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Mathematics > Analysis of PDEs

arXiv:2211.06286 (math)
[Submitted on 11 Nov 2022 (v1), last revised 1 Jun 2023 (this version, v2)]

Title:Traveling wave solutions to the one-phase Muskat problem: existence and stability

Authors:Huy Q. Nguyen, Ian Tice
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Abstract:We study the Muskat problem for one fluid in 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.
Comments: Funding source for H. Q. N. corrected
Subjects: Analysis of PDEs (math.AP); Fluid Dynamics (physics.flu-dyn)
MSC classes: 76S05, 35R35, 35C07, 35S16, 35B40, 76D03
Cite as: arXiv:2211.06286 [math.AP]
  (or arXiv:2211.06286v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.2211.06286

Submission history

From: Huy Nguyen Q [view email]
[v1] Fri, 11 Nov 2022 15:42:48 UTC (45 KB)
[v2] Thu, 1 Jun 2023 15:44:01 UTC (45 KB)
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