1, . . . , wn). The map (u, ux, uxx) 7→ (v, w) is smooth outside

the n manifolds Zi.=

{(v, w) ; vi = wi = 0

}; moreover it is C1,1, i.e. continuously differentiable

with Lipschitz continuous derivatives on a whole neighborhood of the point (u∗, 0, 0).

Proof. Consider the mapping Λ : IRn × IRn × IRn 7→ IR2n defined by

Λ(u, v, w) .=n∑

i=1

Λi(u, vi, wi), (5.43)

Λi(u, vi, wi).=

(vi ri

(u, vi, λ∗i − θ(wi/vi)

)(wi − λ∗i vi) ri

(u, vi, λ∗i − θ(wi/vi)

))

. (5.44)

Given u close to u∗ and (ux, uxx) in a neighborhood of (0, 0) ∈ IRn+n, we need to find vectors v, w

such that

Λ(u, v, w) =(ux, uxx −A(u)ux

). (5.45)

This will be achieved by applying the implicit function theorem.

43

Observe that each map Λi is well defined and continuous also when vi = 0, because in this

case (5.21) implies ri = ri(u). Computing the Jacobian matrix of partial derivatives w.r.t. (vi, wi)

we find

∂Λi

∂(vi, wi)=

(ri 0

−λ∗i ri ri

)

+(

viri,v + (wi/vi)θ′iri,σ −θ′iri,σ

wiri,v − λ∗i viri,v − λ∗i (wi/vi)θ′iri,σ + (wi/vi)2θ′iri,σ λ∗i θ′iri,σ − (wi/vi)θ′iri,σ

)

.= Bi + Bi .

(5.46)

Here and throughout the following, by θi, θ′i we denote the function θ and its derivative, evaluated

at the point s = wi/vi. Notice that the matrix valued function Bi is well defined and continuous

also when vi = 0. Indeed, either |wi/vi| > 2δ1, in which case θ′i = 0, or else |wi| ≤ 2δ1|vi|. In this

second case, by (5.30) we have ri,σ = O(1) · vi. In all cases we have the estimate

Bi(u, vi, wi) = O(1) · vi . (5.47)

According to (5.46), we can split the Jacobian matrix of partial derivatives of Λ as the sum

of two matrices:∂Λ

∂(v, w)= B(u, v, w) + B(u, v, w) . (5.48)

For (v, w) small, B has a uniformly bounded inverse, while B → 0 as (v, w) → 0. Since Λ(u, 0, 0) =

(0, 0) ∈ IR2n, by the implicit function theorem we conclude that the map (v, w) 7→ Λ(u, v, w) is C1

and invertible in a neighborhood of the origin, for |u− u∗| suitably small. We shall write Λ−1 for

this inverse mapping. In other words,

Λ−1(u, ux, ut) = (v, w) if and only if Λ(u, v, w) = (ux, ut) .

Having proved the existence and uniqueness of the decomposition, it now remains to study

its regularity. Since the cutoff function θ vanishes for |s| ≥ 3δ1, it is clear that each Λi is smooth

outside the manifold Zi.=

{(v, w) ; vi = wi = 0

}having codimension 2. We shall not write out

the second derivatives of the maps Λi explicitly. However, it is clear that

∂2Λ∂vi∂vj

=∂2Λ

∂vi∂wj=

∂2Λ∂wi∂wj

= 0 if i 6= j . (5.49)

Moreover, recalling that θi ≡ 0 for |wi/vi| > 2δi and using the bounds (5.30), we have the estimates

∂2Λ∂v2

i

,∂2Λ

∂vi∂wi,

∂2Λ∂w2

i

= O(1) . (5.50)

In other words, all second derivatives exist and are uniformly bounded outside the n manifolds Zi.

Therefore, Λ is continuously differentiable with Lipschitz continuous first derivatives on a whole

neighborhood of the point (u∗, 0, 0). Clearly the same holds for the inverse mapping Λ−1.

44

Remark 5.3. By possibly performing a linear transformation of variables, we can assume that

the matrix A(u∗) is diagonal, hence its eigenvectors r∗1 , . . . r∗n form an orthonormal basis:⟨r∗i , r∗j

⟩= δij . (5.51)

Observing that ∣∣ri(u, vi, σi)− r∗i∣∣ = O(1) · (|u− u∗|+ |vi|

), (5.52)

from (5.21) and the above assumption we deduce⟨ri(u, vi, σi), rj(u, vj , σj)

⟩= δij +O(1) · (|u− u∗|+ |vi|+ |vj |

)

= δij +O(1) · δ0 ,(5.53)

⟨ri, A(u)rj

⟩= O(1) · δ0 j 6= i . (5.54)

Another useful consequence of (5.51)-(5.52) is the following. Choosing the bound on the total

variation δ0 > 0 small enough, the vectors ri will remain close the an orthonormal basis. Hence

the decomposition (5.1) will satisfy

|ux| ≤∑

i

|vi| ≤ 2√

n|ux| . (5.55)

5.3. Bounds on derivative components

Relying on the bounds on the derivatives of u stated in Corollary 4.4, in this last section we

derive some analogous estimates, valid for the components vi, wi and their derivatives.

Lemma 5.4. In the same setting as Proposition 4.2, assume that the bound∥∥ux(t)

∥∥L1 ≤ δ0

holds on a larger time interval [0, T ]. Then for all t ∈ [t, T ], the decomposition (5.41) is well

defined. The components vi, wi satisfy the estimates∥∥vi(t)

∥∥L1 ,

∥∥wi(t)∥∥L1 = O(1) · δ0 , (5.56)

∥∥vi(t)∥∥L∞ ,

∥∥wi(t)∥∥L∞ ,

∥∥vi,x(t)∥∥L1 ,

∥∥wi,x(t)∥∥L1 = O(1) · δ2

0 , (5.57)∥∥vi,x(t)

∥∥L∞ ,

∥∥wi,x(t)∥∥L∞ = O(1) · δ3

0 . (5.58)

Proof. By Lemma 5.2, in a neighborhood of the origin the map (v, w) 7→ Λ(u, v, w) in (5.43) is

well defined, locally invertible, and continuously differentiable with Lipschitz continuous deriva-

tives. Hence, for δ0 > 0 suitably small, the L∞ bounds in (4.29) and (4.30) guarantee that the

decomposition (5.41) is well defined. From the identity (5.45) we deduce

vi, wi = O(1) · (|ux|+ |uxx|).

45

By (4.12) and (4.29)-(4.30) this yields the L1 bounds in (5.56) and the L∞ bounds in (5.57).

Differentiating (5.45) w.r.t. x we obtain

∂Λ∂u

ux +∂Λ

∂(v, w)(vx, wx) =

(uxx, uxxx −A(u)uxx −

(ux •A(u)

)ux

). (5.59)

Using the estimate∂Λ∂u

= O(1) · (|v|+ |w|),

since the derivative ∂Λ/∂(v, w) has bounded inverse, from (5.59) we deduce

(vx, wx) = O(1) ·(|uxx|+ |uxxx|+ |ux|2 + |ux|

(|v|+ |w|)).

Together with the bounds (4.29)–(4.31), this yields the remaining L1 estimates in (5.57) and the

L∞ estimates in (5.58).

Corollary 5.5. With the same assumptions as Lemma 5.2, one has the estimates

∥∥ri,x(t)∥∥L1 ,

∥∥λi,x(t)∥∥L1 = O(1) · δ0 , (5.60)

∥∥ri,x(t)∥∥L∞ ,

∥∥λi,x(t)∥∥L∞ = O(1) · δ2

0 . (5.61)

Proof. Recalling (5.30) we have

ri,x =∑

j

vj ri,urj + vi,xri,v + σi,xri,σ

= O(1) ·∑

j

|vj |+O(1) · |vi,x|+O(1) · |σi,x| |vi|

= O(1) ·∑

j

|vj |+O(1) · (|vi,x|+ |wi,x|).

The estimate for ri,x thus follows from the corresponding bounds in Lemma 5.2. Clearly λi,x =

O(1) · |ri,x|, hence it satisfies the same bounds.

In the remaining part of this section we shall examine the relations between wi and vi,x. Due

to the presence of a cutoff function, the results will depend on the ratio |wi/vi|. It is convenient

to recall here the bounds

|λi − λ∗i | = O(1) · |ri − r∗i | = O(1) · δ0 , |vi| , |wi| = O(1) · δ20 , (5.62)

which follow from (5.52) and (5.57). Moreover, one should keep in mind our choice of the constants

0 < δ0 << δ1 << 1 . (5.63)

46

From the basic relations (5.41) and the identity ut + A(u)ux = uxx it follows

∑

i

wiri +∑

i

(A(u)− λ∗i )viri

=∑

i

vi,xri +∑

ij

viri,u vj rj +∑

i

viri,vvi,x +∑

i

viri,σσi,x .

Taking the inner product with ri and recalling that ri has unit norm and is thus orthogonal to its

derivatives ri,v, ri,urj , we obtain

wi + (λi − λ∗i )vi = vi,x + Θi, (5.64)

where

Θi.=

∑

j 6=i

⟨ri,

(λ∗j −A(u)

)rj

⟩vj +

∑

j 6=i

∑

k

⟨ri, rj,urk

⟩vjvk

+∑

j 6=i

⟨ri, rj,v

⟩vjvj,x +

∑

j 6=i

⟨ri, rj,σ

⟩vjσj,x +

∑

j 6=i

⟨ri, rj

⟩(vj,x − wj)

= O(1) · δ0

∑

j 6=i

(|vj |+ |wj − vj,x|

).

