Seminarof Analysis

Speaker: Prof. ZdzislawBrzezniak (Department of Mathematics, University of York)

Date/time: 14/05/2025 (Wednesday), from 14:15 to 15:15.

Location:  Online via Zoom.

Please findthe details below:

Link: https://videoconf-colibri.zoom.us/j/98576468310?pwd=Onlqd9Yz2ioDQfAAbP4qmdIrvQYSJY.1

MeetingID: 985 76468310

Passcode: 971632

Title:  Weak martingale solutions to stochasticNavier-Stokes-Cahn-Hilliard system with transport noise

Abstract:  In thistalk, we investigate the weak solvability of an initial boundary value problemknown as the Navier-Stokes-Cahn-Hilliard system, which describes the dynamicsof a homogeneous, incompressible and isothermal mixture of two immiscibleNewtonian fluids flowing in a bounded  2Dor 3D domain under stochastic perturbations.

We assume that the density and viscosity of themixture are constants and, to prove the existence result, we consider anapproximation problem and use the Jakubowski-Skorohod Theorem to prove that thelaws of the corresponding solutions on a certain non-metric topological space$Z_T$ have a sequence weakly convergent to a new probability

measure on $Z_T$.

Now, by following the argument of Mikuleviciusand Rozovskii in their paper (Ann. Probab. 33(1) (2005), 137--176) with somemodifications, we prove that the canonical process on the space $Z_T$ is infact a martingale solution of our problem with respect to the new measure.

The approach is quite interesting compared tothe existence approach in the literature, since we combine both theJakubowski-Skorohod theorem and the Mikulevicius and Rozovskii argument to dealwith our problem.

This talk is based on a joint research withAristide Ndongmo Ngana (York).

Seminar of Analysis

Speaker:  Prof. Pierre-A. Vuillermot (Department of Mathematics, University of Lisbon and Institut Élie Cartan de Lorraine).

Date/time:  21/05/2025 (Wednesday), from 14:15 to 15:15.

Location:   Room 4.7, Building VIII.

Title:  On a new class of weighted Orlicz-Sobolev spaces and their embeddings

Abstract:  In this talk, I will introduce a new scale of weighted Orlicz-Sobolev spaces and discuss the existence of various continuous and compact embeddings for them. Continuous embeddings are relatively easy to come by, I will show that the way to compactness is more tortuous as it is based on the combination of the existence of a Schauder basis in the spaces under consideration with a condition on the generating Orlicz functions regarding their local behavior at the origin. I will illustrate the results by means of several examples.