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

TL;DR

This paper proves that smooth solutions to certain three-dimensional compressible magnetohydrodynamic and Navier-Stokes equations with finite mass and energy become singular in finite time, providing energy estimates for solutions with finite momentum.

Contribution

It establishes finite-time blow-up of smooth solutions for these equations and extends results to the multidimensional Navier-Stokes system, including energy estimates.

Findings

Smooth solutions lose regularity in finite time

Finite energy solutions exhibit blow-up behavior

Energy estimates are derived for solutions with finite momentum

Abstract

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 Navier-Stokes system. Moreover, for the solution with a finite momentum of inertia we get the two-sided estimates of different components of total energy.