We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itó-Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first establish the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the L 1 -contraction principle. Finally, the L 1 contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.2000 Mathematics Subject Classification. 45K05, 46S50, 49L20, 49L25, 91A23, 93E20.
Abstract. The stochastic Allen-Cahn equation with multiplicative noise involves the nonlinear drift operator A (x) = ∆x − |x| 2 − 1 x. We use the fact that A (x) = −J ′ (x) satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate supfor all small δ > 0, where X is the strong variational solution of the stochastic Allen-Cahn equation, while Y j : 0 ≤ j ≤ J solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh {tj ; 1 ≤ j ≤ J} of size k > 0 which covers [0, T ].
This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng & Nualart[9] have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a-priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.2000 Mathematics Subject Classification. 60H15, 35L65, 35R60, 60E15.
We are concerned with multidimensional stochastic balance laws driven by Lévy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise only depends on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable.
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