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

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized NavierStokes equation modeling by pðxÞ-power law with Dirichlet boundary condition under the restriction ð3n/ðn + 2Þn + 2Þ < p ðxÞ < ð2ðn + 1ÞÞ/ðn − 1Þ. In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction ð2ðn + 1ÞÞ/ðn + 3Þ < pðxÞ < ð2ðn + 1ÞÞ/ðn − 1Þ, which arises from the nonflatness of domain.