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Unbounded solutions for the Muskat problem

Omar Sánchez

Abstract

We prove the local existence of solutions of the form $x^2+ct+g,$ with $g\in H^s(\mathbb R)$ and $s\geq 3,$ for the Muskat problem in the stable regime. We use energy methods to obtain a bound of $g$ in Sobolev spaces. In the proof we deal with the loss of the Rayleigh-Taylor condition at infinity and a new structure of the kernels in the equation. Remarkably, these solutions grow quadratically at infinity.

Unbounded solutions for the Muskat problem

Abstract

We prove the local existence of solutions of the form with and for the Muskat problem in the stable regime. We use energy methods to obtain a bound of in Sobolev spaces. In the proof we deal with the loss of the Rayleigh-Taylor condition at infinity and a new structure of the kernels in the equation. Remarkably, these solutions grow quadratically at infinity.
Paper Structure (6 sections, 21 theorems, 387 equations, 1 figure)

This paper contains 6 sections, 21 theorems, 387 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous results
  3. Notation and preliminaries
  4. Energy estimates
  5. Bounds on the Kernels
  6. Regularization

Key Result

Theorem 1

Let $s\geq 3$ and $g_0\in H^s(\mathbb R).$ Then there exists a time $T_0=T(\lVert g_0\rVert_{H^s})>0$ and a function $g\in L^\infty([0,T_0]:H^s(\mathbb R))\cap W^{1,\infty}([0,T_0]: H^{s-1}(\mathbb R))$ such that the function solves (MEC) with $h(x,0)=x^2+g_0(x).$

Figures (1)

  • Figure 1: Interface $h(x,t).$

Theorems & Definitions (42)

  • Theorem 1
  • Lemma 1.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 32 more