Local free boundary problem for viscous compressible magnetohydrodynamics

We consider the motion of viscous compressible magnetohydrodynamics fluid in a domain bounded by a free surface. In the external domain, there is electromagnetic field generated by some currents that keeps the magnetohydrodynamics flow in the bounded domain. Then on the free surface, transmission conditions for electromagnetic fields are imposed. In this paper, we prove the existence of local regular solutions by the method of successive approximations. The L₂ approach is used. This helps us to treat the transmission conditions.     

On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω₀×(0,L)∈R³. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L_∞(0,L;L₂(Ω₀)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem.     

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     

Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.      

On a problem of viscous strip deformation with a free boundary

We consider a class of solutions to the 2D Navier-Stokes equations in a strip such that the longitudinal component of velocity is a linear function of the longitudinal coordinate while the transversal component and the pressure do not depend on this coordinate. One of the strip boundaries is free and the other boundary can be a solid wall or free too. We formulate conditions for a global solvability in time of corresponding initial boundary value problems, describe their asymptotic properties, give examples of exact solutions, study blowing up solutions in the case when the both strip boundaries are free.     

On a decay rate for 1D-viscous compressible barotropic fluid equations

The Navier-Stokes equations of a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H ¹ as well as the mass force such that the stationary density is positive. The uniform lower bound for the density is proved. By constructing suitable Lyapunov functionals, decay rate estimates in L ²-norm and H ¹-norm are given. The decay rate is exponential if so the decay rate of the nonstationary part of the mass force is. The results are proved in the Eulerian coordinates for a wide class of increasing state functions including with any γ > 0 as well as functions of arbitrarily fast growth. We also extend the results for equations of a multicomponent compressible barotropic mixture (in the absence of chemical reactions). Received December 20, 2000; accepted February 27, 2001.     

Existence of viscous compressible barotropic flow in a moving domain with free upper surface,via Galerkin method

The previous results on a global in time existence or stability have based on the local time existence in anisotropic Sobolev-Slobodetski spacesW ₂ ^{2+r, 1+r/2}, which are obtained by energy method for weak norm estimates, and by linear theory for higher norm estimates. On the other hand, in the paper of B.J. Jin and M. Padula [2], the global in time existence and stability have been obtained by purely Energy method, where the regularity class is different from anisotropic Sobolev-Slobodetski spacesW ₂ ^{2+r, 1+r/2}. We construct solution local in time of viscous compressible Navier-Stokes equations in a moving domain with free surface, via Galerkin method for the solution of linearized problem and, via iterative procedure for the solution of the nonlinear problem. With this method we obtain local in time solution whose regularity class is the same as the one in [2].

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Sunto I risultati noti sull'esistenza globale o la stabilità sono basati su esistenza locale nel tempo in spazi anisotropici di Sobolev-SlobodetskiW ₂ ^{2+r, 1+r/2}, essi sono ottenuti con il metodo dell'energia per il calcolo di stime di norme deboli, e con la teoria lineare per stime in norme più regolari. D'altra parte, nel lavoro di B.J. Jin e M. Padula [2], esistenza globale int e stabilità sono state ottenute unicamente col metodo dell'energia, in classi di regolarità diverse da quelle degli spazi di Sobolev-Slobodevski. In questa nota costruiamo soluzioni locali int per le equazioni di fluidi di Navier-Stokes viscosi comprimibili in domini con frontiera libera, con il metodo di Galerkin per le soluzioni del problema linearizzato e con una procedura iterativa per le soluzioni del problema nonlineare. Con questo metodo otteniamo soluzioni locali nel tempo avente la stessa classe di regolarità richiesta in [2].

     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain ${\Omega = \mathbb{R}^3_+}$ with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ${\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}$ .     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.     

Evolution free boundary problem for equations of viscous compressible heat‐conducting capillary fluids

In the paper the global motion of a viscous compressible heat conducting capillary fluid in a domain bounded by a free surface is considered. Assuming that the initial data are sufficiently close to a constant state and the external force vanishes we prove the existence of a global‐in‐time solution which is close to the constant state for any moment of time. The solution is obtained in such Sobolev–Slobodetskii spaces that the velocity, the temperature and the density of the fluid have $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end , $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end and $W_2^{1+\alpha,1/2+\alpha/2}$\nopagenumbers\end —regularity with α∈(¾, 1), respectively. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

A free boundary problem for a fluid flow in a heterogeneous porous medium

In this paper we study the problem of seepage of a fluid through a porous medium, assuming the flow governed by a nonlinear Darcy law and nonlinear leaky boundary conditions. We prove the continuity of the free boundary and the existence and uniqueness of minimal and maximal solutions. We also prove the uniqueness of theS ₃-connected solution in various situations.     

On the plane couette flow of a viscous compressible fluid with transpiration cooling

The exact solution for the plane couette flow of a viscous compressible, heat conducting, perfect gas with the same gas injection at the stationary plate and its corresponding removal at the moving plate has been studied. It is found that the gas injection is very helpful in reducing the temperature recovery factor. Effects of injection on the shearing stress at the lower plate, longitudinal velocity profiles and the enthalpy are shown graphically.     

On a free piston problem for potential ideal fluid flow

Our goal was to model and analyze a stationary and evolutionary potential ideal fluid flow through the junction of two pipes in the gravity field. Inside the ‘vertical’ pipe, there is a heavy piston that can freely move along the pipe. In the stationary case, we are interested in the equilibrium position of the piston in dependence on the geometry of junction, and in the evolutionary case, we study motion of the piston also in dependence on geometry. We formulate corresponding initial and boundary value problems and prove the existence results. The problem is nonlinear because the domain is unknown. Furthermore, we study some qualitative properties of the solutions and compare them with the qualitative properties of a free piston problem for Newtonian fluid flow. All theoretical results are illustrated with numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Solvability of a stationary boundary value problem for a model system of the equations of barotropic motion of a mixture of compressible viscous fluids

Under consideration is some boundary value problem for a model system of equations that describes the steady barotropic motion of a homogeneous mixture of compressible viscous fluids in a bounded three-dimensional domain. We prove the existence theorem for weak solutions of the problem, imposing no restrictions on the structure of total viscosity matrix except the standard requirements of positive definiteness.     

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.