Existence of solutions of Navier-Stokes equations, in 2D, with non-local viscosity

Abstract

  In this article, we had studied the existence, uniqueness of solutions and exponential decay of Navier-Stokes equations, in two dimensions, with non-local viscosity, defined as follows: \begin{align*} \left\{ \begin{array}{ll} \dfrac{d \bm{u}(x,t)}{dt} - c(l(u_{1}(x,t)),l(u_{2}(x,t))) \Delta \bm{u}(x,t) + (\bm{u}(x,t) \cdot \nabla )\bm{u}(x,t) + \nabla p(x) = \bm{f}(t,x), & \mbox{in } \Omega \times (0,T),\\ div(\bm{u}(x))=0, & \mbox{in } \Omega,\\ \bm{u}(x,t)=\bm{g}, & \mbox{on } \partial \Omega,\\ \bm{u}(x,0)=\bm{u}_0(x), & \mbox{in } \Omega, \end{array}\right. \end{align*} where $\Omega$ is a domain sufficiently regular and its boundary being $\partial \Omega$ regular, $\bm{u}: \Omega \times [0,T] \rightarrow \mathbb{R}^2$, $l:L^2(\Omega) \rightarrow \mathbb{R},$ is a continuous linear form, $a: \mathbb{R}^2 \rightarrow (0,\infty)$ satisfying following hypoteses: \begin{enumerate} \item $0<c_- \leq c(x_1,x_2) \leq c_+$, \item $|c(x) - c(y)| \leq A_1 |x_1-y_1| + A_2 | x_2 - y_2|$, with $A_1, A_2 > 0.$ \end{enumerate} We start by investigating existence of solutions to weak form of this problem, to achieve this we use Faedo-Galerkin methods and Compactness methods.