Conformal restriction: the radial case

We describe all random sets that satisfy the radial conformal restriction property, therefore providing the analogue in the radial case of results of Lawler, Schramm and Werner in the chordal case.Comment: Accepted for publication by Stochastic Processes and their Application

Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect

We investigate a non-isothermal diffuse-interface model that describes the dynamics of two-phase incompressible flows with thermo-induced Marangoni effect. The governing PDE system consists of the Navier--Stokes equations coupled with convective phase-field and energy transport equations, in which the surface tension, fluid viscosity and thermal diffusivity are temperature dependent functions. First, we establish the existence and uniqueness of local strong solutions when the spatial dimension is two and three. Then in the two dimensional case, assuming that the $L^\infty$-norm of the initial temperature is suitably bounded with respect to the coefficients of the system, we prove the existence of global weak solutions as well as the existence and uniqueness of global strong solutions.Comment: accepted by Euro. J. Appl. Math. in June 201