Papers by Zdzislaw Brzezniak
Stochastic Ferromagnetism, 2013
This monograph examines magnetization dynamics at elevated temperatures which can be described by... more This monograph examines magnetization dynamics at elevated temperatures which can be described by the stochastic Landau-Lifshitz-Gilbert equation (SLLG). The first part of the book studies the role of noise in finite ensembles of nanomagnetic particles: we show geometric ergodicity of a unique invariant measure of Gibbs type and study related properties of approximations of the SLLG, including time discretization and Ginzburg-Landau type penalization. In the second part we propose an implementable space-time discretization using random walks to construct a weak martingale solution of the corresponding stochastic partial differential equation which describes the magnetization process of infinite spin ensembles. The last part of the book is concerned with a macroscopic deterministic equation which describes temperature effects on macro-spins, i.e. expectations of the solutions to the SLLG. Furthermore, comparative computational studies with the stochastic model are included. We use constructive tools such as e.g. finite element methods to derive the theoretical results, which are then used for computational studies. The numerical experiments motivate an interesting interplay between inherent geometric and stochastic effects of the SLLG which still lack a rigorous analytical understanding: the role of space-time white noise, possible finite time blow-up behavior of solutions, long-time asymptotics, and effective dynamics.
Stochastics and Stochastic Reports, 1996
arXiv (Cornell University), Jan 18, 2010
We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach s... more We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.
arXiv (Cornell University), Aug 1, 2020
We show that that the stochastic 3D primitive equations with either the physical boundary conditi... more We show that that the stochastic 3D primitive equations with either the physical boundary conditions or Neumann boundary conditions on the top and bottom and Dirichlet boundary condition on the sides driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in stochastic and PDE sense under certain assumptions on the growth of the noise. For the latter boundary conditions global existence is established using an argument based on decomposition of vertical velocity to barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.
Journal of Differential Equations, 2021
Abstract We show that the stochastic 3D primitive equations with the Neumann boundary condition o... more Abstract We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For the case of the Neumann boundary condition on the bottom, global existence is established by using the decomposition of the vertical velocity to the barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.
Journal of Mathematical Analysis and Applications, 2015
In this paper we first prove the existence and uniqueness of solution to the stochastic Navier-St... more In this paper we first prove the existence and uniqueness of solution to the stochastic Navier-Stokes equation on the rotating 2-dimensional sphere. Then we show the existence of an asymptotically compact random dynamical system associated with the equation. Contents 1. Introduction 2. The Navier-Stokes equations on a rotating unit sphere 2.1. Preliminaries 2.2. The weak formulation 3. Stochastic Navier-Stokes equation on a rotating unit sphere 10 4. Proofs of Theorems 3.2 and 3.3 11 4.1. Proof of Theorem 3.2 11 4.2. Proof of Theorem 3.3 18 5. Ornstein-Uhlenbeck processes 21 5.1. Preliminaries 21 5.2. Ornstein-Uhlenbeck process 24 6. Random dynamical systems generated by the stochastic NSEs on the sphere 27 7. Appendix 35 References 36
Applied Mathematics Research eXpress, 2012
We consider a Landau-Lifshitz-Gilber equation perturbed by a multiplicative spacedependent noise ... more We consider a Landau-Lifshitz-Gilber equation perturbed by a multiplicative spacedependent noise for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere S 2. The regularity of weak solutions is also discussed. Our research is in response to the paper by Kohn et al. "Magnetic elements at finite temperature and large deviation theory." Journal of Nonlinear Science 15 (2005): 223-53, which calls for the study of stochastic equations with spatially varying magnetization.
Indiana University Mathematics Journal, 2021
In this paper, we prove the existence of a unique maximal local strong solutions to a stochastic ... more In this paper, we prove the existence of a unique maximal local strong solutions to a stochastic system for both 2D and 3D penalised nematic liquid crystals driven by multiplicative Gaussian noise. In the 2D case, we show that this solution is global. As a by-product of our investigation, but of independent interest, we present a general method based on fixed point arguments to establish the existence and uniqueness of a maximal local solution of an abstract stochastic evolution equations with coefficients satisfying local Lipschitz condition involving the norms of two different Banach spaces.
Journal of Differential Equations, 2021
We consider a stochastic Camassa-Holm equation driven by a onedimensional Wiener process with a f... more We consider a stochastic Camassa-Holm equation driven by a onedimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.
