Smooth solutions for a stochastic hydrodynamical equation in Heisenberg paramagnet
type: Item
instance of: scholarly article
title: Smooth solutions for a stochastic hydrodynamical equation in Heisenberg paramagnet
zbMATH Open document ID: 1230.35151
author: Yongqian Han
author: Xueke Pu
author: Boling Guo
publication date: 2011-11-04T00:00:00Z
review text: In this paper the authors study the following stochastic hydrodynamical equation with additive noise in Heisenberg paramagnet \(du=(\Delta u+u\times\Delta u)dt+\Phi dW(t)\), \(u(0,x)=u_0(x)\), \(x\in R^2\), where \(u=(u_1,u_2,u_3)\) is a \(R^3\)-valued vector denoting the magnetic moment and \(\times\) denotes the vector cross product in \(R^3\). The Wiener process \(W(t)\) is assumed to be a cylindrical one on \(L^2(R^2)\). It is of the form \(W(t)=\sum_{i}\beta_{i}(t)e_{i}\), where \(\{e_{i}\}_{i}\) is an orthonormal complete basis in the Hilbert space \(L^2(R^2)\) and \(\{\beta_{i}\}_{i}\) is a family of mutually-independent, real-valued standard Wiener processes. The main result of this paper is the following: Suppose \(u_0\in H^2(R^2)\) is \({\mathcal F}_0\)-measurable and \(\Phi\in L_2^{0,2}\) is a Hilbert-Schmidt operator from \(L^2(R^2)\) to \(H^2(R^2)\). Then there exists a unique solution \(u(t)\in{\mathcal C}([0,T];H^2(R^2))\), a.s. of the considered equation for any \(T>0\).
reviewed by: Aleksandr D. Borisenko
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76W05
zbMATH DE Number: 5968225
zbMATH Keywords: stochastic hydrodynamic equation
zbMATH Keywords: Heisenberg paramagnet
zbMATH Keywords: smooth solution
zbMATH Keywords: additive noise
zbMATH Keywords: existence and uniqueness
MaRDI profile type: Publication
full work available at URL: s10114-011-8159-2
OpenAlex ID: W1973696936
cites work: On the stochastic Korteweg-de Vries equation
cites work: The asymptotic behavior of the stochastic Ginzburg–Landau equation with multiplicative noise
cites work: Global smooth solution of hydrodynamical equation for the Heisenberg paramagnet
cites work: Stochastic differential equations. An introduction with applications.
cites work: Q5759616
cites work: On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness
cites work: Partial regularity for two dimensional Landau-Lifshitz equations
cites work: Smooth solution to the 1-dimensional spin equations of antiferromagnets
cites work: On the continuous limit for a system of classical spins
cites work: Regular solutions for multiplicative stochastic Landau-Lifshitz-Gilbert equation and blow-up phenomena
cites work: Stochastic Equations in Infinite Dimensions
cites work: A reduced theory for thin-film micromagnetics
cites work: Stochastic Burgers' equation
DOI: 10.1007/S10114-011-8159-2
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5968225