On the well-posedness of two-dimensional Muskat problem with an elastic interface

TL;DR

This work establishes the well-posedness of the two-dimensional Muskat problem with an elastic interface in both one- and two-phase settings. The authors reformulate the problem using Dirichlet–Neumann operators and perform a paralinearization of the elastic term to reveal a $5^{\text{th}}$-order quasilinear parabolic structure, enabling robust a priori energy estimates and contraction arguments. They prove local well-posedness for arbitrary data in $H^s$, $s\ge 2$, for both the one- and two-phase cases, and derive global well-posedness for small initial data in subcritical spaces with $s>\\tfrac{3}{2}$ in the stable regime $\\rho^+ \\leq \\rho^-$. The global results exploit a fixed-point framework in integral form using fractional heat kernels, together with parabolic spacetime estimates to control nonlinearities. Overall, the paper extends the Muskat theory to elastic interfaces, providing a rigorous foundation with potential implications for poroelasticity, hydroelastic waves, and related free-surface problems.

Abstract

We investigate the two-dimensional Muskat problem with a nonlinear elastic interface, for both one-phase and two-phase scenarios. Following the framework developed by Nguyen [35,36], we demonstrate that the problem is locally well-posed in $H^s$ for $s\geq 2$ for arbitrary initial data. Furthermore, for the one-phase case and the stable two-phase case $(ρ^+ \leq ρ^-)$, we establish global well-posedness for small initial data in $H^s$ when $s> \frac{3}{2}$.

Figures (1)

• Figure 1: Two-phase Muskat problem with an elastic interface

Theorems & Definitions (46)

• Proposition 1.1: MR4131404
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof