Well-posedness of stochastic third grade fluid equation (Q2662492)

The authors consider a bounded, not axisymmetric and simply connected domain \(\mathcal{O}\subset \mathbb{R}^{2}\), with a sufficiently regular boundary \( \Gamma \). They analyze the well-posedness for the stochastic third grade fluid problem written as \[ \begin{aligned} d(\upsilon (Y))&=-\nabla p+\nu \Delta Y-(Y\cdot \nabla )\upsilon -\sum_{j}\upsilon ^{j}\nabla Y^{j}+(\alpha _{1}+\alpha _{2})\operatorname{div}(A^{2})+\beta \operatorname{div}(\left\vert A\right\vert ^{2}A+U)dt+\sigma (t,Y)dW_{t}, \\ \operatorname{div}(Y)&=0, \end{aligned} \] posed in \(\mathcal{O}\times (0,T)\), where \(Y\) is the velocity of the fluid, \(\nabla Y\) its Jacobian matrix, \(D(Y)=\frac{ \nabla Y+(\nabla Y)^{\intercal }}{2}\), \(A=A(Y)=2D(Y)\), and \(\upsilon (Y)=Y-\alpha _{1}Y\). The constants \(\nu \), \(\alpha _{1}\), \(\alpha _{2}\), and \( \beta \) satisfy different properties. The stochastic perturbation \(\sigma (t,Y)d\mathcal{W}_{t}=\sum_{k=1}^{m}\sigma ^{k}(t,Y)d\mathcal{W}_{t}^{k}\), whose diffusion coefficient \(\sigma (t,Y)=(\sigma _{1}(t,Y),\ldots ,\sigma _{m}(t,Y))\) satisfies suitable growth assumptions, involves a standard \( \mathbb{R}^{m}\)-valued Wiener process \(\mathcal{W}_{t}=(\mathcal{W} _{t}^{1},\ldots ,W_{t}^{m})\) defined on a complete probability space \( (\Omega ,\mathcal{F},P)\) endowed with a filtration \(\{F_{t}\}_{t\in \lbrack 0,T]}\) for \(\mathcal{W}_{t}\). The boundary conditions \(Y\cdot n=0\), \((n\cdot D(Y))\cdot \tau =0\) are imposed on \(\Gamma \times (0,T)\) and the solution \(Y\) starts from an initial data \(Y_{0}\) at \(t=0\). The authors define the notion of strong solution as a stochastic process \(Y\in L^{2}(\Omega ,L^{\infty }(0,T;W))\) which satisfies the variational formulation: \[ (\sigma (Y(t)),\phi )=\int_{0}^{t}[-2\nu (D(Y),D(\phi ))+((Y\cdot \nabla )\phi ,\upsilon (Y))-\sum_{j}(\upsilon ^{j}(Y)\nabla Y^{j},\phi )]ds-\int_{0}^{t}((\alpha _{1}+\alpha _{2})A^{2}+\beta (\left\vert A\right\vert ^{2}A),\nabla \phi )ds+(\upsilon (Y(0)),\phi )+\int_{0}^{t}(U(s),\phi )ds+\int_{0}^{t}(\sigma (s,Y(s)),\phi )d\mathcal{W} _{s}, \] for \(U\in L^{2}(\Omega \times (0,T),L^{2}(\mathcal{O}))\) and \( Y_{0}\in L^{2}(\Omega ,W)\), for a.e. \((\omega ,t)\in \Omega \times \lbrack 0,T]\) and all \(\phi \in V=\{y\in H^{1}(\mathcal{O})\mid \operatorname{div}y=0\) in \(\mathcal{ O}\) and \(y\cdot n=0\) on \(\Gamma \}\). Here \(W=\{y\in V\cap H^{2}(\mathcal{O} )\mid (n\cdot D(y))\cdot \tau =0\) on \(\Gamma \}\). Under growth properties on \(\sigma \) and regularity properties on the force \(U\) and the initial data \( Y_{0}\), the authors prove the existence of a unique strong solution \(Y\in L^{p}(\Omega ,L^{\infty }(0,T;W))\), \(p\geq 6\), to this problem. For the proof, the authors apply the Galerkin method. They build an appropriate orthonormal basis for \(W\) and the associated problem. They prove the existence of a unique solution \(Y_{n}\in L^{2}(\Omega ,L^{\infty }(0,T;W))\) to this problem and shows that this solution satisfies uniform estimates, which leads to some weak convergence of \(Y_{n}\). They apply the projection operator on the finite-dimensional space to this weak limit and they prove that the difference between the sequence obtained by projecting the weak limit and the finite-dimensional Galerkin approximations converges strongly to zero, up to some stopping time.