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

Abstract: We study the density patch problem for the two-dimensional inhomogeneous incompressible Navier--Stokes system with vacuum, for initial data consisting of a Lipschitz density patch and a divergence-free velocity field in $L<sup>2(\mathbb{R}<sup>2)$. We establish uniqueness of solutions at the natural energy level, thereby concluding the global well-posedness for $L<sup>2$ data. Furthermore, we prove a log-Lipschitz estimate for the velocity field, extending the classical result of Chemin-Lerner to the inhomogeneous setting. As a consequence, the associated flow belongs to $L^\infty_t C_x<sup>{1-\varepsilon}$. for any $\varepsilon \in (0,1)$, ensuring that the patch boundary remains a continuous curve of Hausdorff dimension $1$, thus preserving its initial dimension for all time.

• The paper establishes uniqueness of Leray–Hopf solutions at the L2 energy level for density patch initial data.
• It introduces a localized energy method and parabolic displacement bounds to derive log-Lipschitz regularity even in the presence of vacuum.
• The work ensures geometric stability by preserving the Hausdorff dimension of patch interfaces, informing numerical schemes for rough initial data.

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

Introduction and Context

This paper addresses the two-dimensional inhomogeneous incompressible Navier–Stokes equations with vacuum. The focus is the density patch problem first raised by Lions, considering initial data where the density is an indicator function of a Lipschitz domain (a "patch") and the velocity field belongs to $L^2(\mathbb{R}^2)$, i.e., the natural energy space.

Classically, solutions in this setting are known to exist, but uniqueness and regularity—especially in the presence of vacuum—are highly nontrivial due to the loss of ellipticity and degeneracy on the vacuum set. The essential questions considered are whether uniqueness can be established at the energy level and how the geometric regularity (such as the boundary Hausdorff dimension) of the evolving patch is maintained by the system’s dynamics.

Main Results

Uniqueness and Well-Posedness in the Energy Space

A central result is the proof of uniqueness of Leray–Hopf solutions—termed “immediately strong solutions”—in the natural energy space $L^2(\mathbb{R}^2)$ for initial data given by a Lipschitz domain density patch and a divergence-free $L^2$ velocity field. Accordingly, the paper resolves global well-posedness for this critical configuration.

The uniqueness proof relies on an adaptation of the relative energy method and parabolic displacement bounds for the associated flows. Critical stability estimates are established for the time evolution of differences between solutions, overcoming the lack of direct $L^2$–$L^2$ controls outside the patch. A localized Ladyzhenskaya-type inequality is derived along the evolving patch, replacing the conventional global estimates unavailable in the vacuum setting.

Log-Lipschitz Regularity and Patch Geometry

The author extends the fundamental Chemin–Lerner log-Lipschitz regularity result—previously established for homogeneous Navier–Stokes—to the inhomogeneous setting with vacuum. Specifically, it is shown that

$$|u(t,x) - u(t,y)| \leq \gamma(t)\,|x-y|\,(-\log|x-y|)^{1-\eta}$$

for all $0<|x-y|<1$, any $\eta \in (0,1/2)$, and almost every $t>0$, with $L^2(\mathbb{R}^2)$0.

Consequently, the associated flow map $L^2(\mathbb{R}^2)$1 belongs to $L^2(\mathbb{R}^2)$2; this regularity is optimal at energy level and guarantees preservation of the Hausdorff dimension of the patch boundary for all $L^2(\mathbb{R}^2)$3. However, it falls short of showing preservation of Lipschitz continuity or perimeter regularity (see implications below).

Techniques and Proof Strategies

Energy Methods and Patch-Adapted Inequalities

The analysis begins with a detailed relative energy inequality for pairs of solutions. Because the vacuum prohibits classical Ladyzhenskaya controls, the approach centers on evolving domain-subsets, leveraging measure-preserving flow maps and parabolic displacement bounds to localize functional inequalities. For the "difference" system, spatial localization is essential: the difference is controlled only on the region where density is nonzero.

