front tracking approximations for slow erosion debora ... · front tracking approximations for slow...

Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XX

FRONT TRACKING APPROXIMATIONS FOR SLOW EROSION

Debora Amadori

Dipartimento di Matematica Pura & Applicata, University of L’Aquila, Italy.

Wen Shen

Mathematics Department, Penn State University, University Park, PA 16802, U.S.A.

(Communicated by the associate editor name)

Abstract. In this paper we study an integro-differential equation describing

slow erosion, in a model of granular flow. In this equation the flux is non local

and depends on x, t. We define approximate solutions by using a front trackingtechnique, adapted to this special equation. Convergence of the approximate

solutions is established by means of suitable a priori estimates. In turn, these

yield the global existence of entropy solutions in BV. Such entropy solutionsare shown to be unique.

We also prove the continuous dependence on initial data and on the erosion

function, for the approximate as well as for the exact solutions. This establishesthe well-posedness of the Cauchy problem.

1. Introduction and preliminaries. Consider the scalar integro-differentialequation

u(t, x)t +

(f(u(t, x)) exp

{∫ ∞x

f(u(t, s)) ds

})x

= 0 , (1)

associated with the initial data

u(0, x) = u(x) , x ∈ R . (2)

This equation was first derived in [2] as the slow erosion limit for a granular flowmodel proposed in [16], with a specific function f . A more general model was laterderived in [23, 3], for a wider class of functions f . Here, the unknown variable udescribes the slope of the standing profile of granular matter, which varies in timedue to the occurrence of small avalanches. The function f is called the erosionfunction, which denotes the erosion rate per unit length in space covered by theavalanche. See [23] for a more detailed derivation of the model.

To simplify notation, we let F denote the integral term, i.e.,

F (x;u) = exp

{∫ ∞x

f(u(t, s)) ds

}, (3)

and we write (1) asut + (fF )x = 0 . (4)

2000 Mathematics Subject Classification. Primary: 35L65; Secondary: 35L67, 35Q70, 35L60,

35L03.Key words and phrases. Granular flow, slow erosion, conservation laws, front tracking, existence

and uniqueness of BV solutions.The second author is partially supported by NSF grant DMS-0908047.

1

2 DEBORA AMADORI AND WEN SHEN

Throughout this paper, we assume that the erosion function f ∈ C2{(0,+∞)}satisfies the conditions (F):

f(1) = 0 , f ′ > 0 , f ′′ < 0 , lims→0+

f(s) = −∞ , lims→+∞

f(s)

s= 0 . (5)

The physical meaning of these assumptions is as follows. (i) At the critical slope u =1 there is no erosion or deposition, hence f(1) = 0. (ii) When the slope approaches0, there is infinitely large deposition. (iii) When the slope is very large, the erosionfunction f grows slower than any linear functions. The class of functions satisfyingthe assumptions (F) includes the logarithm function, as well as f(s) = sa − 1, for0 < a < 1.

We remark that the assumptions in (5) are sharp, in order to prevent blowup ofu. In [23] it is proved that the slope u can become unbounded in finite time, if f(s)approaches a linear asymptote with positive slope, as s→ +∞.

Throughout the paper we will use ‖·‖L1 , ‖·‖L∞ and TV{·} to denote the L1

norm, the L∞ norm and the total variation, respectively, all in the space variable.We use sign(·) to denote the sign function, and C to denote a generic boundedconstant that does not depend on the critical variables.

Solutions of the Cauchy problem will be obtained within the class W consistingof all functions w : R 7→ R satisfying the property

infxw(x) > 0 , w(·) ∈ BV (R) , supp{w(·)− 1} is bounded . (6)

A definition of entropy weak solutions for (1)-(2) is now given.

Definition 1.1. A function u : [0, T ] × R 7→ R is called an entropy weak solutionfor (1) with initial data u(x) ∈ W if

• [0, T ] 3 t→ u(t, ·) ∈ W, is continuous in L1loc and u(0, ·) = u.

• For every test function φ ∈ C∞c (R2), one has the integral identity∫ T

0

∫R

(uφt + f(u)F (x;u)φx) dx dt =

∫R

(u(T, x)φ(T, x)− u(0, x)φ(0, x)

)dx .

• For a.e. x < y, there exists some constant C (that does not depend on x, y ort), such that

u(t, x+)− u(t, y+) ≤ C max {1/t, 1} (y − x) . (7)

Remark 1. Here the definition of entropy solution follows from that of scalarconservation law with variable coefficient

ut + (k(t, x)f(u))x = 0 (8)

where the flux function f is concave and the coefficient k is positive and Lipschitzcontinuous. For (8), the Oleinik inequality (7) would guarantee the uniqueness ofthe solution, by estimates for an adjoint problem, see [18]. In our case, such dualityargument runs into difficulty due to the integral term (3). However, we still usethe same definition, since this inequality guarantees the Lax admissibility of shocks,and will be important in the analysis of the large time behavior of solutions. Analternative definition could have been given in terms of Kruzkov entropy inequalities,as in [3].

Existence of global BV solutions and continuous dependence on initial data fora initial-boundary value problem for (1), were studied in [3]. In that paper, aniteration technique was used, freezing the global term at every time step.

FRONT TRACKING APPROXIMATIONS FOR SLOW EROSION 3

In this paper, we introduce a different approximation technique, based on wave-front tracking. Our approximate solutions are piecewise constant in space and evolvecontinuously in time. A precise description of the algorithm is given in Section 3.1.

A somewhat similar algorithm is used in [23] where a Hamilton-Jacobi equation(with an integral term) for the height of the profile is treated, and piecewise affine(possibly discontinuous) approximate solutions are constructed. Such front trackingalgorithms provide better intuition and control over wave interactions, and allowus to derive a-priori estimates in a more straightforward way. The first part of ouranalysis will establish the global existence and uniqueness of BV solutions.

Theorem 1.2. Let T > 0 and an initial data u ∈ W be given. Then the Cauchyproblem (1)-(2) admits a unique entropy weak solution u = u(t, x) defined for allt ∈ [0, T ], that moreover satisfies

infxu(t, x) ≥ inf

xu(x) , ‖u(t, ·)− 1‖L1 ≤ ‖u− 1‖L1 .

The existence of solutions is proved by compactness on the family of front trackingapproximate solutions. Then the uniqueness is proved using the Oleinik estimate(7), combined with an error formula for the L1 distance of two different solutions,corresponding to the same initial data.

Then we study the continuous dependence of the solutions, on both the initialdata and the erosion function f . By directly comparing the L1 distance between twopiecewise constant approximate solutions, passing to the limit we obtain the con-tinuous dependence for the exact solutions. By uniqueness (stated in Theorem 1.2),the same conclusion holds for any pair of entropy solutions.

More precisely, consider a second Cauchy problem

vt +

(g(v) exp

{∫ ∞x

g(v(t, s)) ds

})x

= 0 , v(0, x) = v(x) ∈ W (9)

with an erosion function g that satisfies the same assumptions (F), i.e,

g(1) = 0 , g′ > 0 , g′′ < 0 , lims→0

g(s) = −∞ , lims→+∞

g(s)

s= 0 . (10)

Note that it is important to have the same critical slope for both erosion functions,otherwise ‖u− v‖L1 would already be unbounded.

We have the following stability Theorem.

