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This paper intends to study an experimentally observed stabilizing phenomenon and to prove a mathematically rigorous stability result on the perturbations near a background magnetic field. Physical experiments and numerical simulations have observed a remarkable phenomenon that a background magnetic field can smooth and stabilize the electrically conducting turbulent fluids. To understand the mechanism of this phenomenon, we focus on a special 2D magnetohydrodynamic (MHD) system with anisotropic dissipation and partial damping and examine the stability near a background magnetic field. Due to the lack of full dissipation and damping, this stability problem is not trivial. Without the presence of a magnetic field, the fluid velocity is governed by the 2D Navier--Stokes equations in the whole space ℝ² with only vertical dissipation, and its stability (near the trivial solution) is still an open problem. However, when coupled to the magnetic field in such an MHD system, we are able to show that any perturbation near a background magnetic field is globally stable in Sobolev space H². This result reflects the observed stabilizing effect of the magnetic field. Mathematically, the MHD system obeyed by the perturbations can be converted to a system of wave equations which exhibits extra smoothing and stabilizing properties. These properties allow us to control the nonlinearity in the anisotropic Navier--Stokes equations and thus establish the desired stability result. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of three-dimensional compressible barotropic Navier--Stokes equations with large external potential forces. We first prove that the Cauchy problem has a unique global strong solution with large oscillations and interior vacuum, provided the initial data are of small energy and the unique steady state is strictly away from vacuum. Next, we show that if the initial data are more regular and satisfy some additional compatibility conditions, the strong solution is indeed a classical one away from the initial time. Finally, as by-products, the global existence and large-time behavior of weak solutions with large external potential forces and discontinuous initial data containing vacuum are also obtained. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of three-dimensional isentropic compressible magnetohydrodynamic equations in the present paper. For regular initial data with small energy but possibly large oscillations, we prove the global well-posedness of classical solution, where the flow density is allowed to contain vacuum states, and the large-time behavior of the solution is also shown. [ABSTRACT FROM AUTHOR]