Dynamical behaviors for 1D compressible Navier-Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T_∗>0, which is compressed at an algebraic rate and vanishes at the time T_∗, meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.     

An adaptive numerical method to handle blow-up in a parabolic system

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments that illustrate the behaviour of the adaptive method.     

Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity

In this paper we study a free boundary problem for the viscous, compressible, heat conducting, one-dimensional real fluids. More precisely, the viscosity is assumed to be a power function of density, i.e., μ(ρ)=ρα, where ρ denotes the density of fluids and α is a positive constant. In addition, the equations of state include and are more general than perfect flows which only depend linearly on temperature. The global existence (uniqueness) of smooth solutions is established with for general, large initial data, which improves the previous results. Moreover, it is also shown that the solutions will not develop vacuum, mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth, and do not contain vacuum.     

A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension

In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.     

Existence and uniqueness of solutions for a class of non-Newtonian fluids with singularity and vacuum

The aims of this paper are to discuss existence and uniqueness of local solutions for a class of non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals. There are two important points in this paper, one is that we allow the initial vacuum; another one is that the viscosity term of momentum equation is with singularity and fully nonlinearity.     

GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid,assuming that it is in the thermodynamically sense perfect and polytropic.The fluid is between a static solid wall and a free boundary connected to a vacuum state.We take the homogeneous boundary conditions for velocity,microrotation and heat flux on the solid border and that the normal stress,heat flux and microrotation are equal to zero on the free boundary.The proof of the global existence of the solution is based on a limit procedure.We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.     

Unique solvability for the density‐dependent non‐Newtonian compressible fluids with vacuum

The paper is devoted to the existence and uniqueness of local solutions for the density‐dependent non‐Newtonian compressible fluids with vacuum in one‐dimensional bounded intervals. The important points in this paper are that the initial density may vanish in an open subset and the viscosity coefficient is nonlinearly dependent of density and shear rate.     

Global weak solutions for a viscous liquid–gas model with transition to single-phase gas flow and vacuum

This work deals with a viscous two-phase liquid–gas model relevant to the flow in wells and pipelines. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. The model is rewritten in terms of Lagrangian coordinates and is studied in a free boundary setting where the liquid and gas masses are of compact support initially, and continuous at the boundary. Consequently, the initial masses involve a transition to single-phase gas flow and vacuum at the boundary. An appropriate balance between pressure and viscous forces is identified which allows obtaining pointwise upper and lower estimates of masses. These estimates rely on the assumption of a certain relation between the rate of degeneracy of the viscosity coefficient and the rate that determines how fast the initial masses are vanishing at the boundary. By combining these estimates with basic energy type of estimates, higher order regularity estimates are obtained. The existence of global weak solutions is then proved by showing compactness for a class of semi-discrete approximations.     

带自由边界的可压缩欧拉与欧拉-泊松方程组径向对称解的爆破北大核心CSCD

本文在R^(N)(N=2,3)中研究描述流向外部真空的可压缩流体的欧拉与欧拉-泊松方程组径向对称解的爆破.在分离流体与真空的连续自由边界条件下考虑其自由边值问题.对于径向对称的欧拉方程组,证明若初始流平均向外流动,则其光滑解将在有限时刻爆破.对于带有斥力与弛豫项的单极与双极径向对称欧拉-泊松方程组,证明若某个与初始动量有关的加权泛函适当大,则其光滑解将在有限时刻爆破。     

Stability of viscous contact wave for compressible navier-stokes system of general gas with free boundary

In this paper, we study the large time behavior of solutions to the nonisentropic Navier-Stokes equations of general gas, where polytropic gas is included as a special case, with a free boundary. First we construct a viscous contact wave which approximates to the contact discontinuity, which is a basic wave pattern of compressible Euler equation, in finite time as the heat conductivity tends to zero. Then we prove the viscous contact wave is asymptotic stable if the initial perturbations and the strength of the contact wave are small. This generalizes our previous result [6] which is only for polytropic gas.     

ON THE EXISTENCE OF LOCAL CLASSICAL SOLUTION FOR A CLASS OF ONE-DIMENSIONAL COMPRESSIBLE NON-NEWTONIAN FLUIDS

In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.     

On global motion of a compressible viscous heat-conducting fluid bounded by a free surface

We consider the motion of a viscous compressible heat-conducting fluid in ³ bounded by a free surface which is under constant exterior pressure. We present the global existence theorems in two cases: when the free surface is under the surface tension and without it.     

Free boundary value problem for a viscous two-phase model with mass-dependent viscosity

In this paper, we study a free boundary value problem for two-phase liquid-gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum with a discontinuity. This is an extension of the paper [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV]. Just as in [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV], the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when β∈(0,1], which have improved the previous result of Evje and Karlsen, and get the asymptotic behavior result, also we obtain the regularity of the solutions by energy method.     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

A finite volume scheme for a model coupling free surface and pressurised flows in pipes

A model is derived for the coupling of transient free surface and pressurized flows. The resulting system of equations is written under a conservative form with discontinuous gradient of pressure. We treat the transition point between the two types of flows as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to such discontinuities in the flux. The validation is performed by comparison with experimental results.     

THE VORTEX BLOB METHOD AS A SECOND-GRADE NON-NEWTONIAN FLUID

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity and initial data in the space of Radon measures M (R ²). The solutions of this regularized PDE, also known as the isotropic Lagrangian averaged Euler or Euler-α equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in M (R ²) such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes as are common in practical vortex computations.     

On the irrotational approximation to steady supersonic flow

Under the hypothesis that the initial perturbation has small BV norm, we prove that in any bounded domain the L¹ norm of the difference between solutions to the isentropic Euler system of steady supersonic flow and the system of steady irrotational supersonic flow with the same initial data can be bounded by the cube of the total variation of the initial perturbation.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

Smooth zero-contact-angle solutions to a thin-film equation around the steady state

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the “complete wetting” regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide “minimal” conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.