Theorem 1.3. Let u and v be entropy weak solutions for the Cauchy problems (1)-(2) and (9), respectively. Denote by κ0 and M a lower and upper bound for both uand v, respectively, and by m0 a bound on ‖u− 1‖L1 , ‖v − 1‖L1 .

Then

‖u(t, ·)− v(t, ·)‖L1 ≤ ‖u− v‖L1 + C

∫ t

0

‖u(s, ·)− v(s, ·)‖L1 ds

+ Ct[‖f − g‖L∞ + ‖f − g‖Lip

](11)

where

‖f − g‖L∞ = maxκ0≤s≤M

|f(s)− g(s)| ,

‖f − g‖Lip = maxκ0≤s≤M

|f ′(s)− g′(s)| ,

and C depends only on κ0, M , m0.


By Gronwall’s Lemma, (11) gives continuous dependence.Other PDE models for granular flow can be found in [14, 20, 4, 11]. For math-

ematical properties of the steady state solutions we refer to [9, 10]. A numericalstudy can be found in [15]. For time-dependent solutions, see the recent results[22, 1, 2, 3, 23]. Other well-known examples of conservation law involving integralterms include the Camassa-Holm equation [8, 6] and a variational wave equation[7]. For some related results on stability for general scalar balance law, we refer to[17, 12].

The rest of the paper is structured in the following way. In Section 2 we givethe basic analysis and some formal arguments. In Section 3 we prove Theorem 1.2by front tracking approximation and uniqueness argument. Finally, in Section 4 weprove Theorem 1.3.

2. Basic analysis. By the method of characteristics, for smooth solutions one has

x = f ′(u)F , u = ut + xux = f2(u)F . (12)

Due to the nonlinearity of the erosion function f , characteristics will merge, whichleads to discontinuities in solutions. We call them shocks or shock waves. To see howthese shocks form, let z = ux, and consider its evolution along the characteristic,

z = zt + f ′Fzx = −f ′′Fz2 + 3ff ′Fz − f3F . (13)

Assuming u, f, f ′ bounded, the first term (−f ′′Fz2) dominates as |z| is large. Since−f ′′F > 0, then z blows up to +∞ in finite time, leading to an upward jump in u.

The traveling speed of the shock waves satisfies the Rankine-Hugoniot jumpcondition. Let u has a jump at x0, with u(x−0 ) = u− and u(x+

0 ) = u+. TheRankine-Hugoniot condition gives

λs = F (x0;u)f(u−)− f(u+)

u− − u+. (14)

Since f is concave, only upward jumps are admissible. Initial downward jumpswill open up into rarefaction waves. This is confirmed by (13), where z blows uponly to +∞, i.e.,

z ≥ −C max{1/t, 1} . (15)

Therefore, an Oleinik-type one-sided entropy inequality (see [19]) holds: for anyt > 0, and x < y, one has

u(t, x)− u(t, y) ≤ (y − x)C max{1/t, 1} . (16)

Wave interactions are determined by the local behavior of the flux, i.e., theerosion function f , which is a concave function. The interactions are similar tothose of a scalar conservation law. When two (or more) admissible shocks interact,they will simply merge into a bigger admissible shock. No new waves would beformed at interactions.

Next is a technical Lemma connecting properties of u with the global term F .

Lemma 2.1. Let u : R 7→ R satisfy

u(x) ≥ κ0 > 0, ‖u(·)− 1‖L1 ≤ m0 .

Then, the function f(u(x)) is absolutely integrable, i.e.,

‖f(u(·))‖L1 =

∫R|f(u(x))| dx = C ≤ f ′(κ0)m0 . (17)


Furthermore, the integral function F (x;u) as defined in (3) satisfies

e−C ≤ F ≤ eC , TV{F} ≤ CeC . (18)

Proof. Since ‖u− 1‖L1 ≤ m0, then the function x 7→ f(u(x)) is absolutely inte-grable, because u 7→ f(u) is uniformly Lipschitz on [κ0,∞] and f(1) = 0. Moreover∫

R|f(u(x))| dx =

∫R|f(u(x))− f(1)| dx ≤ f ′(κ0) ‖u− 1‖L1 .

This gives (17). The upper and lower bound on F is obvious by its definition and(17). Finally, since x 7→ F is Lipschitz continuous, we have

TV{F} = ‖Fx‖L1 = ‖f(u)F‖L1 ≤ ‖F‖L∞ ‖f(u)‖L1 ≤ CeC . (19)

Below we give some formal arguments, which serves as guideline for the a prioriestimates for the approximate solutions.(1). Lower bound on u. By (12), u is non-decreasing along characteristics, thereforethe lower bound follows.(2). Bound on total mass. The trivial solution is u ≡ 1. Equation (1) can bewritten as

(u− 1)t + (f(u)F )x = 0 . (20)

By the assumptions (5) we have sign(u−1) = sign(f(u)). Since F > 0, we concludethat the L1 norm of u− 1 is non-increasing in time.

By Lemma 2.1, F is uniformly bounded from below and above, and has boundedvariation.(3). Bounded support for u− 1. By the lower bound on u, the characteristic speedf ′(u)F is now bounded. Therefore, for t ≤ T , the support for u− 1 is bounded.(4). Upper bound on u. Integrate the conservation law (20) over the region (t, y)with 0 ≤ t ≤ T and y ≤ x(t) where t→ x(t) is a characteristic, we get∣∣∣∣∣

∫ T

0

[(u− 1)f ′(u)− f(u)]F dt

∣∣∣∣∣=

∣∣∣∣∣∫ x(T )

−∞(u(T, x)− 1)dx−

∫ x(0)

−∞(u(0, x)− 1) dx

∣∣∣∣∣ ≤ 2m0 , (21)

thanks to the bound on ‖u− 1‖L1 . Define an auxiliary function

α(u) =u− 1

f(u)if u 6= 1 , α(1) = 1/f ′(1) . (22)

This function is well-defined for all u > 0. At u = 0 we can set α(0) = 0 bycontinuity. The function is nonnegative, α(u) > 0 for u > 0, and is increasing in u,i.e.,

α′(u) =f(u)− (u− 1)f ′(u)

f2(u)> 0 for u > 0 , u 6= 1 . (23)

By the last assumption on f in (5), α(u) grows to +∞ as u→ +∞,

limu→+∞

α(u) = +∞ . (24)

The evolution of α(u) along a characteristic is

d

dtα(u(t, x(t)) = α′(u)u = [f(u)− (u− 1)f ′(u)]F .


By (21), we have, for all T ,

α(u(T, x(T ))) ≤ α(u(0, x(0))) + 2m0 . (25)

By (24) we conclude that u(t, x) remains bounded for all t, x.(5). BV bound on u. Let z = ux. Differentiating (1) in x, one gets

zt + (f ′(u)F (x;u)z)x = (f2(u)F (x;u))x .

Let κ0 > 0 and M be a lower and upper bound for u, respectively. We define

‖f‖L∞ = maxκ0≤s≤M

|f(s)| . (26)

Formally we have

d

dtTV{u} ≤ TV{f2(u)F} ≤ ‖f‖2L∞ TV{F}+ 2 ‖f‖L∞ f ′(κ0)TV{u} ‖F‖L∞

≤ C (1 + TV{u}) .Therefore, TV{u} can grow exponentially, but remains bounded for finite time.

3. Front tracking approximate solutions. In this section we prove Theorem1.2. The algorithm for the piecewise constant front tracking approximation is de-scribed in Section 3.1. Then we establish a priori estimates in Section 3.2. Allestimates are used in Section 3.3 to achieve compactness, which leads to existenceof entropy weak solutions. Finally, uniqueness of entropy weak solutions is treatedin Section 3.4, completing the proof.

