Abstract
In this paper, we establish the solvability of martingale solutions for the stochastic Navier-Stokes equations with Ito-Levy noise in bounded and unbounded domains in $ \mathbb{R} ^d$,$d=2,3.$ The tightness criteria for the laws of a sequence of semimartingales is obtained from a theorem of Rebolledo as formulated by Metivier for the Lusin space valued processes. The existence of martingale solutions (in the sense of Stroock and Varadhan) relies on a generalization of Minty-Browder technique to stochastic case obtained from the local monotonicity of the drift term.
Highlights
Martingale solutions provide a characterization of the spacetime statistical solutions for stochastic system
The concept of martingale solutions approach for finite dimensional diffusion processes was pioneered in the works of Stroock and Varadhan [35, 36] while this approach for some class of infinitedimensional problems in Lusin spaces was formulated in Metivier [22]
The stochastic Navier-Stokes equations (SNSEs) driven by Gaussian noise has been extensively studied over the past decades
Summary
Martingale solutions provide a characterization of the spacetime statistical solutions for stochastic system. We establish the existence of martingale solutions supported on the set of all solutions of the Navier-Stokes equations perturbed by Gaussian and Levy noises. Stochastic Navier-Stokes equations, martingale solutions, Levy noise. By making use of the stochastic Minty-Browder argument involving the locally monotone operators, we prove the existence of martingale solutions for SNSEs with multiplicative Ito-Levy noises in bounded and unbounded domains. Let u, v ∈ D(0, T ; V )∩L∞(0, T ; H)∩L2(0, T ; V) be the two paths defined on a same probability space (Ω, F , Ft, P ) with same Q-Wiener process W and Poisson measure πk, k = 1, 2 · · · satisfying the system (2.1).
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