The existence of strong solution for generalized Navier-Stokes equations with \(p(x)\)-power law under Dirichlet boundary conditions (Q2064711)

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