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On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise. (English) Zbl 1424.60077

A stochastic Navier-Stokes equation \[du+((u\cdot\nabla)u-\nu\Delta u+\nabla p)\,dt=f\,dt+g(u)\,\mathcal{W},\quad\nabla\cdot u=0,\quad u|_{\partial\mathcal{O}}=0,\quad u(0)=u_0\] on a bounded open set \(\mathcal{O}\subseteq\mathbb{R}^2\) with the smooth boundary is considered. Here, \(p\) denotes the pressure, \(\nu\) the viscosity and \(\mathcal{W}\) is a cylindrical Wiener process on a Hilbert space \(U\). Standardly, \(H\) and \(V\) denote the closures of the set of smooth, compactly supported and divergence free vector fields in \(L^2(\mathcal{O})\) and \(H^1(\mathcal{O})\), respectively, \(\mathcal{P}_H\) denotes the Leray-Hopf projector of \(L^2(\mathcal{O})\) onto \(H\), \(B(u,v)=\mathcal{P}_H(u\cdot\nabla v)\), \(A=-\mathcal{P}_H\Delta\) denotes the Stokes operator on the domain \(V\cap H^2(\mathcal{O})\), \(\{e_k\}\) is the orthonormal basis of eigenvectors of \(A\) in \(H\) and \(H_n\) denotes the linear span of \(\{e_1,\dots,e_n\}\). It is assumed that \[\|g(t,x)\|_{L_2(U,\mathcal{D}(A^{j/2}))}\le K_j(1+\|x\|_{\mathcal{D}(A^{j/2})}),\] \[\|g(t,x)-g(t,y)\|_{L_2(U,\mathcal{D}(A^{j/2}))}\le K_j\|x-y\|_{\mathcal{D}(A^{j/2})}\] for \(j\in\{0,1,2\}\). The drift \(f\) is assumed to be a predictable process belonging to \(L^1(\Omega;L^4(0,T;V^\prime))\) and \(u_0\) an \(\mathcal{F}_0\)-measurable random variable belonging to \(L^4(\Omega;H)\cap L^2(\Omega;V)\). A solution \(u\) is understood in the strong PDE and probabilistic sense, with paths in \(C([0,T];V)\cap L^2(0,T;\mathcal{D}(A))\). The \(H_n\) valued processes \(u^n\) are assumed to solve the Galerkin system \[du^n+(\nu Au^n+P_nB(u^n))\,dt=P_nf\,dt+P_ng(u^n)\,d\mathcal{W},\quad u^n(0)=P_nu_0.\] It is known that under the standing assumptions, global solutions \(u\) and \(u^n\) exist for every \(n\in\mathbb{N}\). The authors prove that \[\mathbb{E}\,\left[\sup_{t\in[0,T]}\phi\left(\|u(t)\|^2_V\right)\right]\le C(f,g,u_0,T)\] where \(\phi(x)=\log(1+x)\), and that \[\lim_{n\to\infty}\mathbb{E}\,\left[\sup_{t\in[0,T]}\phi\left(\|u(t)-u^n(t)\|^2_V\right)\right]^{1-\varepsilon}=0\] holds for every \(\varepsilon\in(0,1)\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q30 Navier-Stokes equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
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