Blow up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy

We prove that the smooth solutions to the Cauchy problem for the three-dimensional compressible barotropic magnetohydrodynamic equations with conserved total mass and finite total energy lose the initial smoothness within a finite time. Further, we show that the same result holds for the solution to the Cauchy problem for the multidimensional compressible NavierStokes system. Moreover, for the solution with a finite momentum of inertia we get the two-sided estimates of different components of total energy. 1. The finite time blow up result The set of equations which describe compressible viscous magnetohydrodynamics are a combination of the compressible Navier-Stokes equations of fluid dynamics and Maxwells equations of electromagnetism. We consider the system of partial differential equations for the three-dimensional viscous compressible magnetohydrodynamic flows in the Eulerian coordinates [1] for barotropic case: ∂tρ+ divx(ρu) = 0, (1.1) ∂t(ρu) + Divx(ρu⊗ u) +∇xp(ρ) = (curlxH)×H +DivxT, (1.2) ∂tH − curlx(u×H) = −curlx(ν curlxH), divxH = 0, (1.3) where ρ, u = (u1, u2, u3), p, H = (H1, H2, H3), denote the density, velocity, pressure and magnetic field. We denote Div and div the divergency of tensor and vector, respectively. Here T is the stress tensor given by the Newton law T = Tij = μ (∂iuj + ∂jui) + λdivu δij , (1.4) where the constants μ and λ are the coefficient of viscosity and the second coefficient of viscosity, ν ≥ 0 is the coefficient of diffusion of the magnetic field. We assume that μ > 0, λ+ 2 3μ > 0. The state equation has the form p = Aρ . (1.5) 1991 Mathematics Subject Classification. 76W05, 35Q36.