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

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 $L_t^{\mathrm{\infty }}{C}_x^{1-\epsilon }$. for any $\epsilon \in (0,1)$, ensuring that the patch boundary remains a continuous curve of Hausdorff dimension $1$, thus preserving its initial dimension for all time.