3.1. The algorithm. Let ε be the approximation parameter, and uε be the ε-approximate solution that we now construct. For a given initial data u ∈ W, onecan construct a piecewise constant approximation, call it uε, such that uε → u inL1loc, and uε ∈ W. The approximation could be achieved by a suitable sampling in

u. This will be the discrete initial data for the algorithm, i.e., uε(0, x) = uε(x) andchoose it such that uε(x) ≤ u(x). Let m0, κ0 satisfy

uε(0, x) ≥ κ0 > 0 ∀ x , ‖uε − 1‖L1 ≤ m0 , ε > 0.

Let xi (i = 0, · · · , N) be the points where uε has jumps, which we call fronts, andwrite

ui+ 12(t) = uε(t, x) for x ∈ [xi, xi+1) .

The algorithm will result in a set of ODEs that govern the evolution of xi and ui+ 12

in t.The approximation to the initial data must satisfy the following requirements.

• The downward jumps should be small because they are not admissible. Intro-duce the quantities

ηi(t) = ui− 12(t)− ui+ 1

2(t) , η(t) = max

iηi(t) . (27)

Note that η(t) measures the size of the downward jumps at t. We require that

η(0) ≤ ε . (28)

This ensures that possible initial (big) downward jumps will open up into afan of small downward jumps, each one of size ≤ ε. We refer to downwardjumps as rarefaction fronts, and upward jumps as shock fronts.

• Whenever u crosses 1 with negative gradient, we will make sure that u = 1 issampled. This will lead to a clean a priori L1 estimate.


• Denote the discrete version of the integral term as

F ε(t, x) = F (x;uε) = exp

{∫ ∞x

f(uε(t, y)) dy

}. (29)

For accuracy and convergence of F ε, we define the quantities

ζi+ 12(t) = (xi+1(t)− xi(t)) · |f(ui+ 1

2(t))| , ζ(t) = max

iζi+ 1

2(t) , (30)

and require

ζ(0) ≤ ε . (31)

Then, F ε satisfies

e−ε ≤ F ε(0, xi+1)

F ε(0, xi)≤ eε . (32)

Therefore, as ε→ 0+, we have the convergences of F ε(0, ·)

F ε(0, ·)→ F (0, ·) , F εx(0, ·)→ Fx(0, ·) , in L1loc . (33)

Now we describe the algorithm. All fronts will travel with Rankine-Hugoniotspeed, i.e.,

xi = F ε(xi)f(ui+ 1

2)− f(ui− 1

2)

ui+ 12− ui− 1

2

, (34)

and ui+ 12(t) evolves as

ui+ 12

= −f(ui+ 12)F ε(xi+1)− F ε(xi)

xi+1 − xi. (35)

Since F ε is smooth on the interval [xi, xi+1], by the Mean Value Theorem we have

F ε(xi+1)− F ε(xi)xi+1 − xi

= −f(ui+ 12)F ε(xi+ 1

2) , for some xi+ 1

2∈ (xi, xi+1) . (36)

This leads to

ui+ 12

= f2(ui+ 12)F ε(xi+ 1

2) , xi+ 1

2∈ (xi, xi+1) . (37)

Merging of fronts. Since the fronts travel with different speeds, nearby fronts couldapproach each other as they travel. If this happens, all approaching fronts willmerge into one, and we rearrange the indices. The new front will then travel withthe new Rankine-Hugoniot speed. Total number of fronts will decrease in time.

Remark 2. The choice of ui+ 12

in (35) is motivated as follows. In order to keep uε

constant on the interval [xi, xi+1), uεt must be piecewise constant. This leads to apiecewise constant approximation for F εx , by a finite difference of the form

F εx(x) ≈ F ε(xi+1)− F ε(xi)xi+1 − xi

, x ∈ [xi, xi+1) . (38)

In the end, the ε-approximate solution uε is a weak solution of the approximateequation

uεt + (f(uε)F ε)x = 0 , (39)

where for every given t, F ε is a linear interpolation of F ε in x through nodal points,i.e.,

F ε(x) = F ε(xi)xi+1 − xxi+1 − xi

+ F ε(xi+1)x− xi

xi+1 − xi, for x ∈ [xi, xi+1] . (40)


Remark 3. A front may change type in time, even without interaction. This isdifferent from standard front tracking approximation for conservation laws. Indeed,we have

ui− 12(t)− ui+ 1

2(t)

= −f(ui− 12)F ε(xi)− F ε(xi−1)

xi − xi−1+ f(ui+ 1

2)F ε(xi+1)− F ε(xi)

xi+1 − xi= F ε(xi)

{f2(ui− 1

2) er − f2(ui+ 1

2) es}

for some values r and s that satisfy sign(r) = sign(ui− 12−1) , sign(s) = sign(1−ui+ 1

2)

and |r| ≤ ζi− 12

, |s| ≤ ζi+ 12

. Front type can change in the following cases.

• For a rarefaction front (i.e., a downward jump) with 1 > ui− 12> ui+ 1

2, we have

ηi(t) < 0. This front may evolve into a shock front with 1 > ui+ 12> ui− 1

2.

• For a shock front (i.e., an upward jump) with 1 < ui− 12< ui+ 1

2, it may change

into a rarefaction front.

Remark 4. Two rarefaction fronts would never approach each other. If two nearbyfronts approach, say xi(t) = xi+1(t), then one of them must be a shock front, andwe must have

xi(t) ≥ xi+1(t) , ⇒ ui− 12(t) ≤ ui+ 3

2(t) ,

so the out-coming front must be a shock front. If more than two fronts merge simul-taneously, say xi(t) = · · · = xj(t) with i < j− 1, then between each two rarefactionfronts there must be at least one shock front. We can pair each rarefaction frontwith a neighboring shock front, possibly leaving the first front xi unpaired. By thediscussion above, each pair must result in a shock front. Then, if the unpaired frontxi is a rarefaction, the final out-coming front could be a rarefaction, but with sizesmaller than that of xi. This implies that the maximum size of rarefaction frontsη(t) does not increase at any merging time.

3.2. A priori estimates. All the a priori estimates are summarized in the nextLemma.

Lemma 3.1. Let uε be an ε-approximate solution with initial data uε ∈ W thatsatisfies (28) and (31). Then, for any t ∈ [0, T ], we have x→ uε(t, x) ∈ W. For εsufficiently small we have

η(t) ≤ Cε , ζ(t) ≤ Cε , (41)

for some constant C independent of ε.

Proof. (1). Lower bound for uε. By (37) we clearly have ui+ 12≥ 0. The lower

bound follows,

infxuε(t, x) ≥ inf

xuε(0, x) ≥ κ0 .

(2). Bound on ‖uε − 1‖L1 . This follows from the facts that all fronts travelwith Rankine-Hugoniot speed and u = 1 is always sampled when uε crosses 1 withnegative slope. In more detail, since uε is piecewise constant, we have

‖uε − 1‖L1 =∑i

∣∣∣ui+ 12− 1∣∣∣ (xi+1 − xi) .