(5.65)

The above estimate on Θi is obtained using (5.54) together with the L∞ bounds in (5.62) and the

bound rj,σ = O(1) · vi in (5.30). Summing (5.64) over i = 1, . . . , n and using (5.61)-(5.62), from

(5.64) we obtain ∑

i

|wi − vi,x| = O(1) · δ0

∑

j

|vj | . (5.66)

This yields the implications

|wi| < 3δ1|vi| =⇒ vi,x = O(1) · vi +O(1) · δ0

∑

j 6=i

|vj | , (5.67)

|wi| > δ1|vi| =⇒ vi = O(1) · vi,x +O(1) · δ0

∑

j 6=i

|vj | . (5.68)

The following technical lemma will be used in the estimate of the sources due to cut-off terms.

Lemma 5.6. If |wi/vi| ≥ 3δ1/5, then

|wi| ≤ 2|vi,x|+O(1) · δ0

∑

j 6=i

|vj |, |vi| ≤ 52δ1

|vi,x|+O(1) · δ0

∑

j 6=i

|vj | . (5.69)

On the other hand, if |wi/vi| ≤ 4δ1/5, then

|vi,x| ≤ δ1|vi|+O(1) · δ0

∑

j 6=i

|vj | . (5.70)

47

Proof. The analysis is based on the formula (5.64). By (5.62)-(5.63), from the condition |wi/vi| ≥3δ1/5 two cases can arise. On one hand, if

|Θi| ≤ δ1

10|vi| , (5.71)

then

|vi,x| ≥ 3δ1

5|vi| − O(1) · δ0|vi| − δ1

10|vi| ≥ 2δ1

5|vi| ,

and hence

|vi| ≤ 52δ1

|vi,x|, |wi| ≤ |vi,x|+O(1) · δ0|vi|+ δ1

10|vi| ≤ 2|vi,x| . (5.72)

On the other hand, if (5.71) fails, using (5.65)-(5.66) and then (5.62) we find

|vi| = O(1) · δ0

∑

j 6=i

|vj | ,∣∣λi − λ∗i

∣∣ |vi| = O(1) · δ20

∑

j 6=i

|vj | . (5.73)

In both cases, the estimates in (5.69) hold.

Next, if |wi/vi| ≤ 4δ1/5, from (5.64)–(5.66) we deduce

|vi,x| ≤ 4δ1

5|vi|+O(1) · δ0 |vi|+O(1) · δ0

∑

j 6=i

|vj | .

By (5.63) we can assume that O(1) · δ0 ≤ δ1/10. If (5.71) holds, we now conclude |vi,x| ≤ δ1|vi|. If

(5.71) fails, then (5.73) is valid. In both cases we have (5.70).

6 - Interaction of Viscous Waves

Let u = u(t, x) be a solution of the system

ut + A(u)ux = uxx . (6.1)

According to the analysis of the previous chapter, at each point we can decompose the gradient ux

in terms of viscous travelling waves. Assuming that ux, uxx remain sufficiently small, by Lemma

51 the corresponding components vi, wi in (5.41) are then well defined. The equations governing

the evolution of these 2n components can be written in the form

{vi,t + (λivi)x − vi,xx = φi ,

wi,t + (λiwi)x − wi,xx = ψi ,(6.2)

48

with λi.=

⟨ri , A(u)ri

⟩, as in (5.26). Here φi, ψi are appropriate functions involving u, v, w and

their first first order derivatives. Writing out their explicit form requires lengthy calculations. We

shall only outline the basic procedure.

Differentiating (6.1) one finds that the vector (ux, ut) = Λ(u, v, w) satisfies the evolution

equation(

ux

ut

)

t

+([

A(u) 00 A(u)

](ux

ut

))

x

−(

ux

ut

)

xx

=(

0(ux •A(u)

)ut −

(ut •A(u)

)ux

). (6.3)

Observe that, in the conservative case A(u) = Df(u), the right hand side vanishes because

(ux •A(u)

)ut =

(ut •A(u)

)ux = D2f(u) (ux ⊗ ut) .

For notational convenience, we introduce the variable z.= (v, w) and write λ for the 2n× 2n

diagonal matrix with entries λi defined at (5.26):

λ.=

(diag(λi) 0

0 diag(λi)

).

From (6.3) it now follows

∂Λ∂u

ut+∂Λ∂z

(vw

)

t

+([

A(u) 00 A(u)

]Λ

)

x

− ∂Λ∂z

(vw

)

xx

− ∂Λ∂u

uxx

− ∂2Λ∂u[2]

(ux ⊗ ux)− ∂2Λ∂z[2]

·(

vx

wx

)⊗

(vx

wx

)− 2

∂2Λ∂u ∂z

ux ⊗(

vx

wx

)

=(

0(ux •A(u)

)ut −

(ut •A(u)

)ux

).

Therefore,

∂Λ∂z

[(vw

)

t

+(

λ

(vw

))

x

−(

vw

)

xx

]

=∂Λ∂z

(λ

(vw

))

x

−([

A(u) 00 A(u)

]Λ

)

x

+∂Λ∂u

A(u)ux +(

0(ux •A(u)

)ut −

(ut •A(u)

)ux

)

+∂2Λ∂u[2]

(ux ⊗ ux) +∂2Λ∂z[2]

·(

vx

wx

)⊗

(vx

wx

)+ 2

∂2Λ∂u ∂z

ux ⊗(

vx

wx

)

.= E(6.4)

Multiplying both sides of (6.4) by the inverse of the matrix differential ∂Λ/∂z, one obtains an

expression for the source terms φi, ψi in (6.2). For our purposes, however, we only need an upper

bound for the norms ‖φi‖L1 and ‖ψi‖L1 . In this direction we observe that, since ∂Λ/∂z has

uniformly bounded inverse, one has the bounds

φi = O(1) · |E| , ψi = O(1) · |E|, i = 1, . . . , n . (6.5)

49

The right hand side of (6.4) can be computed by differentiating the map Λ in (5.43) and using the

identities (5.39) and (5.28).

Before giving a precise estimate on the size of the source terms, we provide an intuitive

explanation of how they arise. Consider first the special case where u is precisely one of the

travelling wave profiles on the center manifold (fig. 16a), say u(t, x) = Uj(x− σjt). We then have

ux = vj rj , ut = (wj − λ∗jvj)rj , vi = wi = 0 for i 6= j ,

and therefore {vi,t + (λivi)x − vi,xx = 0 ,

wi,t + (λiwi)x − wi,xx = 0 .

Indeed, this is obvious when i 6= j. The identity φj = 0 was proved in (5.35), while the relation

wj = (λ∗j − σj)vj implies ψj = 0.

jU

Uk

’xx

u =Uj

figure 16a

x ’x

u

UU

x0

j

Uk

Uj

figure 16b

Next, consider the case of a general solution u = u(t, x). The sources on the right hand sides

of (6.2) arise for three different reasons (fig. 16b).

1. The ratio |wj/vj | is large and hence the cutoff function θ in (5.40) is active. Typically, this

will happen near a point x0 where ux = 0 but ut = uxx 6= 0. In this case the identity (5.35)

50

fails because of a “wrong” choice of the speed: σj 6= λ∗j − (wj/vj). The difference in the speed is∣∣θj − (wj/vj)∣∣, where θj denotes the value of θ at s = wj/vj . In the corresponding source terms, a

detailed analysis will show that this cutoff error always enters multiplied by a factor vjvj,x or by

vjwj,x. Its strength is thus of order O(1) · (|vj,x|+ |wj,x|) ∣∣wj − θjvj

∣∣.

2. Waves of two different families j 6= k are present at a given point x. These will produce

quadratic source terms, due to transversal interactions. The strength of these terms is estimated

by the product of different components of v, w and of their first order derivatives. For example:

|vjvk|, |vjwk|, |vj,xwk| . . .

3. Since the decomposition (5.39) is defined pointwise, it may well happen that the travelling j-

wave profile Uj at a point x is not the same as the profile Uj at a nearby point x′. This is the case

whenever these two travelling waves have different speeds. It is the rate of change in this speed,

i.e. σj,x, that determines the infinitesimal interaction between nearby waves of the same family.

A detailed analysis will show that the corresponding source terms can only be linear or quadratic

w.r.t. σj,x, with the square of the strength of the wave always appearing as a factor. These terms

can thus be estimated as O(1) ·v2j σj,x +O(1) ·v2

j σ2j,x. Observing that σj,x = (wj/vj)xθ′j and θ′j = 0

when |wj/vj | ≥ 3, this motivates the presence of the remaining terms in the following lemma.

Lemma 6.1. The source terms in (6.2) satisfy the bounds

φi, ψi =O(1) ·∑

j

(|vj,x|+ |wj,x|) · |wj − θjvj | (cutoff error)

+O(1) ·∑

j

|vj,xwj − vjwj,x| (change in speed, linear)

+O(1) ·∑

j

∣∣∣∣vj

(wj

vj

)

x

∣∣∣∣2

· χ{|wj/vj |<3δ1

} (change in speed, quadratic)

+O(1) ·∑

j 6=k

(|vjvk|+ |vj,xvk|+ |vjwk|+ |vj,xwk|+ |vjwk,x|+ |wjwk|)

(interaction of waves of different families)

A proof of this Lemma, involving lengthy computations, is given in [BiB].

