Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension
Let \(A\) be a self-adjoint, positive operator on a Hilbert space \(H\), such that \(H^{-1}\) is a compact operator. Let \(H^{2s}\) denote the domain of \(A^ s\) with the graph norm \((s \geq 0)\) and \(H^{-s}\) denote the dual of \(H^ s\). Assume that for some \(s \geq 1\) we are given a \(B \in C(H; H^{-s})\) such that for any \(u \in H^ s\), \((B(u),u) = 0\) and \(| B(u)|_{-s} \leq | u |^ 2\), where \(| \cdot |_ r\) is the norm in \(H^ r\) \((r \in R)\) and \(| \cdot | = | \cdot |_ 0\). Let \(F \in C (H; H^{-1})\) and \(| F(u) |_{-1} \leq C(1 + | u |)\). Finally, fix another Hilbert space \(U\) and let \(G\) be a continuous function on \(H\) with values in the space of Hilbert-Schmidt operators from \(U\) into \(H\), with at most linear growth. The stochastic equation \[ \begin{aligned} & dX(t) = \biggl[ - AX(t) + B \bigl( X(t) \bigr) + F \bigl( X(t) \bigr) \biggr] dt + G \bigl( X(t) \bigr) dW(t),\\ & X(0) = x_ 0 \in H,\end{aligned} \] is proved to have a weak solution, where \(W\) is a cylindrical Wiener process in \(U\). This solution is in \(L^ 2 (\Omega \times [0,T]\); \(H^ 1) \cap L^ p (\Omega\); \(C([0,T]\); \(H^ \gamma)) \cap L^ p (\Omega\); \(L^ \infty ([0,T]; H))\) for any \(T\), \(\gamma < 2 - s\) and \(p \geq 1\). The proof employs the factorization method. This result is then applied to obtain the existence of a solution of a Navier-Stokes equation.
scientific article; zbMATH DE number 704517