Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state

We consider the Navier-Stokes system describing motions of viscous compressible heat-conducting and “self-gravitating” media. We use the state function of the form p(u,θ)=p₀(u)+p₁(u)θ linear with respect to the temperature θ, but we admit rather general nonmonotone functions p₀ and p₁ of u, which allows us to treat various physical models of nuclear fluids (for which p and u are the pressure and the specific volume) or thermoviscoelastic solids. For solutions to an associated initial-boundary value problem with “fixed-free” boundary conditions and arbitrarily large data, we prove a collection of estimates independent of time interval, including uniform two-sided bounds for u, and describe asymptotic behavior as t→∞. Namely, we establish the stabilization pointwisely and in L^q for u, in L² for θ, and in L^q for v (the velocity), for any q∈[2,∞). For completeness, the proof of the corresponding global existence theorem is also included.     

Remark on the stabilization of a viscous barotropic medium with a nonmonotonic equation of state

We consider the compressible barotropic Navier-Stokes system in one dimension, with a nonmonotonic equation of state. The associated free boundary problem is investigated, and we prove asymptotic properties of the unique globally defined solution for large time.

We also make some comments on a related model of quantum fluid describing the dynamics of cold nuclear matter (zero temperature).     

On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

In this paper,we study the stability of solutions of the Cauchy problem for 1-D compressible NarvierStokes equations with general initial data.The asymptotic limit of solution is found,under some conditions.The results in this paper imply the case that the limit function of solution as t →∞ is a viscous contact wave in the sense,which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero.As a by-product,the decay rates of the solution for the fast diffusion equations are also obtained.The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.     

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the generalized invariant regions and the modified Godunov scheme

We study the motion of isentropic gas in nozzles. This is a major subject in fluid dynamics. In fact, the nozzle is utilized to increase the thrust of rocket engines. Moreover, the nozzle flow is closely related to astrophysics. These phenomena are governed by the compressible Euler equations, which are one of crucial equations in inhomogeneous conservation laws.In this paper, we consider its unsteady flow and devote to proving the global existence and stability of solutions to the Cauchy problem for the general nozzle. The theorem has been proved in Tsuge (2013). However, this result is limited to small data. Our aim in the present paper is to remove this restriction, that is, we consider large data. Although the subject is important in Mathematics, Physics and engineering, it remained open for a long time. The problem seems to rely on a bounded estimate of approximate solutions, because we have only method to investigate the behavior with respect to the time variable. To solve this, we first introduce a generalized invariant region. Compared with the existing ones, its upper and lower bounds are extended constants to functions of the space variable. However, we cannot apply the new invariant region to the traditional difference method. Therefore, we invent the modified Godunov scheme. The approximate solutions consist of some functions corresponding to the upper and lower bounds of the invariant regions. These methods enable us to investigate the behavior of approximate solutions with respect to the space variable. The ideas are also applicable to other nonlinear problems involving similar difficulties.     

Existence and stability of solutions to the compressible Euler equations with an outer force

We study the compressible Euler equation with an outer force. The global existence theorem has been proved in many papers, provided that the outer force is bounded. However, the stability of their solutions has not yet been obtained until now. Our goal in this paper is to prove the existence of a global solution without such an assumption as boundedness. Moreover, we deduce a uniformly bounded estimate with respect to the time. This yields the stability of the solution.When we prove the global existence, the most difficult point is to obtain the bounded estimate for approximate solutions. To overcome this, we employ an invariant region, which depends on both space and time variables. To use the invariant region, we introduce a modified difference scheme. To prove their convergence, we apply the compensated compactness framework.     

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.     

A new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in a bounded domain

This paper is to derive a new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in terms of the density $\rho$ and the pressure $P$. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm $\left|\right|\rho {\left|\right|}_{L^{\infty }(0,t;L^{\infty })}+\left|\right|P{\left|\right|}_{L^{p_0}(0,t;L^{\infty })}<\infty$ for some constant  $p_0$ satisfying $1<p_0\le 2$. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as $\left|\right|\rho {\left|\right|}_{L^{\infty }(0,t;L^{\infty })}+\left|\right|P{\left|\right|}_{L^{\infty }(0,t;L^{\infty })}<\infty$.     

Global spherically symmetric flows for a viscous radiative and reactive gas in an exterior domain

We consider the equations for a viscous, compressible, radiative and reactive gas (pressure $P=R\rho \theta +a{\theta }^4/3$, internal energy $e=c_v\theta +a{\theta }^4/\rho$) over an unbounded exterior domain in ${\mathbb{R}}^n$, where $n\ge 2$ is the space dimension. The existence, uniqueness, and large-time behavior of global spherically symmetric solutions are established for large initial data. The key point in the analysis is to deduce certain uniform a priori estimates on the solutions, especially on lower and upper bounds of the specific volume and temperature.     

Free boundary problem for a viscous compressible flow associated with oxidation of silicon

This paper is concerned with a free boundary problem describing the oxidation process of silicon. Its mathematical model is a compressible Navier-Stokes equations coupling a parabolic equation and a hyperbolic one. Surface tension is involved at the free boundary and density equation is non-homogeneous. It is proved that for arbitrary data satisfying only natural consistency conditions the problem is uniquely solvable on some finite time interval. Supported by National Natural Science Foundation of China     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

Effect of viscous dissipation on natural convection heat and mass transfer from vertical cone in a non-Newtonian fluid saturated non-Darcy porous medium

In this paper we investigate the influence of viscous dissipation and Soret effect on natural convection heat and mass transfer from vertical cone in a non-Darcy porous media saturated with non-Newtonian fluid. The surface of the cone and the ambient medium are maintained at constant but different levels of temperature and concentration. The Ostwald-de Waele power law model is used to characterize the non-Newtonian fluid behavior. The governing equations are non-dimensionalized into non-similar form and then solved numerically by local non-similarity method. The effect of non-Darcy parameter, viscous dissipation parameter, Soret parameter, buoyancy ratio, Lewis number and the power-law index parameter on the temperature and concentration field as well as on the heat and mass transfer coefficients is analyzed.     

Global classical solution to the 3D isentropic compressible Navier–Stokes equations with general initial data and a density‐dependent viscosity coefficient

In this paper, we study the global existence of classical solutions to the three‐dimensional compressible Navier–Stokes equations with a density‐dependent viscosity coefficient (λ = λ(ρ)). For the general initial data, which could be either vacuum or non‐vacuum, we prove the global existence of classical solutions, under the assumption that the viscosity coefficient μ is large enough. Copyright © 2014 John Wiley & Sons, Ltd.     

Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data

This paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in L^p and the velocity u converges to the steady state velocity in W^1,p for any 1p<∞ as time goes to infinity; furthermore, we show that if the initial density contains vacuum at least at one point, then the derivatives of the density must blow up as time goes to infinity.     

Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.When we prove the existence of the time periodic solution, we face with two problems. One is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To overcome this, we employ the generalized invariant region, which depends on the space variables. This enable us to investigate the behavior of solutions in detail. Second is to construct a continuous map. We apply a fixed point theorem to the map from initial data to solutions after one period. Then, the map needs to be continuous. To construct this, we introduce the modified Lax–Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. The formula yields the desired map. Moreover, the invariant region grantees that it maps a compact convex set to itself. In virtue of the fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution. Finally, we apply the compensated compactness framework to prove the convergence of our approximate solutions.