6.1. Transversal wave interactions

In this subsection we establish an a priori bound on the total amount of interactions between

waves of different families. More precisely, let u = u(t, x) be a solution of the parabolic system

(6.1) and assume that ∥∥ux(t)∥∥L1 ≤ δ0 t ∈ [0, T ] . (6.6)

51

In this case, for t ≥ t, by Corollary 4.3 all higher derivatives will be suitably small and we can thus

define the components vi, wi according to (5.39). These will satisfy the linear evolution equation

(6.2), with source terms φi, ψi estimated in Lemma 6.1. Assuming that

∫ T

t

∫ {∣∣φi(t, x)∣∣ +

∣∣ψi(t, x)∣∣}

dxdt ≤ δ0 , i = 1, . . . , n, (6.7)

and relying on the bounds (5.56)–(5.58) on vi, wi and their derivatives, we shall prove the sharper

estimate

∫ T

t

∫ ∑

j 6=k

(|vjvk|+ |vj,xvk|+ |vjwk|+ |vj,xwk|+ |vjwk,x|+ |wjwk|)dxdt = O(1) · δ2

0 . (6.8)

As a preliminary, we establish a more general estimate on solutions of two independent linear

parabolic equations, with strictly different drifts.

Lemma 6.2. Let z, z] be solutions of the two independent scalar equations

{zt +

(λ(t, x)z

)x− zxx = ϕ(t, x) ,

z]t +

(λ](t, x)z]

)x− z]

xx = ϕ](t, x) ,(6.9)

defined for t ∈ [0, T ]. Assume that

inft,x

λ](t, x) − supt,x

λ(t, x) ≥ c > 0 . (6.10)

Then

∫ T

0

∫ ∣∣z(t, x)∣∣ ∣∣z](t, x)

∣∣ dxdt

≤ 1c

(∫ ∣∣z(0, x)∣∣ dx +

∫ T

0

∫ ∣∣ϕ(t, x)∣∣ dxdt

)(∫ ∣∣z](0, x)∣∣ dx +

∫ T

0

∫ ∣∣ϕ](t, x)∣∣ dxdt

).

(6.11)

Proof. We consider first the homogeneous case, where ϕ = ϕ] = 0. Define the interaction potential

Q(z, z]) .=∫∫

K(x− y)∣∣z(x)

∣∣ ∣∣z](y)∣∣ dxdy , (6.12)

with

K(s) .={

1/c if s ≥ 0,1/c · ecs/2 if s < 0. (6.13)

52

Computing the distributional derivatives of the kernel K we find that cK ′ − 2K ′′ is precisely the

Dirac distribution, i.e. a unit mass at the origin. A direct computations now yields

d

dtQ

(z(t), z](t)

)

=d

dt

∫∫K(x− y)

∣∣z(x)∣∣ ∣∣z](y)

∣∣ dxdy

=∫∫

K(x− y){(

zxx − (λz)x

)sgnz(x)

∣∣z](y)∣∣ +

∣∣z(x)∣∣(z]

yy − (λ]z])y

)sgnz](y)

}dxdy

≤∫∫

K ′(x− y){

λ∣∣z(x)

∣∣ ∣∣z](y)∣∣− λ]

∣∣z(x)∣∣ ∣∣z](y)

∣∣}

dxdy

+∫∫

K ′′(x− y){∣∣z(x)

∣∣ ∣∣z](y)∣∣ +

∣∣z(x)∣∣ ∣∣z](y)

∣∣}

dxdy

≤ −∫∫ (

cK ′ − 2K ′′)∣∣z(x)∣∣ ∣∣z](y)

∣∣ dxdy

= −∫ ∣∣z(x)

∣∣ ∣∣z](x)∣∣ dx

Therefore∫ T

0

∫ ∣∣z(t, x)∣∣ ∣∣z](t, x)

∣∣ dxdt ≤ Q(z(0), z](0)

) ≤ 1c

∥∥z(0)∥∥L1

∥∥z](0)∥∥L1 , (6.14)

proving the lemma in the homogeneous case.

To handle the general case, call Γ, Γ] the Green functions for the corresponding linear ho-

mogenous systems. The general solution of (6.9) can thus be written in the form

z(t, x) =∫

Γ(t, x, 0, y)z(0, y)dy +∫ t

0

∫Γ(t, x, s, y)ϕ(s, y)dyds,

z](t, x) =∫

Γ](t, x, 0, y)z](0, y)dy +∫ t

0

∫Γ](t, x, s, y)ϕ](s, y)dyds.

(6.15)

From (6.14) it follows∫ T

max{s,s′}

∫Γ(t, x, s, y) · Γ](t, x, s′, y′) dxdt ≤ 1

c(6.16)

for every couple of initial points (s, y) and (s′, y′). The estimate (6.11) now follows from (6.16)

and the representation formula (6.15).

Remark 6.3. Exactly the same estimate (6.11) would be true also for a system without viscosity.

In particular, if

zt +(λ(t, x)z

)x

= 0, z]t +

(λ](t, x)z]

)x

= 0,

and if the speeds satisfy the gap condition (6.10), then

d

dt

[1c

∫∫

x<y

∣∣z](t, x)z(t, y)∣∣ dxdy

]≤ −

∫ ∣∣z(t, x)∣∣ ∣∣z](t, x)

∣∣ dx .

53

Notice that the left hand side can be written as (d/dt)Q(z, z]), with Q as in (6.12) but now

K(s) .={

1/c if s ≥ 0,0 if s < 0.

In the case where viscosity is present, our definition (6.12)-(6.13) thus provides a natural counter-

part to the Glimm interaction potential between waves of different families, introduced in [G] for

strictly hyperbolic systems.

Lemma 6.2 allows us to estimate the integral of the terms |vivk|, |vjwk| and |wjwk| in (6.8).

We now work toward an estimate of the remaining terms |vj,xvk|, |vj,xwk| and |vjwk,x|, containing

one derivative w.r.t. x.

Lemma 6.4. Let z, z] be solutions of (6.9) and assume that (6.10) holds, together with the esti-

mates∫ T

0

∫ ∣∣ϕ(t, x)∣∣ dxdt ≤ δ0,

∫ T

0

∫ ∣∣ϕ](t, x)∣∣ dxdt ≤ δ0 , (6.17)

∥∥z(t)∥∥L1 ,

∥∥z](t)∥∥L1 ≤ δ0 ,

∥∥zx(t)∥∥L1 ,

∥∥z](t)∥∥L∞ ≤ C∗δ2

0 , (6.18)

∥∥λx(t)∥∥L∞ ,

∥∥λx(t)∥∥L1 ≤ C∗δ0 , lim

x→−∞λ(t, x) = 0 (6.19)

for all t ∈ [0, T ]. Then one has the bound

∫ T

0

∫ ∣∣zx(t, x)∣∣ ∣∣z](t, x)

∣∣ dxdt = O(1) · δ20 . (6.20)

Proof. The left hand side of (6.20) is clearly bounded by the quantity

I(T ) .= sup(τ,ξ)∈[0,T ]×IR

∫ T−τ

0

∫ ∣∣zx(t, x)z](t + τ, x + ξ)∣∣ dxdt ≤ (C∗δ2

0)2 · T ,

the last inequality being a consequence of (6.18). For t > 1 we can write zx in the form

zx(t, x) =∫

Gx(1, y)z(t− 1, x− y) dy +∫ 1

0

∫Gx(s, y)

[ϕ− (λz)x

](t− s, x− y) dyds ,

54

where G(t, x) .= exp{−x2/4t}/2√

πt is the standard heat kernel. Using (6.11) we obtain

∫ T−τ

1

∫ ∣∣zx(t, x) z](t + τ, x + ξ)∣∣ dxdt

≤∫ T−τ

1

∫∫ ∣∣∣Gx(1, y) z(t− 1, x− y) z](t + τ, x + ξ)∣∣∣ dydxdt

+∫ T−τ

1

∫ ∫ 1

0

∫ ∥∥λx‖L∞∣∣∣Gx(s, y) z(t− s, x− y) z](t + τ, x + ξ)

∣∣∣ dydsdxdt

+∫ T−τ

0

∫ ∫ 1

0

∫ ∥∥λ‖L∞∣∣∣Gx(s, y) zx(t− s, x− y) z](t + τ, x + ξ)

∣∣∣ dydsdxdt

+∫ T−τ

1

∫ ∫ t

t−1

∫ ∣∣∣Gx(t− s, x− y)ϕ(s, y) z](t + τ, x + ξ)∣∣∣ dydsdxdt

≤(∫ ∣∣Gx(1, y)

∣∣ dy + ‖λx‖L∞∫ 1

0

∫ ∣∣Gx(s, y)∣∣ dyds

)

· sups,y,τ,ξ

(∫ T−τ

1

∫ ∣∣z(t− s, x− y)∣∣ ∣∣z](t + τ, x + ξ)

∣∣ dxdt

)

+(‖λ‖L∞ ·

∫ 1

0

∫ ∣∣Gx(s, y)∣∣ dyds

)·(

sups,y,τ,ξ

∫ T−τ

1

∫ ∣∣zx(t− s, x− y)∣∣ ∣∣z](t + τ, x + ξ)

∣∣ dxdt

)

+ ‖z]‖L∞ ·∫ 1

0

∫ ∣∣Gx(s, y)∣∣ dsdy ·

∫ T

0

∫ ∣∣ϕ(t, x)∣∣ dxdt

≤(

1√π

+ ‖λx‖L∞ 2√π

)4δ2

0

c+ ‖λ‖L∞ 2√

πI(T ) + C∗δ2

0

2√π

δ0 .

