On the small time asymptotics of the two-dimensional stochastic Navier-Stokes equations
Strong solutions of the \(2\)-dimensional incompressible Navier-Stokes equation \[ du-\nu \Delta u\,dt+(u\cdot\nabla)u\,dt=f(t)\,dt+\sigma(t,u)\,dW \] with the Dirichlet boundary condition on a bounded domain \(D\) are considered. If \(V\) denotes the divergence free elements of \(H^1_0(D;\mathbb R^2)\), \(H\) is the closure of \(V\) in \(L^2(D;\mathbb R^2)\), \(W\) is an \(H\)-valued Wiener process with a trace-class covariance operator \(Q\) on \(H\), \(u(0)\in L^4(\Omega;H)\), \(f\in L^4_{loc}(\mathbb R_+;V^\prime)\) and \(\sigma\) is Lipschitz (on a suitable space) then unique global strong solutions in \(L^2(\Omega;C(\mathbb R_+;H))\cap L^2_{loc}(\Omega\times\mathbb R_+;V)\) are known to exist. Let \(\mu^\varepsilon_x\) denote the law of \(u^\varepsilon(t):=u(\varepsilon t)\) on \(C([0,1];H)\) where \(u(0)=x\in H\). Given an absolutely continuous \(g\in C([0,T];H)\), define \[ I(g)=\inf\,\{\frac 12\int_0^1\|Q^{-1/2}h^\prime(t)\|^2_H\,dt:h\in AC([0,1];H),\,g^\prime=\sigma(\cdot,g)h^\prime\}. \] The large deviation principle with the rate function \(I\) is proved to hold, i.e. \[ \limsup_{\varepsilon\to 0,\,\,x_n\to x}\varepsilon\log\mu^\varepsilon_{x_n}(F)\leq- i_{F,x},\qquad\liminf_{\varepsilon\to 0,\,\,x_n\to x}\varepsilon\log\mu^\varepsilon_{x_n}(G)\geq-i_{G,x} \] holds for every open set \(G\subset C([0,1];H)\) and every closed set \(F\subset C([0,1];H)\) where \(i_{L,x}=\inf\,\{I(g):g\in L,\,g(0)=x\}\). The sharp constant in the Burkholder-Gundy-Davis inequality for real continuous martingales is a substantial ingredient in the proof.
scientific article; zbMATH DE number 5758867
scientific article