Stochastic scalar conservation laws

By Jin Feng, David Nualart. 2008

157 citations recorded by the source corpus.

We introduce a notion of stochastic entropic solution a la Kruzkov, but with Ito's calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L1 contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/52a2ccc091000590306a9aa6ab05fe38e4a6af5c

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In Applied Mathematics

Citations in the corpus

Degenerate parabolic stochastic partial differential equations (2013, background): “The concept of kinetic solution was first introduced by Lions, Perthame, Tadmor in [24] for deterministic scalar conservation laws and applies to more general situations than the one of entropy solution as considered for example in [4,12,21].”
Scalar conservation laws with stochastic forcing (2010, methodology): “to be equivalent to the notion of entropy solution used in [Kim03, FN08, VW09], provided the convergence of the vanishing viscosity method has been proved, hence in the context of [Kim03, VW09] or in [FN08] in dimension 11. In fact, we prove that our notion of kinetic solution is also equivalent to ”
Scalar conservation laws with rough (stochastic) fluxes (2013, background): “It is natural to ask whether the approach we are putting forward for (5) also applies to SSCL with semilinear rough path dependence like Recently Debussche and Vovelle [4] (see also Feng and Nualart [5] and Chen, Ding and Karlsen [1], Debussche, Hofmanova and Vovelle [3], Hofmanova [7, 8]) put forwa”
Degenerate parabolic stochastic partial differential equations: Quasilinear case (2013, background): “Also in the stochastic setting there are several papers concerned with entropy solutions for hyperbolic conservation laws, the first one being [16] then [2, 10, 27].”
The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation (2014, background): “Nualart introduced in [18] a notion of strong entropy solution in order to prove the uniqueness of the entropy solution.”
Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, background): “Debussche and Vovelle [19], Debussche, Hofmanova and Vovelle [17], Feng and Nualart [25], Hofmanova [33], Holden and Risebro [34], etc.”
Stochastic non-isotropic degenerate parabolic–hyperbolic equations (2016, result): “[16] J. Feng and D. Nualart.”
On Nonlinear Stochastic Balance Laws (2011, methodology): “Now the same arguments as in Section 4 of Feng-Nualart [9] yield that there exists an Ft–adapted stochastic process u = u(t) ∈ C([0,∞);L2(Rd)) satisfying almost surely that (i) E î ‖uε(t, ·)‖(2)Hs(Rd) ó <∞ for all t > 0; (ii) ∂xixju (t, ·) ∈ C(R) for all i, j = 1, .”
A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “1), further developed in the stochastic setting in [2,5,21,48,62].”
THE CAUCHY PROBLEM FOR CONSERVATION LAWS WITH A MULTIPLICATIVE STOCHASTIC PERTURBATION (2012, background): “The notion of strong entropy condition of Feng and Nualart does not seem to be required to capture the noise–noise interaction, instead we recover that term through a comparison with the vanishing viscosity solution.”