Stochastic fractional conservation laws

In this paper, we consider the Cauchy problem for the nonlinear fractional con-servation laws with stochastic forcing. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanish-ing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper considers the Cauchy problem for the nonlinear fractional conservation laws with stochastic forcing. The existence of desired kinetic solution and the convergence of the approximate viscous solutions to a kinetic solution are shown. Furthermore, the existence of an invariant measure under a nonlinearity-diffusivity condition is proved, and the uniqueness and ergodicity of the invariant measure are demonstrated.