A direct computation, by using summation-by-parts, gives

d

dt‖uε − 1‖L1 =

∑i

sign(ui+ 12− 1)ui+ 1

2(xi+1 − xi) +

∣∣∣ui+ 12− 1∣∣∣ (xi+1 − xi)

=∑i

F ε(xi)Ii ,

where

Ii =∣∣∣f(ui+ 1

2)∣∣∣− ∣∣∣f(ui− 1

2)∣∣∣+

f(ui− 12)− f(ui+ 1

2)

ui− 12− ui+ 1

2

(∣∣∣ui− 12− 1∣∣∣− ∣∣∣ui+ 1

2− 1∣∣∣) .

There are several situations.

• If sign(ui− 12− 1) = sign(ui+ 1

2− 1), then Ii = 0;

• If sign(ui− 12−1) 6= sign(ui+ 1

2−1) and ui− 1

2≥ 1 ≥ ui+ 1

2, then by construction

we must have either ui+ 12

= 1 or ui− 12

= 1. In either case we have Ii = 0.

• If sign(ui− 12− 1) 6= sign(ui+ 1

2− 1) and ui− 1

2≤ 1 ≤ ui+ 1

2, then by concavity

of f we have∣∣∣f(ui+ 12)∣∣∣ ≤ f(ui− 1

2)− f(ui+ 1

2)

ui− 12− ui+ 1

2

∣∣∣ui+ 12− 1∣∣∣ ,

∣∣∣f(ui− 12)∣∣∣ ≥ f(ui− 1

2)− f(ui+ 1

2)

ui− 12− ui+ 1

2

∣∣∣ui− 12− 1∣∣∣ .

Therefore Ii ≤ 0.

In conclusion, we have for all t ≥ 0,

d

dt‖uε(t, ·)− 1‖L1 ≤ 0 , ⇒ ‖uε(t, ·)− 1‖L1 ≤ ‖uε(0, ·)− 1‖L1 ≤ m0 . (42)

Furthermore, by Lemma 2.1, the integral term F ε satisfies

e−C ≤ F ε ≤ eC , TV{F ε} ≤ CeC , (43)

where

C = f ′(κ0)m0 ≥ supt‖f(uε(t, ·))‖L1 .

(3). Bound on the support for uε − 1. This is obvious since the speeds for the firstand last fronts are bounded, thanks to the lower bound on uε.

(4). A priori bounds on η and ζ. Assume a priori that, for some t ≤ T and M > 0

ui+ 12(t) < M , for t < t , ∀ i . (44)

We have

ηi(t) = ˙ui− 12(t)− ˙ui+ 1

2(t) = f2(ui− 1

2)F ε(xi− 1

2)− f2(ui+ 1

2)F ε(xi+ 1

2)

=(f2(ui− 1

2)− f2(ui+ 1

2))F ε(xi− 1

2) + f2(ui+ 1

2)(F ε(xi− 1

2)− F ε(xi+ 1

2))

≤ C1Mη + C2M2ζ (45)


for some C1, C2 depending on f , κ0, ‖F ε‖L∞ . For ζ(t), we have

ζi+ 12(t)

= (xi+1 − xi)∣∣∣f(ui+ 1

2)∣∣∣+ (xi+1 − xi)sign(f(ui+ 1

2))f ′(ui+ 1

2) ˙ui+ 1

2

= (xi+1 − xi)∣∣∣f(ui+ 1

2)∣∣∣− f ′(ui+ 1

2)∣∣∣f(ui+ 1

2)∣∣∣ (F ε(t, xi+1)− F ε(t, xi))

=∣∣∣f(ui+ 1

2)∣∣∣ · {[xi+1 − f ′(ui+ 1

2)F ε(t, xi+1)

]−[xi − f ′(ui+ 1

2)F ε(t, xi)

]}.

These two terms are negative if xi, xi+1 are both shock fronts. If one of them is ararefaction front, say ui+ 1

2≤ ui− 1

2, then we have

f ′(ui+ 12)F ε(t, xi)− xi ≤ ‖F ε‖L∞ sup

1≤s≤M|f ′′(s)| (ui− 1

2− ui+ 1

2) ≤ C(M)η .

This implies

ζi+ 12≤ C(M)Mη , (46)

for some C possibly depending on M . Taking supreme over all i in (45) and (46),we get

η ≤ C1Mη + C2M2ζ, ζ ≤ C(M)Mη . (47)

Notice that ζ is continuous when fronts merge, while η may be discontinuous butit does not increase (see Remark 4). Hence by standard comparison argument wearrive at

η(t) ≤ C3ε, ζ(t) ≤ C3ε , for all t ∈ [0, t ] (48)

for some C3 = C3(M,T ).

(5). Upper bound for uε. We define a discrete version of the backward characteristiccurve, in the spirit of Dafermos [13].

Definition 3.2. Let T > 0. On [0, T ] × R we define a family of ε-approximatecharacteristics as a family of continuous curves with the following properties.

(i) For every (t, x) ∈ [0, T ] × R, the continuous map [0, t] 3 t 7→ x(t) satisfiesx(t) = x and

x(t) = f ′ (uε(t, x+))F ε(t, x(t)) +O(ε) , |O(ε)| ≤ Cε . (49)

where the constant C does not depend on t, t or x. Furthermore, the curvedoes not cross any nodal curve for 0 < t < t.

(ii) Given any two points (t, x1) and (t, x2) with x1 < x2, the two correspondingcurves x1(t) and x2(t) do not cross, i.e., x1(t) < x2(t) for all t ∈ (0, t].

The existence of ε-approximate characteristics is obvious once we have the esti-mate (48). Indeed, let (t, x) be a point on an ε-approximate characteristics. Then,if x is sufficiently away from any rarefaction front, we chose C = 0 in (49). Oth-erwise, if x gets very close to a rarefaction front, say xi, then we can choose xto be some intermediate value between f ′ (uε(t, x))F ε(t, x) and xi, such that x(t)approaches xi(t) but does not cross it as t decreases. Furthermore, if x is betweentwo rarefaction fronts xi and xi+1 and it gets very close to both fronts, then wechoose x to be some intermediate value between xi and xi+1, such that x(t) doesnot cross either xi or xi+1 as t decreases. Finally, if (t, x) is already on a rarefaction

front, say (t, xi), then we can choose x = f ′(ui− 1

2))F ε(t, x(t)) for t ∈ [t− δε, t] for

some small values of δ, such that x(t) departs from xi(t) on the right as t decreases.Thanks to the estimate (48), the property (49) holds.


We now rewrite (39) as

(uε − 1)t + (f(uε)F ε)x = 0 . (50)

Consider a point (t, x) such that xi(t) < x < xi+1(t) for some i. Let t → x(t)be a ε-approximate characteristic curve as in Definition 3.2, and denote by i =i(t) the index for the interval (xi, xi+1) where the curve remains. Integrating theconservation law (50) over the region in (t, y) where 0 ≤ t ≤ t ≤ T , y ≤ x(t), we get∣∣∣∣∣

∫ t

0

(ui+ 1

2(t)− 1)x(t)− f(ui+ 1

2(t))F ε(t, x(t)

)dt

∣∣∣∣∣=

∣∣∣∣∣∫ x(t)

−∞(uε(t, y)− 1) dy −

∫ x(0)

−∞(uε(0, y)− 1) dy

∣∣∣∣∣ ≤ 2m0 , (51)

where the last inequality holds thanks to the bound on ‖uε − 1‖L1 .Recall the auxiliary function α(u) in (22). By (23) and (37), its evolution along

the ε-approximate characteristic curve satisfies

d

dtα(ui+ 1

2(t)) = α′(ui+ 1

2) ˙ui+ 1

2=[f(ui+ 1

2)− (ui+ 1

2− 1)f ′(ui+ 1

2)]F ε(xi+ 1

2) , (52)

where xi+ 12

is defined in (36). We have

α(ui+ 12(t))− α(ui+ 1

2(0))

