Stochastic perturbation of scalar conservation laws

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English

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

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Publié par

profil-urra-2012

Langue

English

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

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

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 LetTNbe theN Let-dimensional torus.A∈C2(R;RN),u0∈L∞(TN) and W¯∈C1(TN×R consider the equation). We ∂tu(x, t) + div(A(u))(x, t) =W¯(x, t), x∈TN, t >0 (1) with initial condition

u(x,0) =u0(x), x∈TN.(2) Eq. (1) is easily solved in the linear caseA(u) =au,a∈RN it. Indeed, ¯ rewrites as the transport equation (∂t+a∙ r)u=W ,whose solution t u(x, t) =u0(x−ta) +Z0W(¯x−(t−s)a, s)ds

satisﬁes (2).

1.1 Characteristic curves

1.1.1 From solution to characteristic curves LetT >0. Suppose thatu∈C1(TN×[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¯TN×(0, T),(3) which is a non-linear transport equation with source term. Herea(u) := A0(u integral curves). Theγof the vector ﬁeld (1, a(u))Tare the curves (s(σ), ξ(σ))Tsatisfying 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∈TN,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)

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