Dissipative Solutions to the Stochastic Euler Equations

We study the three-dimensional incompressible Euler equations subject to stochastic forcing. We develop a concept of dissipative martingale solutions, where the nonlinear terms are described by generalised Young measures. We construct these solutions as the vanishing viscosity limit of solutions to the corresponding stochastic Navier–Stokes equations. This requires a refined stochastic compactness method incorporating the generalised Young measures. As a main novelty, our solutions satisfy a form of the energy inequality which gives rise to a weak–strong uniqueness result (pathwise and in law). A dissipative martingale solution coincides (pathwise or in law) with the strong solution as soon as the latter exists.

Martingale solutions to a stochastic smectic-A liquid crystal model with multiplicative noise of jump type

In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic smectic-A liquid crystal system driven by a pure jump noise in both 2D and 3D bounded domains. We prove the existence of a global weak martingale solution under some non-Lipschitz assumptions on the coefficients. The construction of the solution is based on a Faedo–Galerkin approximation, a compactness method and the Skorokhod representation theorem. In the two-dimensional case, we prove the pathwise uniqueness of the weak solution, which implies the existence of a unique probabilistic strong solution.

On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects

<p style='text-indent:20px;'>We consider a stochastic Allen-Cahn-Navier-Stokes equations with inertial effects in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ D\subset\mathbb{R}^{d} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d = 2, 3 $\end{document}</tex-math></inline-formula>, driven by a multiplicative noise. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients. The construction of the solution is based on the Faedo-Galerkin approximation, compactness method and the Skorokhod representation theorem.</p>

Fully discrete finite element approximation of the stochastic Cahn–Hilliard–Navier–Stokes system

In this paper we study the numerical approximation of the stochastic Cahn–Hilliard–Navier–Stokes system on a bounded polygonal domain of $\mathbb{R}^{d}$, $d=2,3$. We propose and analyze an algorithm based on the finite element method and a semiimplicit Euler scheme in time for a fully discretization. We prove that the proposed numerical scheme satisfies the discrete mass conservative law, has finite energies and constructs a weak martingale solution of the stochastic Cahn–Hilliard–Navier–Stokes system when the discretization step (both in time and in space) tends to zero.

Splitting-up scheme for the stochastic Cahn–Hilliard Navier–Stokes model

In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of [Formula: see text] [Formula: see text]. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.