Zhouping Xin

On the weak solutions to a shallow water equation

We obtain the existence of global-in-time weak solutions to the Cauchy problem for a one-dimensional shallow-water equation that is formally integrable and can be obtained by approximating directly the Hamiltonian for Eulers equation in the shallow-water regime. The solution is obtained as a limit of viscous approximation. The key elements in our analysis are some new a priori one-sided supernorm and space-time higher-norm estimates on the first-order derivatives.

Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.

Interface behavior of compressible Navier-Stokes equations with vacuum

In this paper, we study a one-dimensional motion of viscous gas near vacuum with (or without) gravity. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier--Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum, across which the density changes continuously. The regularity and behavior of the solutions near the interfaces and expanding rate of the interfaces are studied. Smoothness of the solutions is discussed. The uniqueness of the weak solutions to the free boundary problem is also proved.

On the global existence of solutions to the Prandtl's system

Abstract In this paper we establish a global existence of weak solutions to the two-dimensional Prandtls system for unsteady boundary layers in the class considered by Oleinik (J. Appl. Math. Mech. 30 (1966) 951) provided that the pressure is favourable. This generalizes the local well-posedness results due to Oleinik (1966; Mathematical Models in Boundary Layer Theory, Chapman & Hall, London, 1999). For the proof, we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations.

Spherically Symmetric Isentropic Compressible Flows with Density-Dependent Viscosity Coefficients

We prove the existence of global weak solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions. We focus on the case where those coefficients vanish on vacuum. The solutions are obtained as limits of solutions in annular regions between two balls, and the equations hold in the sense of distribution in the entire space-time domain. In particular, we prove the existence of spherically symmetric solutions to the Saint–Venant model for shallow water.

Serrin-Type Criterion for the Three-Dimensional Viscous Compressible Flows

We extend the well-known Serrins blowup criterion for the three-dimensional (3D) incompressible Navier–Stokes equations to the 3D viscous compressible cases. It is shown that for the Cauchy problem of the 3D compressible Navier–Stokes equations in the whole space, the strong or smooth solution exists globally if the velocity satisfies the Serrins condition and either the supernorm of the density or the

Blowup Criterion for Viscous Baratropic Flows with Vacuum States

L^1(0,T;L^\infty)

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

-norm of the divergence of the velocity is bounded. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, the Serrins condition on the velocity can be removed in this criterion.

Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases.

EXISTENCE OF GLOBAL STEADY SUBSONIC EULER FLOWS THROUGH INFINITELY LONG NOZZLES

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.