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

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.Comment: 8 page

Stochastic perturbations method for a system of Riemann invariants

Basing on our results [1] on a representation of solutions to the Cauchy problem for multidimensional non-viscous Burgers equation obtained by a method of stochastic perturbation of the associated Langevin system, we deduce an explicit asymptotic formula for smooth solutions to the Cauchy problem for any genuinely nonlinear hyperbolic system of equations written in the Riemann invariants.Comment: 7 page

Note on the typhoon eye trajectory

We consider a model of typhoon based on the three-dimensional baroclinic compressible equations of atmosphere dynamics averaged over hight and describe a qualitative behavior of the vortex and possible trajectories of the typhoon eye.Comment: 20 pages. 6 figure