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Shock Waves for Conservation Laws With Physical Viscosity

Tai-Ping Liu

Academia Sinica, Taiwan Stanford University

In Honor of 60th Birthday of Alberto Bressan June 13-June 17, 2016, SISSA

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Conservation laws with physical viscosity ut + f(u)x = (B(u)ux)x, u ∈ Rn. Examples: p-system, n = 2, vt − ux = 0 ut + p(v)x = (ε(v)ux)x. Compressible Navier-Stokes equations, n = 3, ρt + (ρu)x = 0, continuity equation, (ρu)t + (ρu2 + p)x = (µux)x, momentum equations. (ρE)t + (ρEu + pu)x = (µuux + κθx)x, energy equation. MHD, Visco-elasticy, n = 7.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Conservation laws with physical viscosity ut + f(u)x = (B(u)ux)x. Kawashima-Shizuta Systems: Assumption 1: The system has a strictly convex entropy η.That is, there exists an entropy pair (η(u), F(u)), such that η(u) is strictly convex, M0(u) ≡ ∇2η(u) > 0, (∇η)f ′ = ∇F, and (∇2η)B ≥ 0. The above physical systems are endowed with convex

• entropy. This is needed for the local existence theory.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

ut + f(u)x = (B(u)ux)x. Kawashima-Shizuta Systems: Assumption 2: The viscosity matrix B(u) is nonzero and there exists a smooth one-to-one mapping u = g(˜ u) such that the null space K of ˜ B(˜ u) ≡ B(g(˜ u))g′(˜ u) is independent of ˜

• u. Also, K⊥ is invariant under

g′(˜ u)tM0(g(˜ u)), and ˜ B(˜ u) maps Rn to K⊥. That is, by change of variables a new viscosity matrix ˜ B has a special block structure.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

ut + f(u)x = (B(u)ux)x. Kawashima Systems: Assumption 3: Any right eigenvector of f ′(u) is not in the null space of B(u). Assumption 3, together with Assumption 1, implies that the system is fully dissipative, even though the viscosity matrix B is rank deficient. Here we use the term full dissipation to indicate the situation where small solutions decay in time, rather than where solutions become regular

• immediately. Indeed, discontinuities from initial data are

permanent in the solution when B is degenerate.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Kawashima-Shizuta Systems: ut + f(u)x = (B(u)ux)x. Perturbation of constant state u0 = 0: u(x, 0) ∈ Hs, s ≥ 2. Theorem (Kawashima Dissertation 1984) Golbal existence and decay of solutions.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Hyperbolic characteristics: f ′(u)rj(u) = λj(u)rj(u), lj(u)f ′(u) = λj(u)lj(u), ljrk = δjk, j, k = 1, 2, · · · , n. Assuming that the p-characteristic is genuinely nonlinear, ∇λp(u) · rp(u) = 0, then there exists p-shock waves u(x, t) = φ(x − st): −s(φ − u−) + f(φ) − f(u−) = B(φ)φ′, satisfying the Rankine-Hugoniot condition s(u+ − u−) = f(u+) − f(u−), φ(±∞) = u±, and the Lax entropy condition λp(u−) > s > λp(u+).

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

For a small perturbation of a constant state, say u0 = 0, there are accurate approximate solutions the j-diffusion waves θj(x, t)rj(0): θj are self-similar θi = θi(x, t) = (t + 1)−1/2ζ((x − λj(0)(t + 1))/ √ t + 1) and solutions to θjt + λj(0)θjx + Cii(θ2

j )x = (lt j Brj)(0)θixx.

The solution is unique if we require the total mass to be fixed, ∞

−∞ θj(x, t) dx = dj. Here the equation is Burgers if the

j-characteristic is genuinely nonlinear Cjj = 1

2(lj(0))tf ′′(0)(rj(0)rj(0)) = 0 and heat equation if it is

linearly degenerate Cjj = 0. The viscosity coefficient (lt

j Brj)(0)

is postive under Assumption 1 and Assumption 3.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Eigen decompostion of initial perturbation: ∞

−∞

u(x, 0)dx =

n

• j=1

djrj(0). Theorem (Liu-Zeng 1997) u(x, t) →

n

• j=1

θj(x, t)rj(0), as t → ∞. Proof: Explicit construction of Green’s function for the linearized systems. Duhamel’s principle for pointwise estimates of lower differentials. Kawashima type energy estimates for higher differentials.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Linearized system, around a constant state: ut + Aux = Buxx, A ≡ f ′(0), B ≡ B(0). Green’s function (matrix) G(x, t): Gt + AGx = BGxx, G(x, 0) = δ(x)I. Green’s function G∗(x, t) for artificial viscosity, µj ≡ ljBrj > 0, j = 1, · · · , n,: Ht+diag(λ1, · · · , λn)Hx = diag(µ1, · · · , µn)Hxx, H(x, 0) = δ(x)I, H(x, t) = diag( 1 √4πµ1t e

− (x−λ1t)2

4µ1t

, · · · , 1 √4πµnt e− (x−λnt)2

4µnt

). G∗(x, t) ≡

• r1

, · · · , rn ∗ H(x, t)       l1 . . . ln      

∗

.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Theorem G(x, t) = G∗(x, t)+O(1)(t+1)−1/2t−1/2

n

• j=1

e−

(x−λj t)2 4Ct

+

m

• k=1

δ(x−βkt)Pk. Proof: Fourier transform ˆ G(η, t). Long-Short wave decomposition. Long wave, |η| small, ˆ G(η, t) well approximated by ˆ G∗, Invert the Fourier transform for the long wave part by complex analytic method to yield heat kernel and faster decaying terms. Short wave, |η| large, expansion to yield δ-functions, intermediate waves by weighted energy and spectral gap. Perturbation theory of Kato.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Perturbation of a pth Lax shock: |u(x, 0) − φ(x)| = O(1)ε(1 + |x|)−3/2, λpu−) > s > λp(u+). Eigen decomposition ∞

