Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space

Abstract

We consider a generalization of the compressible barotropic Navier–Stokes equations to the case of non-Newtonian fluid in the whole space. The viscosity tensor is assumed to be coercive with an exponent q > 1 . We prove that if the total mass and momentum of the system are conserved, then one can find a constant q γ > 1 depending on the dimension of space n and the heat ratio γ such that for q ∈ [ q γ , n ) there exists no global in time smooth solution to the Cauchy problem. We prove also an analogous result for solutions to equations of magnetohydrodynamic non-Newtonian fluid in 3D space.