Stochastic perturbation of scalar conservation laws

Description

Stochastic perturbation of scalar conservation laws? J. Vovelle April 21, 2009 Abstract In this course devoted to the study of stochastic perturbations of scalar conservation laws, we expose the first part of the paper [EKMS00]. After a short introduction to scalar conservation laws and stochastic differential equations, we give the proof of the existence of an invariant measure for the stoschastic inviscid periodic Burgers' Equation in one dimension according to [EKMS00]. Contents 1 Scalar conservation laws with source term 2 1.1 Characteristic curves . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 From solution to characteristic curves . . . . . . . . . 2 1.1.2 From characteristic curves to solution . . . . . . . . . 3 1.1.3 The one-dimensional Burgers' Equation . . . . . . . . 5 1.2 Entropy solutions . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Lax-Oleinik Formula . . . . . . . . . . . . . . . . . . . 8 2 Stochastic differential equations 9 2.0.2 Stochastic integral . . . . . . . . . . . . . . . . . . . . 9 2.0.3 Stochastic differential equation .

• equations


• surface then


• stochastic differential


• let u0 ?


• free surface


• w¯ ?


• scalar conservation

Sujets

• Équation
• Stochastic differential equation
• Free surface

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 9
• Langue: English

Extrait

conservation

perturbation of scalar

Abstract

In this course devoted to the study of stochastic perturbations of scalar conservation laws, we expose the first part of the paper [EKMS00]. After a short introduction to scalar conservation laws and stochastic differential equations, we give the proof of the existence of an invariant measure for the stoschastic inviscid periodic Burgers’ Equation in one dimension according to [EKMS00].

Scalar conservation laws with source term 1.1 Characteristic curves . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 From solution to characteristic curves . . . . . . . . . 1.1.2 From characteristic curves to solution . . . . . . . . . 1.1.3 The one-dimensional Burgers’ Equation . . . . . . . . 1.2 Entropy solutions . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Lax-Oleinik Formula . . . . . . . . . . . . . . . . . . .

2 2 2 3 5 6 8

laws ∗

Stochastic

J. Vovelle

April 21, 2009

3 Scalar conservation laws with stochastic forcing 14 3.1 Entropy solution . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Burgers’ Equation . . . . . . . . . . . . . . . . . . . . . . . . 15

∗ Course given at the Spring SchoolAnalytical and Numerical Aspects of Evolution EquationsT.U. Berlin, 30/03/09–03/04/09

Contents

1

1

Stochastic differential equations 2.0.2 Stochastic integral . . . . . . . . . . . . . . . . . . . . 2.0.3 Stochastic differential equation . . . . . . . . . . . . . 2.0.4 Transition semi-group . . . . . . . . . . . . . . . . . . 2.0.5 Invariant measure . . . . . . . . . . . . . . . . . . . . 2.0.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 11 11 12

2

4

Invariant measure for the stochastic Burgers’ Equation 4.1 One-sided minimizers . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Bound on velocities . . . . . . . . . . . . . . . . . . . 4.1.2 Intersection of minimizers . . . . . . . . . . . . . . . . 4.1.3 Invariant solution . . . . . . . . . . . . . . . . . . . . .

16 16 18 20 23

1 Scalar conservation laws with source term LetT N be theN Let-dimensional torus.A∈C 2 (R;R N ),u 0 ∈L ∞ (T N ) and W ¯ ∈C 1 (T N ×R consider the equation). We ∂ t u(x, t) + div(A(u))(x, t) =W ¯ (x, t), x∈T N , t >0 (1) with initial condition

u(x,0) =u 0 (x), x∈T N .(2) Eq. (1) is easily solved in the linear caseA(u) =au,a∈R N  it. Indeed, ¯ rewrites as the transport equation (∂ t +a∙ r)u=W ,whose solution t u(x, t) =u 0 (x−ta) + Z 0 W( ¯ x−(t−s)a, s)ds

satisfies (2).

1.1 Characteristic curves

1.1.1 From solution to characteristic curves LetT >0. Suppose thatu∈C 1 (T N ×[0, T]) is a regular solution to the scalar conservation law (1). By the chain-rule formula, we then have (∂ t +a(u)∙ r)u=Wni ¯ T N ×(0, T),(3) which is a non-linear transport equation with source term. Herea(u) := A 0 (u integral curves). Theγof the vector field (1, a(u)) T are the curves (s(σ), ξ(σ)) T satisfying the equation ˙ = 1,  s ˙ (σ = ) a(u(ξ(σ), σ)), (4) ξ(σ)

¯ and (3) asserts that the derivative ofualong any suchγisW(γ). More technically, forx∈T N ,t∈(0, T], takings=σ(by (4)), letξ(s;t, x) be the solution to ˙ ξ(s) =a(u(ξ(s), s)), ξ(t) =x.(5)

2