(6.21)

On the initial time interval [0, 1], by (6.18) one has

∫ 1

0

∫ ∣∣zx(t, x) z](t + τ, x + ξ)∣∣ dxdt ≤

∫ 1

0

∥∥zx(t)∥∥L1

∥∥z](t + τ)∥∥L∞ dt ≤ (C∗δ2

0)2 . (6.22)

Moreover, (6.19) implies

‖λ‖L∞ ≤ ‖λx‖L1 ≤ C∗δ0 << 1 .

From (6.21) and (6.22) it thus follows

I(T ) ≤ (C∗δ20)2 +

4δ20

c+

12I(T ) + C∗δ3

0 .

For δ0 sufficiently small, this implies I(T ) ≤ 9δ20/c, proving the lemma.

Using the two previous lemmas we now prove the estimate (6.8). Setting z.= vj , z] .= vk,

λ.= λj , λ] .= λk, an application of Lemma 6.2 yields the desired bound on the integral of |vjvk|.

Moreover, Lemma 6.4 allows us to estimate the integral of |vj,xvk|. Notice that the assumptions in

(6.18) are a consequence of (5.56)-(5.57), while (6.19) follows from (5.60)-(5.61). The simplifying

condition λ(t,−∞) = 0 in (6.19) can be easily achieved, using a new space coordinate x′ .= x−λ∗j t.

55

The other terms |vjwk|, |wjwk|, |vj,xwk| and |vjwk,x| are handled similarly.

6.3. Interaction of waves of the same family

We now study the interaction of viscous waves of the same family. As in the previous section,

let u = u(t, x) be a solution of the parabolic system (6.1) whose total variation remains bounded

according to (6.6). Assume that the components vi, wi satisfy the evolution equation (6.2), with

source terms φi, ψi bounded as in (6.7). Using the bounds (5.56)–(5.58), for each i = 1, . . . , n we

shall prove the estimates

∫ T

t

∫|wi,xvi − wivi,x| dxdt = O(1) · δ2

0 , (6.23)

∫ T

t

∫

|wi/vi|<3δ1

|vi|2∣∣∣∣(

wi

vi

)

x

∣∣∣∣2

dxdt = O(1) · δ30 . (6.24)

The above integrals will be controlled in terms of two functionals, related to shortening curves.

Consider a parameterized curve in the plane γ : IR 7→ IR2. Assuming that γ is sufficiently smooth,

its length is computed by

L(γ) .=∫ ∣∣γx(x)

∣∣ dx . (6.25)

We can also define the area functional as the integral of a wedge product:

A(γ) .=12

∫∫

x<y

∣∣γx(x) ∧ γx(y)∣∣ dxdy . (6.26)

To understand its geometrical meaning, observe that if γ is a closed curve, the integral

12

∫γ(y) ∧ γx(y) dy =

12

∫∫

x<y

γx(x) ∧ γx(y) dx dy

yields the sum of the areas of the regions enclosed by the curve γ, multiplied by the corresponding

winding number. In general, the quantity A(γ) provides un upper bound for the area of the convex

hull of γ.

Let now γ = γ(t, x) be a planar curve which evolves in time, according to the vector equation

γt + λγx = γxx . (6.27)

Here λ = λ(t, x) is a sufficiently smooth scalar function. It is then clear that the length L(γ(t)

)

of the curve is a decreasing function of time. One can show that also the area functional A(γ(t)

)

is monotonically decreasing. Moreover, the amount of decrease dominates the area swept by the

56

curve during its motion. An intutive way to see this is the following. In the special case where γ

is a polygonal line, with vertices at the points P0, . . . , Pm, the integral in (6.26) reduces to a sum:

A(γ) =12

∑

i<j

∣∣vi ∧ vj

∣∣ , vi.= Pi − Pi−1 .

If we now replace γ by a new curve γ′ obtained by replacing two consecutive edges vh, vk by one

single edge (fig. 17b), the area between γ and γ′ is precisely |vh∧vk|/2, while an easy computation

yields

A(γ′) ≤ A(γ)− 12|vh ∧ vk| .

The estimate on the area swept by a smooth curve (fig. 17a) is now obtained by approximating a

shortening curve γ by a sequence of polygonals, each obtained from the previous one by replacing

two consecutive edges by a single segment.

γ

γ’

vγ

v

γ’k

h

figure 17a figure 17b

We shall apply the previous geometric considerations toward a proof of the estimates of (6.23)-

(6.24). Let v, w be two scalar functions, satisfying

vt + (λv)x − vxx = φ ,

wt + (λw)x − wxx = ψ .(6.28)

Define the planar curve γ by setting

γ(t, x) =(∫ x

−∞v(t, y)dy,

∫ x

−∞w(t, y)dy

). (6.29)

Integrating (6.27) w.r.t. x, one finds the corresponding evolution equation for γ :

γt + λγx − γxx = Φ(t, x) .=(∫ x

−∞φ(t, y)dy,

∫ x

−∞ψ(t, y)dy

). (6.30)

In particular, if no sources were present, the motion of the curve would reduce to (6.27). At each

fixed time t, we now define the Length Functional as

L(t) = L(γ(t)

)=

∫ √v2(t, x) + w2(t, x) dx (6.31)

57

and the Area Functional as

A(t) = A(γ(t)

)=

12

∫∫

x<y

∣∣v(t, x)w(t, y)− v(t, y)w(t, x)∣∣ dxdy . (6.32)

We now estimate the time derivative of the above functionals, in the general case when sources

are present.

Lemma 6.5 Let v, w be solutions of (6.2), defined for t ∈ [0, T ]. For each t, assume that the maps

x 7→ v(t, x) x 7→ w(t, x) and x 7→ λ(t, x) are C1,1, i.e. continuously differentiable with Lipschitz

derivative. Then the corresponding area functional (6.32) satisfies

d

dtA(t) ≤ −

∫ ∣∣∣vx(t, x)w(t, x)−v(t, x)wx(t, x)∣∣∣dx+

∥∥v(t)∥∥

L1

∥∥ψ(t)∥∥

L1 +∥∥w(t)

∥∥L1

∥∥φ(t)∥∥

L1 . (6.33)

Proof. In the following, given a curve γ, at each point x where γx 6= 0 we define the unit normal

n = n(x) oriented so that γx(x) ∧ n =∣∣γx(x)

∣∣ > 0. For every vector v ∈ IR2 this implies

γx(x) ∧ v =∣∣γx(x)

∣∣ ⟨n , v

⟩.

Given a unit vector n, we shall also consider the projection of γ along n, namely

y 7→ χn(y) .=⟨n , γ(y)

⟩.

If γ = γ(t, x) is any smooth curve evolving in time, the time derivative of the area functional in

(6.26) can be computed as

dAdt

=12

∫ ∫

x<y

sgn(γx(x) ∧ γx(y)

){γxt(x) ∧ γx(y) + γx(x) ∧ γxt(y)

}dxdy

=12

∫ ∫sgn

(γx(x) ∧ γx(y)

) {γx(x) ∧ γxt(y)

}dydx

=12

∫ ∣∣γx(x)∣∣(

d

dt

∫ ∣∣∣⟨n , γx(y)

⟩∣∣∣ dy

)dx

=12

∫ ∣∣γx(x)∣∣(∫

sgn⟨n , γx(y)

⟩ · ⟨n , γxt(y)⟩dy

)dx

=12

∫ ∣∣γx(x)∣∣ d

dt

(Tot.Var.{χn}

)dx

For each x, we are here choosing the unit normal n = n(x) to the curve γ at the point x. To

compute the derivative of the total variation, assume that the function y 7→ χn(x)(y) has a finite

number of local maxima and minima, say attained at the points (fig. 18)

y−p < · · · < y−1 < y0 = x < y1 < · · · < yq .

58

Assume, in addition, that its derivative dχn/dy changes sign across every such point. Then

d

dt

(Tot.Var.{χn}

)= −sgn

⟨n , γxx(x)

⟩ · 2∑

−p≤α≤q

(−1)α⟨n , γt(yα)

⟩. (6.34)

Notice the sign factor in front of (6.34). If the inner product⟨n , γxx(x)

⟩is positive, the even

indices α correspond to local minima and the odd indices to local maxima. The opposite is true if

the inner product is negative. Using (6.34) and observing that

⟨n , γx(x)

⟩= 0 , sgn

⟨n , γxx(yα)

⟩= (−1)α · sgn

⟨n , γxx(x)

⟩,

we obtain

dAdt

= −∫ ∣∣γx(x)

∣∣sgn⟨n , γxx(x)

⟩ · ∑

−p≤α≤q

(−1)α⟨n , γt(yα)

⟩ dx

≤ −∫ ∑

α

∣∣γx(x) ∧ γxx(yα)∣∣ dx +

∫ ∣∣γx(x)∣∣ ·

∣∣∣∣∣∑α

(−1)α⟨n , Φ(yα)

⟩∣∣∣∣∣ dx

≤ −∫ ∣∣γx(x) ∧ γxx(x)

∣∣ dx +∫∫ ∣∣γx(x) ∧ Φx(y)

∣∣ dydx .

