The Existence of Strong Solution for Generalized Navier-Stokes Equations with p x -Power Law under Dirichlet Boundary Conditions

Abstract

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized Navier-Stokes equation modeling by $p\left(x\right)$ -power law with Dirichlet boundary condition under the restriction $\left(3n/\left(n+2\right)n+2\right)<p\left(x\right)<\left(2\left(n+1\right)\right)/\left(n-1\right)$ . In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction $\left(2\left(n+1\right)\right)/\left(n+3\right)<p\left(x\right)<\left(2\left(n+1\right)\right)/\left(n-1\right)$ , which arises from the nonflatness of domain.