hopf-cole transformation - brown university · 2017. 3. 7. · hopf-cole transformation hopf-cole...

Hopf-Cole Transformation

Tai-Ping Liu

Academia Sinica, TaiwanStanford University

March 20, 2016Brown University

Tai-Ping Liu Hopf-Cole Transformation


Hopf, Eberhard The partial differential equationut + uux = µuxx . Comm. Pure Appl. Math. 3, (1950),201-230.Julian D. Cole On a quasi-linear parabolic equationoccurring in aerodynamics. Quart. Appl. Math. 9, (1951),225-236.

Hopf-Cole transformation:

ut + uux = κuxx Burgers equation,⇒ Bt + (Bx )2

2 = κBxx , Bx = u, Hamilton-Jacobi equation,Introduce Hopf-Cole relation B(x , t) = −2κ log[φ(x , t)],⇒ φt = κφxx , Heat equation.



Solution formula for initial value problem for Burgers equation:

u(x , t) = u(x , t , κ) =

∫∞−∞

x−yt e−

(x−y)2

4κt −1

2κ

∫ y0− u(z,0)dzdy∫∞

−∞ e−(x−y)2

4κt −1

2κ

∫ y0− u(z,0)dzdy

.



Burgers equation:Bateman proposed it for considering shock profile:Bateman, H. Some recent researches on the motion offluids, Monthly Weather Review 43, (1915), 163-170.Burgers proposed it for the study of turbulence:Burgers, J. M. Application of a model system to illustratesome points of the statistical theory of free turbulence.Nederl. Akad. Wetensch., Proc. 43, (1940), 2-12.Cole derived the Burgers equation from gas dynamics:

∂w∂t

+ β∂w∂x

=43ν∗∂2w∂x2 .

”for w = excess of flow velocity over a sonic velocity, whereβ = (γ + 1)/2, ν∗ = the kinematic viscosity at soniccondition”



Hopf-Cole Transformation:

Hopf: ”The reduction of (1) to the heat equation was knownto me since the end of 1946. However, it was not until 1949that I became sufficiently acquainted with the recentdevelopment of fluid dynamics to be convinced that atheory of (1) could serve as an instructive introduction intosome of the mathematical problems involved.”Friedrichs, K. O. Formation and decay of shock waves.Communications on Appl. Math. 1, (1948). 211245.Hopf was inspired by the works of Burgers on turbulenceand Friedrichs’ theory of N-waves.Forsyth, A.R. Theory of Differential Equations, Vol. VI,Cambridge University Press (1906), Page 102, Ex. 3.The Hopf-Cole transformation is embedded in this exercisein Forsyth’s book.



Hopf:Solution formula for the inviscid Burgers equationut + (u2/2)x = 0 in the zero dissipation limit κ→ 0+:

F (x , y , t) = (x−y)2

2t +∫ y

0− u(z,0)dz,miny F (x , y , t) = F (ξ, t),limκ→0+ u(x , t , κ) = u(ξ,0).

Metastable states:

limt→∞

limκ→0+

u(x , t , κ) 6= limκ→0+

limt→∞

u(x , t , κ).

Modern theory of hyperbolic conservation laws.



Hopf:

bt + (b2

2)x = κbxx , b(x ,0) = Aδ(x), Burgers kernel

By Hopf-Cole transformation,

b(x , t ; A) =

√κ√t(e

A2κ − 1)e−

x24κt

√π +

∫∞x√4κt

(eA

2κ − 1)e−y2dy.

For initial data with finite mass, a solution of the Burgersequation tends to Burgers kernel:∫ ∞−∞|u(x , t)−b(x , t ; A)|dx = O(1)t−

12 , as t →∞, A =

∫ ∞−∞

u(x ,0)dx .



Hopf:On the other hand, for inviscid Burgers equation, the solutiontends to N-waves:∫∞−∞ |u(x , t)− N(x , t ; p,q)| = O(1)t−

12 ,

p = minx∫ x−∞ u(x ,0)dx , q = maxx

∫∞x u(x ,0)dx , two time invariants,

N(x , t ; p,q) =

{xt , for −

√−2pt < x <

√2qt ;

0, otherwise.

limt→∞

limκ→0+

u(x , t , κ) = N − waves, two time invariants;

6= limκ→0+

limt→∞

u(x , t , κ) one time invariant.

N-waves represent metastable states for the Burgers solutions.



Outside of gas dynamics:Miura, R. M. Korteweg-de Vries equation andgeneralizations. I. A remarkable explicit nonlineartransformation. J. Mathematical Phys. 9 (1968),1202-1204.

Vt − 6VVx + Vxxx = 0, KdV ⇒ φt − 6φ2φx + φxxx = 0, Modified KdV,V = φ2 ± φx , Miura transformation.