(6.35)

If we specialize this formula to the case where γ is the curve in (6.29) and Φ the function in (6.30),

we finddAdt

≤ −∫ ∣∣∣∣

(v(x)w(x)

)∧

(vx(x)wx(x)

)∣∣∣∣ dx +∫∫ ∣∣∣∣

(v(x)w(x)

)∧

(φ(y)ψ(y)

)∣∣∣∣ dydx ,

from which (6.33) clearly follows. Notice that, by an approximation argument, we can assume that

the functions χn(x) have the required regularity for almost every (t, x) ∈ IR2.

γ(x)

γ(y )

γ(y )

n

1

2

γ

figure 18

59

Lemma 6.6. Together with the hypotheses of Lemma 6.5, at a fixed time t assume that γx(t, x) 6= 0

for every x. Then

d

dtL(t) ≤ − 1

(1 + 9δ21)3/2

∫

|w/v|≤3δ1

|v|∣∣∣(w

v

)x

∣∣∣2

dx +∥∥φ(t)

∥∥L1 +

∥∥ψ(t)∥∥L1 . (6.36)

Proof. As a preliminary, recalling that

γx = (v, w) , γxt + (λγx)x − γxxx = (φ, ψ) ,

we derive the identities

|γxx|2 |γx|2 −⟨γx, γxx

⟩2 =(v2

x + w2x

)(v2 + w2)− (

vvx + wwx

)2

=(vwx − vxw

)2 = v4∣∣(w/v)x

∣∣2,

|v|3|γx|3 =

1(1 + (w/v)2

)3/2.

Thanks to the assumption that γx never vanishes, we can now integrate by parts and obtain

d

dtL(t) =

∫ ⟨γx, γxt

⟩√⟨

γx, γx

⟩ dx =∫ {⟨

γx, γxxx

⟩

|γx| −⟨γx, (λγx)x

⟩

|γx| +

⟨γx, (φ, ψ)

⟩

|γx|

}dx

=∫ {

|γx|xx −(λ |γx|

)x− |γxx|2 −

⟨γx/|γx|, γxx

⟩2

|γx|

}dx +

∫ ⟨γx, (φ, ψ)

⟩

|γx| dx

≤ −∫ |v| ∣∣(w/v)x

∣∣2(1 + (w/v)2

)3/2dx +

∥∥φ(t)∥∥L1 +

∥∥ψ(t)∥∥L1 .

Since the integrand is non-negative, the last inequality clearly implies (6.36).

Remark 6.7. Let u = u(t, x) be a solution to a scalar, viscous conservation law

ut + f(u)x − uxx = 0 ,

and consider the planar curve γ.=

(u, f(u)−ux

)whose components are respectively the conserved

quantity and the flux (fig. 19). If λ.= f ′, the components v

.= ux and w.= −ut evolve according

to (6.27). Defining the speed s(x) .= −ut(x)/ux(x), the area functional A(γ) in (6.26) can now be

written asA(γ) =

12

∫∫

x<y

∣∣ux(x)ut(y)− ut(x)ux(y)∣∣ dxdy

=12

∫∫

x<y

∣∣ux(x) dx∣∣ ·

∣∣ux(y) dy∣∣ ·

∣∣s(x)− s(y)∣∣

=12

∫∫

x<y

[wave at x]× [wave at y]× [difference in speeds] .

60

c

f

a c db

γ

x

d

b

a

u

figure 19

It now becomes clear that the area functional can be regarded as an interaction potential between

waves of the same family. In the case where viscosity is present, this provides a counterpart to the

interaction functional introduced by Liu in [L4] for general hyperbolic systems.

Remark 6.8. In the case where u is precisely a viscous travelling wave, the curve γ =(u, f(u)−ux

)

reduces to a segment. Assume now that the flux f is genuinely nonlinear, say with f ′′ ≥ c > 0.

Consider a solution u which initially consists of two viscous travelling waves, far apart from each

other (fig. 20). To fix the ideas, let the first wave join a left state a with a middle state b, and

the second wave join the middle state b with a right state c, with a > b > c. The strength of the

two waves can be measured as s = a− b, s′ = b− c. The corresponding curve γ is approximately

given by two segments, joining the points P.=

(a, f(a)

), Q

.=(b, f(b)

), P ′ .=

(c, f(c)

). After a

long time τ >> 0, the two shocks will interact, merging into one single viscous shock. The curve

γ(τ) will then reduce to one single segment, joining P with P ′. The area swept by the curve is

approximately the area of the triangle PQP ′. The assumption of genuine nonlinearity implies

area swept = O(1) · |ss′| (|s|+ |s′|).

In this case, the decrease in the area functional is of cubic order w.r.t. the strengths s, s′ of the

interacting waves. This is indeed the correct order of magnitude needed to control the strength of

new waves generated by the interaction in the genuinely nonlinear case. It is remarkable that this

area functional gives the correct order of magnitude of waves generated by an interaction also for

a general flux function f , not necessarily convex.

We now return to the proof of the crucial estimates (6.23)-(6.24). Recalling that the compo-

nents vi, wi satisfy the equations (6.2), we can apply the previous lemmas with v.= vi, w

.= wi,

λ.= λi, φ

.= φi, ψ.= ψi, calling Li and Ai the corresponding length and area functionals. For

t ∈ [t, T ], the bounds (5.56)-(5.57) yield

Ai(t) ≤∥∥vi(t)

∥∥L∞ · ∥∥wi(t)

∥∥L1 = O(1) · δ3

0 , (6.37)

61

γ(0)

γ(τ)

u(0)

u( )τs

s

Q

P

P

u

f

f(u)−u

b

b

x

’

c a

a

c

’

figure 20

Li(t) ≤∥∥vi(t)

∥∥L1 +

∥∥wi(t)∥∥L1 = O(1) · δ0. (6.38)

Using (6.33) we now obtain∫ T

t

∫ ∣∣wi,xvi − wivi,x

∣∣ dxdt ≤∫ T

t

∣∣∣∣d

dtAi(t)

∣∣∣∣ dt +∫ T

t

(∥∥vi(t)∥∥

L1

∥∥ψi(t)∥∥

L1 +∥∥wi(t)

∥∥L1

∥∥φi(t)∥∥

L1

)dt

≤ Ai(t) + supt∈[t,T ]

(∥∥vi(t)∥∥L1 +

∥∥wi(t)∥∥L1

)·∫ T

t

∫ (∣∣φi(t, x)∣∣ +

∣∣ψi(t, x)∣∣)

dxdt

= O(1) · δ20 ,

(6.39)

proving (6.23). To establish (6.24), we first observe that, by an approximation argument, it is not

restrictive to assume that the set of points in the t-x plane where vi,x(t, x) = wi,x(t, x) = 0 is at

most countable. In this case, for almost every t ∈ [t, T ] the inequality (6.36) is valid, and hence∫ T

t

∫

|wi/vi|<3δ1

|vi|∣∣∣∣(

wi

vi

)

x

∣∣∣∣2

dxdt ≤∫ T

t

∣∣∣∣d

dtLi(t)

∣∣∣∣ dt +∫ T

t

(∥∥φi(t)∥∥

L1 +∥∥ψi(t)

∥∥L1

)dt

≤ Li(t) +∫ T

t

∫ (∣∣φi(t, x)∣∣ +

∣∣ψi(t, x)∣∣)dxdt

= O(1) · δ0.

(6.40)

The estimate (6.24) is now a consequence of (6.40) and of the bound ‖vi(t)‖L∞ = O(1) · δ30 , given

at (5.57).

62

6.4. Bounds on cutoff terms

In this section we provide an estimate on the total strength of source terms due to the cutoff

function. In the same setting as the two previous sections, in particular assuming that (6.7) holds,

we claim that ∫ T

t

∫ (|vi,x|+ |wi,x|) |wi − θivi| dxdt = O(1) · δ2

0 . (6.41)

We recall that θi.= θ(wi/vi), where θ is the cut-off function introduced at (5.40). Notice that the

integrand can be 6= 0 only when |wi/vi| > δ1.

Consider another cut-off function η : IR 7→ [0, 1] such that (fig. 15)

η(s) ={

0 if |s| ≤ 3δ1/5 ,1 if |s| ≥ 4δ1/5 . (6.42)

We can assume that η is a smooth even function, such that

|η′| ≤ 21/δ1 , |η′′| ≤ 101/δ21 .

For convenience, we shall write ηi.= η(wi/vi).

Toward a proof of the estimate (6.41), we first reduce the integrand to a more tractable

expression. Since the term |wi − θivi| vanishes when |wi/vi| ≤ δ1, and is ≤ |wi| otherwise, by

(5.69) we always have the bound

|wi − θivi| ≤ |ηiwi| ≤ ηi

2|vi,x|+O(1) · δ0

∑

j 6=i

|vj | .

