Stochastic two dimensional Euler equations
The main result of the paper, stated in Section 2, is the following. Consider an open \({\mathcal O}\subset{\mathbb R}^{2}\) satisfying several regularity assumptions (named A), \(q\in [2,\infty)\), the \(W^{1,q}\) spaces of \({\mathcal O}\) and \(H^{1,q}\) the intersection of \(W^{1,q}\) with the closure in \(L^{q}\) of \(C_{\text{sol}}^{\infty }\) defined as the set of all \(v\in C_{0}^{\infty }(\overline {\mathcal O} ,{\mathbb R}^{2})\) with \(\text{div }v=0\), \(\langle v,n\rangle=0\) on \(\partial {\mathcal O}\). Consider, both jointly measurable, an \(F\) which takes \([0,T]\times H^{1,2}\) in \(W^{1,2}\) and \([0,T]\times H^{1,q}\) in \(W^{1,q}\), a \(G\) which takes \([0,T]\times H^{1,2}\) in \(L_{(\text{HS})}({\mathcal H},W^{1,2})\) and \([0,T]\times H^{1,q}\) in \(R({\mathcal H},W^{1,q})\), where \({\mathcal H}\) is a separable Hilbert space, (HS) meaning Hilbert Schmidt operators, \(T\in R\) meaning the convergence in \(L^{2}(P,W^{1,q})\) of \(\sum_{k}\beta_{k}Te_{k}\) for an orthonormal basis \(e_{k}\) and for independent standard Gaussian r.v. \(\beta_{k}\) on the probability space \(P\). Suppose that there exist \(a_{1}\in L^{1}(0,T)\), \(a_{2}\in L^{r}(0,T)\), \(r>2\), such that for \(v\in H^{1,2}\cap H^{1,q}\) the four considered maps satisfy corresponding norm inequalities \(\|F(t,v)\|\leq a_{1}(t)(1+\|\cdot \|)\), \(\|G(t,v)\|\leq a_{2}(t)(1+\|v\|)\). Suppose also that \(\langle F(t,v),z\rangle\) and \(|\langle G^{\ast }(t,v)z\rangle|\) are continuous in \(v\) on H\(_{L^{2}}^{1,2}\), for every \(z\in C_{\text{sol}}^{\infty }\). Then, for every \(u_{0}\in H^{1,2}\cap H^{1,q}\), there exists a martingale \(H^{1,2}\cap H^{1,q}\) valued solution of \[ \partial_{t}u+\langle u,\nabla\rangle u+\nabla p=F(t,u)+G(t,u)\dot W, \] \(\text{div } u=0\), \(\langle u,n\rangle=0\) on \(\partial {\mathcal O}\) (stochastic incompressible Euler equations). This means a filtration on a probability space \(P\), an adapted cylindrical Wiener process \(W(t)\) (i.e. \(W(t) : {\mathcal H}\rightarrow L^{2}(P)\) are linear bounded, \(E(W(t)\varphi W(t)\psi)=t\langle\psi ,\varphi\rangle\) and \(W(t)\psi\) are adapted Wiener processes) and an adapted measurable \(u(t)\in H^{1,2}\cap H^{1,q}\) such that \(u\in L^{p}(P,L^{\infty }(0,T,H^{1,2}\cap H^{1,q}))\) for all \(p\in [1,\infty)\) and \[ \langle u(t),z\rangle=\langle u_{0},z\rangle+\int_0^t\left(\sum_{i=1}^{2}\langle u_{i}(s)u(s),\nabla z_{i}\rangle +\langle F(s,u(s)),z\rangle\right)ds+\left\langle\int_{_{0}}^{^{t}}G(s,u(s))dW(s),z\right\rangle \] for all \(z\in C_{\text{sol}}^{\infty }\). The proof (in Section 6) goes by using the compactness of families of solutions of equations (linearized modified Navier-Stokes) obtained by regularizing the given one, studied in Section 5. In Section 3 (analytical preliminaries) a generator of an analytic semigroup on a Banach space is constructed and the term \(\langle u,\nabla\rangle u\) is studied. In Section 4 (probabilistic preliminaries) the cylindrical Wiener process is studied; propositions establishing the finiteness of an \(R\)-norm (proof in an Appendix), an inequality involving such norms and the Itô formula for \(|\cdot |_{q}^{p}\) are stated. In Sections 6 and 7 two applications of the main theorem are proved, in which \(F(t,u)(x)=f(t,x,u(x))\) and \(G(t,u)\dot W\) is replaced by a finite sum of \(g(t,x,u(x))\dot W\), the \(W\)'s being independent and being ``spatially homogeneous Gaussian random fields'' as functions of \((t,x)\).
scientific article; zbMATH DE number 1906041
scientific article