Scalar conservation laws with stochastic forcing

Debussche, Arnaud; Vovelle, Julien

doi:10.1016/j.jfa.2010.02.016

Cited by 128 publications

(312 citation statements)

References 17 publications

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“…[17]), only discontinuous solutions to the Cauchy problem for (1) can be rigorously defined for large time (discontinuities appear even if the initial condition is smooth), and a notion of entropic solution is necessary for uniqueness. In the case that g is stochastic, further notions of solutions are necessary [9,10].…”

Section: Introductionmentioning

confidence: 99%

“…Additionally, let us mention [14] where stochastic entropic solutions to one-dimensional scalar conservation laws have been univoquely defined. In [9], kinetic solutions have been recently defined in any dimension (they coincide with the former in the case of space dimension one). Moreover we refer to [10] for a Hamilton-Jacobi reformulation of Burgers equation.…”

Section: Introductionmentioning

confidence: 99%

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Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing

et al. 2015

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Abstract. We perform numerical simulations in the one-dimensional torus for the first order Burgers equation forced by a stochastic source term with zero spatial integral. We suppose that this source term is a white noise in time, and consider various regularities in space. For the numerical tests, we apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time. Our Monte-Carlo simulations are analyzed in bounded time intervals as well as in the large time limit, for various regularities in space. The empirical mean always converges to the space-average of the (deterministic) initial condition as t → ∞, just as the solution of the deterministic problem without source term, even if the stochastic source term is very rough. The empirical variance also stablizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise.Résumé. Nous effectuons uneétude numérique de l'équation de Burgers non visqueuse en dimension un d'espace, avec des conditions aux limites périodiques et un terme source stochastique de moyenne spatiale nulle. Ce terme source possède la régularité d'un bruit blanc en temps, tandis que nous considérons différentes régularités en espace. Pour les tests numériques, nous utilisons un schéma de volumes finis combinant une intégration en temps de type Euler-Maruyama avec le flux numérique de Godunov. Nous effectuons des simulations avec la méthode de Monte-Carlo et analysons les résultats pour différentes régularités en accordant une attention particulière au comportement en temps long. Il apparaît que la moyenne empirique des réalisations converge toujours vers la moyenne en espace de la condition initiale (déterministe) quand t → ∞, comme c'est le cas pour la solution du problème sans terme source, même dans le cas où le terme stochastique est peu régulier en espace. Par ailleurs, la variance empirique converge elle aussi en temps long, vers une valeur qui dépend de la régularité et de l'amplitude du terme stochastique.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing

et al. 2015

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“…Also in the stochastic setting there are several papers concerned with entropy solutions for hyperbolic conservation laws, the first one being [16] then DEGENERATE PARABOLIC SPDE'S: QUASILINEAR CASE 3 [2,10,27]. The first work dealing with kinetic solutions in the stochastic setting was given by Debussche and Vovelle [7]. Their concept was then further generalized to the case of semilinear degenerate parabolic SPDEs by Hofmanová [12].…”

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confidence: 99%

“…In order to explain these recent developments more precisely, let us recall the basic ideas of the proofs in [7] and [12].…”

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confidence: 99%

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Degenerate parabolic stochastic partial differential equations: Quasilinear case

Debussche¹,

Hofmanová²,

Vovelle³

2016

Ann. Probab.

113

160

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In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an L 1 -contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized Itô formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

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Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws

Gess

Souganidis

2016

Comm Pure Appl Math

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Abstract. We study the long-time behavior and regularity of the pathwise entropy solutions to stochastic scalar conservation laws with random in time spatially homogeneous fluxes and periodic initial data. We prove that the solutions converge to their spatial average, which is the unique invariant measure of the associated random dynamical system, and provide a rate of convergence, the latter being new even in the deterministic case for dimensions higher than two. The main tool is a new regularization result in the spirit of averaging lemmata for scalar conservation laws, which, in particular, implies a regularization by noise-type result for pathwise quasi-solutions.

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Scalar conservation laws with stochastic forcing

Cited by 128 publications

References 17 publications

Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing

Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing

Degenerate parabolic stochastic partial differential equations: Quasilinear case

Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws

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