第144回

日時 Time

9月11日（金）16:30–18:00

場所 Place

金沢大学自然科学5号館数学棟4階コロキウム＋Zoom（ハイブリッド）
　　　Zoom 登録リンク：https://kanazawa-university.zoom.us/meeting/register/4uVM-yNuTh2tAB_GuiWgrQ

講演者 Speaker

Georg Prokert (Eindhoven University of Technology)

タイトル Title

The Multidimensional Muskat problem driven by gravity

概要 Abstract

The classical Muskat problem describes the motion of two immiscible, incompressible liquids in a porous medium, separated by a sharp moving interface. We discuss this problem in an unbounded geometry in arbitrary space dimensions. The problem is recast as a nonlinear, nonlocal evolution equation for the function describing the interface. This evolution equation involves singular integral operators arising from the classical double layer potential for the Laplacian. As we consider gravity as the only driving mechanism, well-posedness of the problem is equivalent to a nontrivial Rayleigh-Taylor condition. We show that the evolution given by the Muskat problem is a smooth flow on the set of interfaces satisfying this condition, in all subcritical L_2-based Sobolev spaces. For this flow we additionally obtain parabolic smoothing. The main technical tools are a class of nonlinear generalizations of Riesz transforms arising in the evolution equation. Their properties concerning smoothness, commutators, and localizations enable us to apply abstract results on fully nonlinear parabolic evolution equations to our evolution equation. This class of operators seems to be of broader interest as it appears in other problems as well. (joint work with B. Matioc, Regensburg University)