arXiv subjects

Z. Brzeźniak

Publications and source records attributed to Z. Brzeźniak.

7 recordsLinked to original sources

arXiv2026

Weak solutions to the 2D or 3D stochastic NSCHEs

We consider a diffuse interface model for the mixture of two incompressible fluids driven by transport noise. Under suitable abstract assumptions, we prove the existence of a global weak martingale solution as well as the pathwise uniqueness of a global strong solution in the two dimensional case. This is the first result addressing a unified framework derived from the works by Brze{ź}niak \& Motyl and Mikulevicius \& Rozovskii, on the study of stochastic partial differential equations.

math.PR↗

arXiv2026

On a stochastic Cahn-Hilliard-Brinkman model

In this paper, we consider a stochastic version of the Cahn-Hilliard-Brinkman model in a smooth two- or three-dimensional domain with dynamical boundary conditions. The system describes creeping two-phase flows and is basically a coupling of the Brinkman equation for the velocity field that governs the flow through the porous media coupled with convective Cahn-Hilliard equations for the phase field, both with two independent sources of randomness given by general multiplicative-type Wiener noises in the Cahn-Hilliard equations. The existence of a weak solution, both in the probabilistic and PDEs sense, is proved. Our construction of a solution is based on the classical Faedo-Galerkin approximation, the Yosida approximation and uses a compactness method. Our paper is the first attempt to generalize the paper \cite{Colli+Knopf+Schimperna+Signor_2024} to a stochastic setting.

math.PR↗

arXiv2021

Ergodicity for stochastic equation of Navier--Stokes type

In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier--Stokes equations system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant probability measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite-dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic N.S.E., and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.

math.PR↗

arXiv2016

Large deviations and transitions between equilibria for stochastic Landau-Lifshitz-Gilbert equation

We study a stochastic Landau-Lifshitz equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is compactness, or weak to strong continuity, of the solution map for a deterministic Landau-Lifschitz equation, when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications of ferromagnetic nanowires to the fabrication of magnetic memories. This is an updated version of the previous version of this paper.

math.PR↗

arXiv2013

Global evolution of random vortex filament equation

We prove the existence of a global solution for the filament equation with inital condition given by a geometric rough path in the sense of Lyons (1998).Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli (2007) showed the existence of global solution for a smooth initial condition while Bessaih, Gubinelli, Russo (2005) proved the existence of a local solution for a general initial condition given by a rough path.

math.PR↗

arXiv2013

Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains

We show that the stochastic flow generated by the Stochastic Navier-Stokes equations in a 2-dimensional Poincaré domain has a unique random attractor. This result complements a recent result by Brzeźniak and Li [10] who showed that the flow is asymptotically compact and generalizes a recent result by Caraballo et al. [12] who proved existence of a unique pullback attractor for the time-dependent deterministic Navier-Stokes equations in a 2-dimensional Poincaré domain.

math.PR↗

arXiv2009

Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs

The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For example, we will consider linear parabolic SPDEs with gradient noise and stochastic NSEs with multiplicative noise. Our results generalize the results proved in \cite{[Ghidaglia-1986]} for deterministic PDEs.

math.AP↗