On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model
Let \(M\) be a bounded domain in \(\mathbb R^2\) and let \(T>0\) be a constant. Consider the following stochastic Cahn-Hilliard-Navier-Stokes equation on \(M\times [0,T]\): \[ \begin{aligned} &\partial_t v-\theta_1\Delta v +(v\cdot \nabla)v+\nabla p- \kappa \mu \nabla \phi = g_1(t,v)+g_2(t,v)\dot W_t,\\ & \mathrm{div } v=0,\\ &\partial_t \phi +v\cdot \nabla \phi-\theta_3\mu=0,\\ &\mu=-\theta_2\Delta\phi +\alpha f(\phi),\end{aligned} \] where \(\theta_i,\kappa,\alpha\) are positive constants, \(W_t\) is a cylindrical Brwonian motion on a reference Hilbert space, \(g_i, f\) are suitable maps. Under suitable regularity and growth conditions on \(g_i\) and \(f\), the author proves the existence and uniqueness of variational solutions for the unknown velocity \(v=(v_1v_2)\) and order parameter \(\phi\).
scientific article; zbMATH DE number 6712370
scientific article