Hyperbolic Conservation Laws with Memory Cleopatra Christoforou - - PowerPoint PPT Presentation

Hyperbolic Conservation Laws with Memory

Cleopatra Christoforou Northwestern University USA July, 2006

Eleventh International Conference on Hyperbolic Problems Theory, Numerics, Applications Lyon, France

Hyperbolic Conservation Laws with Memory

July, 06

Conservation Laws in one-space dimension: Ut + F(U)x = 0 Elastic medium: the flux F is determined by the value U(x, t). Viscoelastic medium: the flux depends also on the past history

• f the medium U(x, τ) for τ < t.

Materials with fading memory that correspond to constitutive relations with flux of the form: F(U(x, t)) +

t

0 k(t − τ)G(U(x, τ)) dτ

(1) i.e. Ut + F(U)x +

t

0 k(t − τ)G(U(x, τ))x dτ = 0

(2)

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Classical solutions: Dafermos, Hrusa, MacCamy, Nohel, Re-

nardy, Slemrod, Staffans... Main results:

• If the initial data are “small” and sufficiently smooth, then

there exists a unique global smooth solution to (2) that decays to equilibrium as t → +∞.

in constrast to elastic media.

• If the initial data are “large”, then singularities develop in finite

time.

as in elastic media. Weak solutions:

G.-Q. Chen, Dafermos, Nohel, Rogers, Tzavaras... Summary of results: Existence of global weak solutions in L∞ (bounded measurable functions) is established, for special equations by the method

• f compansated compactness.

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The aim is to treat entropy weak solutions of bounded variation (BV ). Motivation: Hyperbolic Conservation Laws with Fading Memory in one-space dimension Ut + F(U)x +

t

0 k(t − τ)G(U(x, τ))x dτ = 0

(3) can be viewed as a linear Volterra equation under suitable choice

• f F and G. This was first observed by MacCamy [M] and later

employed in Dafermos [D] and Nohel–Rogers–Tzavaras [NRT].

  

Ut + A(U)Ux + g(U) = H(t) ¯ U −

t

0 K(t − τ)U(τ) dτ

U(0, x) = ¯ U(x) , (4) where, x ∈ R, U(t, x) ∈ Rn, A(U) ∈ Mn×n, g : Rn → Rn and H, K : [0, +∞) → Mn×n.

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Hyperbolic Conservation Laws with Memory

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The Vanishing Viscosity Method.

  

Uε

t + A(Uε)Uε x + g(Uε) = H(t) ¯

U −

t

0 K(t − τ)Uε(τ) dτ + εUε xx

Uε(0, x) = ¯ U(x) , (5) ⇒ Uε → U in L1

loc

as ε → 0+ ♦ Scalar conservation law: g ≡ 0, H ≡ 0, K ≡ 0

• Oleinik [O], 1957: One-space dimension
• Kruzkov [K], 1970: Several space dimensions

♦ Systems of conservation laws in one-space dimension:

• Bianchini and Bressan [BiB], 2005: g ≡ 0, H ≡ 0, K ≡ 0.
• Christoforou [C], 2006: g = 0 and H ≡ 0, K ≡ 0.

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Assumptions (⋆): Let U∗ be a constant equilibrium solution to the hyperbolic problem (4),

• 1. ˜

B(U∗) . = [R(U∗)]−1Dg(U∗)R(U∗) is strictly column diagonally dominant, i.e. ˜ Bii(U∗) −

• j=i

| ˜ Bji(U∗)| ≥ β > 0 i = 1, ..., n. 2. ˜ K(s) . = R(U∗)−1K(s)R(U∗) ∈ L1[0, +∞) is absolutely domi- nated by ˜ B, i.e. there exists a positive constant κ ≥ 0, such that for each i = 1, . . . , n

n

• j=1

+∞

| ˜ Kji(s)| ds < κ, and 0 ≤ κ < β.

• 3. H(·) ∈ L1[0, +∞).

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Theorem 1. (G-Q. Chen, Christoforou) Consider the Cauchy problem Uε

t +A(Uε) Uε x+g(Uε) = H(t) ¯

U−

t

0 K(t−τ) Uε(τ, x) dτ+εUε xx (6)

Uε(0, x) = ¯ U(x). (7) Assume that the system is strictly hyperbolic. Under Assump- tions (⋆), there exists a constant δ0 > 0 such that if ¯ U − U∗ ∈ L1 and TV {¯ U} < δ0, then for each ε > 0, (6)-(7) has a unique solution Uε, defined for all t ≥ 0, that satisfies TV {Uε(t, ·)} +

t

0 TV {Uε(s, ·)} ds ≤ C TV {¯

U}, (8) where C is a positive constant that is independent of t and ε.

