Passage to the limit in a chain of the muskat problems (I)

Abstract

We justify a weak passage to the limit in a chain of the Muskat problems. We consider some smooth approximations [4] of the classical Muskat free boundary problem as a system of nonlinear parabolic equations with a small parameter ∈ which corresponds to the width of transition layers. In spherically symmetric case we prove that for some class of initial data, generated by a chain of the Muskat problems, the solutions of smooth approximate problems converge to a weak solution of the Muskat problem. This weak solution can be regarded as interpretation of so-called finger phenomenon.

Keywords:

• Muskat problem
• finger phenomenon
• free boundary problem

^*Osaka University, Osaka, Japan

^†MOSCOW State University, Moscow, Russia

^*Osaka University, Osaka, Japan

^†MOSCOW State University, Moscow, Russia

Notes

^*Osaka University, Osaka, Japan

^†MOSCOW State University, Moscow, Russia