arXiv subjects

Zhouping Xin

Publications and source records attributed to Zhouping Xin.

At least 73 records · Page 4Linked to original sources

arXiv2015

On Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem

This paper proves the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant $γ$ lies in the stability range $(4/3, 2)$. It is shown that for small perturbations of a Lane-Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary provided that $γ$ lies in $(4/3, 2)$. The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane-Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary.

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arXiv2015

Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities

This paper concerns the existence of global weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients. We construct suitable approximate system which has smooth solutions satisfying the energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate. Then, after adapting the compactness results due to Mellet-Vasseur [Comm. Partial Differential Equations 32 (2007)], we obtain the global existence of weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in two or three dimensional periodic domains or whole space for large initial data. This, in particular, solved an open problem in [P. L. Lions. Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford University Press, 1998].

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arXiv2015

Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities

The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent $γ$ lies in the stability regime $(4/3, 2)$, by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. The results obtained in this paper extend those in \cite{LXZ} of the authors for the constant viscosities to the case of density dependent viscosities which are degenerate at vacuum states.

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arXiv2015

Non-uniqueness of Admissible Weak Solutions to Compressible Euler Systems with Source Terms

We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as rotations, for general piecewise Lipschitz initial densities and some suitably constructed initial momentum, we obtain infinitely many global admissible weak solutions. Furthermore, we construct a class of finite-states admissible weak solutions to the Euler system with anti-symmetric sources. Under the additional smallness assumption on the initial densities, we also obtain multiple global-in-time admissible weak solutions for more general sources including damping. The basic framework are based on the convex integration method developed by De~Lellis and Székelyhidi \cite{dLSz1,dLSz2} for the Euler system. One of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term.

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arXiv2015

Uniform regularity for the free surface compressible Navier-Stokes equations with or without surface tension

In this paper, we investigate the uniform regularity of solutions to the 3-dimensional isentropic compressible Navier-Stokes system with free surfaces and study the corresponding asymptotic limits of such solutions to that of the compressible Euler system for vanishing viscosity and surface tension. It is shown that there exists an unique strong solution to the free boundary problem for the compressible Navier-Stokes system in a finite time interval which is independent of the viscosity and the surface tension. The solution is uniform bounded both in $W^{1,\infty}$ and a conormal Sobolev space. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. Based on such uniform estimates, the asymptotic limits to the free boundary problem for the ideal compressible Euler system with or without surface tension as both the viscosity and the surface tension tend to zero, are established by a strong convergence argument.

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arXiv2015

On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity

It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomenon, it remain to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also give an affirmative answer to a strong version of conjecture proposed by J.Nash in 1950s

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arXiv2015

On Liouville Type of Theorems to the 3-D Incompressible Axisymmetric Navier-Stokes Equations

Liouville type of theorems play a key role in the blow-up approach to study the global regularity of the three-dimensional Navier-Stokes equations. In this paper, we will prove Liouville type of theorems to the 3-D axisymmetric Navier-Stokes equations with swirls under some suitable assumptions on swirl component velocity $u_θ$ which are scaling invariant. It is known that $ru_θ$ satisfies the maximum principle. The assumptions on $u_θ$ will be natural and useful to make further studies on the global regularity to the three-dimensional incompressible axisymmetric Navier-Stokes equations.

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arXiv2014

Global Smooth Supersonic Flows in Infinite Expanding Nozzles

This paper concerns smooth supersonic flows with Lipschitz continuous speed in two-dimensional infinite expanding nozzles, which are governed by a quasilinear hyperbolic equation being singular at the sonic and vacuum state. The flow satisfies the slip condition on the walls and the flow velocity is prescribed at the inlet. First, it is proved that if the incoming flow is away from the sonic and vacuum state and its streamlines are rarefactive at the inlet, then a flow in a straight nozzle never approaches the sonic and vacuum state in any bounded region. Furthermore, a sufficient and necessary condition of the incoming flow at the inlet is derived for the existence of a global smooth supersonic flow in a straight nozzle. Then, it is shown that for each incoming flow satisfying this condition, there exists uniquely a global smooth supersonic flow in a symmetric nozzle with convex upper wall. It is noted that such a flow may contain a vacuum. If there is a vacuum for a global smooth transonic flow in a symmetric nozzle with convex upper wall, it is proved that for the symmetric upper part of the flow, the first vacuum point along the symmetric axis must be located at the upper wall and the set of vacuum points is the closed domain bounded by the tangent half-line of the upper wall at this point to downstream and the upper wall after this point. Moreover, the flow speed is globally Lipschitz continuous in the nozzle, and on the boundary between the gas and the vacuum, the flow velocity is along this boundary and the normal derivatives of the flow speed and the square of the sound speed both are zero. As an immediate consequence, the local smooth transonic flow obtained in [10] can be extended into a global smooth transonic flow in a symmetric nozzle whose upper wall after the local flow is convex.