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Moreover, if V ε is another solution of (6) with initial data ¯ V , then Uε(t) − V ε(t)L1 +

t

0 Uε(τ) − V ε(τ)L1dτ ≤ L ¯

U − ¯ V L1. (9) Furthermore, the continuous dependence property with respect to time holds, i.e. Uε(t) − Uε(s)L1 ≤ L′(|t − s| + √ε| √ t − √s|), (10) for t, s > 0. Finally, as ε ↓ 0+, Uε → U in L1

loc ,

where U is the admissible weak solution of the hyperbolic system with memory (4). The proof follows closely the fundamental ideas in Bianchini– Bressan and the techniques in Christoforou in order to treat the source g(u). Additional estimates are employed to handle the integral term as lower order perturbation in terms of the damping effect of g due to Assumptions (⋆).

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Theorem 2. (G.-Q. Chen, Christoforou) Scalar equation:

  

ut + (f(u))x +

t

0 k(t − τ)(f(u(τ)))x dτ = 0

u(0, x) = u0(x) , (11) Let r be the resolvent kernel associated with k: r + r ∗ k = −k. Assume r is nonnegative, nonincreasing in L1(R+), then for each ε > 0, consider uε

t+f(uε)x+

t

0 k(t−τ)f(uε(τ))x dτ = ε

• uε +

t

0 k(t − τ)uε(τ) dτ

• xx

. There exists a unique solution uε defined globally with a uniform BV bound.

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Then as ε → 0+, uε converges in L1

loc to an entropy solution

u ∈ BV to

  

ut + (f(u))x +

t

0 k(t − τ)(f(u(τ)))x dτ = 0

u(0, x) = u0(x) , which satisfies: TV {u(t)} +

t

0 r(t − τ)TV {u(τ)} dτ ≤ L M(u0)

(12) u(t) − u(s)L1 ≤ C M(u0) |t − s|, (13) uL∞(R2

+) ≤ 2u0L∞(R),

(14) where L = 1 + rL1(R+), M(u0) = TV {u0} + 2u0L∞(R) and C is a constant independent of ε and TV {u0}.

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Proof: By employing the resolvent kernel r, (11) can be written as follows ut + f(u)x + r(0)u = r(t)u0 −

t

0 r′(t − τ)u(τ) dτ

Note the similarity of the above equation with the one in the case of systems, (4). From here and on, the techniques are motivated by Vol’pert [V]- Kruzkov [K]. Let v = ux, vt + (f′(u)v)x + εr(0)v = εr(εt)ux − ε2

t

0 r′(ε(t − τ))v(τ) dτ + vxx

d dt(v(t)L1) + εr(0)v(t)L1 ≤εr(εt)TV {u0} − ε2

t

0 r′(ε(t − τ))v(τ)L1 dτ 10

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Integrating over t ∈ [0, T] and changing the order of integration v(T)L1 + εr(0)

T

0 v(t)L1 dt ≤ TV {u0} + ε

T

0 r(εt) dt · TV {u0}

− ε

T

0 [r(ε(T − τ)) − r(0)]v(τ)L1 dτ.

Thus, v(t)L1+ε

t

0 r(ε(t−τ))v(τ)L1 dτ ≤ TV {u0}·

• 1 + rL1[0,+∞)
• .

Similarly, if w = ut, one can establish the L1 time dependence estimate.

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Hyperbolic Conservation Laws with Memory

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Theorem 3. (G.-Q. Chen, Christoforou). Uniqueness and Stability in L1. Let the resolvent kernel r(t) associated with k be a nonnegative and non-increasing function in L1(R+). Let u, v ∈ BV (R2

+) be

entropy solutions to (11) with initial data u0, v0 ∈ BV (R), re- spectively. Then u(t)−v(t)L1(R)+

t

0 r(t−τ)u(τ)−v(τ)L1(R) dτ ≤ L u0−v0L1(R).

That is, any entropy solution in BV to (11) is unique and stable in L1. As a consequence, if u0 is only in L∞, not necessarily in BV (R), there exists a global entropy solution u ∈ L∞ to (11).

