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Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime

Qasim Khan, Anthony Suen, Bao Quoc Tang

Abstract

We address a generalised three-dimensional $α$-Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when $α\in[0,1)$ and also prove global-in-time existence for strong solutions when $α\in[0,\frac{1}{2})$ with initial data controlled by explicit constants. We obtain maximum principles for the $L^{\infty}$-norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these $L^{\infty}$-norms. Finally, some convergence results for strong solutions as $α\to0^+$ are also proved.

Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime

Abstract

We address a generalised three-dimensional -Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when and also prove global-in-time existence for strong solutions when with initial data controlled by explicit constants. We obtain maximum principles for the -norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these -norms. Finally, some convergence results for strong solutions as are also proved.
Paper Structure (8 sections, 14 theorems, 198 equations)

This paper contains 8 sections, 14 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Local-in-time well-posedness
  3. Global-in-time existence with bounded data
  4. Maximum principle and asymptotic behaviour of solutions
  5. Decay of $L^\infty$-norm of $f$
  6. Decay of $L^\infty$-norm of $\nabla f$
  7. Convergence of solutions as $\alpha\to0^{+}$
  8. Existence of $k_0(\alpha)$

Key Result

Theorem 2.1

Let $f_0\in H^k(\mathbb R^2)$ for $k\ge4$. Then for any $\alpha\in[0,1)$, there exists a time $T>0$ such that the contour equation contour eqn possess a unique solution in $C^1([0,T];H^k(\mathbb R^2))$ with $f(x,0)=f_0(x)$.

Theorems & Definitions (35)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 25 more