密度斑块问题在\(L^2(\mathbb{R}^2)\)中的适定性和对数Lipschitz正则性
$L^2(\mathbb{R}^2)$ Well-Posedness and Logarithmic Lipschitz Regularity for the Density Patch Problem
AI总结：
研究二维非齐次不可压缩带真空的Navier-Stokes系统的密度斑块问题，通过建立解的唯一性得出\(L^2\)数据全局适定性，证明速度场的对数Lipschitz估计，使相关流满足特定条件，保持斑块边界初始维数。
AI中文摘要：
我们研究二维非齐次不可压缩带真空的Navier-Stokes系统的密度斑块问题，其初始数据由Lipschitz密度斑块和\(L^2(\mathbb{R}^2)\)中的无散速度场组成。我们在自然能量水平上建立了解的唯一性，从而得出\(L^2\)数据的全局适定性。此外，我们证明了速度场的对数Lipschitz估计，将Chemin-Lerner的经典结果扩展到非齐次情形。结果表明，相关流属于\(L^\infty_t C_x^{1-\varepsilon}\)，保证斑块边界始终是Hausdorff维数为1的连续曲线，保持其初始维数。
英文摘要：
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.
