Global well-posedness for the one-phase Muskat problem

Abstract

The free boundary problem for a two-dimensional fluid filtered in porous media is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L_t^{\infty }{L}_x^2$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.

arXiv comment40 pages

References in corpus (8)

• Regularity theory for fully nonlinear integro-differential equations
• On the global existence for the Muskat problem
• Breakdown of smoothness for the Muskat problem
• Absence of splash singularities for SQG sharp fronts and the Muskat problem
• On The Muskat Problem With Viscosity Jump: Global In Time Results
• Absence of squirt singularities for the multi-phase Muskat problem
• Global wellposedness for the 3D Muskat problem with medium size slope
• Global well-posedness for the three dimensional Muskat problem in the critical Sobolev space

Cited by in corpus (1)

• Global solutions for the Muskat problem in the scaling invariant Besov space ${\dot{B}}_{\infty ,1}^1$