带真空无磁扩散不可压磁流体方程柯西问题的局部适定性

该文讨论了在真空远场的密度条件下,二维不可压零磁耗散磁流体力学方程组柯西问题的局部适定性.在初始密度和磁场具有一定的衰减性时,证明了磁流体方程具有唯一的局部强解.当初值满足兼容性条件和适当的正则性条件时,该强解就是经典解.除此之外,文中还给出了一个仅与磁场有关的爆破准则.     

On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations with vacuum

This paper concerns the Cauchy problem of the barotropic compressible Navier–Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier–Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.     

Global classical solutions to the compressible micropolar viscous fluids with large oscillations and vacuum

In this paper, we consider the three dimensional Cauchy problem of the compressible micropolar viscous flows. We prove the existence of unique global classical solution for smooth initial data with small initial energy but possibly large oscillations and the initial density may allowed to contain the interior and far field vacuum states. Furthermore, the large time behavior of the solution is obtained as well.     

Blowup of smooth solution for non‐isentropic magnetohydrodynamic equations without heat conductivity

In this paper, we establish a blow‐up criterion for the three‐dimentional viscous, compressible magnetohydrodynamic flows. It is shown that for the Cauchy problem and the initial‐boundary‐value problem with initial density allowed to vanish, the strong or smooth solution for the three‐dimentional magnetohydrodynamic flows exists globally if the density, temperature, and magnetic field is bounded from above. Copyright © 2016 John Wiley & Sons, Ltd.     

Well‐posedness for the density‐dependent incompressible flow of liquid crystals

In this paper, we are concerned with the Cauchy problem for the density‐dependent incompressible flow of liquid crystals in thewhole space (N ≥ 2).We prove the localwell‐posedness for large initial velocity field and director field of the system in critical Besov spaces if the initial density is close to a positive constant. We show also the global well‐posedness for this system under a smallness assumption on initial data. In particular, this result allows us to work in Besov space with negative regularity indices, where the initial velocity becomes small in the presence of the strong oscillations. Copyright © 2014 John Wiley & Sons, Ltd.     

On the restricted evolution problem of the Einstein relativistic mechanics

This work is focused on the interior Cauchy problem for the Einstein’s field equations. Precisely, in the relativistic study of the evolution of a continuum reversible system, the Cauchy problem is broken into two separate problems: the initial data problem and the restricted problem of evolution.     

Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations

In this paper, we study the Cauchy problem for the three-dimensional compressible magnetohydrodynamics equations. We establish a blowup criterion for global regularity of strong solutions, which depends only on density and magnetic field. In addition, the initial data can be arbitrarily large and contain vacuum states. The proof is based on the new a priori estimates for three-dimensional compressible magnetohydrodynamics equations.     

A Blow-up Criterion for 2D Compressible Nematic Liquid Crystal Flows in Terms of Density

This paper is concerned with the Cauchy problem for compressible nematic liquid crystal flows in the two-dimensional space (2D). We establish a blow-up criterion in terms of the density only, provided the macroscopic average of the nematic liquid crystal orientation field satisfies a geometric condition. In particular, the initial vacuum is allowed and the compatibility condition is removed.     

Global well‐posedness of classical solutions with large oscillations and vacuum to the three‐dimensional isentropic compressible Navier‐Stokes equations

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier‐Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. The initial density is allowed to vanish, and the spatial measure of the set of vacuum can be arbitrarily large; in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc.     

Global weak solutions to the cometary flow equation with a self-generated electric field

We study the Cauchy problem of a cometary flow equation with a self-generated electric field. This kinetic model originates from the theory of astrophysical plasmas and can be viewed as a perturbation, by a wave-particle collision operator, of the classical Vlasov-Poisson system. By asymptotic methods in kinetic theory, global existence of nonnegative weak solutions to the Cauchy problem in three space variables is established for bounded initial data having finite second order velocity moments.     

