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Xiaoding Shi

Publications and source records attributed to Xiaoding Shi.

28 records · Page 2Linked to original sources
arXiv2018

Stability of degenerate stationary solution to the outflow problem for full Navier-Stokes equations

This paper is concerned with the large-time behavior of solutions to the outflow problem of full compressible Navier-Stokes equations in the half line. This is one of the series of papers by the authors on the stability of nonlinear waves to the outflow problem. We show the time asymptotic stability of degenerate (transonic) stationary solution for the general gas including ideal polytropic gas. The key point of the proof is based the new property of the degenerate stationary solution and the delicate energy estimates.

math.AP↗
arXiv2018

Global Solution for Gas-Liquid Flow of 1-D van der Waals Equation of State with Large Initial Data

This paper is concerned with a diffuse interface model for the gas-liquid phase transition. The model consists the compressible Navier-Stokes equations with van der Waals equation of state and a modified Allen-Cahn equation. The global existence and uniqueness of strong solution with the periodic boundary condition (or the mixed boundary condition) in one dimensional space is proved for large initial data. Furthermore, the phase variable and the density of the gas-liquid mixture are proved to stay in the physical reasonable interval. The proofs are based on the elementary energy method and the maximum principle, but with new development, where some techniques are introduced to establish the uniform bounds of the density and to treat the non-convexity of the pressure function.

math.AP↗
arXiv2018

Global Strong Solutions to Magnetohydrodynamics with Density-Dependent Viscosity and Degenerate Heat-Conductivity

We deal with the equations of a planar magnetohydrodynamic compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity proportional to a positive power of the temperature. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov (1987)], we obtain the global existence and uniqueness of strong solutions which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the motion of the flow has large oscillations and the interaction between the hydrodynamic and magnetodynamic effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant viscosity and heat conductivity case to that of nonlinear viscosity and degenerate heat-conductivity.

math.AP↗
arXiv2018

Global Strong Solutions to Compressible Navier-Stokes System with Degenerate Heat Conductivity and Density-Depending Viscosity

We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)], we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant heat conductivity case to the degenerate and nonlinear one under stress-free and thermally insulated boundary conditions.

math.AP↗
arXiv2018

Nonlinearly Exponential Stability of Compressible Navier-Stokes System with Degenerate Heat-Conductivity

We study the large-time behavior of strong solutions to the one-dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas, when the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature. Both the specific volume and the temperature are proved to be bounded from below and above independently of time. Moreover, it is shown that the global solution is nonlinearly exponentially stable as time tends to infinity. Note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977); Kazhikhov. Boundary Value Problems for Hydrodynamical Equations, 50 (1981)] and can be arbitrarily large. Therefore, our result can be regarded as a natural generalization of the Kazhikhov's ones for the constant heat conductivity case to the degenerate and nonlinear one.

math.AP↗
arXiv2018

Asymptotic Stability of Solutions for 1-D Compressible Navier-Stokes-Cahn-Hilliard system

This paper is concerned with the evolution of the periodic boundary value problem and the mixed boundary value problem for a compressible mixture of binary fluids modeled by the Navier-Stokes-Cahn-Hilliard system in one dimensional space. The global existence and the large time behavior of the strong solutions for these two systems are studied. The solutions are proved to be asymptotically stable even for the large initial disturbance of the density and the large velocity data. We show that the average concentration difference for the two components of the initial state determines the long time behavior of the diffusive interface for the two-phase flow.

math.AP↗
arXiv2018

Exponential Decay for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations in 3D Bounded Domains

For barotropic compressible Navier-Stokes equations in three-dimensional (3D) bounded domains, we prove that any finite energy weak solution obtained by Lions [Mathematical topics in fluid mechanics, Vol. 2. Compressible models(1998)] and Feireisl-Novotný-Petzeltová [J. Math. Fluid Mech. 3(2001), 358-392] decays exponentially to the equilibrium state. This result is established by both using the extra integrability of the density due to Lions and constructing a suitable Lyapunov functional just under the framework of Lions-Feireisl's weak solutions.

math.AP↗
arXiv2015

Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Navier-Stokes Equations with Vacuum

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier-Stokes equations on the whole space $\mathbb{R}^2$ admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the spatial gradients of the velocity and the pressure which are the same as those of the homogeneous case.

math.AP↗
arXiv2014

Global well-posedness and large time asymptotic behavior of strong solutions to the 2-D compressible magnetohydrodynamic equations with vacuum

The authors study the Cauchy problem of the magnetohydrodynamic equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For two spatial dimensions, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) provided the smooth initial data are of small total energy, and obtain some a priori decay with rates (in large time) for the pressure, the spatial gradient of both the velocity field and the magnetic field. Moreover, for three spatial dimensions case, some similar decay rates are also obtained.

math.AP↗