≤

∣∣∣∣∣∫ t

0

[f(ui+ 1

2)− (ui+ 1

2− 1)f ′(ui+ 1

2)]F ε(xi+ 1

2) dt

∣∣∣∣∣ .Thanks to (48), the values F ε(xi+ 1

2), F ε(t, x(t)) and F ε(t, x(t)) are very close to

each other, by a factor of C = eC3ε. We now have

α(ui+ 12(t))− α(ui+ 1

2(0))

≤ C

∣∣∣∣∣∫ t

0

[f(ui+ 1

2)− (ui+ 1

2− 1)f ′(ui+ 1

2)]F ε(t, x(t)) dt

∣∣∣∣∣≤ C

∣∣∣∣∣∫ t

0

[f(ui+ 1

2)F ε(t, x(t))− (ui+ 1

2− 1)x(t)

]dt

∣∣∣∣∣+ C

∣∣∣∣∣∫ t

0

(ui+ 12− 1)

[x(t)− f ′(ui+ 1

2)F ε(t, x(t))

]dt

∣∣∣∣∣≤ 2Cm0 + C

∣∣∣∣∣∫ t

0

I(t) dt

∣∣∣∣∣ .In the last inequality we used (51), and for I(t) we use (49) and (48) to get

I(t) = (ui+ 12(t)− 1)

[x(t)− f ′(ui+ 1

2(t))F ε(t, x(t))

]= (ui+ 1

2(t)− 1)

[(x(t)− f ′(ui+ 1

2(t))F ε(t, x(t)))

+(F ε(t, x(t))− F ε(t, x(t)))]

≤ (ui+ 12− 1) · Cε , (53)


for some C does not depend on t, x, ε. Now, let M1 be a finite constant such that

α(M1) = α

(supxu(x)

)+ 2Cm0 , (54)

and let t be the first time in [0, T ] that α(ui+ 12) = α(M1 + 1), so

ui+ 12(t) < M1 + 1 , for t < t . (55)

Now consider (44) with M = M1 + 1. From (53) we have |I(t)| ≤ M1Cε where Cnow depends on M1. Then

α(M1 + 1) ≤ α(M1) + C

∣∣∣∣∣∫ t

0

I(t) dt

∣∣∣∣∣ ≤ α(M1) + CM1Cεt ,

which gives

t ≥ α(M1 + 1)− α(M1)

CM1 C· 1

ε=

C

ε.

By choosing ε small, t can be arbitrarily large, leading to the upper bound for uε

for any finite T .In turn, this gives the uniform bounds in (41) on η and ζ for t ∈ [0, T ].

(6). BV bound for uε. For the ε-approximate solution uε we have

d

dtTV{uε} =

d

dt

∑i

∣∣∣ui+ 12− ui− 1

2

∣∣∣ =∑i

sign(ui+ 12− ui− 1

2)[

˙ui+ 12− ˙ui− 1

2

]=

∑i

sign(ui+ 12− ui− 1

2)[f2(ui+ 1

2)F ε(xi+ 1

2)− f2(ui− 1

2)F ε(xi− 1

2)]

≤∑i

∣∣∣f2(ui+ 12)− f2(ui− 1

2)∣∣∣F ε(xi+ 1

2)

+f2(ui− 12)∣∣∣F ε(xi+ 1

2)− F ε(xi− 1

2)∣∣∣

≤ 2 ‖f‖L∞ f ′(κ0)‖F ε‖L∞TV{uε}+ ‖f‖2L∞TV{F ε}≤ C · TV{uε}+ C . (56)

Therefore, total variation of uε can grow exponentially in time, but remains boundedfor finite time t ≤ T , completing the proof.

Remark 5. By Lemma 3.1 and the fact that fronts can only merge, the set ofODEs for xi(t) in (34) and for ui+ 1

2(t) in (35) are well-posed, generating unique

approximate solutions.

Remark 6. The L1 continuity in time for uε and F ε follows by a standard argu-ment, as a consequence of the a priori bounds in Lemma 3.1. We omit the details.

In next Lemma we establish the discrete version of the Oleinik inequality.

Lemma 3.3. A discrete version of a one-sided entropy inequality holds for uε,

uε(t, x+)− uε(t, y+) ≤ C max{1/t, 1}(y − x) , y > x+ ε (57)

where the constant C does not depend on x, y, t or ε.


Proof. Fix τ > 0, with τ ≤ T , and two points y1, y2 such that y1 + ε < y2. Thenthere exist two ε-approximate characteristics x1(·) and x2(·) defined on [0, τ ], with

x1(τ) = y1 , x2(τ) = y2 , x1(t) < x2(t) ∀ t ∈ [0, τ ].

Recalling (37), along any ε-approximate characteristic curve x(·) one has

d

dtuε(t, x(t)) = f2(uε(t, x))F ε(t, x) +O(ε) . (58)

By Lemma 3.1, there exist positive constants L, M and c, such that |O(ε)| ≤ Mεand

|F ε(t, x)− F ε(t, y)| ≤ L |x− y| , f ′′(u) ≤ −c < 0 ,

0 ≤ f2(u) · F ≤M , |f(u)| , |f ′(u)| ≤M ,1

M≤ F ≤M .

Let u(τ, y1+) = u1 and u(τ, y2+) = u2, and consider u1 > u2 (otherwise theestimate holds trivially). Then there exists τ0 with 0 ≤ τ0 < τ such that u(t, x1(t))−u(t, x2(t)) ≥ 0 for t ∈ [τ0, τ ], possibly vanishing for t = τ0. In this time interval wehave

d

dt(x2(t)− x1(t))

= f ′(u(t, x2(t))F ε(t, x2(t))− f ′(u(t, x1(t))F ε(t, x1(t)) +O(ε)

= [f ′(u(t, x2)− f ′(u(t, x1)]F ε(t, x2) + f ′(u(t, x1) [F ε(t, x2)− F ε(t, x1)] +O(ε)

≥ c

M[u(t, x1(t)− u(t, x2(t)]−ML(x2(t)− x1(t))− |O(ε)| . (59)

Now assume in addition

x2(t)− x1(t) ≤ y2 − y1 ∀ t ∈ [t0, τ ] , (60)

where t0 ≥ τ0 will be specified later. From (59) we deduce

d

dt(x2(t)− x1(t))

≥ c

M[u(t, x1(t)− u(t, x2(t)]−ML(y2 − y1)− |O(ε)|

≥ c

M[u(t, x1(t)− u(t, x2(t)]−M1(y2 − y1) (61)

where M1 depends on M and L. Moreover, from (58) we deduce

d

dt(u(t, x1(t))− u(t, x2(t)))

≤ L1 (u(t, x1(t))− u(t, x2(t))) +M2 (x2(t)− x1(t)) +O(ε)

≤ L1 (u(t, x1(t))− u(t, x2(t))) +M3(y2 − y1) (62)

for some constants L1, M2, and M3 depending on L and M . This yields

u(t, x1(t))− u(t, x2(t))

≥ e−L1(τ−t)(u1 − u2)−M3(y2 − y1)(τ − t)≥ e−L1T (u1 − u2)−M3(y2 − y1)(τ − t) . (63)

We now consider two cases.Case 1. If

c e−L1T

M· (u1 − u2) ≤ 4M1 (y2 − y1) ,


this immediately implies

u1 − u2 ≤4MM1

ceL1T (y2 − y1) . (64)

Case 2. If instead we have

c e−L1T

M· (u1 − u2) > 4M1 (y2 − y1) , (65)

thend

dt(x2(τ)− x1(τ)) > 0 ,

and by continuity there exists a time interval [t0, τ ] such that (60) holds. Indeed,we can choose t0 ≥ τ0 such that

c

M

[e−L1T (u1 − u2)−M3(y2 − y1)(τ − t)

]≥ 3M1(y2 − y1) , t ∈ [t0, T ] . (66)

From (63), we find that

c

3M

[u(t, x1(t))− u(t, x2(t))

]≥ M1(y2 − y1) .

