Conservation laws driven by Lévy white noise

Abstract

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.

Continuous dependence estimate for conservation laws with Lévy noise

“…The work in [2] establishes existence and uniqueness of entropy solution for the multidimensional Cauchy problem (1.1).…”

“…Roughly speaking, the theory developed in [16] covers quasi-linear parabolic equations driven by Lévy noise and typically the solutions of such equations enjoy regularizing effect. A comprehensive entropy solution theory, within L p -solution framework, for (1.1) is made available by Biswas et al [2] very recently. We also mention that Dong and Xu [10] established the global well-posedness of strong, weak and mild solutions for one-dimensional viscous Burger's equation driven by Poisson process with Dirichlet boundary condition.…”

“…In view of results in [2], this problem has unique entropy solution and, following [6], we derive a fractional BV estimate.…”

“…The existence of global smooth solutions for (2.1) is detailed in [2], and the following proposition holds. …”

“…[2]) could be viewed as continuous dependence on the initial data for problems of type (1.1). A continuous dependence result involving the flux function and the noise coefficient is a subtle one and the proof requires the regularized problem: (s, y)) ds = |z|>0 σ (v (s, y); z)Ñ(dz, ds) + yy v (s, y) ds, (s, y) ∈ T , v (0, y) = v 0 (y), y ∈ R d ;…”