Singularity formation to the Cauchy problem of the two-dimensional non-baratropic magnetohydrodynamic equations without heat conductivity

We study the singularity formation of strong solutions to the two-dimensional (2D) Cauchy problem of the non-baratropic compressible magnetohydrodynamic equations without heat conductivity. It is proved that the strong solution exists globally if the density and the pressure are bounded from above. In particular, the criterion is independent of the magnetic field and is just the same as that of the compressible Navier-Stokes equations. Our method relies on weighted energy estimates and a Hardy-type inequality.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1705.05161, arXiv:1705.06606; text overlap with arXiv:1801.0758

Strong solutions to the Cauchy problem of the two-dimensional non-baratropic non-resistive magnetohydrodynamic equations with zero heat conduction

This paper concerns the Cauchy problem of the non-baratropic non-resistive magnetohydrodynamic (MHD) equations with zero heat conduction on the whole two-dimensional (2D) space with vacuum as far field density. By delicate weighted energy estimates, we prove that there exists a local strong solution provided the initial density and the initial magnetic decay not too slow at infinity.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1506.02156; text overlap with arXiv:1306.4752, arXiv:1608.08876 by other author

Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum

This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity $\|\rho_0\|_{L^\infty}+\|b_0\|_{L^3}$ is suitably small and the viscosity coefficients satisfy $3\mu>\lambda$. Here, the initial velocity and initial temperature could be large. The assumption on the initial density do not exclude that the initial density may vanish in a subset of $\mathbb{R}^3$ and that it can be of a nontrivially compact support. Our result is an extension of the works of Fan and Yu \cite{FY09} and Li et al. \cite{LXZ13}, where the local strong solutions in three dimensions and the global strong solutions for isentropic case were obtained, respectively. The analysis is based on some new mathematical techniques and some new useful energy estimates. This paper can be viewed as the first result concerning the global existence of strong solutions with vacuum at infinity in some classes of large data in higher dimension.Comment: 23 page

Global Strong Solutions to the Compressible Magnetohydrodynamic Equations with Slip Boundary Conditions in 3D Bounded Domains

We deal with the barotropic compressible magnetohydrodynamic equations in three-dimensional (3D) bounded domain with slip boundary condition and vacuum. By a series of a priori estimates, especially the boundary estimates, we prove the global well-posedness of classical solution and the exponential decay rate to the initial-boundary-value problem of this system for the regular initial data with small energy but possibly large oscillations. The initial density of such a classical solution is allowed to contain vacuum states. Moreover, it is also shown that the oscillation of the density will grow unboundedly with an exponential rate when the initial state contains vacuum.Comment: 42 pages. arXiv admin note: text overlap with arXiv:2102.0634