Therefore, on this time interval, from (61) we have

d

dt(x2(t)− x1(t)) ≥ 2c

3M[u(t, x1(t)− u(t, x2(t)] (67)

hence (60) is valid on [t0, T ]. Integrating (67) and using (63), we now have

y2 − y1 ≥ 2c

3M

∫ τ

t0

[e−L1T (u1 − u2)−M3(y2 − y1)(τ − t)

]dt

≥ 2c

3M

{e−L1T (τ − t0)(u1 − u2)−M3(y2 − y1)

(τ − t0)2

2

},

so that (1 +

cM3

3M(τ − t0)2

)(y2 − y1) ≥ 2c

3Me−L1T (τ − t0)(u1 − u2) .

We require that t0 satisfies, in addition to (66),

cM3

3M(τ − t0)2 ≤ 1 . (68)

Therefore we deduce that

y2 − y1 ≥ce−L1T

3M(τ − t0)(u1 − u2) . (69)

If t0 = 0, then we have

u1 − u2 ≤ 3MeL1T

c· 1

τ· (y2 − y1) .

If t0 = τ0 > 0, then u(τ0, x1(τ0)) = u(τ0, x2(τ0)) and from (63) we simply have

u1 − u2 ≤ TeL1TM3(y2 − y1) .

Finally if t0 > τ0, then the definition of t0 implies that either

τ − t0 =

√M

cM3,


or else by (65)

M3(y2 − y1)(τ − t0) = e−L1T (u1 − u2)− 3MM1

c(y2 − y1)

≥ MM1

c(y2 − y1) .

Putting together the above two cases, we can choose a possibly larger t0 such that

τ − t0 = min

{√M

cM3,MM1

cM3

}=M4.

Since the bound above is independent on y1, y2, ε and τ ∈ [0, T ], from (69) wededuce the desired estimate

u1 − u2 ≤ max

{3M

cτ,

3M

cM4, TM3

}eL1T (y2 − y1) . (70)

Together with (64), we complete the proof.

3.3. Convergence of the approximate solutions and existence of entropyweak solutions. Since all fronts xi travel with the Rankine-Hugoniot speed, ourε-approximate solution uε provides a weak solution to the modified conservationlaw (39). Rewrite it as

uεt + (f(uε)F ε)x = Eε , where Eε(t, x) =[f(uε)(F ε − F ε)

]x. (71)

The following discrete weak formulation holds for all test functions φ ∈ C∞c (R2),∫ T

0

∫R

(uεφt + f(uε)F εφx) dx dt

=

∫R

(uε(T, x)φ(T, x)− uε(0, x)φ(0, x)

)dx−

∫ T

0

∫REεφdx dt . (72)

To achieve existence of weak solutions, we observe that, thanks to the a prioriestimates in Lemma 3.1, there exist some limit functions u(t, x) and F (t, x) suchthat, by extracting a subsequence ε→ 0, one has

(i) uε(0, ·)→ u(0, ·) and uε(T, ·)→ u(T, ·) in L1loc(R);

(ii) uε → u and F ε → F in L1loc([0, T ]× R);

(iii) For any given a < b, one has∫ b

a

|Eε(t, x)| dx ≤ TV{f(uε)}∥∥F ε − F ε∥∥

L∞ + ‖f‖L∞ TV{F ε − F ε} .

Since F ε interpolates linearly F ε through nodal points, and by Lemma 3.1,one has ∥∥F ε − F ε∥∥

L∞ ≤ Cεand

TV{F ε − F ε}

=∥∥(F ε − F ε)x

∥∥L1 =

∑i

∫ xi+1

xi

∣∣∣f(ui+ 12)∣∣∣ · ∣∣∣F ε(x)− F ε(xi+ 1

2)∣∣∣ dx

≤∑i

∣∣∣f(ui+ 12)∣∣∣ (xi+1 − xi)TV{F ε; (xi, xi+1)}

≤ ζ(t)TV{F ε(t, ·)} ≤ Cε .


Therefore ∫ b

a

|Eε(t, x)| dx→ 0

uniformly for t ∈ [0, T ].(iv) Since uε and f(uε) are uniformly bounded, the identity (29) holds in the limit,

i.e.,

F (t, x) = exp

{∫ ∞x

f(u(t, y)) dy

}a.e. (t, x) ∈ [0, T ]× R .

Furthermore, by taking the limit ε → 0 in (57), the entropy inequality holds.The existence of entropy weak solutions follows.

3.4. Uniqueness of entropy weak solutions. Here we complete the proof ofTheorem 1.2, proving that any two entropy weak solutions of (1), correspond-ing to the same initial data, must coincide. Therefore, each sequence {uε} of ε-approximate solutions converges to a unique limit as ε→ 0.

Let T > 0 and u ∈ W. Let u(t, x) be an entropy weak solution according toDefinition 1.1. Hence u is the unique entropy weak solution of the problem

vt + [k(t, x)f(v)]x = 0 , k(t, x) = exp

{∫ ∞x

f(u(t, s)) ds

}, v(0, ·) = u .

This fact can be proved by adapting the proof by duality argument valid for (con-cave) conservation laws without inhomogeneity ([19, 21]; see also [18]).

Now let u(t, x) be another entropy weak solution of (1) for the same initial datau. It will be the unique solution of

vt +[k(t, x)f(v)

]x

= 0 , k(t, x) = exp

{∫ ∞x

f(u(t, s)) ds

}, v(0, ·) = u .

We claim that there exists a constant L > 0 such that

‖u(t, ·)− u(t, ·)‖L1(R) ≤ L

∫ t

0

‖u(τ, ·)− u(τ, ·)‖L1(R) dτ (73)

for all t ∈ [0, T ]. By Gronwall Lemma, this implies that u(t, ·) = u(t, ·).To prove (73), we proceed as in [3, Subsect.3.3] and rely on the error formula

(see [5])

‖u(t, ·)− u(t, ·)‖L1(R) ≤∫ t

0

E(τ) dτ

with

E(τ) = lim suph→0+

‖u(τ + h, ·)− uτ (τ + h, ·)‖L1(R)

h(74)

and where uτ (t, x), defined for t ≥ τ , is the entropy weak solution of

vt +[k(t, x)f(v)

]x

= 0 , v(τ, x) = u(τ, x) .

To estimate the error term E(τ), we proceed as in [3] by evaluating the L1 distance

of solutions to equations with different coefficients k and k. We end up with theestimate

E(τ) ≤ L ‖u(τ, ·)− u(τ, ·)‖L1(R)


where the constant L depends on

‖u‖L∞ , ‖u‖L∞ , TV{u(t, ·)} , TV{u(t, ·)} , TV{k(t, ·)} , TV{k(t, ·)}

that are all bounded uniformly in time t ∈ [0, T ]. We omit the details.

