Self-similar solutions of the three-dimensional Muskat problem with surface tension

Technical Summary: Self-Similar Solutions of the Three-Dimensional Muskat Problem with Surface Tension

Problem Formulation
The paper addresses the three-dimensional one-phase Muskat problem for an incompressible fluid in a porous medium with a free upper boundary subject to surface tension. The interface $\Sigma_t$ is modeled as the graph of a function $\eta(t, x)$ over $\mathbb{R}^2$. The governing equations combine Darcy's law, incompressibility, and a dynamic boundary condition where the pressure jump is balanced by surface tension $\sigma \kappa(\eta)$. This system reduces to a third-order quasilinear parabolic equation for the interface height:
$$\partial_t \eta = -\frac{\sigma}{\mu} G(\eta)\kappa(\eta),$$
where $G(\eta)$ is the Dirichlet–Neumann operator and $\kappa(\eta)$ is the mean curvature.

The authors seek self-similar solutions emanating from unbounded, conical initial data $\eta(0, x) = \varepsilon|x|$. Such solutions take the form $\eta_\varepsilon(t, x) = t^{1/3}U_\varepsilon(t^{-1/3}x)$, where the profile $U_\varepsilon$ satisfies a stationary equation involving a non-translation-invariant dilation term $x \cdot \nabla$. The central challenge is constructing these solutions for small $\varepsilon$ when the initial data is unbounded and possesses a singularity at the origin, falling outside the scope of standard small-data Cauchy theories in bounded or critical Sobolev spaces.

Methodology and Strategy
The proof employs a perturbative approach, decomposing the profile $U_\varepsilon$ into a leading linear term and a correction:
$$U_\varepsilon = \varepsilon u_* + v_\varepsilon,$$
where $u_*$ is the "regularized linear cone" defined as $u_* = e^{-\frac{\sigma}{\mu}|D|^3}|x|$. The correction $v_\varepsilon$ is sought in a specific function space $X_r = \dot{H}^r(\mathbb{R}^2) \cap \dot{H}^{r+3}(\mathbb{R}^2)$ with $3/2 < r < 2$.

The methodology relies on three primary technical pillars:

1. Linear Analysis of the Similarity Operator:
   The linear operator $L = \frac{3\sigma}{\mu}|D|^3 - x \cdot \nabla + 1$ is analyzed via its Fourier transform along rays. The authors construct a raywise inverse $L^{-1}$ using a weighted Hardy estimate. This inverse gains three derivatives at high frequencies while preserving the conical low-frequency structure, allowing the inversion of the profile equation modulo constants.

2. Favorable Quadratic Cancellation:
   A critical difficulty in handling the nonlinearity $G(\eta)\kappa(\eta)$ with unbounded data is the behavior of the quadratic term. The authors identify a specific algebraic cancellation in the symbol of the bilinear form $B(f, g) = \frac{1}{2}(D_\eta G(0)[f]\Delta g + D_\eta G(0)[g]\Delta f)$. In the "high–high-to-low" frequency interaction region (where input frequencies are large but the output frequency is small), the symbol satisfies an estimate $|b(\alpha, \beta)| \lesssim |\alpha+\beta||\beta|^3$. This extra gain in output frequency ensures the convergence of the homogeneous low-frequency summation, which would otherwise diverge for conical data.

3. Finite-Order Cone-Adapted Dirichlet–Neumann Estimates:
   Standard Dirichlet–Neumann estimates fail for unbounded graphs because high Sobolev norms do not control low-frequency energy. The authors introduce mixed half-space norms ($X_r$ and $Y_r$) that impose high Sobolev weights only on positive horizontal frequencies. They establish tame estimates for the Dirichlet–Neumann operator and its shape derivatives on these spaces, proving that the operator is $C^3$ in a neighborhood of the conical profile. This allows for the control of the cubic remainder terms in the nonlinear expansion.

Key Results
The paper establishes the following main theorem (Theorem 1.3):

• Existence: For sufficiently small $\varepsilon$, there exists a self-similar solution $\eta_\varepsilon(t, x) = t^{1/3}U_\varepsilon(t^{-1/3}x)$ to the Muskat problem with surface tension.
• Profile Structure: The profile $U_\varepsilon$ is a perturbation of the regularized cone: $U_\varepsilon = \varepsilon u_* + v_\varepsilon$, where the correction $v_\varepsilon$ satisfies $\|v_\varepsilon\|_{X_r} \leq C \varepsilon^2$.
• Second-Order Expansion: The authors identify the leading quadratic correction $v^{(2)} = \frac{3\sigma}{\mu} L^{-1} B(u_*, u_*)$. The solution admits the expansion $U_\varepsilon = \varepsilon u_* + \varepsilon^2 v^{(2)} + O_{X_r}(|\varepsilon|^3)$ as $\varepsilon \to 0$.
• Initial Trace and Regularization: The solution attains the initial trace $\eta_\varepsilon(0, x) = \varepsilon|x|$ locally uniformly. Despite the conical singularity at $t=0$, the solution is instantaneously smoothed for any $t > 0$; the profile $U_\varepsilon$ is $C^1$, and the singularity is "rounded" on a spatial scale of $O(t^{1/3})$.

Significance and Claims
The paper claims to provide the first construction of self-similar solutions for the three-dimensional Muskat problem with surface tension emanating from exact conical initial data. The significance lies in overcoming the limitations of standard Cauchy theory, which typically requires bounded or critical Sobolev initial data. By separating the unbounded conical component and utilizing a specialized function space adapted to the geometry of the problem, the authors demonstrate that the singular cone is instantaneously regularized by surface tension.

The work extends previous results on gravity-driven Muskat problems (where the principal operator is first-order) to the surface tension case (third-order operator). The authors note that while the algebraic cancellation used is dimension-independent, the specific choice of Sobolev norms and the range of the parameter $r$ depend on the spatial dimension. The paper concludes that the conical singularity is not a barrier to well-posedness in the self-similar regime, provided the initial cone is sufficiently small.