Existence and exponential behavior for the stochastic 2D Cahn-Hilliard-Oldroyd model of order one (Q2070474)

The authors consider the stochastic Cahn-Hilliard-Oldroyd model of order one, for the motion of an incompressible isothermal mixture of two immiscible non-Newtonian fluids written as: \[ du(t)+[-\nu _{1}\Delta u+(\beta \ast \Delta u)(t)+(u\cdot \nabla )u+\nabla p-\mathcal{K}\mu \nabla \phi ]dt=\sigma _{1}(t,u,\phi )dW_{t}^{1}+\int_{Z}\gamma (t,u(t^{-}),\phi (t),z) \widetilde{\pi }(dt,dz), \] \[ d\phi (t)=[\nu _{2}\Delta \mu -u\cdot \nabla \phi ]dt+\sigma _{2}(t,u,\phi )dW_{t}^{2}, \] \(\mu =-\varepsilon \Delta \phi +\alpha f(\phi )\), \(\operatorname{div}(u)=0\), posed in \((0,T)\times \mathcal{M}\), where \(T>0 \) and \(\mathcal{M}\) is a bounded and open domain in \(\mathbb{R}^{2}\) with a smooth boundary \(\partial \mathcal{M}\). Here \(\beta (t)=\gamma e^{-\delta t}\), with \(\gamma =\frac{1}{\varsigma }(\nu -\frac{\kappa }{\varsigma })>0\), \( \nu _{1}=\frac{\kappa }{\varsigma }\), \(\delta =\frac{1}{\varsigma }>0\), \( \varsigma,\kappa >0\), \(W_{t}^{i}\), \(i=1,2\), are two cylindrical Wiener processes in a separable Hilbert space \(U\) defined on the probability space \( (\Omega,\mathcal{F},\mathbb{P})\), \(Z\) is a measurable subspace of some Hilbert space, \(\widetilde{\pi }(dt,dz)=\pi (dt,dz)-\lambda (dz)dt\) is a compensated Poisson random measure, \(\lambda (dz)\) being a \(\sigma \)-finite Lévy measure on the Hilbert space with an associated Poisson random measure \(\pi (dt,dz)\) such that \(\mathbb{E}[\pi (dt,dz)]=\lambda (dz)dt\), \( \mu \) is the chemical potential of the binary mixture which is given by the variational derivative of the free energy functional \(\mathcal{E}_{0}(\phi )=\int_{D}(\frac{\varepsilon }{2}\left\vert \nabla \phi \right\vert ^{2}+\alpha F(\phi ))dx\), where \(F(r)=\int_{0}^{r}f(\zeta )d\zeta \) is a double-well potential, \(\nu _{2}\) and \(\kappa \) are positive constants that correspond to the mobility constant and capillarity (stress) coefficient, respectively, \(\varepsilon \) and \(\alpha \) are two positive parameters describing the interactions between the two phases. The processes \(W_{t}^{i}\), \(i=1,2\), and \(\widetilde{\pi }\) are mutually independent. The boundary conditions \(\partial _{\eta }\phi =\partial _{\eta }\Delta \phi =0=u\) are imposed on \((0,T)\times \partial \mathcal{M}\), together with the initial conditions \((u,\phi )(0)=(u_{0},\phi _{0})\) in \(\mathcal{M}\). The authors introduce a variational formulation of the problem in appropriate spaces and the notion of global strong solution. Assuming appropriate hypotheses on the data, \(\sigma _{1}(\cdot,0,0)\in L^{p}(\Omega,\mathcal{F},\mathbb{P} ;L^{2}(0,T;\mathcal{L}^{2}(U;H^{1})))\) and \((u_{0},\phi _{0})\in L^{p}(\Omega,\mathcal{F},\mathbb{P},\mathbb{H})\) with \(\mathbb{E}\mathcal{E} ^{p}(u_{0},\phi _{0})<\infty \), for all \(p\geq 2\), they prove that the problem has a unique strong solution. For the proof, the authors introduce a finite dimensional Galerkin approximation on which they prove uniform estimates. For the uniqueness of the strong solution, they argue by contradiction. In the last part of their paper, the authors prove an exponential stability result. Under appropriate hypotheses, the strong solution to a stochastic 2D Cahn-Hilliard-Oldroyd model converges to the unique stationary solution to a stationary equation, which is exponentially stable.