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Global Regularity for Gravity Unstable Muskat Bubbles
About this Title
Francisco Gancedo, Eduardo García-Juárez, Neel Patel and Robert M. Strain
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 292, Number 1455
ISBNs: 978-1-4704-6764-7 (print); 978-1-4704-7700-4 (online)
DOI: https://doi.org/10.1090/memo/1455
Published electronically: November 21, 2023
Keywords: Fluid interface,
Muskat problem,
Global regularity,
Bubble,
Equilibria,
Unstable,
Viscosity jump,
Surface tension
Table of Contents
Chapters
- 1. Introduction
- 2. Contour dynamics formulation
- 3. Notation and main results
- 4. Implicit function theorem
- 5. Fourier multiplier estimates
- 6. A priori estimates on the vorticity strength
- 7. Global existence and instant analyticity
- 8. Uniqueness
Abstract
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.- Thomas Alazard and Omar Lazar, Paralinearization of the Muskat equation and application to the Cauchy problem, Arch. Ration. Mech. Anal. 237 (2020), no. 2, 545–583. MR 4097324, DOI 10.1007/s00205-020-01514-6
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