A blow-up criterion for strong solutions to three-dimensional compressible magnetohydrodynamic equations

We are concerned with an initial boundary value problem for the compressible magnetohydrodynamic equations with viscosity depending on the density. It is show that for the initial density away from vacuum, the strong solution to the problem exists globally if the gradient of velocity satisfies $\|\nabla\mathbf{u}\|_{L^{2}(0,T;L^\infty)}<\infty$. Our method relies upon the delicate energy estimates and elliptic estimates.Comment: 17 page

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

On formation of singularity of the full compressible magnetohydrodynamic equations with zero heat conduction

We are concerned with the formation of singularity and breakdown of strong solutions to the Cauchy problem of the three-dimensional full compressible magnetohydrodynamic equations with zero heat conduction. It is proved that for the initial density allowing vacuum, the strong solution exists globally if the deformation tensor $\mathfrak{D}(\mathbf{u})$ and the pressure $P$ satisfy $\|\mathfrak{D}(\mathbf{u})\|_{L^{1}(0,T;L^\infty)}+\|P\|_{L^{\infty}(0,T;L^\infty)}<\infty$. In particular, the criterion is independent of the magnetic field. The logarithm-type estimate for the Lam{\'e} system and some delicate energy estimates play a crucial role in the proof.Comment: to appear in Indiana University Mathematics Journal. arXiv admin note: text overlap with arXiv:1705.0516

Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies the Serrin's condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or vacuum appears in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin's blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.Comment: 25 page

A blow-up criterion of strong solutions to the 2D compressible magnetohydrodynamic equations

This paper establishes a blow-up criterion of strong solutions to the two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion depends on the density, but is independent of the velocity and the magnetic field. More precisely, once the strong solutions blow up, the $L^{\infty}$-norm for the density tends to infinity. In particular, the vacuum in the solutions is allowed.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1402.4851, arXiv:1310.1673, arXiv:1210.5930 by other author

Singularity formation to the 2D Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction

The formation of singularity and breakdown of strong solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction are considered. It is shown that for the initial density allowing vacuum, the strong solution exists globally if the density $\rho$ and the pressure $P$ satisfy $\|\rho\|_{L^{\infty}(0,T;L^\infty)}+\|P\|_{L^{\infty}(0,T;L^\infty)}<\infty$. In addition, the initial density can even have compact support. The logarithm-type estimate for the Lam{\'e} system and some weighted estimates play a crucial role in the proof.Comment: 15 page

Blow-up criterion for the 33D non-resistive compressible Magnetohydrodynamic equations

In this paper, we prove a blow-up criterion in terms of the magnetic field $H$ and the mass density $\rho$ for the strong solutions to the $3$D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the $L^\infty$ norms of $(H,\rho)$ control the possible blow-up (see \cite{olga}\cite{zx}) for strong solutions, which means that if a solution of the compressible isentropic non-resistive MHD equations is initially smooth and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the $L^\infty$ norm of $H$ or $\rho$ as the critical time approaches.Comment: 22 pages. arXiv admin note: text overlap with arXiv:1401.270

Blow-up criterion for the compressible magnetohydrodynamic equations with vacuum

In this paper, the 3-D compressible MHD equations with initial vacuum or infinity electric conductivity is considered. We prove that the $L^\infty$ norms of the deformation tensor $D(u)$ and the absolute temperature $\theta$ control the possible blow-up (see [5][18][20]) for strong solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $D(u)$ and $\theta$ as the critical time approaches. The viscosity coefficients are only restricted by the physical conditions. Our criterion (see (\ref{eq:2.911})) is similar to [17] for $3$-D incompressible Euler equations, [10] for $3$-D compressible isentropic Navier-stokes equations and [22]for $3$-D compressible isentropic MHD equations.Comment: 21pages. arXiv admin note: substantial text overlap with arXiv:1401.270

Global Existence of Strong Solutions to Incompressible MHD

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for incompressible MHD equations in a bounded smooth domain of three spatial dimensions with initial density being allowed to have vacuum, in particular, the initial density can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $|\sqrt\rho_0u_0|_{L^2(\Omega)}^2+|H_0|_{L^2(\Omega)}^2$ and $|\nabla u_0|_{L^2(\Omega)}^2+|\nabla H_0|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.Comment: 10 pages. Communications on Pure and Applied Analysis, 201

Global strong solutions to the 3D full compressible Navier-Stokes system with vacuum in a bounded domain

In this short paper we establish the global well-posedness of strong solutions to the 3D full compressible Navier-Stokes system with vacuum in a bounded domain $\Omega\subset \mathbb{R}^3$ by the bootstrap argument provided that the viscosity coefficients $\lambda$ and $\mu$ satisfy that $7\lambda>9\mu$ and the initial data $\rho_0$ and $u_0$ satisfy that $\|\rho_0\|_{L^\infty(\Omega)}$ and $\|\rho_0|u_0|^5\|_{L^1(\Omega)}$ are sufficient small