Key is the establishment of a localized $L^2(\mathbb{R}^2)$4 estimate for differences on the supports of the evolving patches and transported subdomains—controlled by parabolic displacement rates—to close Grönwall-type estimates and conclude uniqueness.

Atomic Decomposition and Dynamic Interpolation

Log-Lipschitz regularity is achieved using an atomic decomposition of initial data in the real interpolation space between $L^2(\mathbb{R}^2)$5 and $L^2(\mathbb{R}^2)$6, for $L^2(\mathbb{R}^2)$7. Each atom is evolved under the linearized system with the solution's flow. Parabolic decay rates for these atoms, together with scale-invariant bounds for the Lipschitz norm, enable summation across scales to produce the logarithmic modulus of continuity for the full velocity field.

Compared to frequency-based approaches exploiting the Fourier transform, this argument is robust to the inhomogeneous system (and vacuum) and does not require global smoothness or translation-invariance.

Numerical and Analytical Claims

• Strong uniqueness is shown for immediately strong solutions in the energy space. This extends previous results requiring higher regularity ($L^2(\mathbb{R}^2)$8 or Besov) and advances uniqueness to the critical energy level for the density patch, including in the presence of vacuum.
• Dimension preservation: The Hausdorff dimension of the boundary remains 1 for all times under the dynamics, despite the loss of classical Lipschitz or $L^2(\mathbb{R}^2)$9 regularity at energy level. This is a sharp result for weak (e.g., fractal-continuous) propagation.
• Sharp modulus of continuity: The flow is shown to have a $L^2(\mathbb{R}^2)$0 modulus, confirming propagation of rough but controlled interface continuity.

Implications and Future Directions

Theoretical Implications

This work closes the uniqueness question for 2D inhomogeneous incompressible Navier–Stokes with density patches at the $L^2(\mathbb{R}^2)$1 energy level, attributing precise regularity information to the density patch interfaces and velocity field. The techniques developed provide a blueprint for future analysis of degenerate transport–diffusion PDEs with non-smooth data, notably in the presence of vacuum or free boundaries.

The results sharpen the understanding of geometric stability for Eulerian interfaces in viscous flows, quantifying the sharp threshold below which Lipschitz (or higher) regularity is lost, but continuity (and dimensionality) is retained.

Practical and Computational Impact

While the main contributions are theoretical, they directly inform computational schemes by identifying precisely which regularity properties are to be expected and which might be broken in the numerical approximation of such problems. In particular, simulation of inhomogeneous flows with rough initial data must anticipate loss of Lipschitz interface regularity but can rely on continuous moduli and dimensional regularity for the patch.

Open Problems and Future Work

Several avenues remain:

• Interface curvature and perimeter control: Whether the perimeter (or BV-norm) of the interface remains finite globally in time is open at the $L^2(\mathbb{R}^2)$2 level and is directly tied to whether cusp or filamentation phenomena (spontaneous generation of fractal-like singularities) can arise dynamically.
• Extension to 3D: While analogous questions in 3D are known to be harder and less is known about the propagation of regularity or even existence, these methods and results may inform analogous structure for degenerate inhomogeneous models.
• Optimality of regularity: Verification that the log-Lipschitz modulus is indeed optimal for $L^2(\mathbb{R}^2)$3 (and possibly improved for higher regularity) would further clarify the geometric evolution in these systems.

Conclusion

This paper rigorously establishes global well-posedness and uniqueness of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes equations with density patch initial data and $L^2(\mathbb{R}^2)$4 velocity, including configurations with vacuum. The analysis provides a sharp modulus of continuity for the velocity and flow, guaranteeing topological stability (in the sense of Hausdorff dimension) of the evolving density patch interface. The strategies introduced offer new tools for studying degenerate PDEs with transport and diffusion mechanisms and lay the groundwork for further advances in both mathematical fluid dynamics and the analysis of non-smooth, nonlocal, or free boundary PDEs.

For the full technical development and proofs, see "$L^2(\mathbb{R}^2)$5 Well-Posedness and Logarithmic Lipschitz Regularity for the Density Patch Problem" (2607.10676).