Free Boundary Problem For Equations Research Articles

Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition

AbstractWe study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well‐posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics. For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local‐in‐time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5, 12]. It contains, in particular, the reduction to a fixed domain, using the “good unknown” of Alinhac [1], and a suitable Nash‐Moser‐type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity, and we briefly discuss its extension to general relativity. © 2009 Wiley Periodicals, Inc.

On an inequality for a free boundary problem for equations of a viscous compressible heat-conducting capillary fluid

We derive an inequality for a local solution of a free boundary problem for a viscous compressible heat-conducting capillary fluid. This inequality is crucial to proving the global existence of solutions belonging to certain anisotropic Sobolev–Slob

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

AbstractIn 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.

Local Existence of Solutions of a Free Boundary Problem for Equations of Compressible Viscous Heat-Conducting Capillary Fluids

Local Existence of Solutions of a Free Boundary Problem for Equations of Compressible Viscous Heat-Conducting Capillary Fluids

Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting fluids

The local existence and the uniqueness of solutions for equations describing the motion of viscous compressible heat-conducting fluids in a domain bounded by a free surface is proved. First, we prove the existence of solutions of some auxiliary problems b

On the Global Existence Theorem for a Free Boundary Problem for Equations of a Viscous Compressible Heat Conducting Capillary Fluid

On the Global Existence Theorem for a Free Boundary Problem for Equations of a Viscous Compressible Heat Conducting Capillary Fluid

On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid

We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are