Discrete & Continuous Dynamical Systems - B, 2017
In this note we prove the existence and uniqueness of local maximal smooth solution of the stocha... more In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.
SIAM Journal on Mathematical Analysis, 2019
Stochastic fractionally dissipative quasi-geostrophic type equation on $R^d$ with a multiplicativ... more Stochastic fractionally dissipative quasi-geostrophic type equation on $R^d$ with a multiplicative Gaussian noise is considered. We prove the existence of a martingale solution. 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 nonmetric spaces. In the 2D sub-critical case we prove also the pathwise uniqueness of the solutions.
Stochastics and Partial Differential Equations: Analysis and Computations, 2019
In this paper, we prove several mathematical results related to a system of highly nonlinear stoc... more In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting we replace the Ginzburg–Landau function$$\mathbb {1}_{|{\mathbf {n}}|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}$$1|n|≤1(|n|2-1)nby an appropriate polynomial$$f({\mathbf {n}})$$f(n)and we give sufficient conditions on the polynomialffor these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider$$f({\mathbf {n}})=\mathbb {1}_{|d|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}$$f(n)=1|d|≤1(|n|2-1)nand if the initial condition$${\...
Probability Theory and Related Fields, 2018
We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract ... more We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H 1 on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo-Galerkin-method based on the Littlewood-Paleydecomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2017
We consider a Banach space (E, •) such that, for some q ≥ 2, the function x → x q is of C 2 class... more We consider a Banach space (E, •) such that, for some q ≥ 2, the function x → x q is of C 2 class and its kth, k = 1, 2, Fréchet derivatives are bounded by some constant multiples of the (q − k)th power of the norm. We also consider a C 0-semigroup S of contraction type on (E, •). Finally we consider a compensated Poisson random measureÑ on a measurable space (Z, Z). We study the following stochastic convolution process u(t) = t 0 Z 1 Partially supported by the National Natural Science Foundation of China (No. 11501509).
Stochastic Partial Differential Equations: Analysis and Computations, 2014
In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier-... more In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier-Stokes equations on the torus suggested by the Lie-Trotter product formulas for stochastic differential equations of parabolic type. The stochastic system is split into two problems which are simpler for numerical computations. An estimate of the approximation error is given for periodic boundary conditions. In particular, we prove that the strong speed of the convergence in probability is almost 1/2. This is shown by means of an L 2 (, P) convergence localized on a set of arbitrary large probability. The assumptions on the diffusion coefficient depend on the fact that some multiple of the Laplace operator is present or not with the multiplicative stochastic term. Note that if one of the splitting steps only contains the stochastic integral, then the diffusion coefficient may not contain any gradient of the solution.
Using the approach of the splitting method developed by I. Gyöngy and N. Krylov for parabolic qua... more Using the approach of the splitting method developed by I. Gyöngy and N. Krylov for parabolic quasi linear equations, we study the speed of convergence for general complex-valued stochastic evolution equations. The approximation is given in general Sobolev spaces and the model considered here contains both the parabolic quasi-linear equations under some (non strict) stochastic parabolicity condition as well as linear Schrödinger equations
Stochastic Processes and their Applications, 2013
The solution Xn to a nonlinear stochastic differential equation of the form dXn(t) + An(t)Xn(t) d... more The solution Xn to a nonlinear stochastic differential equation of the form dXn(t) + An(t)Xn(t) dt − 1 2 N j=1 (B n j (t)) 2 Xn(t) dt = N j=1 B n j (t)Xn(t)dβ n j (t) + fn(t) dt, Xn(0) = x, where β n j is a regular approximation of a Brownian motion βj , B n j (t) is a family of linear continuous operators from V to H strongly convergent to Bj (t), An(t) → A(t), {An(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 − 1 2 N j=1 B 2 j (t)X(t) dt = N j=1 Bj (t)X(t) dβj(t) + f (t) dt, X(0) = x. Here V ⊂ H ∼ = H ′ ⊂ 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 = N j=1 Bj(t)Y (t)•dβj (t)+f (t) dt.
Probability Theory and Related Fields, 2004
An extensible beam equation with a stochastic force of a white noise type is studied, Lyapunov fu... more An extensible beam equation with a stochastic force of a white noise type is studied, Lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution.
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Papers by Zdzislaw Brzezniak