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.     

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.     

Global weak solutions to a class of compressible non-Newtonian fluids with vacuum

The global weak solution of an initial-boundary value problem for a compressible non-Newtonian fluid is studied in a three-dimensional bounded domain. By the techniques of artificial pressure, a solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global weak solution to the three-dimensional compressible non-Newtonian fluid with vacuum and large data is established.     

Global existence of classical solution for a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum

In this work, we obtain the global existence and uniqueness of classical solutions to a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum in one dimension, where the initial vacuum is allowed. We get the upper and lower bounds of gas and liquid masses n and m by the continuity methods which we use to study the compressible Navier-Stokes equations.     

Global classical solution to 1D compressible Navier–Stokes equations with no vacuum at infinity

C. Miao In this paper, we are concerned with the 1D Cauchy problem of the compressible Navier–Stokes equations with the viscosity μ(ρ) = 1+ρ^β(β≥0). The initial density can be arbitrarily large and keep a non‐vacuum state at far fields. We will establish the global existence of the classical solution for 0≤β < γ via a priori estimates when the initial density contains vacuum in interior interval or is away from the vacuum. We will show that the solution will not develop vacuum in any finite time if the initial density is away from the vacuum. To study the well‐posedness of the problem, it is crucial to obtain the upper bound of the density. Some new weighted estimates are applied to obtain our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum

We study the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in a bounded annulus Ω of R3. And a result on the existence and uniqueness of global spherically symmetric classical solutions is obtained. Here the initial data could be large and initial vacuum is allowed.     

Global existence of classical solutions to two‐dimensional Navier–Stokes equations with Cauchy data containing vacuum

In this paper, the Cauchy problem to the two‐dimensional isentropic compressible Navier–Stokes equations with smooth initial data containing vacuum is investigated. If the initial data are of small energy but possibly large oscillations, we obtain the global well‐posedness of classical solutions in the case of initially nonvacuum far fields. In particular, the smallness of the energy only depends on the norm of the initial velocity, where β can be arbitrary close to 0. In the case of compactly supported initial density, a blow‐up example is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Global solutions to the micropolar compressible flow with constant coefficients and vacuum

We prove the global existence and uniqueness of strong solutions for an initial boundary value problem modeling the motion of the compressible micropolar fluids in one dimensional space. Compared with former studies, we are concerned with the nonisentropic case with constant transport coefficients and the initial density is allowed to have vacuum. Our analysis is based on the nonlinear energy method and the crucial step is to derive the uniform upper and lower bounds on the ratio of the density to its initial value.     

Global classical solution for a three-dimensional viscous liquid-gas two-fluid flow model with vacuum

This is a continuation of the paper （J. Math. Phys., 52（2011）, 093102）. We consider the Cauchy problem to the three-dimensional viscous liquid-gas two-fluid flow model. The global existence of classical solution is proved, where the initial vacuum is allowed.     

Global existence of classical solutions for the 3D generalized compressible Oldroyd-B model

In this paper, we prove the global existence of small classical solutions to the 3D generalized compressible Oldroyd-B system. It can be seen as compressible Euler equations coupling the evolution of stress tensor τ. The result mainly shows that singularity of solutions to compressible Euler equations can be prevented by the coupling of viscoelastic stress tensor. Moreover, unlike most complex fluids containing compressible Euler equations, the irrotational condition ∇×u=0 would not be proposed here to achieve the global well-posedness.     

On the existence and stability of solutions for the micropolar fluids in exterior domains

We prove the existence of a global strong solution in some class of Marcinkiewicz spaces for the micropolar fluid in an exterior domain of R³, with initial conditions being a non‐smooth disturbance of a steady solution. We also analyse the large time behaviour of those solutions and apply our results in the context of the Navier–Stokes equations. Copyright © 2007 John Wiley & Sons, Ltd.     

Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data

In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.     

Global strong solutions for a class of compressible non‐Newtonian fluids with vacuum

We study a class of compressible non‐Newtonian fluids in one space dimension. We prove, by using iterative method, the global time existence and uniqueness of strong solutions provided that the initial data satisfy a compatibility condition and the initial density is small in its H¹‐norm. The main difficulty is due to the strong nonlinearity of the system and the initial vacuum. Copyright © 2010 John Wiley & Sons, Ltd.     

Global classical solutions of viscous liquid–gas two‐phase flow model

In this paper, we consider the global existence and uniqueness of the classical solutions for the three‐dimensional where the existence of global classical solutions to the compressible Navier–Stokes equations was obtained by using the continuity methods under the assumption that the initial energy is sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.     

Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.     

Global weak solutions to one-dimensional compressible viscous hydrodynamic equations

In this article, we are concerned with the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction δ~2ρ((φ(ρ))xxφ′(ρ))x withφ(ρ) = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions. The diffusion term εu xx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in[1](α =1/2) to 0 α≤1. In addition, we perform the limit ε→ 0 with respect to 0 α≤1/2.     

A blow-up criterion for classical solutions to the compressible Navier-Stokes equations

In this paper, we obtain a blow-up criterion for classical solutions to the 3-D compressible Navier-Stokes equations just in terms of the gradient of the velocity, analogous to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, the initial vacuum is allowed in our case.