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Hyperbolic Conservation Laws with Memory

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Application: The kernel is a relaxation kernel (i.e. kν depends

• n a small parameter ν > 0).

Theorem 4. (G-Q. Chen, Christoforou) Let uν be the unique entropy solution to:

• uν

t + f(uν)x +

t

0 kν(t − τ)(f(uν(τ)))x dτ = 0

uν(0, x) = u0(x). (15) uν,ε → uν as ε → 0+ Assume that rν is uniformly bounded in L1 independent of ν, rνL1 ≤ M, = ⇒ TV {uν} ≤ C TV {u0}. If kν(t) ⇀ (α − 1) δ(t) as ν → 0+, = ⇒ uν → u in L1

loc

(16) local conservation law

• ut + α(f(u))x = 0

u(0, x) = u0(x).

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There is a rich family of kernels kν that satisfy the assumptions

• f the Theorems.

Set of Kernels: (i) k′

ν(t) ≥ 0 and kνL1(R+) ≤ K for some constant K indepen-

dent of ν > 0; (ii) det(1 + ˆ kν(z)) = 0 for any z with Re(z) ≥ 0, and ˆ kν(t)(1 + ˆ kν(t)) ≤ 0 for the Laplace transform ˆ kν of kν; (iii) supω∈R|(1 + ˆ kν(iω))−1| ≤ q for some constant q independent

• f ν;

(vi) There exist positive numbers T ∼ ν and τ ∼ ν such that

• |s|≥T |kν(t)| ≤

1 12q, sup

0<s<η

• R |kν(t) − kν(t − s)| dt ≤ 1

4.

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• Example. kν(t) = (α−1)

ν

e− t

ν

resolvent kernel: rν(t) = (1−α)

ν

e−α

ν t, for 0 < α < 1.

  

ut + (f(u))x + (α−1)

ν

t

0 e−(t−τ)

ν

(f(u(τ)))x dτ = 0 u(0, x) = u0(x) , (17)

• r equivalently a system of two equations with relaxation:

        

ut + (f(u) − v)x = 0 vt = (1 − α)f(u) − v ν (u(0, x), v(0, x)) = (u0, v0) (18) characteristic condition: 0 < α < 1. Thus, the result of Theorem 4 applying this special case is equivalent to the zero relaxation limit as first considered sys- tematically in Chen-Levermore-Liu [CLL]; also see [LN,STW,Yo] for the model.

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Some of the References: [BiB] S. Bianchini, A. Bresssan, Vanishing viscosity of nonlinear hyperbolic systems, Ann. of Math. 161 (2005), (1) , 223-342. [CC] G.-Q. Chen, C. Christoforou, Solutions for a nonlocal conservation law with fading memory, (preprint). [CLL] G.-Q. Chen, D. Levermore, and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), 787–830. [C] C. Christoforou, Hyperbolic systems of balance laws via vanishing viscos- ity, J. Diff. Eqs, 221/2 (2006), 470–541. [C1] C. Christoforou, Uniqueness and sharp estimates on solutions to hyper- bolic systems with dissipative source, Comm. PDE (accepted). [D1] C. M. Dafermos, Dissipation in materials with memory, Viscoelasticity and rheology, (Madison, Wis., 1984), 221–234, Academic Press, Orlando, FL, 1985. 16

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Hyperbolic Conservation Laws with Memory

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[D2] C. M. Dafermos, Hyperbolic conservation laws with memory, Differen- tial equations (Xanthi, 1987), 157-166. [DN] C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyper- bolic volterra integrodifferential equations, Comm. PDE 4 (3) 1979, 219-278. [K] S. Kruzkov, First-order quasilinear equations with several space variables,

• Mat. Sbornik 123 (1970), 228-255.

[Lx] P. D. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl.

• Math. 10 (1957), 537-566.

[M] R. C. MacCamy, An integrodifferential equation with applications in heat flow, Q. Appl. Math. 35 (1977), 1-19. [NRT] J. A. Nohel, R. C. Rogers, A. E. Tzavaras, Weak solutions for a nonlinear system in viscoelasticity. Comm. PDE 13 (1988) no. 1, 97-127. [O] O. A. Oleinik, Discontinuous solutions of non-linear differential equa-

• tions. Usp. Mat. Nauk 12 (1957), 3–73.

[RHN] M. Renardy, W. Hrusa and J. A. Nohel, Mathematical Problemsin Viscoelasticity, Longman, New York, 1987. 17

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