Smooth solutions for a stochastic hydrodynamical equation in Heisenberg paramagnet
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 \(T>0\).
scientific article; zbMATH DE number 5968225