Blow‐up criterion for 3D viscous‐resistive compressible magnetohydrodynamic equations

In this paper, we establish a blow‐up criterion of strong solutions for 3D viscous‐resistive compressible magnetohydrodynamic equations, which depends only on and . Our result improves the previous criterion in Lu's paper (Journal of Mathematical Analysis and Applications 2011; 379: 425–438.) for compressible magnetohydrodynamic equations by removing a stringent condition on the viscous coefficients μ > 4λ. In addition, initial vacuum states are also allowed in our cases. Copyright © 2012 John Wiley & Sons, Ltd.     

Global well-posedness of the non-isentropic full compressible magnetohydrodynamic equations

In this paper, we are concerned with Cauchy problem for the multi-dimensional (N ≥ 3) non-isentropic full compressible magnetohydrodynamic equations. We prove the existence and uniqueness of a global strong solution to the system for the initial data close to a stable equilibrium state in critical Besov spaces. Our method is mainly based on the uniform estimates in Besov spaces for the proper linearized system with convective terms.     

Strong solutions to 3D compressible magnetohydrodynamic equations with Navier‐slip condition

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier‐slip condition. The Navier‐slip condition differs in many aspects from no‐slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier‐slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L^∞ norm of the deformation tensor D(u). Copyright © 2015 John Wiley & Sons, Ltd.     

Blow-up criterion for compressible MHD equations

In this paper, we study the 3D compressible magnetohydrodynamic equations. We obtain a blow up criterion for the local strong solutions just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion (see J.T. Beal, T. Kato and A. Majda (1984) [1]) for the ideal incompressible flow. In addition, initial vacuum is allowed in our case.     

Global well‐posedness of classical solutions to 3D compressible magnetohydrodynamic equations with large potential force

We consider the Cauchy problem of isentropic compressible magnetohydrodynamic equations with large potential force in . When the initial data (ρ₀,u₀,H₀) is of small energy, we investigate the global well‐posedness of classical solutions where the flow density is allowed to contain vacuum states.     

A Beale–Kato–Majda criterion for the 3D viscous magnetohydrodynamic equations

In this paper, we investigate the Cauchy problem for the 3D viscous incompressible magnetohydrodynamic equations and establish a Beale–Kato–Majda regularity criterion of smooth solutions in terms of the velocity vector in the homogeneous bounded mean oscillations space. Copyright © 2014 John Wiley & Sons, Ltd.     

A regularity criterion for two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion

We study the global regularity of classical solution to two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow‐up can be controlled by the L^∞‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.     

Local existence and blow‐up criterion for the generalized Boussinesq equations in Besov spaces

In this paper, we consider the three‐dimensional generalized Boussinesq equations, a system of equations resulting from replacing the Laplacian ? Δ in the usual Boussinesq equations by a fractional Laplacian ( ? Δ)^α. We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood–Paley theory and Bony's paradifferential calculus. The results in this paper can be regarded as an extension to the Serrin‐type criteria for Navier–Stokes equations and magnetohydrodynamics equations, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

An improved blow‐up criterion for smooth solutions of the two‐dimensional MHD equations

Our main object is to establish a regularity criterion with p≥q > 1 for the incompressible magnetohydrodynamics equations with zero magnetic diffusivity. Copyright © 2016 John Wiley & Sons, Ltd.     

A regularity criterion for the density‐dependent magnetohydrodynamic equations

In this paper, regularity criterion for the 3‐D density‐dependent magnetohydrodynamic equation is considered. It is proved that the solution keeps smoothness only under an integrable condition on the velocity field in multiplier spaces. Hence, it turns out that the velocity field plays a dominant role in the regularity criteria of the weak solutions to this nonlinear coupling problem even with density. Copyright © 2009 John Wiley & Sons, Ltd.     

On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space

In this paper, we consider sufficient conditions for the regularity of Leray–Hopf solutions of the 3D incompressible magnetohydrodynamic equations via two components of the velocity and magnetic fields in terms of BMO spaces. We prove that if belongs to the space , then the solution (u,b) is regular. This extends recent results contained by Gala, Ji E and Lee J. Copyright © 2013 John Wiley & Sons, Ltd.     

L∞ continuation principle to the non‐baratropic non‐resistive magnetohydrodynamic equations without heat conductivity

The aim of this paper is to establish a continuation principle for strong solutions to the full compressible magnetohydrodynamic system without resistivity and heat conductivity. We prove that if the solution loses its regularity in finite time, the dominated part is due to the hyperbolic effect. More precisely, it is essentially shown that the strong solution exists globally if the density, temperature, and magnetic field are bounded from above, where vacuum is allowed to exist. This verifies a problem proposed by D.Serre. Copyright © 2016 John Wiley & Sons, Ltd.     

A new blowup criterion for strong solutions to the three‐dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain

In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three‐dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in BMO norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO‐type estimate for the Lam system 2.6 and a variant of the Brezis‐Waigner's inequality 2.3 play a critical role in the proof. Copyright © 2017 John Wiley & Sons, Ltd.     

On the blow‐up criterion of 3D Navier‐Stokes equation in

In this paper, we prove two results about the blow‐up criterion of the three‐dimensional incompressible Navier‐Stokes equation in the Sobolev space . The first one improves the result of Cortissoz et al. The second deals with the relationship of the blow up in and some critical spaces. Fourier analysis and standard techniques are used.     

Zero Mach number limit to compressible quantum magnetohydrodynamic equations with vanishing viscosity coefficients

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero.     

Singularity formation to the 2D Cauchy problem of nonbarotropic magnetohydrodynamic equations without heat conductivity

The question of whether the two-dimensional (2D) nonbarotropic compressible magnetohydrodynamic (MHD) equations with zero heat conduction can develop a finite-time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. Such a problem is interesting in studying global well-posedness of solutions. In this paper, we proved that, for the initial density allowing vacuum states, the strong solution exists globally if the density and the pressure are bounded from above. Our method relies on weighted energy estimates and a Hardy-type inequality.     

A blow-up criterion for 3-D non-resistive compressible heat-conductive magnetohydrodynamic equations with initial vacuum

In this paper, we prove a blow-up criterion of strong solutions to the 3-D viscous and non-resistive magnetohydrodynamic equations for compressible heat-conducting flows with initial vacuum. This blow-up criterion depends only on the gradient of velocity and the temperature, which is similar to the one for compressible Navier-Stokes equations.     

A new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in a bounded domain

This paper is to derive a new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in terms of the density $\rho$ and the pressure $P$. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm $\left|\right|\rho {\left|\right|}_{L^{\infty }(0,t;L^{\infty })}+\left|\right|P{\left|\right|}_{L^{p_0}(0,t;L^{\infty })}<\infty$ for some constant  $p_0$ satisfying $1<p_0\le 2$. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as $\left|\right|\rho {\left|\right|}_{L^{\infty }(0,t;L^{\infty })}+\left|\right|P{\left|\right|}_{L^{\infty }(0,t;L^{\infty })}<\infty$.