Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data. (English) Zbl 1503.35147
The paper deals with the inhomogeneous Muskat problem, which is a free-boundary problem modeling the evolution of an internal wave at the interface between two incompressible homogeneous fluids filling an inhomogeneous porous medium where the permeability is a step function. The main goal is to study the long-time dynamics of the Lipschitz norm of smooth solutions (typically in \(H^3\)): the authors prove that, for some specific initial data, such solutions decay exponentially fast in \(W^{1, \infty}\) towards the equilibrium state. Moreover, the authors provide the global existence of solutions with Lipschitz regularity and detect the evolution in time of such Lipschitz solutions, which are showed to decay. Since some solutions to the Muskat problem experience turning singularities (finite time blow up of the derivative of the interface) and splash singularities (self intersection of the interface), the Lipschitz class is particularly relevant for such problem.
Reviewer: Roberta Bianchini (Roma)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35R35 | Free boundary problems for PDEs |
| 76B07 | Free-surface potential flows for incompressible inviscid fluids |
| 76B55 | Internal waves for incompressible inviscid fluids |
| 76S05 | Flows in porous media; filtration; seepage |
| 35B45 | A priori estimates in context of PDEs |
| 35B50 | Maximum principles in context of PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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