Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Well-posedness for viscosity solutions of the one-phase Muskat problem in all dimensions

Russell Schwab, Son Tu, Olga Turanova

TL;DR

This work proves global well-posedness for viscosity solutions of the one-phase Muskat problem in all dimensions by recasting the graph-based evolution as a nonlocal Hamilton-Jacobi-Bellman equation driven by the Hele-Shaw operator $H$. Central to the approach is establishing a robust viscosity-solution framework for the equation $\partial_t f = H(f)$, including a precise global comparison property, pointwise evaluation at regular points, and a Perron-based existence theory. The key technical contribution is the detailed analysis of $H$—its Lipschitz continuity, splitting property, and monotone/translation-invariant structure—so that the Muskat dynamics can be controlled from initial data in $BUC(\mathbb{R}^d)$. By relating $M(f) = H(f) - 1$, the authors transfer the Hele-Shaw well-posedness to the Muskat problem, yielding a unique viscosity solution that preserves modulus of continuity in time. The results unify and extend prior dimension-specific and regularity-restricted outcomes, providing a dimension-agnostic, robust framework for the one-phase Muskat evolution via nonlocal integro-differential operators.

Abstract

In this article, we apply the viscosity solutions theory for integro-differential equations to the \emph{one-phase} Muskat equation (also known as the Hele-Shaw problem with gravity). We prove global well-posedness for the corresponding Hamilton-Jacobi-Bellmann equation with bounded, uniformly continuous initial data, in all dimensions.

Well-posedness for viscosity solutions of the one-phase Muskat problem in all dimensions

TL;DR

This work proves global well-posedness for viscosity solutions of the one-phase Muskat problem in all dimensions by recasting the graph-based evolution as a nonlocal Hamilton-Jacobi-Bellman equation driven by the Hele-Shaw operator . Central to the approach is establishing a robust viscosity-solution framework for the equation , including a precise global comparison property, pointwise evaluation at regular points, and a Perron-based existence theory. The key technical contribution is the detailed analysis of —its Lipschitz continuity, splitting property, and monotone/translation-invariant structure—so that the Muskat dynamics can be controlled from initial data in . By relating , the authors transfer the Hele-Shaw well-posedness to the Muskat problem, yielding a unique viscosity solution that preserves modulus of continuity in time. The results unify and extend prior dimension-specific and regularity-restricted outcomes, providing a dimension-agnostic, robust framework for the one-phase Muskat evolution via nonlocal integro-differential operators.

Abstract

In this article, we apply the viscosity solutions theory for integro-differential equations to the \emph{one-phase} Muskat equation (also known as the Hele-Shaw problem with gravity). We prove global well-posedness for the corresponding Hamilton-Jacobi-Bellmann equation with bounded, uniformly continuous initial data, in all dimensions.
Paper Structure (36 sections, 40 theorems, 288 equations)

This paper contains 36 sections, 40 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Method of proof
  3. Outline of the paper
  4. Literature and earlier results
  5. Viscosity solutions and integro-differential equations
  6. The Hele-Shaw problem without gravity
  7. The one-phase Muskat problem
  8. The two-phase Muskat problem
  9. Notation
  10. The Hele-Shaw operator, $H$, and its properties
  11. Function spaces
  12. The basic properties
  13. Pointwise evaluation of $H$
  14. The Lipschitz property
  15. Step 0.
  16. ...and 21 more sections

Key Result

theorem 1

Given any $f_0\in BUC(\mathbb R^d)$, there exists a unique viscosity solution, $f\in C^0(\mathbb R^d\times [0,T))$, to (eqIntro:MuskatHJB). Furthermore, if $f_0$ has a modulus of continuity, $\omega$, then for each $t\in[0,T)$, $f(\cdot, t)$ has the same modulus of continuity.

Theorems & Definitions (81)

  • theorem 1
  • definition 3
  • definition 4
  • definition 6
  • proposition 7
  • definition 8: Global comparison property
  • proposition 9: Basic Properties
  • proof : Proof of Proposition \ref{['propAuxOp:HProperties']}
  • lemma 10
  • lemma 12
  • ...and 71 more