A stochastic Allen-Cahn-Navier-Stokes system with singular potential

Di Primio Andrea, Politecnico di Milano

The Allen-Cahn equations, as proposed by S. M. Allen and J. W. Cahn in 1979, are usually seen as a second-order relaxation of the fourth-order Cahn–Hilliard system, yet having independent interest in a number of contexts (e.g. ordering of atoms during separation phenomena). In order to take hydrodynamical effects into account, the Allen-Cahn system is usually coupled with the celebrated Navier-Stokes equations. Moreover, two independent cylindrical stochastic perturbations, which account for thermodynamical effects (e.g. microscopic collisions) can be introduced. The resulting problem, namely the stochastic Allen-Cahn–Navier-Stokes system, starting from random initial data in a smooth domain of the d-dimensional Euclidean space, is the topic of this talk. A singular potential, as prescribed by the classical thermodynamical derivation of the model, is considered in the Allen–Cahn system. The problem is endowed with a no-slip conditon for the Navier–Stokes velocity field, as well as homogeneous Neumann conditions for the Allen–Cahn order parameter and chemical potential. In this talk, I will give some insight on the work carried out jointly with M. Grasselli (Politecnico di Milano) and L. Scarpa (Politecnico di Milano) towards showing the existence of analytically-weak martingale solutions in two and three spatial dimensions, as well as probabilistically-strong solutions in two dimensions. Moreover, the existence of a unique pressure field is also achieved and will be discussed. Open problems and future directions of research will also be presented.

Area: CS5 - Stochastic PDEs for physical models (Margherita Zanella)

Keywords: Allen-Cahn equations; Navier-Stokes equations; stochastic PDEs; martingale solutions; logarithmic potential

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