In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution is based on the classical Faedo-Galerkin approximation, compactness method and the Jakubowski's version of the Skorokhod representation theorem for non-metric spaces. We prove the solution is pathwise unique and further establish the existence of a strong solution in the 2-D case.
In this paper we establish local-in-time existence and uniqueness of strong solutions in H s for s > n 2 to the viscous, zero thermaldiffusive Boussinesq equations in R n , n = 2, 3. Beale-Kato-Majda type blow-up criterion has been established in three-dimensions with respect to the BM O-norm of the vorticity. We further prove the local-in-time existence and blow-up criterion for non-viscous and fully ideal Boussinesq systems. Commutator estimates due to Kato and Ponce (1988) [23] and Fefferman et. al. (2014) [18] play important roles in the calculations.Mathematics Subject Classification (2010). Primary 76D05; Secondary 76D03.
In this paper, a higher order time-discretization scheme is proposed, where the iterates approximate the solution of the stochastic semilinear wave equation driven by multiplicative noise with general drift and diffusion. We employ variational method for its error analysis and prove an improved convergence order of 3 2 for the approximates of the solution. The core of the analysis is Hölder continuity in time and moment bounds for the solutions of the continuous and the discrete problem. Computational experiments are also presented.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.