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Zdzisław Brzeźniak

Publications and source records attributed to Zdzisław Brzeźniak.

51 records · Page 3Linked to original sources
arXiv2017

Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains

Stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for non-metric spaces. Moreover, some compactness and tightness criteria in non-metric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.

math.PR↗
arXiv2016

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

Strong existence and pathwise uniqueness of solutions with $L^{\infty}$-vorticity of 2D stochastic Euler equations is proved. The noise is multiplicative and involves first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. Stability under regularization is also proved.

math.PR↗
arXiv2016

Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains

Building upon a recent work by two of the authours and J. Seidler on bw-Feller property for stochastic nonlinear beam and wave equations, we prove the existence of an invariant measure to stochastic 2-D Navier-Stokes (with multiplicative noise) equations in unbounded domains. This answers an open question left after the first authour and Y. Li proved a corresponding result in the case of an additive noise.

math.PR↗
arXiv2016

2D Constrained Navier-Stokes Equations

We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We also show that the solution of constrained Navier-Stokes converges to the solution of Euler equation as viscosity $ν$ vanishes.

math.AP↗
arXiv2015

Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces

Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fréchet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)ξ(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $ξ:[0,\infty)\timesΩ\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a càdlàg modification a $\tilde{u}$ of the process $u$ which satisfies the following maximal inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E} \left(\int_0^t\int_Z \|ξ(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1<p\leq 2$ with $C=C(q,p)$.

math.PR↗
arXiv2015

Regularity of Ornstein-Uhlenbeck processes driven by a L{é}vy white noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by Lévy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.

math.PR↗
arXiv2014

Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering the space variable-independent solutions only. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and we obtain also results on attractivity towards an invariant measure. We also present a suitable numerical scheme for approximating the solutions subject to a sphere constraint.

math.PR↗
arXiv2013

Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise

Motivated by applications to a manifold of semilinear and quasilinear stochastic partial differential equations (SPDEs) we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Lévy processes. We illustrate the main result of our paper by showing how it can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.

math.AP↗
arXiv2013

Uniqueness in Law of the stochastic convolution process driven by Lévy noise

We will give a proof of the following fact. If $\mathfrak{A}_1$ and $\mathfrak{A}_2$, $\tilde η_1$ and $\tilde η_2$, $ξ_1$ and $ξ_2$ are two examples of filtered probability spaces, time homogeneous compensated Poisson random measures, and progressively measurable Banach space valued processes such that the laws on $L^p([0,T],{L}^{p}(Z,ν;E))\times \CM_I([0,T]\times Z)$ of the pairs $(ξ_1,η_1)$ and $(ξ_2,η_2)$ %, $i=1,2$, are equal, and $u_1$ and $u_2$ are the corresponding stochastic convolution processes, then the laws on $ (\DD([0,T];X)\cap L^p([0,T];B)) \times L^p([0,T],{L}^{p}(Z,ν;E))\times \CM_I([0,T]\times Z) $, where $B \subset E \subset X$, of the triples $(u_i,ξ_i,η_i)$, $i=1,2$, are equal as well. By $\DD([0,T];X)$ we denote the Skorokhod space of $X$-valued processes.

math.PR↗
arXiv2012

Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise

The solution $X_n$ to a nonlinear stochastic differential equation of the form $dX_n(t)+A_n(t)X_n(t)\,dt-\tfrac12\sum_{j=1}^N(B_j^n(t))^2X_n(t)\,dt=\sum_{j=1}^N B_j^n(t)X_n(t)dβ_j^n(t)+f_n(t)\,dt$, $X_n(0)=x$, where $β_j^n$ is a regular approximation of a Brownian motion $β_j$, $B_j^n(t)$ is a family of linear continuous operators from $V$ to $H$ strongly convergent to $B_j(t)$, $A_n(t)\to A(t)$, $\{A_n(t)\}$ is a family of maximal monotone nonlinear operators of subgradient type from $V$ to $V'$, is convergent to the solution to the stochastic differential equation $dX(t)+A(t)X(t)\,dt-\frac12\sum_{j=1}^NB_j^2(t)X(t)\,dt=\sum_{j=1}^NB_j(t)X(t)\,dβ_j(t)+f(t) \,dt$, $X(0)=x$. Here $V\subset H\cong H'\subset V'$ where $V$ is a reflexive Banach space with dual $V'$ and $H$ is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation $dY(t)+A(t)Y(t)\,dt=\sum_{j=1}^NB_j(t)Y(t)\circ dβ_j(t)+f(t)\,dt$.

math.PR↗
arXiv2012

Multidimensional stochastic Burgers equation

We consider multidimensional stochastic Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. In both cases we show that for positive viscosity $ν>0$ there exists a unique strong global solution in $L^p$ for $p>d$. In the case of torus we also establish a uniform in $ν$ a priori estimate and consider a limit $ν\todown 0$ for potential solutions. In the case of $\Rd$ uniform with respect to $ν$ a priori estimate established if a Beale-Kato-Majda type condition is satisfied.

math-ph↗
arXiv2011

Ergodic properties of Fractional Stochastic Burgers Equation

We prove the existence and uniqueness of invariant measures for the fractional stochastic Burgers equation (FSBE) driven by fractional power of the Laplacian and space-time white noise. We show also that the transition measures of the solution converge to the invariant measure in the norm of total variation. To this end we show first two results which are of independent interest: that the semigroup corresponding to the solution of the FSBE is strong Feller and irreducible.

math.PR↗
arXiv2011

Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

Let $M$ be a compact Riemannian homogeneous space (e.g. a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation \mathbf D_t\partial_tu=\sum_{k=1}^d\mathbf D_{x_k}\partial_{x_k}u+f_u(Du)+g_u(Du)\,\dot W$ in any dimension $d\ge 1$, where $f$ and $g$ are continuous multilinear mappings and $W$ is a spatially homogeneous Wiener process on $\mathbb R^d$ with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed.

math.PR↗