−∞

u(x, 0)dx =

• j<p

djrj(u−) +

• j>p

djrj(u+) + x0(u+ − u−). Theorem (Liu-Zeng 2013) The solution exists and tends to the combination of diffusion waves and shock wave as t → ∞: u(x, t) → φ(x + x0 − st) +

• j<p

θj(x, t)rj(u−) +

• j>p

θj(x, t)rj(u+).

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

System of hyperbolic conservation laws: ut + f(u)x = 0. f ′(u)ri(u) = λi(u)ri(u), li(u)f ′(u) = λi(u)li(u). Nonlinear coupling: Hyperbolic nonlinearity Cj

kl(u) ≡ lj(u)f ′′(u)(rk(u), rl(u)), j, k, l = 1, 2, · · · , n.

The i-th characteristic field genuinely nonlinear iff Ci

ii = 0,

representing shock forming, [Lax]. Cj

kl, j = k or j = k, measure nonlinear inviscid coupling.

ljBrk, coupling due to viscosity. The nonlinear coupling yields nonlinear wave interactions and the wave patterns of the following kinds:

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Coupling waves: ψi(x, t) ≡ [(x − λ0

i (t + 1))2 + t + 1]− 1

2 ,

i = p, ¯ ψi(x, t) ≡ [|x − λ0

i (t + 1)|3 + (t + 1)2]− 1

3 ,

i = p, χi(x, t) ≡ min{ε− 1

2 ψ 3 2

i (x, t), ε

1 2 (t + 1)− 1 2 ψ 1 2

i (x, t)}chari(x, t),

i = p, chari(x, t) ≡      1, if 0 < x < λ0

i (t + 1) and i > p

1, if λ0

i (t + 1) < x < 0 and i < p

0, otherwise , ψp(x, t) ≡ [(|x| + ε(t + 1))2 + t + 1]− 1

2 ,

¯ ψp(x, t) ≡ [(|x| + ε(t + 1))3 + (t + 1)2]− 1

3 ,

η(x, t) ≡ min{ε(x2 + 1)− 1

2 (|x| + t + 1)− 1 2 , (x2 + 1)− 3 4 (|x| + t + 1)− 1 2 }. Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

For the combined energy estimates (for higher order differentiations) and pointwise estimates (for lower order differentiations) we require that the initial perturbation U(x, 0) = u(x, 0) − φ(x + x0) ∈ H13(R). To effectviely describe the wave coupling, we further require the pointwise decay of the perturbation. Thus we assume that δ0 ≡ sup

x∈R

• (|x| + 1)

3 2

3

• j=0
• ∂j

∂xj U(x, 0)

• + (|x| + 1)

5 4

• ∂4

∂x4 U(x, 0)

• +(|x|+1)

6

• j=5
• ∂j

∂xj U(x, 0)

• +(|x|+1)

1 2

8

• j=7
• ∂j

∂xj U(x, 0)

• +U(x, 0)H13

is bounded.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Theorem Suppose that the magnitude δ0 of the initial data and the shock strength ε are sufficiently small. Then the solution u(x, t) of the Kawashima-Shizuta system exists globally in time and |¯ vi(x, t)| ≤ Cδ0

• ψ

3 2

i (x, t) +

• k=i

¯ ψ

3 2

k (x, t) + χi(x, t)

+ ε2e−ε|x|/µψ

1 2

p (x, t) + ε

1 2 η(x, t)

• , i = p,

|¯ vp(x, t)| ≤ Cδ0

• ψ

3 2

p (x, t) +

• k=p

¯ ψ

3 2

k (x, t) + εe−ε|x|/µψ

1 2

p (x, t) + η(x, t)

• ,

n

i=1 ¯

vi(x, t)rj(φ) ≡ u(x, t) − φ(x + x0 − st) −

• j<p

θj(x, t)rj(u−) −

• j>p

θj(x, t)rj(u+).

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Proof: Explicit construction of accurate approximate Green’s function for system linearized around the shock profile: interpolation of the Green’s function about the constant states u±. For the p-field, use Burgers weights; for the transversal fields, use simple partition of unity. Assessment of the accuracy of the approximation. Sharp estimate of the Green’s function, particularly the coupling components of shock with other waves and the dependence on the shock strength ε. Study of wave interactions through the Duhamel’s principle, making use of the explicit form of the Green’s

• function. The leading coupling waves are identified.

Close the analysis by energy method for higher differentials.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity

Physical Viscosity

Conclusion:

1

Coupling gives rise to sub-scale waves. The coupling waves around the shock profile decay slower in L∞(x) but faster in L1(x) than the coupling wave between the main diffusion waves.

2

How to extend the Bianchini-Bressan theory of zero dissipation limit to system with physical viscosity? How about the zero mean free path limit for the Boltzmann equation in the kinetic theory?

3

One basic problem to address these highly singular limits is the Reimann problem. A general solution to the Riemann problem contains rich coupling due to the very distinct nature of shock waves and rarefaction waves.

Tai-Ping Liu Shock Waves for Conservation Laws With Physical Viscosity