$L^2(\mathbb{R}^2)$ Well-Posedness and Logarithmic Lipschitz Regularity for the Density Patch Problem

Technical Summary: $L^2(\mathbb{R}^2)$ Well-Posedness and Logarithmic Lipschitz Regularity for the Density Patch Problem

Problem Statement
This work addresses the density patch problem for the two-dimensional inhomogeneous incompressible Navier–Stokes system with vacuum. The initial configuration consists of a density $\rho_0 = \mathbf{1}_D$, the indicator function of a bounded Lipschitz domain $D \subset \mathbb{R}^2$, representing a patch of homogeneous fluid surrounded by vacuum, and a divergence-free velocity field $u_0 \in L^2_\sigma(\mathbb{R}^2)$.

The central questions are:

1. Uniqueness: Does a unique solution exist at the natural energy level ($u_0 \in L^2$) for this degenerate system where the momentum equation loses ellipticity in vacuum regions?
2. Geometric Regularity: Does the geometric regularity of the initial patch boundary $\partial D$ persist over time? Specifically, does the patch boundary remain a curve of Hausdorff dimension 1, or does it develop singularities (e.g., cusps) that increase its dimension?

Methodology

1. Uniqueness via Relative Energy and Flow Distortion Control
To prove uniqueness (Theorem A), the author employs a relative energy argument adapted for low-regularity data. The core difficulty lies in the vacuum setting, where standard Ladyzhenskaya-type inequalities fail because the weighted norm $\|\sqrt{\rho} \delta u\|_{L^2}$ only controls the velocity difference on the support of the density.

• Localized Estimates: Instead of global estimates, the proof utilizes a localized $L^4$ estimate on the evolving patch $X(t, D)$. This relies on the "parabolic displacement" property $|X(t, x) - x| \lesssim \sqrt{t}$, which holds for immediately strong solutions.
• Handling Vacuum Mismatch: A critical term in the energy difference involves the mismatch between two transported patches. The author introduces a composed flow map $Y_\tau(s, x) = X_2(s; \tau, X_1(\tau, x))$ to track the interaction between two potential solutions.
• Flow Distortion: Since the velocity field is not Lipschitz (only $L^2$), the flow is not uniformly Lipschitz. The author proves that the distortion of the flow satisfies a specific bound involving a weight function $w_M(t, s) = \exp(M\sqrt{\log(t/s)})$. This allows for the control of the transport of the density mismatch without requiring $\nabla u \in L^1_t L^\infty_x$.
• Gronwall Argument: By combining these localized estimates with time-weighted energy functionals and a duality formulation for the transport equation, the author closes a Gronwall argument to establish uniqueness.

2. Log-Lipschitz Regularity via Atomic Decomposition
To establish the regularity of the velocity field (Theorem B), the paper extends the classical Chemin–Lerner result to the inhomogeneous setting with vacuum.

• Atomic Decomposition: The initial data $u_0 \in L^2$ is decomposed into a series of divergence-free atoms $\{u_{j,0}\}$ using real interpolation between homogeneous Sobolev spaces $\dot{H}^{-s}$ and $\dot{H}^s$.
• Linearized Evolution: Each atom evolves according to a linearized system driven by the full velocity field. The author utilizes known decay estimates for these linearized solutions (derived in prior works [9, 23]) which show parabolic decay in time.
• Summation Strategy: The velocity difference $|u(t, x) - u(t, y)|$ is estimated by splitting the sum of atoms into low and high frequencies. By choosing a cutoff frequency $N$ dependent on $|x-y|$, the author derives a log-Lipschitz estimate of the form:
  $$|u(t, x) - u(t, y)| \leq \gamma(t) |x-y| (-\log |x-y|)^{1-\eta}$$
  for $\eta \in (0, 1/2)$. This proof avoids frequency localization techniques (Fourier analysis) in favor of an atomic decomposition, making the argument robust for settings where Fourier approaches are unavailable.

Key Contributions and Results

• Theorem A (Uniqueness): The paper establishes the uniqueness of "immediately strong solutions" for initial data $(\rho_0, u_0)$ where $\rho_0 = \mathbf{1}_D$ and $u_0 \in L^2_\sigma(\mathbb{R}^2)$. Combined with the existence result from [23], this concludes the global well-posedness of the density patch problem in the natural energy space $L^2$.
• Theorem B (Log-Lipschitz Regularity): For both strictly positive densities and the density patch with vacuum, the velocity field satisfies a log-Lipschitz estimate with a time-dependent coefficient $\gamma(t) \in L^1_{loc}$. This extends the classical result of Chemin and Lerner [5] to the inhomogeneous case with vacuum.
• Corollary 1.3 (Dimension Preservation): As a consequence of the log-Lipschitz regularity, the associated flow map $X(t, \cdot)$ belongs to $L^\infty_{loc}([0, \infty); C^{1-\varepsilon}(\mathbb{R}^2))$ for any $\varepsilon \in (0, 1)$. Consequently, the Hausdorff dimension of the patch boundary is preserved for all time:
  $$\dim_H(\partial D(t)) = 1 \quad \forall t \geq 0.$$

Significance and Claims

The paper claims to resolve the uniqueness problem for the density patch problem at the critical energy level $L^2$, a regime previously open due to the degeneracy of the momentum equation in vacuum. The author notes that while previous results required higher regularity (e.g., $u_0 \in H^1$ or Besov spaces $\dot{B}^0_{2,1}$), this work operates at the natural energy level.

Regarding geometric regularity, the paper demonstrates that while the boundary remains a 1-dimensional object (Hausdorff dimension 1), the lack of Lipschitz regularity in the velocity field (inherent to $L^2$ data) prevents the preservation of the interface as a Lipschitz curve. The author explicitly states that the possibility of cusp formation remains open, and it is not yet known whether the perimeter of the patch becomes infinite over time. The significance of the $\eta > 0$ condition in the log-Lipschitz estimate is highlighted as the precise threshold required to prevent the dimensional growth of the boundary, distinguishing this result from the classical $\eta=0$ case where the dimension could theoretically grow.

The methodology is presented as a robust alternative to Fourier-based frequency localization, utilizing atomic decomposition and flow distortion estimates that may be applicable to other settings where Fourier methods are not available.