Well-posedness and stability results for a quasilinear periodic Muskat problem

Anca-Voichita Matioc; Bogdan-Vasile Matioc

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arXiv · 1706.09260

Well-posedness and stability results for a quasilinear periodic Muskat problem

Anca-Voichita Matioc·Bogdan-Vasile Matioc

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Abstract

We study the Muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two-dimensional geometry. We first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown. Secondly, we show that this evolution equation has a quasilinear structure, which is at a formal level not obvious, and we also disclose the parabolic character of the equation. Exploiting these aspects, we establish the local well-posedness of the problem for arbitrary initial data in $H^s(\mathbb{S})$, with $s\in(3/2,2)$, determine a new criterion for the global existence of solutions, and uncover a parabolic smoothing property. Besides, we prove that the zero steady-state solution is exponentially stable.

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Anca-Voichita Matioc, Bogdan-Vasile Matioc. 2017-06-28. Well-posedness and stability results for a quasilinear periodic Muskat problem. https://arxiv.org/abs/1706.09260

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