4. Stability of entropy weak solutions. In this section we prove Theorem 1.3.Let u, v be entropy weak solutions for the Cauchy problems (1)-(2) and (9), respec-tively, and let {uε}ε>0 and {vε}ε>0 be sequences of corresponding ε-approximatesolutions, generated by the algorithm in Section 3.

By the uniqueness of entropy weak solutions, it will be sufficient to prove thestability of front tracking approximations. In the following, we aim at proving that

‖uε(t, ·)− vε(t, ·)‖L1 ≤ ‖uε − vε‖L1 + C

∫ t

0

‖uε(s, ·)− vε(s, ·)‖L1 ds

+ Ct[‖f − g‖L∞ + ‖f − g‖Lip + ε

](75)

for a suitable constant C depending only on the bounds on the solutions. Then, bytaking the limit ε→ 0+ in (75) and using the fact that uε → u and vε → v in L1

loc

for all t, we obtain (11), proving Theorem 1.3.Introducing the notation

G(v;x) = exp

{∫ ∞x

g(v(t, y)) dy

}, (76)

we can rewrite the equation for v as

vt + (gG)x = 0 . (77)

Let κ0, M and m0 be constants as in Theorem 1.3 for both uε and vε, i.e.,

κ0 ≤ uε(t, x) ≤M , κ0 ≤ vε(t, x) ≤M ,

‖uε(t, ·)− 1‖L1 ≤ m0 , ‖vε(t, ·)− 1‖L1 ≤ m0 .

Recalling Lemma 2.1 and (56), the following quantities are uniformly bounded,

‖F ε‖L∞ , TV{F ε} , TV{uε} , ‖f(uε)‖L∞ , ‖f(uε(t, ·))‖L1 ,

‖Gε‖L∞ , TV{Gε} , TV{vε} , ‖g(vε)‖L∞ , ‖g(vε(t, ·))‖L1 .

For a given ε > 0, let xi (i = 0, · · ·N) be the points where either uε or vε has ajump. We have

‖uε(t, ·)− vε(t, ·)‖L1 =∑∣∣∣ui+ 1

2(t)− vi+ 1

2(t)∣∣∣ (xi+1(t)− xi(t)) . (78)

Here and in the rest the summation∑

is always over i, unless if it is stated other-wise. Differentiating (78) w.r.t. t, we have

d

dt‖uε(t, ·)− vε(t, ·)‖L1 = A+B (79)

where

A =∑

sign(ui+ 12− vi+ 1

2)(ui+ 1

2− vi+ 1

2)(xi+1 − xi) , (80)

B =∑∣∣∣ui+ 1

2− vi+ 1

2

∣∣∣ (xi+1 − xi) . (81)


Estimates on A. Recall ˙ui+ 12

in (35), namely

ui+ 12

= −f(ui+ 12)Φi+ 1

2, Φi+ 1

2=F ε(xi+1+m)− F ε(xi−n)

xi+1+m − xi−n, (82)

where xi−n, xi+1+m are the two nearby fronts for uε, and for vi+ 12,

vi+ 12

= −g(vi+ 12)Ψi+ 1

2, Ψi+ 1

2=Gε(xi+1+l)−Gε(xi−k)

xi+1+l − xi−k, (83)

where xi−k, xi+1+l are the two nearby fronts for vε. Then

ui+ 12− vi+ 1

2= −(f(ui+ 1

2)− f(vi+ 1

2))Φi+ 1

2− (f(vi+ 1

2)− g(vi+ 1

2))Φi+ 1

2

−g(vi+ 12)(Φi+ 1

2−Ψi+ 1

2) .

We can writeA = A1 +A2 +A3

where

A1 = −∑∣∣∣f(ui+ 1

2)− f(vi+ 1

2)∣∣∣Φi+ 1

2(xi+1 − xi) , (84)

A2 = −∑


2)(f(vi+ 1

2)− g(vi+ 1

2))Φi+ 1

2(xi+1 − xi) , (85)

A3 = −∑


2)g(vi+ 1

2)(Φi+ 1

2−Ψi+ 1

2)(xi+1 − xi) . (86)

Note that Φi+ 12

and Ψi+ 12

are approximations to F εx and Gεx respectively on the

interval [xi, xi+1). By our construction we have∣∣∣F εx(x)− Φi+ 12

∣∣∣ ≤ Cε , ∣∣∣Gεx(x)−Ψi+ 12

∣∣∣ ≤ Cε , ∀x ∈ [xi, xi+1) . (87)

We immediately have the estimates for A2 and A3

A2 ≤ TV{F ε} · ‖f − g‖L∞ + Cε , (88)

A3 ≤ ‖g‖L∞ TV{F ε −Gε}+ Cε . (89)

The estimate for A1 will be established later in (101), together with terms from B.Estimates on B. By summation-by-parts, we have

B =∑(∣∣∣ui− 1

2− vi− 1

2

∣∣∣− ∣∣∣ui+ 12− vi+ 1

2

∣∣∣) xi . (90)

At every xi, we define the artificial speeds si, si as follows. If uε has a jumps at xi,we let si = si = xi. Otherwise, if vε has a jump at xi, we let

si = F ε(xi) ·f(vi+ 1

2)− f(vi− 1

2)

vi+ 12− vi− 1

2

, si = F ε(xi) ·g(vi+ 1

2)− g(vi− 1

2)

vi+ 12− vi− 1

2

. (91)

Note that we use the F ε for the global term in all these speeds. We now have

B = B1 +B2 +B3

where

B1 =∑

si(∣∣∣ui− 1

2− vi− 1

2

∣∣∣− ∣∣∣ui+ 12− vi+ 1

2

∣∣∣) , (92)

B2 =∑

(si − si)(∣∣∣ui− 1

2− vi− 1

2

∣∣∣− ∣∣∣ui+ 12− vi+ 1

2

∣∣∣) , (93)

B3 =∑

(xi − si)(∣∣∣ui− 1

2− vi− 1

2

∣∣∣− ∣∣∣ui+ 12− vi+ 1

2

∣∣∣) . (94)


Here in B2 and B3 we only need to sum over all jumps in vε.Now consider B1. At every point xi, we define λ−i and λ+

i as

λ−i =f(ui− 1

2)− f(vi− 1

2)

ui− 12− vi− 1

2

F ε(xi) , λ+i =

f(ui+ 12)− f(vi+ 1

2)

ui+ 12− vi+ 1

2

F ε(xi) . (95)

The term B1 can be written as

B1 = B1,a +B1,b ,

where

B1,a =∑∣∣∣ui+ 1

2− vi+ 1

2

∣∣∣ (λ+i − si)−

∣∣∣ui− 12− vi− 1

2

∣∣∣ (λ−i − si) , (96)

B1,b =∑∣∣∣ui− 1

2− vi− 1

2

∣∣∣λ−i − ∣∣∣ui+ 12− vi+ 1

2

∣∣∣λ+i . (97)

Consider B1,a and write B1,a =∑B1,a,i. There are various situations. Let’s

first consider if uε has a jump at xi so vi− 12

= vi+ 12. There are several cases.

• If ui− 12≥ vi− 1

2and ui+ 1

2≥ vi+ 1

2, we have

B1,a,i = (ui+ 12− vi+ 1

2)λ+i − (ui− 1

2− vi− 1

2)λ−i − (ui+ 1

2− ui− 1

2)si = 0 .

