In this paper, the one-dimensional quasilinear Verigin problem and the relative diffraction problem are studied. This free boundary problem describes a motion of the interface between two groups of animals. The existence and uniqueness of a classical solution locally in time are obtained. The continuous dependence of the solution on the internal boundary and the smoothness of the free boundary are also given.

A free boundary problem for a parabolic system describing an ecological model

Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy’s law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an $$L_p$$
 -setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh–Taylor instability.

The Rayleigh–Taylor instability for the Verigin problem with and without phase transition

Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria are identified and their thermodynamical stability is investigated by means of a variational approach. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. It is further shown that a non-degenerate equilibrium is dynamically stable in the natural state manifold if and only if it is thermodynamically stable. Finally, it is shown that a solution which does not develop singularities exists globally and converges to an equilibrium in the state manifold.

The Verigin problem with and without phase transition

We study the two-phase Muskat--Verigin free-boundary problem for elliptic equations with nonlinear sources. The existence of a smooth solution and a smooth free boundary is proved locally in time by applying the parabolic regularization of a condition on the free boundary.

The existence of a smooth interface in the evolutionary elliptic Muskat--Verigin problem with nonlinear source

In this paper a Verigin problem with kinetic condition is considered. The existence and uniqueness of a classical solution locally in time of this problem are obtained.

Xu Longfeng

Papers

A verigin problem with kinetic condition