Cauchy problem and averaging in Fock space

We investigate a Cauchy problem in the Fock space for a system consisting of a two-level atom, a quantum field, and a classical field. A solution estimate is obtained for the Cauchy problem with initial data from a special class. This class is invariant with respect to the dynamic semigroup of the system. We propose an averaging method for solving the Cauchy problem in the case where the Hamiltonian parameters differ greatly in the order of magnitude. An estimate of the averaging error is obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 1, pp. 92–106, October, 1998.     

Vlasov–Poisson equations for a two-component plasma in a half-space

The Vlasov–Poisson equations for a two-component high-temperature plasma with an external magnetic field in a half-space are considered. The electric field potential satisfies the Dirichlet condition on the boundary, and the initial density distributions of charged particles satisfy the Cauchy conditions. Sufficient conditions for the induction of the external magnetic field and the initial charged-particle density distributions are obtained that guarantee the existence of a classical solution for which the supports of the charged-particle density distributions are located at some distance from the boundary.     

奇数维可压缩液晶流方程解的逐点估计

首先, 本文利用标准的能量估计方法得到高维(3 维及以上) 的液晶流方程组小初值经典解的整体存在性. 然后, 本文运用Green 函数方法, 得到奇数维情形(3 维及以上) 该解的逐点估计. 该结果表明, 密度ρ和动量m同Navier-Stokes 方程组一样满足一般Huygens 原理, 而单位向量场d则没有这种现象, 其有着与热方程的解类似的时空估计.     

Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

We consider the Cauchy problem for one-dimensional(1D) barotropic compressible Navier-Stokes equations with density-depending viscosity and large external forces. Under a general assumption on the densitydepending viscosity, we prove that the Cauchy problem admits a unique global strong(classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently.As a consequence, we obtain the large time behavior of the solution without external forces.     

Global existence of strong solutions to the Cauchy problem for a 1D radiative gas

We consider a one-dimensional radiation hydrodynamics model in the case of the equilibrium diffusion approximation which is described by the compressible Navier-Stokes system with the additional terms in the pressure and internal energy respectively, which embody the effect of radiation. Under the physical growth conditions on the heat conductivity, we establish the existence and uniqueness of strong solutions to the Cauchy problem with large initial data, where the initial density and velocity may have differing constant states at infinity. Moreover, we show that if there is no vacuum in the initial density, then, the vacuum and concentration of the density will never occur in any finite time.     

耦合Schrdinger-KdV方程组Cauchy问题的适定性

本文研究了耦合Schrodinger－KdV方程组的Cauchy问题，此耦合方程组刻化了一维Langmuir和离子声波相互作用的非线性动力学行为．本文建立了此问题在Hk×Hk中的整体适定性理论（k∈Z+）．     

A priori estimates in terms of the maximum norm for the solutions of the Navier-Stokes equations

In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field. For illustrative purposes, we first derive corresponding a priori estimates for certain parabolic systems. Because of the pressure term, the case of the Navier-Stokes equations is more difficult, however.     

Global solvability for quasilinear parabolic equation with inhomogeneous density and a source

We study the Cauchy problem for quasilinear parabolic equation with inhomogeneous density and a source. We show that this problem has a global solution under the assumptions that initial datum is small enough in the integral sense and the source term has overcritical behaviour. The sharp estimates of a solution is obtained as well.     

Entropy solutions to the Verigin ultraparabolic problem

We study the Cauchy problem for the two-dimensional ultraparabolic model of filtration of a viscous incompressible fluid containing an admixture, with diffusion of the admixture in a porous medium taken into account. The porous medium consists of the fibers directed along some vector field n ^⊥. We prove that if the nonlinearity in the equations of the model and the geometric structure of fibers satisfy some additional “genuine nonlinearity” condition then the Cauchy problem with bounded initial data has at least one entropy solution and the fast oscillating regimes possible in the initial data are promptly suppressed in the entropy solutions. The proofs base on the introduction and systematic study of the kinetic equation associated with the problem as well as on application of the modification of Tartar H-measures which was proposed by Panov.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.