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arXiv2014

Two Dimensional Subsonic Euler Flow. Past a Wall or a Symmetric Body

The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal veloicty in the upstream, there exists a critical value $ρ_{cr}$ such that if the incoming density in the upstream is larger than $ρ_{cr}$, then there exists a subsonic flow past a wall. Furthermore, $ρ_{cr}$ is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than $ρ_{cr}$. The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the velocity field and its far field states.

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arXiv2014

On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb R^2$

This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen-Leslie system of liquid crystal models in $\mathbb R^2$, with both general Leslie stress tensors and general Oseen-Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant, which solves an interesting open problem. One of the main ideas of our proof is to perform suitable energy estimates at level one order lower than the natural basic energy estimates for the Ericksen-Leslie system.

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arXiv2014

Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data

The low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data is rigorously justified in the whole space $\mathbb{R}^3$. The uniform in Mach number estimates of the solutions in a Sobolev space are obtained on a time interval independent of the Mach number. The limit is proved by using the established uniform estimates and a theorem due to Métiver and Schochet [Arch. Ration. Mech. Anal. 158 (2001), 61-90] for the Euler equations that gives the local energy decay of the acoustic wave equations.

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arXiv2014

Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

This paper concerns the well-posedness theory of the motion of physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in \cite{10',7,16'} by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.

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arXiv2013

Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D

In this paper, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system on $\mathbb T^2$, based on a new approximate system which is different from the classical Ginzburg-Landau approximation. Local energy inequalities are employed to recover the estimates on the second order spacial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. It is shown that these weak solutions conserve the total energy and while the kinetic and potential energies transfer to the heat energy precisely. Furthermore, it is also established that these weak solutions have at most finite many singular times at which the energy concentration occurs, and as a result, the temperature must increase suddenly at each singular time on some part of $\mathbb T^2$.

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arXiv2013

Remarks on Blow-up of Smooth Solutions to the Compressible Fluid with Constant and Degenerate Viscosities

In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contain vacuum in any finite regions. Moreover, a simplified and unified proof on the blow-up results to the classical solutions of the full compressible Navier-Stokes equations without heat conduction by Xin \cite{Xin} and with heat conduction by Cho-Bin \cite{CJ} will be given.

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arXiv2013

Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum

This paper concerns the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For strong and classical solutions, some a priori decay with rates (in large time) for both the pressure and the spatial gradient of the velocity field are obtained provided that the initial total energy is suitably {small.} Moreover, by using these key decay rates and some analysis on the expansion rates of the essential support of the density, we establish the global existence and uniqueness of classical solutions (which may be of possibly large oscillations) in two spatial dimensions, provided the smooth initial data are of small total energy. In addition, the initial density can even have compact support. This, in particular, yields the global regularity and uniqueness of the re-normalized weak solutions of Lions-Feireisl to the two-dimensional compressible barotropic flows for all adiabatic number $γ>1$ provided that the initial total energy is small.

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arXiv2013

Smooth Transonic Flows in De Laval Nozzles

This paper concerns smooth transonic flows of Meyer type in finite de Laval nozzles, which are governed by an equation of mixed type with degeneracy and singularity at the sonic state. First we study the properties of sonic curves. For any $C^2$ transonic flow of Meyer type, the set of exceptional points is shown to be a closed line segment (may be empty or only one point). Furthermore, it is proved that a flow with nonexceptional points is unstable for a $C^1$ small perturbation in the shape of the nozzle. Then we seek smooth transonic flows of Meyer type which satisfy physical boundary conditions and whose sonic points are exceptional. For such a flow, its sonic curve must be located at the throat of the nozzle and it is strongly singular in the sense that the sonic curve is a characteristic degenerate boundary in the subsonic-sonic region, while in the sonic-supersonic region all characteristics from sonic points coincide, which are the sonic curve and never approach the supersonic region. It is proved that there exists uniquely such a smooth transonic flow near the throat of the nozzle, whose acceleration is Lipschitz continuous, if the wall of the nozzle is sufficiently flat.

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arXiv2013

Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$

In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$ for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on the other one; (iv) a new one, which characterizes the maximal existence time of the strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.

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arXiv2013

On the Inviscid Limit of the 3D Navier-Stokes Equations with Generalized Navier-slip Boundary Conditions

In this paper, we investigate the vanishing viscosity limit problem for the 3-dimensional (3D) incompressible Navier-Stokes equations in a general bounded smooth domain of $R^3$ with the generalized Navier-slip boundary conditions (\ref{VSg}). Some uniform estimates on rates of convergence in $C([0,T],L^2(Ω))$ and $C([0,T],H^1(Ω))$ of the solutions to the corresponding solutions of the idea Euler equations with the standard slip boundary condition are obtained.

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