Therefore

(|vi,x|+ |wi,x|) · |wi − θivi| ≤

(|vi,x|+ |wi,x|)ηi

2|vi,x|+O(1) · δ0

∑

j 6=i

|vj |

≤ 2ηiv2i,x + 2ηi|vi,xwi,x|+

∑

j 6=i

(|vjvi,x|+ |vjwi,x|)

≤ 3ηiv2i,x + ηiw

2i,x +

∑

j 6=i

(|vjvi,x|+ |vjwi,x|).

(6.43)

Since we already proved the bounds (6.8) on the integrals of transversal terms, to prove (6.41) we

only need to consider the integrals of v2i,x and w2

i,x, in the region where ηi 6= 0. In both cases,

energy type estimates can be used.

63

We work out in detail the bound for v2i,x. Multiplying the first equation in (6.1) by ηivi and

integrating by parts, we obtain∫

ηiviφi dx =∫ {

ηivivi,t + ηivi(λivi)x − ηivivi,xx

}dx

=∫ {

ηi(v2i /2)t − ηiλivivi,x − ηi,xλiv

2i + ηiv

2i,x + ηi,xvi,xvi

}dx

=∫ {(

ηiv2i /2

)t+ (λiηi)x(v2

i /2)− (ηi,t + 2λiηi,x − ηi,xx

)(v2

i /2) + ηiv2i,x + 2ηi,xvivi,x

}dx .

Therefore ∫ηiv

2i,x dx = − d

dt

[∫ηiv

2i /2 dx

]+

∫ (ηi,t + λiηi,x − ηi,xx

)(v2

i /2) dx

−∫

λi,xηi(v2i /2) dx− 2

∫ηi,xvivi,x dx +

∫ηiviφi dx .

(6.44)

A direct computation yields

ηi,t + λiηi,x − ηi,xx = η′i

(wi,t

vi− vi,twi

v2i

)+ λiη

′i

(wi,x

vi− vi,xwi

v2i

)

− η′′i

(wi

vi

)2

x

− η′i

(wi,xx

vi− vi,xxwi

v2i

− 2vi,xwi,x

v2i

+ 2v2

i,xwi

v3i

)

=[η′i

(wi,t + (λiwi)x − wi,xx

)/vi − η′iwi

(vi,t + (λivi)x − vi,xx

)/v2

i

]

+ 2vi,xη′i/vi · (wi/vi)x − η′′i (wi/vi)2x

= η′i

(ψi

vi− wi

vi

φi

vi

)+ 2η′i

vi,x

vi

(wi

vi

)

x

− η′′i

(wi

vi

)2

x

.

(6.45)

Since λi,x = (λi − λ∗i )x, integrating by parts and using the second estimate in (5.69) one obtains∣∣∣∣∫

λi,xηi(v2i /2) dx

∣∣∣∣ =∣∣∣∣∫

(λi − λ∗i )(ηi,xv2

i /2 + ηivivi,x

)dx

∣∣∣∣

≤ ‖λi − λ∗i ‖L∞ ·

12

∫ ∣∣η′i∣∣ |wi,xvi − vi,xwi| dx +

52δ1

∫ηiv

2i,x dx +O(1) · δ0

∫ ∑

j 6=i

|vi,xvj | dx

≤∫|wi,xvi − vi,xwi| dx +

12

∫ηiv

2i,x dx + δ0

∫ ∑

j 6=i

|vi,xvj | dx .

(6.46)

Indeed, |λi − λ∗i | = O(1) · δ0 << δ1. Using (6.45)-(6.46) in (6.44) we now obtain

12

∫ηi v2

i,x dx ≤ − d

dt

[∫ηiv

2i

2dx

]+

12

∫ ∣∣η′i∣∣(|viψi|+ |wiφi|

)dx +

∫ ∣∣∣∣η′ivivi,x

(wi

vi

)

x

∣∣∣∣ dx

+12

∫ ∣∣∣∣∣η′′i v2

i

(wi

vi

)2

x

∣∣∣∣∣ dx +∫ ∣∣wi,xvi − wivi,x

∣∣ dx

+ δ0

∫ ∑

j 6=i

|vi,xvj | dx + 2∫|ηi,xvivi,x| dx +

∫|viφi| dx .

(6.47)

64

Recalling the definition of ηi, on regions where η′i 6= 0 one has |wi/vi| < δ1, hence the bound (5.70)

holds. In turn, this implies

∣∣ηi,xvivi,x

∣∣ =∣∣∣∣η′ivivi,x

(wi

vi

)

x

∣∣∣∣

≤ 2∣∣∣∣δ1η

′iv

2i

(wi

vi

)

x

∣∣∣∣ +O(1) · δ0

∑

j 6=i

∣∣∣∣η′ivivj

(wi

vi

)

x

∣∣∣∣

≤ 2∣∣δ1η

′i

∣∣ |wi,xvi − wivi,x|+O(1) · δ0|η′i|∑

j 6=i

(|vjwi,x|+ |vjvi,x|

∣∣∣∣wi

vi

∣∣∣∣)

.

(6.48)

Using the bounds (5.56)-(5.57), (6.7)-(6.8) and (6.23)-(6.24), from (6.47) we conclude∫ T

t

∫ηi v2

i,x dxdt ≤∫

ηiv2i (t, x) dx +O(1) ·

∫ T

t

∫ (|viψi|+ |wiφi|)dxdt

+O(1) ·∫ T

t

∫|wi,xvi − wivi,x| dxdt +O(1) · δ0

∫ T

t

∫ ∑

j 6=i

(|vjwi,x|+ |vjvi,x|)dxdt

+O(1) ·∫ T

t

∫

|wi/vi|<δ1

∣∣vi(wi/vi)x

∣∣2 dxdt + 2δ0

∫ T

t

∫ ∑

j 6=i

|vi,xvj | dxdt + 2∫ T

t

∫|viφi| dxdt

= O(1) · δ20 .

(6.49)

In a similar way (see [BiB]) one establishes the bound∫ T

t

∫ηi w2

i,x dxdt = O(1) · δ20 . (6.50)

6.5. Proof of the BV estimates

We now conclude the proof of the uniform BV bounds, for solutions of the Cauchy problem

ut + A(u)ux = uxx , u(0, x) = u(x) . (6.51)

Assume that the initial data u : IR 7→ IRn has suitably small total variation, so that

Tot.Var.{u} ≤ δ0

8√

nκ, lim

x→−∞u(x) = u∗ . (6.52)

We recall that κ is the constant introduced at (4.7), related to the Green kernel G∗ of the linearized

equation (4.6). This constant actually depends on the matrix A(u∗), but it is clear that it remains

uniformly bounded when u∗ varies in a compact subset of IRn.

An application of Proposition 4.5 yields the existence of the solution to the Cauchy problem

(6.51) on an initial interval [0, t], satisfying the bound

∥∥ux(t)∥∥L1 ≤

δ0

4√

n. (6.53)

65

This solution can be prolonged in time as long as its total variation remains small. By the previous

analysis, for t ≥ t we can decompose the vectors ux, ux along a basis of unit vectors ri, as in (5.41).

The corresponding components vi, wi then satisfy the system of evolution equations (6.2). Define

the time

T.= sup

{τ ;

∑

i

∫ τ

t

∫ ∣∣φi(t, x)∣∣ +

∣∣ψi(t, x)∣∣ dxdt ≤ δ0

2

}. (6.54)

If T < ∞, a contradiction is obtained as follows. By (5.55) and (6.53), for all t ∈ [t, T ] one has∥∥ux(t)

∥∥L1 ≤

∑

i

∥∥vi(t)∥∥L1

≤∑

i

(∥∥vi(t)

∥∥L1 +

∫ T

t

∫ ∣∣φi(t, x)∣∣ dxdt

)

≤ 2√

n∥∥ux(t)

∥∥L1 +

δ0

2≤ δ0 .

Using Lemma 6.1 and the bounds (6.7), (6.23), (6.24) and (6.41) we now obtain

∑

i

∫ T

t

∫ ∣∣φi(t, x)∣∣ +

∣∣ψi(t, x)∣∣ dxdt = O(1) · δ2

0 <δ0

2, (6.55)

provided that δ0 was chosen suitably small. Therefore T cannot be a supremum. This contradiction

shows that the total variation remains < δ0 for all t ∈ [t, ∞[ . In particular, the solution u is globally

defined.

7 - Stability of viscous solutions

By the homotopy argument (3.18), to show the global stability of every small BV solution of

the viscous system (3.10) it suffices to prove the bound∥∥z(t, ·)

∥∥L1 ≤ L

∥∥z(0, ·)∥∥L1 (7.1)

on first order perturbations. Toward this goal, consider a solution u = u(t, x) of (3.10). The

linearized evolution equation (3.16) for a first order perturbation can be written as

zt +(A(u)z

)x− zxx =

(ux •A(u)

)zx −

(z •A(u)

)ux . (7.2)

In connection with with (7.2), we introduce the quantity Υ .= zx −A(u)z , related to the flux of z.

