Global-in-time estimates for the 2D one-phase Muskat problem with contact points

1. Problem Statement

The paper investigates the dynamics of a two-dimensional, incompressible viscous fluid evolving in a porous medium (or Hele-Shaw cell) under the influence of gravity and surface tension. The specific physical scenario involves a fluid confined within a vessel with vertical walls and a curved bottom, where the fluid is in contact with a dry region.

Key Features of the Model:

• Governing Equations: The flow is modeled by Darcy's Law (more singular than Stokes or Navier-Stokes), coupled with the incompressibility condition.
• Free Boundary: The interface between the fluid and the dry region is a free boundary $\Gamma(t)$ described by a height function $h(t, x)$.
• Contact Points: A critical feature is the presence of contact points where the fluid, the solid vessel wall, and the dry region meet. Unlike previous studies that often assumed fixed contact points or small contact angles, this work allows for dynamic contact points with arbitrary angles determined by the stationary state.
• Boundary Conditions:
  • Dynamic Boundary ($\Gamma(t)$): Satisfies the Laplace-Young condition (pressure jump due to surface tension) and a kinematic condition.
  • Fixed Boundary ($\Gamma_w$): Satisfies a non-penetration condition (Neumann condition for the velocity potential).
  • Contact Point Evolution: The motion of the contact points is governed by a specific evolution law relating the normal velocity of the contact point to the deviation of the dynamic contact angle from the equilibrium angle (Young's law).

2. Methodology

The authors employ a rigorous analytical framework to derive global-in-time a priori estimates for solutions close to a stationary equilibrium.

A. Formulation and Linearization:

• Potential Formulation: The velocity field is expressed as the gradient of a potential $\phi$ (since $\nabla \cdot u = 0$ and Darcy's law implies $u = -\nabla \phi$). The problem is reformulated as a Dirichlet-Neumann elliptic problem for $\phi$.
• Fixed-Boundary Transformation: To handle the moving domain $\Omega(t)$, the authors introduce a diffeomorphism $\phi_\eta$ that maps the time-dependent domain to a fixed reference domain $\Omega_s$ (the stationary domain). The perturbation is defined as $\eta = h - h_s$, where $h_s$ is the stationary profile.
• Perturbation System: The equations are rewritten in terms of the perturbation $(\eta, \Phi)$, where $\Phi$ is the perturbation of the potential.

B. Energy-Dissipation Scheme:
The core of the proof relies on constructing a hierarchy of energy and dissipation functionals:

1. Basic Energy ($E_\parallel$) and Dissipation ($D_\parallel$): Derived from the fundamental physical energy (gravity + surface tension + contact point energy). These control low-order norms (e.g., $H^1$).
2. Improved Energy ($E$) and Dissipation ($D$): To close the estimates for higher regularity, the authors introduce "improved" functionals. This involves adding higher-order spatial norms (e.g., $H^{3/2+}$, $H^{5/2}$) to the energy and corresponding higher-order terms to the dissipation.
3. Bootstrap Argument: The strategy involves:
   • Controlling time derivatives using the basic energy-dissipation equality.
   • Using elliptic estimates to gain spatial regularity from the time derivatives.
   • Showing that the nonlinear source terms in the energy inequalities can be bounded by the product of the energy and dissipation ($|N(t)| \leq C E^\theta D$), allowing for a global-in-time bound via a continuity argument.

C. Key Technical Innovations:

• Neumann Problem Analysis: Unlike Stokes/Navier-Stokes models which often require Navier-slip conditions to allow contact point movement, the Darcy flow naturally satisfies a zero Neumann condition (non-penetration) on the fixed wall. This allows the contact points to move freely without additional boundary conditions.
• Non-Weighted Sobolev Spaces: Previous works on similar contact point problems (e.g., for Stokes flow) required weighted Sobolev spaces to handle singularities at the corners, imposing restrictions on the contact angle. This paper successfully performs the analysis in non-weighted $L^2$-based Sobolev spaces without restricting the contact angle $\omega \in (0, \pi)$.
• Trace Theorems: The authors utilize a specific trace theorem for homogeneous Sobolev spaces ($\dot{H}^1$) on bounded domains to control the $L^2$-norm of the potential (which lacks a zero Dirichlet condition) using the dissipation.
• Elliptic Regularity in Non-Smooth Domains: The stationary domain $\Omega_s$ has corners at the contact points. The authors leverage spectral properties of the pencil operator associated with the Laplacian in curvilinear polygons to prove $H^2$ regularity for the potential, provided the contact angle is in $(0, \pi)$.

3. Key Contributions

1. Global-in-Time A Priori Estimates: The paper proves that for initial data sufficiently close to the stationary state, the solution exists globally in time and satisfies uniform a priori bounds.
2. Arbitrary Contact Angles: The analysis holds for any contact angle $\omega \in (0, \pi)$ determined by the equilibrium state, removing the need for small-angle assumptions or weighted spaces used in previous literature.
3. Darcy Flow with Moving Contact Points: It extends the theory of moving contact points from viscous fluids (Stokes/Navier-Stokes) to the more singular Darcy flow, which is physically relevant for porous media and Hele-Shaw cells.
4. Exponential Decay: The solution is shown to converge exponentially to the stationary state as $t \to \infty$.

4. Main Results

Theorem 1.1 (Main Result):
Let $h(t, x) = h_s(x) + \eta(t, x)$ be a smooth solution to the system. If the initial energy $E(0)$ is sufficiently small, then for any $T > 0$:
$$\sup_{t \in [0, T]} E(t) + \int_0^T D(t) \, dt \leq C E(0)$$
where $E(t)$ includes norms of $\eta$ and its time derivatives in $H^{3/2+\delta}$ and $H^1$, and $D(t)$ includes higher-order dissipation terms.

Furthermore, there exists a decay rate $\lambda > 0$ such that:
$$\sup_{t \in [0, T]} \left( E_\parallel(t) + \int_{\Omega(t)} |u|^2 + (\partial_t \eta)^2|_{\text{contact}} \right) \leq C E_\parallel(0) e^{-\lambda T}$$

Implications:

• Stability: The stationary solution is non-linearly stable.
• Regularity: The solution maintains high spatial regularity ($H^{3/2+}$ and higher) globally in time.
• Well-posedness: While the paper focuses on a priori estimates, the authors note that these results, combined with local well-posedness theory (to be established in a future work), imply global well-posedness for small data.

5. Significance

This work represents a significant advancement in the mathematical theory of free boundary problems involving contact lines.

• Physical Relevance: It addresses a realistic scenario (fluid in a vessel with moving contact lines) using the correct physical law for porous media (Darcy's Law), which is more singular than the Navier-Stokes equations.
• Mathematical Breakthrough: By avoiding weighted spaces and restrictions on contact angles, the authors provide a more robust and general framework for analyzing contact point dynamics. This bridges a gap between the mathematical treatment of Stokes/Navier-Stokes flows and Darcy flows.
• Methodological Impact: The techniques developed, particularly the use of Neumann problems in non-smooth domains to gain regularity without weights, offer a powerful toolkit for future studies of free boundary problems with corners and moving contact lines.

In summary, the paper establishes the long-term stability and regularity of the Muskat problem with moving contact points, proving that small perturbations from equilibrium decay exponentially without developing singularities, regardless of the specific contact angle (as long as it is strictly between 0 and $\pi$).