Global martingale weak solutions for the three-dimensional stochastic chemotaxis-Navier-Stokes system with Lévy processes.

This paper considers the three-dimensional stochastic chemotaxis-Navier-Stokes (SCNS) system subjected to a Lévy-type random external force in a bounded domain. Until recently, the existed results concerning global solvability of SCNS system mainly concentrated on the case of two spatial dimensions, little is known about the SCNS system in dimension three. We prove in the present work that the initial-boundary value problem for the three-dimensional SCNS system has at least one global martingale solution under proper assumptions, which is weak in both the analytical and stochastic sense. A new stochastic analogue of entropy-energy inequality and an uniform boundedness estimate are derived, which enable us to construct global-in-time approximate solutions from a suitably regularized SCNS system via the Contraction Mapping Principle. The proof of the existence of martingale solution is based on a stochastic compactness method and an elaborate identification of the limits procedure, where the Jakubowski-Skorokhod Theorem is applied to deal with the phase spaces equipped with weak topology. [ABSTRACT FROM AUTHOR]