”It is rare and surprising to find a transformation betweentwo simple nonlinear partial differential equations ofindependent interest. One is reminded of the Hopf-Coletransformation of quadratically nonlinear Burgers equationinto the heat conduction (diffusion) equation. A number ofinvestigators (including us) have attempted unsuccessfullyto find a similar simple linearizing transformation for theKdV equation, but a complicated one will be given in VI.”



Outside of gas dynamics:Kardar, M.; Parisi, G.; Zhang, Y.-C. Dynamic Scaling ofGrowing Interfaces Phys. Rev. Lett. Vol. 56, Iss. 9 -3March (1986), 889-892.Evolution of the profile of a growing interface: theHamilton-Jacobi equation plus a noise η:

∂h∂t

= ν∇2h +λ

2(∇h)2 + η(x , t).

Hopf-Cole transform⇒ linear equation with a source:{∂W∂t = ν∇2W + λ

2ν η(x , t)W ,

W (x , t) = eλ2ν h(x ,t).

New scaling, distinct from deterministic dissipationequations comes up due to the noise.”We thus have an intriguing connection between evolutionsof a hydrodynamic and a growth pattern!”



Scalar convex hyperbolic conservation law, f ′′(u) 6= 0,

ut +f (u)x = 0 ⇒ λt + λλx = 0, inviscid Burgers , λ = f ′(u).

System of hyperbolic conservation laws, e.g. Eulerequations in gas dynamics

ut + f (u)x = 0, u ∈ Rn, f ′(u)r j(u) = λj(u)r j(u),l j(u)f ′(u) = λj(u)l j(u), l j(u) · rk (u) = δjk , j , k = 1,2, · · · ,n.

”Convexity”: ∇uλj(u) · r j(u) 6= 0 ”genuine nonlinear” field,e.g. acoustic waves.j-simple waves: u(x , t) moves along integral curve of r j(u).

λt + λλx = 0, λ(x , t) = λj(u(x , t)) ⇒ ut + f (u)x = 0.



Viscous conservation laws, e.g. CompressibleNavier-Stokes equations

ut + f (u)x = (B(u,ν)ux )x .

Dissipation parameters, e.g. ν = (µ, κ) viscosity and heatconductivity.The Burgers equation is used for construction ofapproximate j-simple waves for each genuinely nonlinearfield

λt + λλx = κλxx , λ(x , t) = λj(u(x , t)).

Burgers dissipation parameter κ is the diagonal element ofthe viscosity matrix B in the characteristic coordinates ofthe hyperbolic part:

κ = l j(u)B(u,ν)r j(u).



ut + f (u)x = 0 hyperbolic conservation lawsSolutions of finite mass tends to N-waves at the rate oft−1/4 in L1(x) as consequence of pointwise estimate.Liu, T.-P. Pointwise convergence to N-waves for solutionsof hyperbolic conservation laws. Bull. Inst. Math. Acad.Sinica 15, (1987), no. 1, 1-17.ut + f (u)x = (B(u,ν)ux )x , viscous conservation laws.Solutions of finite mass tends to Burgers and heat kernelsalso at the rate of t−1/4 in L1(x) and as consequence ofpointwise estimate..Liu, T.-P.; Zeng, Y. Large time behavior of solutions forgeneral quasilinear hyperbolic-parabolic systems ofconservation laws. Mem. Amer. Math. Soc. 125 (1997),no. 599, viii+120 pp.Open problem: Metastability.



Hopf-Cole transformation is used for finding exactexpression of Burgers Nonlinear waves.The Burgers nonlinear waves is used for construction ofapproximate nonlinear waves for system of conservationlaws.The linearized Hopf-Cole transformation is used for theexplicit construction of Green’s function for Burgersequation linearized around a nonlinear wave.The construction is Green’s function for systems is basedon the Burgers Green’s function.This is essential for the study of shock, initial layers forsystem of conservation laws.



Burgers shock formation, λ0 > 0, using Hopf-Cole:(uS)t + uS(uS)x = κ(bS)xx ,

uS(x ,0) =

{λ0, for x < 0,−λ0, for x > 0,

us(x , t) = −λ0

Erfc(−x−λ0t√4κt

)− e−λ0xκ Erfc(x−λ0t√

4κt)

Erfc(−x−λ0t√4κt

) + e−λ0xκ Erfc(x−λ0t√

4κt).

The thickness T0 of the initial layer to form Burgers shock profilebS, the time when the error function Erfc approaches

√π,

bS(x) = limt→∞

uS(x , t) = −λ0 tanh(λ0x2κ

).

λ0T0√4κT0

= O(1), or T0 = O(1)κ

(λ0)2 .