Differentiating (3.10) we derive an evolution equation for Υ:

Υt +(A(u)Υ

)x−Υxx =

[(ux •A(u)

)z − (

z •A(u))ux

]x−A(u)

[(ux •A(u)

)z − (

z •A(u))ux

]

+(ux •A(u)

)Υ− (

ut •A(u))z .

(7.3)

66

Notice that z = ux and Υ = ut yield a particular solution to (7.2)-(7.3). In this special case, (7.1)

reduces to the bound on the total variation,

In general, the estimate (7.1) will still be obtained following a similar strategy as for the

bounds on the total variation. By (4.34) we already know that the desired estimate holds on the

initial time interval [0, t]. To obtain a uniform bound valid for all t > 0, we decompose the vector

z along a basis of unit vectors ri and derive an evolution equation for these scalar components. At

first sight, it looks promising to write

z =∑

i

ziri(u, vi, σi),

where r1, . . . , rn are the same vectors used in the decomposition of ux at (5.41)-(5.42). Unfortu-

nately, this choice leads to an evolution equation for the components zi with non-integrable source

terms. Instead, we shall use a different basis of unit vectors, depending not only on the reference

solution u but also on the perturbation z. As shown in [BiB], a decomposition leading to successful

estimates is

z =∑

i

hiri

(u, vi, λ

∗i − θ(gi/hi)

),

Υ =∑

i

(gi − λ∗i hi)ri

(u, vi, λ

∗i − θ(gi/hi)

),

(7.4)

where θ is the cutoff function introduced at (5.40). Here v1, . . . , vn are the wave strengths in the

decomposition (5.41) of the gradient ux. The next result, analogous to Lemma 5.2, provides the

existence and regularity of the decomposition (7.4). For a proof, see [BiB].

Lemma 7.1 Let |u − u∗| and |v| be sufficiently small. Then for all z, Υ ∈ IRn the system of 2n

equations (7.4) has a unique solution (h1, . . . , hn, g1, . . . , gn). The map (z, Υ) 7→ (h, g) is Lipschitz

continuous. Moreover, it is smooth outside the n manifolds Ni.= {hi = gi = 0}.

Remark 7.2. If we had chosen a basis of unit vectors {r1, . . . , rn} depending only on the reference

solution u, the map (z, Υ) 7→ (h, g) would be linear. This approach seems very natural, because

z, Υ both satisfy linear evolution equations. However, it would not achieve the desired estimates

(7.1). On the other hand, our decomposition (7.4) yields a mapping (z, Υ) 7→ (h, g) which is

homogeneous of degree one, but not linear. Indeed, the unit vectors ri are here related to viscous

travelling waves Ui whose signed strength vi depends on the reference solution u, but whose speed

σi = θ(hi/gi) depends on the perturbation z.

Differentiating (7.2)-(7.3) and using (7.4) we obtain a system of evolution equations for the

scalar components hi, gi : {hi,t + (λihi)x − hi,xx = φi ,

gi,t + (λigi)x − gi,xx = ψi .(7.5)

67

Since the left hand sides are in conservation form, we have

∥∥z(t, ·)∥∥L1 ≤

∑

i

∥∥hi(t)∥∥L1 ≤

∑

i

(∥∥hi(t)∥∥ +

∫ t

t

∫ ∣∣φi(s, x)∣∣ dxds

). (7.6)

Relying on the same techniques described in Sections 6-7, a careful analysis of the source terms

φi, ψi eventually leads to the bounds (7.1). For all details we refer to [BiB].

7.2. Time dependence.

The continuity w.r.t. time of a solution of the viscous system (3.10) easily follows from the

bounds (4.24) and (4.29). Indeed

∥∥ut(t)∥∥L1 ≤ O(1) ·

∥∥ux(t)∥∥L1 +

∥∥uxx(t)∥∥L1

= O(1) · δ0 +O(1) ·max{

δ0√t, δ2

0

}.

For a suitable constant L′ = O(1) · Tot.Var.{u(0, ·)}, the above implies

∥∥u(t)− u(s)∥∥L1 ≤ v(t, x) =

∫ t

s

∥∥ut(t)∥∥L1 dt ≤ L′

(|t− s|+

∣∣√t−√s∣∣)

. (7.7)

7.3. Propagation speed.

It is well known that, for a parabolic equation, a localized perturbation can be instantly

propagated over the whole domain. However, one can show that the amount of perturbation which

is transported with large speed is actually very small.

Example 7.3. Fix a constant λ > 0 and consider the scalar equation

ut + λux = uxx . (7.8)

Consider an initial data such that u(0, x) = u(x) = 0 for all x > 0. In general, for t > 0 one may

well find a solution with u(t, x) 6= 0 for every x ∈ IR. However, we claim that the values of u(t, x)

are exponentially small for large x. Indeed, our assumption implies

∣∣u(x, 0)∣∣ ≤

∥∥u‖L∞ · e−x .= v(x) .

One easily checks that the solution of (7.8) with the above initial data v is

v(t, x) =∥∥u‖L∞ · e(λ+1)t−x.

68

Applying a standard comparison argument to both u and −u, we thus obtain

∣∣u(t, x)∣∣ ≤ ∥∥u‖L∞ · e(λ+1)t−x. (7.9)

The estimate (7.9) is rather crude. However, it already shows that for x > (λ + 1)t the size of the

solution u is exponentially small.

Estimates of the same type can be proved, more generally, on the difference between any two

solutions of the viscous system (3.10). The following result will be useful in order to prove the

finite propagation speed of solutions obtained as vanishing viscosity limits.

Lemma 7.4. For some constants α, β > 0 the following holds. Let u, v be solutions of (3.10) with

small total variation, whose initial data satisfy

u(0, x) = v(0, x) x ∈ [a, b] . (7.10)

Then one has

∣∣u(t, x)− v(t, x)∣∣ ≤

∥∥u(0)− v(0)∥∥L∞ ·

(αeβt−(x−a) + αeβt+(x−b)

). (7.11)

A proof can be found in [BiB].

8 - The vanishing viscosity limit

All the prevous analysis has been concerned with solutions of the parabolic system (3.10) with

unit viscosity. These results can be easily restated in connection with the Cauchy problem

uεt + A(uε)uε

x = ε uεxx , uε(0, x) = u(x) (8.1)

for any ε > 0. Indeed, a function uε is a solution of (8.1) if and only if

uε(t, x) = u(t/ε, x/ε), (8.2)

where u is the solution of

ut + A(u)ux = uxx , u(0, x) = u(εx) . (8.3)

Since the rescaling (8.2) does not change the total variation, our earlier analysis yields the first

part of Theorem 5. Namely, for every initial data u with sufficiently small total variation, the

69

corresponding solution uε(t) .= Sεt u is well defined for all times t ≥ 0. Recalling (3.18) and (7.7),

the bounds (3.4)–(3.6) are derived as follows.

Tot.Var.{uε(t)

}= Tot.Var.

{u(t/ε)

} ≤ C Tot.Var.{u}, (8.4)

∥∥uε(t)− vε(t)∥∥L1 = ε

∥∥u(t)− v(t)∥∥L1 ≤ εL

∥∥u(0)− v(0)∥∥L1 = εL

1ε‖u− v‖L1 , (8.5)

∥∥uε(t)− uε(s)∥∥L1 ≤ ε

∥∥u(t/ε)− u(s/ε)∥∥L1 ≤ εL′

(∣∣∣ t

ε− s

ε

∣∣∣ +

∣∣∣∣∣

√t

ε−

√s

ε

∣∣∣∣∣

). (8.6)

Moreover, if u(x) = v(x) for x ∈ [a, b], then by (7.11) the corresponding solutions uε, vε of (8.1)

satisfy

∣∣uε(t, x)− vε(t, x)∣∣ ≤ ‖u− v‖L∞ ·

{α exp

(βt− (x− a)ε

)+ α exp

(βt + (x− b)ε

)}. (8.7)

We now consider the vanishing viscosity limit. Call U ⊂ L1loc the set of all functions u : IR 7→

IRn with small total variation, satisfying (3.3). For each t ≥ 0 and every initial condition u ∈ U ,

call Sεt u

.= uε(t, ·) the corresponding solution of (8.1). Thanks to the uniform BV bounds (8.4),

we can apply Helly’s compactness theorem and obtain a sequence εν → 0 such that

limν→∞

uεν (t, ·) = u(t, ·) in L1loc . (8.8)

holds for some BV function u(t, ·). By extracting further subsequences and then using a standard

diagonalization procedure, we can assume that the limit in (8.8) exists for all rational times t and

all solutions uε with initial data in a countable dense set U∗ ⊂ U . Adopting a semigroup notation,

we thus define

Stu.= lim

m→∞Sεm

t u in L1loc , (8.9)

for some particular subsequence εm → 0. By the uniform continuity of the maps (t, u) 7→ uε(t, ·) .=

Sεt u , stated in (8.5)-(8.6), the set of couples (t, u) for which the limit (8.9) exists must be closed

in IR+×U . Therefore, this limit is well defined for all u ∈ U and t ≥ 0. Notice that (8.4) and (8.9)

imply a uniform BV bound on the vanishing viscosity limits:

Tot.Var.{Stu

} ≤ lim supm→∞

Tot.Var.{uεm(t)

} ≤ C Tot.Var.{u} . (8.10)

To complete the proof of Theorem 5, we need to show that the map S defined at (8.9) is

a semigroup, satisfies the continuity properties (3.7) and does not depend on the choice of the

subsequence {εm}. We give here a proof in the special case of a conservative system satisfying the

additional hypotheses (H).

