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

Alessandro Violini

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arXiv · 2607.10676

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

Alessandro Violini

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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 $L^2(\mathbb{R}^2)$. We establish uniqueness of solutions at the natural energy level, thereby concluding the global well-posedness for $L^2$ 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 $L^\infty_t C_x^{1-\varepsilon}$. 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.

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Alessandro Violini. 2026-07-12. $L^2(\mathbb{R}^2)$ Well-Posedness and Logarithmic Lipschitz Regularity for the Density Patch Problem. https://arxiv.org/abs/2607.10676

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