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

In this paper, we prove a blow-up criterion of strong solutions to the 3D viscous and non-resistive isentropic compressible magnetohydrodynamic equations with initial vacuum. This blow-up criterion depends only on the gradient of velocity, which is analogous to the one for the compressible Navier–Stokes equations (cf. Huang et al. (2010) [40]).     

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.     

The curvature tensor and the einstein equations for a four-dimensional nonholonomic distribution

The space of possible particle velocities is a four-dimensional nonholonomic distribution on a manifold of higher dimension, say, M ⁴ × ?¹. This distribution is determined by the 4-potential of the electromagnetic field. The equations of admissible (horizontal) geodesics for this distribution are the same as those of the motion of a charged particle in general relativity theory. On the distribution, a metric tensor with Lorentzian signature (+, ?, ?, ?) is defined, which gives rise to the causal structure, as in general relativity theory. Covariant differentiation (a linear connection) and the curvature tensor for this distribution are introduced. The Einstein equations are obtained from the variational principle for the scalar curvature of the distribution. It is proved that the Dirac operator for the four-dimensional distribution can be extended to functions defined on the manifold M ⁴ × S ¹, where S ¹ is the circle. For such functions, electric charges are topologically quantized.     

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.     

Blow-up criterion for three-dimensional compressible viscoelastic fluids with vacuum

In this paper, we study a Cauchy problem for the equations of 3D compressible viscoelastic fluids with vacuum. We establish a blow-up criterion for the local strong solutions in terms of the upper bound of the density and deformation gradient.     

Refined gluing for vacuum Einstein constraint equations

We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in every case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.     

A symmetry completeness criterion for second-order differential equations

A simple criterion for the completeness of an infinitesimal automorphism of a second-order differential equation is given.

     

An oscillation criterion for nonhomogeneous half-linear differential equations

By means of a classical inequality and an averaging technique, we obtain an oscillation criterion for nonhomogeneous half-linear differential equations. Our result is much more general than a recent result of Wong [1] and makes use of the oscillatory behavior of the forcing terms on intervals.     

On regularity criterion for the dissipative quasi-geostrophic equations

In this paper we consider the 2D dissipative quasi-geostrophic equations and study the regularity criterion of the solutions. By means of a commutator estimate based on frequency localization and Bony's paraproduct decomposition, we obtain a regularity criterion

     

Blowup criterion for viscous, compressible micropolar fluids with vacuum

We prove that the maximum norm of velocity gradients controls the possible breakdown of smooth (strong) solutions for the 3-dimensional viscous, compressible micropolar fluids. More precisely, if a solution of the system is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without the bound of the velocity gradients as the critical time approaches. Our result is a generalization of Huang et al. (2011) [13] from viscous barotropic flows to the viscous, compressible micropolar fluids. In addition, initial vacuum states are also allowed in our result.     

Error estimates for semi-discrete gauge methods for the Navier-Stokes equations

The gauge formulation of the Navier-Stokes equations for incompressible fluids is a new projection method. It splits the velocity in terms of auxiliary (nonphysical) variables and and replaces the momentum equation by a heat-like equation for and the incompressibility constraint by a diffusion equation for . This paper studies two time-discrete algorithms based on this splitting and the backward Euler method for with explicit boundary conditions and shows their stability and rates of convergence for both velocity and pressure. The analyses are variational and hinge on realistic regularity requirements on the exact solution and data. Both Neumann and Dirichlet boundary conditions are, in principle, admissible for but a compatibility restriction for the latter is uncovered which limits its applicability.

     

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.     

Blow-up criterion for 2-D Boussinesq equations in bounded domain

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U _t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control and by and . Furthermore, can control by using vorticity transportation equations. At last, can control . Thus, we can find a blow-up criterion in the form of .      

A delay-dependent stability criterion for nonlinear stochastic delay-integro-differential equations

A type of complex systems under both random influence and memory effects is considered. The systems are modeled by a class of nonlinear stochastic delay-integrodifferential equations. A delay-dependent stability criterion for such equations is derived under the condition that the time lags are small enough. Numerical simulations are presented to illustrate the theoretical result.