70

1. (Continuous dependence) Let S be the map defined by (8.9). Then

∥∥Stu− Stv∥∥L1 = sup

r>0

∫ r

−r

∣∣∣(Stu

)(x)− (

Stv)(x)

∣∣∣ dx .

For every r > 0, the convergence in L1loc implies

∫ r

−r

∣∣∣(Stu

)(x)− (

Stv)(x)

∣∣∣ dx = limm→∞

∫ r

−r

∣∣∣(Sεm

t u)(x)− (

Sεmt v

)(x)

∣∣∣ dx ≤ L∥∥u− v

∥∥L1 .

because of (8.5). This yields the Lipschitz continuous dependence w.r.t. the initial data:

∥∥Stu− Stv∥∥L1 ≤ L

∥∥u− v∥∥L1 . (8.11)

The continuous dependence w.r.t. time is proved in a similar way. By (8.6), for every r > 0

we have∫ r

−r

∣∣∣(Stv

)(x)− (

Ssv)(x)

∣∣∣ dx = limm→∞

∫ r

−r

∣∣∣(Sεm

t v)(x)− (

Sεms v

)(x)

∣∣∣ dx

= limm→∞

εmL′(∣∣∣ t

εm− s

εm

∣∣∣ +∣∣∣∣√

t

εm−

√s

εm

∣∣∣∣)

= L′|t− s| .Hence ∥∥Stv − Ssv

∥∥L1 ≤ L′|t− s| . (8.12)

From (8.11) and (8.12) it now follows (3.7).

2. (Finite propagation speed) Consider any interval [a, b] and two initial data u, v, with

u(x) = v(x) for x ∈ [a, b]. By (8.7), for every t ≥ 0 and x ∈ ]a + βt, b− βt[ one has∣∣∣(Stu

)(x)− (

Stv)(x)

∣∣∣ ≤ lim supm→∞

∣∣∣(Sεm

t u)(x)− (

Sεmt v

)(x)

∣∣∣

≤ limm→∞

‖u− v‖L∞ ·{

α exp(βt− (x− a)

εm

)+ α exp

(βt + (x− b)εm

)}

= 0 .(8.13)

In other words, the restriction of the function Stu ∈ L1loc to a given interval [a′, b′] depends only

on the values of the initial data u on the interval [a′ − βt, b′ + βt]. Using (8.13), we now prove a

sharper version of the continuous dependence estimate (8.11), namely∫ b

a

∣∣∣(Stu

)(x)− (

Stv)(x)

∣∣∣ dx ≤ L ·∫ b+βt

a−βt

∣∣u(x)− v(x)∣∣ dx , (8.14)

valid for every u, v and t ≥ 0. For this purpose, define the auxiliary function

w(x) ={

u(x) if x ∈ [a− βt, b + βt] ,v(x) if x /∈ [a− βt, b + βt] .

71

Using the finite propagation speed, we now have

∫ b

a

∣∣∣(Stu

)(x)− (

Stv)(x)

∣∣∣ dx =∫ b

a

∣∣∣(Stw

)(x)− (

Stv)(x)

∣∣∣ dx

≤ L ‖w − v‖L1 = L ·∫ b+βt

a−βt

∣∣u(x)− v(x)∣∣ dx .

3. (Semigroup property) We claim that the map (t, u) 7→ Stu is a semigroup, i.e.

S0u = u , SsStu = Ss+tu . (8.15)

Since every Sε is a semigroup, the first equality in (8.15) is a trivial consequence of the definition

(8.9). To prove the second equality, we observe that

Ss+tu = limm→∞

Sεms Sεm

t u , SsStu = limm→∞

Sεms Stu . (8.16)

We can assume s > 0. Fix any r > 0 and consider the function

u]m(x) .=

{ (Stu

)(x) if |x| > r + 2βs ,(

Sεmt u

)(x) if |x| < r + 2βs .

Observing that Sεmt u → Stu in L1

loc and hence u]m → Stu in L1, we can use (8.7) and (8.5) and

obtain

lim supm→∞

∫ r

−r

∣∣∣(Sεm

s Sεmt u

)(x)− (

Sεms Stu

)(x)

∣∣∣ dx

≤ limm→∞

2r · sup|x|<r

∣∣∣(Sεm

s Sεmt u

)(x)− (

Sεms u]

m

)(x)

∣∣∣ + limm→∞

∥∥Sεms u]

m − Sεms Stu

∥∥L1

≤ limm→∞

2r∥∥Sεm

t u− u]m

∥∥L∞ · 2αe−βs/εm + lim

m→∞L · ∥∥u]

m − Stu∥∥L1

= 0 .

By (8.16), this proves the second identity in (8.15).

4. (Tame Oscillation) Next, we prove that every function u(t, x) =(Stu

)(x) satisfies the tame

oscillation property introduced at (2.27). Indeed, let a, b, τ be given, together with an initial data

u. By the semigroup property, it is not restrictive to assume τ = 0. Consider the auxiliary initial

condition

v(x) .=

u(x) if a < x < b,u(a+) if x ≤ a,u(b−) if x ≥ b,

and call v(t, x) .=(Stv

)(x) the corresponding trajectory.

Observe that

limx→−∞

v(t, x) = u(a+)

72

for every t ≥ 0. Using (8.4) and the finite propagation speed, we can thus write

Osc.{u ; ∆τ

a,b

}= Osc.

{v ; ∆τ

a,b

} ≤ 2 supt

(Tot.Var.

{Stv

})

≤ 2C · Tot.Var.{v} = 2C · Tot.Var.{u(τ) ; ]a, b[

},

proving (2.27) with C ′ = 2C.

5. (Conservation equations) Assume that the system (8.1) is in conservation form, i.e. A(u) =

Df(u) for some flux function f . In this special case, we claim that every vanishing viscosity limit is

a weak solution of the system of conservation laws (1.1). Indeed, if φ is a C2 function with compact

support contained in the half plane {x ∈ IR, t > 0}, integrating by parts we find∫ ∫ [

uφt + f(u)φx

]dxdt = lim

m→∞

∫ ∫ [uεm φt + f(uεm)φx

]dxdt

= − limm→∞

∫ ∫ [uεm

t φ + f(uεm)xφ]dxdt = − lim

m→∞

∫ ∫εm uεm

xx φdxdt

= − limm→∞

∫ ∫εm uεmφxx dxdt = 0 .

An approximation argument shows that the identity (1.3) still holds if we only assume φ ∈ C1c .

6. (Approximate jumps) From the uniform bound on the total variation and the Lipschitz

continuity w.r.t. time, it follows that each function u(t, x) =(Stu

)(x) is a BV function, jointly

w.r.t. the two variables t, x. In particular, an application of Theorem 2.6 in [B3] yields the existence

of a set of times N ⊂ IR+ of measure zero such that, for every (τ, ξ) ∈ IR+ × IR with τ /∈ N , the

following holds. Calling

u− .= limx→ξ−

u(τ, x), u+ .= limx→ξ+

u(τ, x),

there exists a finite speed λ such that the function

U(t, x) .={

u− if x < λt,u+ if x > λt,

for every constant κ > 0 satisfies

limr→0+

1r2

∫ r

−r

∫ κr

−κr

∣∣u(τ + t, ξ + x)− U(t, x)∣∣ dxdt = 0,

limr→0+

1r

∫ κr

−κr

∣∣u(τ + r, ξ + x)− U(r, x)∣∣ dx = 0.

In the case where u− 6= u+, we say that (τ, ξ) is a point of approximate jump for the function

u. On the other hand, if u− = u+ (and hence λ can be chosen arbitrarily), we say that u is

approximately continuous at (τ, ξ). The above result can thus be restated as follows: with the

73

exception of a null set N of “interaction times”, the solution u is either approximately continuous

or has an approximate jump discontinuity at each point (τ, ξ).

7. (Shock conditions) Assume again that the system is in conservative form. Consider a

semigroup trajectory u(t, ·) = Stu and a point (τ, ξ) where u has an approximate jump. Since u

is a weak solution, the states u−, u+ and the speed λ in (8.19) must satisfy the Rankine-Hugoniot

equations (2.4). If u is a limit of vanishing viscosity approximations, the same is true of the solution

U in (8.19). In particular (see [D2]), the Liu admissibility condition (2.15) must hold.

8. (Uniqueness) Assume that the system is in conservation form and that each characterictic

field is either linearly degenerate or genuinely nonlinear. By the previous steps, the semigroup

trajectory u(t, ·) = Stu provides a weak solution to the Cauchy problem (1.1)-(1.2) which satisfies

the Tame Oscillation and the Lax shock conditions. By a theorem in [BG], [B3], such a weak

solution is unique and coincides with the limit of front tracking approximations. In particular, it

does not depend on the choice of the subsequence {εm} :

Stu = limε→0+

Sεt u ,

i.e. the same limit actually holds over all real values of ε.

The above discussion has established Theorem 5 in the special case where the system is in

conservation form and each characteristic field is either linearly degenerate or genuinely nonlinear

For a proof in the general case of a non-conservative hyperbolic system we again refer to [BiB]..

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