• If ui− 12≤ vi− 1

2and ui+ 1

2≤ vi+ 1

2, it is completely similar. We have B1,a,i = 0.

• If ui− 12≤ vi− 1

2= vi+ 1

2≤ ui+ 1

2, then the front is a shock. We have

λ+i ≤ si ≤ λ

−i , therefore B1,a,i ≤ 0 .

• If ui− 12≥ vi− 1

2= vi+ 1

2≥ ui+ 1

2, the front is a rarefaction, therefore it is small.

We have

λ+i − si ≤ Cε , si − λ−i ≤ Cε , therefore B1,a,i ≤ C

∣∣∣ui+ 12− ui− 1

2

∣∣∣ ε .The cases where vε has a jump at xi is completely similar. In summary, we have

B1,a ≤ Cε∑∣∣∣ui+ 1

2− ui− 1

2

∣∣∣+∣∣∣vi+ 1

2− vi− 1

2

∣∣∣ ≤ Cε [TV{uε}+ TV{vε}] . (98)

For B1,b, summation-by-parts again gives

B1,b =∑∣∣∣f(ui+ 1

2)− f(vi+ 1

2)∣∣∣ (F ε(xi+1)− F ε(xi)) . (99)

We compare B1,b with the term A1 in (84). By construction we have (similar to(87)) ∣∣∣∣F ε(xi+1)− F ε(xi)

xi+1 − xi− Φi+ 1

2

∣∣∣∣ ≤ Cε . (100)

So B1,b and A1 are close in value, but with opposite signs. We have

A1 +B1,b

=∑∣∣∣f(ui+ 1

2)− f(vi+ 1

2)∣∣∣ (xi+1 − xi)

(F ε(xi+1)− F ε(xi)

xi+1 − xi− Φi+ 1

2

)≤ Cε ·

∑∣∣∣f(ui+ 12)− f(vi+ 1

2)∣∣∣ (xi+1 − xi)

≤ Cε [‖f(uε(t, ·))‖L1 + ‖f(vε(t, ·))‖L1 ]

≤ C ′ε . (101)

The last inequality holds because ‖f(uε(t, ·))‖L1 and ‖f(vε(t, ·))‖L1 are bothbounded for all t.


Now, consider the term B2. Since we sum over all i where vε has a jump at xi,we have ui+ 1

2= ui− 1

2, therefore∣∣∣ui− 12− vi− 1

2

∣∣∣− ∣∣∣ui+ 12− vi+ 1

2

∣∣∣ ≤ ∣∣∣vi+ 12− vi− 1

2

∣∣∣ . (102)

And,

|si − si| = F ε(xi)

∣∣∣∣∣∣(f(vi+ 1

2)− g(vi+ 1

2))−(f(vi− 1

2)− g(vi− 1

2))

vi+ 12− vi− 1

2

∣∣∣∣∣∣≤ ‖F ε‖L∞ ‖f − g‖Lip .

Therefore,

B2 ≤ ‖F ε‖L∞ ‖f − g‖Lip∑∣∣∣vi+ 1

2− vi− 1

2

∣∣∣ ≤ ‖F ε‖L∞ ‖f − g‖Lip TV{vε} . (103)

Finally, consider the term B3. We have

|xi − si| =g(vi+ 1

2)− g(vi− 1

2)

vi+ 12− vi− 1

2

|F ε(xi)−Gε(xi)| .

Combining this with (102), we get

B3 ≤∑∣∣∣g(vi+ 1

2)− g(vi− 1

2)∣∣∣ · |F ε(xi)−Gε(xi)|

≤ ‖F ε −Gε‖L∞ g′(κ0)TV{vε} . (104)

Concluding the estimate. Putting the estimates (88), (89), (98), (101), (103) and(104) back into (79), we get

d

dt‖uε − vε‖L1

≤ ‖F ε‖L∞ ‖f − g‖Lip TV{vε} + ‖F ε −Gε‖L∞ g′(κ0)TV{vε}+ TV{F ε} ‖f − g‖L∞ + ‖g‖L∞ TV{F ε −Gε} + Cε . (105)

By symmetry we also have

d

dt‖uε − vε‖L1

≤ ‖Gε‖L∞ ‖f − g‖Lip TV{uε} + ‖F ε −Gε‖L∞ f ′(κ0)TV{uε}+ TV{Gε} ‖f − g‖L∞ + ‖f‖L∞ TV{F ε −Gε} + Cε . (106)

Now let’s estimate the terms in (105)-(106). We have

‖f(vε)− g(vε)‖L1 = ‖[f(vε)− f(1)]− [g(vε)− g(1)]‖L1

= ‖vε − 1‖L1 ‖f − g‖Lip . (107)

Note that it is important to have f(1) = g(1) = 0 to obtain (107). For ‖F ε −Gε‖L∞

we have

‖F ε −Gε‖L∞ ≤ max{‖F ε‖L∞ , ‖Gε‖L∞}∫ ∞−∞|f(uε)− g(vε)| dy

≤ max{‖F ε‖L∞ , ‖Gε‖L∞}∫ ∞−∞|f(uε)− f(vε)|+ |f(vε)− g(vε)| dy

≤ max{‖F ε‖L∞ , ‖Gε‖L∞} [f ′(κ0) ‖uε − vε‖L1 + ‖f(vε)− g(vε)‖L1 ] (108)


and for TV{F ε −Gε} we have

TV{F ε −Gε} = ‖F εx −Gεx‖L1 =

∫|f(uε(t, x))F ε − g(vε(t, x))Gε| dx

≤∫|f(uε)− g(vε)|F ε + |g(vε)| |F ε −Gε| dx

≤∫|f(uε)− f(vε)|F ε + |f(vε)− g(vε)|F ε + |g(vε)| |F ε −Gε| dx

≤ ‖F ε‖L∞ [f ′(κ0) ‖uε − vε‖L1 + ‖f(vε)− g(vε)‖L1 ]

+ ‖g(vε)‖L1 ‖F ε −Gε‖L∞ . (109)

By using (107), (108) and (109), the estimates (105) and (106) become

d

dt‖uε(t, ·)− vε(t, ·)‖L1

≤ C[‖uε(t, ·)− vε(t, ·)‖L1 + ‖f − g‖L∞ + ‖f − g‖Lip + ε

], (110)

for some bounded constant C that depends only on κ0, M and m0. Integrating(110) from 0 to t, we obtain the integral estimate (75), completing the proof.

Remark 7. The estimates (105) and (106) are very similar to the ones in [17],Theorem 1.3, where the authors study a scalar conservation law with variable coef-ficients in multi space dimension

ut +∇ · (k(x)f(u)) = ∆A(u) .

Continuous dependence on initial data, on the coefficient k and on the flux functionf is established with very similar results, by using Kruzkov inequality and a variabledoubling technique. However, their coefficient k(x) is local and does not depend ont.

On the other hand, the front tracking algorithm proposed here can be easilyextended to conservation laws with variable coefficient in one space dimension

ut + (k(t, x)f(u))x = 0 ,

for k under suitable assumptions, such as in [3], Theorem 2. Existence and contin-uous dependence on initial data, on the coefficient k and on the function f wouldfollow in a similar way.

Acknowledgements. The authors wish to thank the referees for their useful com-ments, which lead to the improvement of the manuscript.

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E-mail address: [email protected]

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