Burgers rarefaction wave(hR)t + ( (hR)2

2 )x = 0,

hR(x ,0) =

{−λ0, for x < 0,λ0, for x > 0;

bR(x , t) = λ0

eλ0x2κ Erfc(−x+λ0t√

4κt)− e−

λ0x2κ Erfc(x+λ0t√

4κt)

eλ0x2κ Erfc(−x+λ0t√

4κt) + e−

λ0x2κ Erfc(x+λ0t√

4κt).

Within the hyperbolic rarefaction wave region,x ∈ (−λ0t + M

√4κt , λ0t −M

√4κt), and after initial layer time,

the difference of the Burgers rarefaction wave bR and theinviscid rarefaction wave x/t :

bR(x , t)− xt

= O(1)[1

|x − λ0t |+

1|x + λ0t |

], t > O(1)κ

(λ0)2 .



Linear Hopf-Cole transformationBurgers equation linearized around a given solution u(x , t):

ut + ( u2

2 )x = κuxxUx = u, U(x , t) = −2κ log[φ(x , t)],vt + (uv)x = κvxx , Burgers equation linearized around u(x , t).

Linearize the Hopf-Cole relation V + U = −2κ log[φ+ ζ]:

V = −2κζ

φ, linearized Hopf-Cole relation, ⇒

ζt = κζxx and the solution representation to the solution of thelinearized Burgers equation:

v(x , t) =∂

∂x

∫∞−∞[ 1√

4πκte−

(x−y)2

4κt φ(y ,0)V (y ,0)]dy

φ(x , t).



Green’s function for shock profileGreen’s function GS(x , t ; x0, t − t0) for the shock profile bS(x)using linearized Hopf-Cole:

(GS)t + bS(GS)x = κ(GS)xx , GS(x ,0) = δ(x − x0);

GS(x , t ; x0) =1√

4πκte−

(x−x0)2

4κte

λ0x02κ + e−

λ0x02κ

e−λ0x2κ + e−

λ0x2κ

e(λ0)2t

4κ .

The Green’s function as weighted combination of the heatkernel with speeds ±λ0:

GS(x , t ; x0) =1 + e−

λ0|x0|κ

1 + e−λ0|x|

κ

H(x + λ0t , t), for x > 0, x0 > 0;

e−λ0|x|

κ H(x + λ0t , t), for x < 0, x0 > 0;

H(x − λ0t , t), for x < 0, x0 < 0;

e−λ0|x|

κ H(x − λ0t , t), for x > 0, x0 < 0.



Green’s function for rarefaction waves:

GR(x , t ; x0, t0) = e−[x−x0−(λ0(t−t0)]2

4κ(t−t0)

Erfc(−x0+λ0t0√4κt0

) + Erfc(x0+λ0t0√4κt0

)

Erfc(−x+λ0t√4κt

) + Erfc(x+λ0t√4κt

).

The propagation of waves is around the zero line of theexponential, along inviscid characteristics x = x0 + λ0(t − t0).The essential support of the information is in the region givenby

t(x0 − t0x/t)2

4κt0(t − t0)= O(1), or |x − t

t0x0| = O(1)

√κ(t − t0)

tt0,

varying from sub-linear, dissipatve scale√

t − t0 for t − t0 small,to linear, hyperbolic scale t for t − t0 large.



Open problem: Riemann problem

ut + f (u)x = (B(u,ν)ux )x ,

u(x ,0) =

{ul , x < 0,ur , x > 0.

t →∞ ⇒ ν → 0, zero dissipation limit,

ut + f (u)x = (B(u,ν)ux )x ⇒ ut + f (u)x = 0.

Hoff, D.; Liu, T.-P. The inviscid limit for the Navier-Stokesequations of compressible, isentropic flow with shock data.Indiana Univ. Math. J. 38 (1989), no. 4, 861-915. singleshock, zero mass, using Hopf-Cole for initial layerBianchini, S.; Bressan, A. Vanishing viscosity solutions ofnonlinear hyperbolic systems. Ann. of Math. (2) 161(2005), no. 1, 223-342. general initial values, artificialviscosity, generalized Glimm.



Boltzmann equation

∂t f(x , t , ξ) + ξ · ∂x f(x , t , ξ) =1k

Q(f, f)(x , t , ξ)

Open problem: Riemann problem

f(x , t , ξ) =

{Ml(ξ), x < 0,Mr (ξ), x < 0.

t →∞ ⇒ k → 0, zero mean free path,

Boltzmann solutions ⇒ Euler solutions,

Yu, S.-H. Initial and shock layers for Boltzmann equation.Arch. Ration. Mech. Anal. 211 (2014), no. 1, 1-60. singleshock, nonzero mass, use Boltzmann Green’s function,Hopf-Cole, etc.


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