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

TL;DR

This paper establishes nonexistence of global smooth solutions for certain compressible non-Newtonian fluids with magnetic effects, under specific conditions on viscosity and initial data, in the whole space.

Contribution

It extends nonexistence results to compressible non-Newtonian fluids with magnetic effects, identifying critical viscosity exponents for solution blow-up.

Findings

No global smooth solutions exist for certain viscosity exponents.

Results apply to both non-Newtonian fluid and magnetohydrodynamic cases.

Conditions depend on space dimension and heat ratio.

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 $q_0>1$ depending on the dimension of space $n$ and the heat ratio $\gamma$ such that for $q\in [q_0,n)$ 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.