skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Global well‐posedness for the one‐phase Muskat problem
Abstract The free boundary problem for a two‐dimensional fluid permeating a porous medium 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 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.  more » « less
Award ID(s):
2205734 2055244
PAR ID:
10491949
Author(s) / Creator(s):
; ;
Publisher / Repository:
Wiley
Date Published:
Journal Name:
Communications on Pure and Applied Mathematics
Volume:
76
Issue:
12
ISSN:
0010-3640
Page Range / eLocation ID:
3912 to 3967
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Archive for Rational Mechanics and Analysis)
    We study the Muskat problem for one fluid in an arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that, for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem. 
    more » « less
  2. ; ; ; (, Memoirs of the American Mathematical Society)
    In this paper, we study the dynamics of fluids in porous media governed by Darcy’s law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle. 
    more » « less
  3. (, Communications on Pure and Applied Mathematics)
    Abstract We apply new results on free boundary regularity to obtain a quantitative convergence rate for the shape optimizers of the first Dirichlet eigenvalue in periodic homogenization. We obtain a linear (with logarithmic factors) convergence rate for the optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost minimizers is used to apply the optimal homogenization theory in Lipschitz domains of Kenig et al. A key idea, to deal with the hard constraint on the volume, is a combination of a large scale almost dilation invariance with a selection principle argument. 
    more » « less
  4. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein’s problem for minimal surfaces.As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity. 
    more » « less
  5. ; ; ; (, RIMS kokyuroku bessatsu)
    Takayoshi Ogawa; Keiichi Kato; Mishio Kawashita (Ed.)
    This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $$A(\Omega)$$, which can be expected to play a similar role to $$L^\infty$$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email