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CuD: Current developments in the theory and numerics of hyperbolic balance laws and related PDEs Location: A701 Session Chair: Simon Markfelder | |
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The density patch problem for the inhomogeneous, incompressible $2D$ Navier-Stokes equations Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany We are interested in the density patch problem for the inhomogeneous, incompressible Navier-Stokes equations in $\mathbb{R}^2$ at a critical level of regularity. More precisely, we assume that the initial density of the fluid is the indicator function of a bounded Lipschitz domain and that the initial velocity $u_0$ lies in the Besov space $\dot{B}^0_{2,1}(\mathbb{R}^2)$. We first introduce a suitable class of solutions and give an overview of the most important a priori estimates. This allows us to prove the global existence and uniqueness of solutions in the critical regularity framework above and to conclude that the Lipschitz regularity of the patch is preserved over time. Compared to previous works related to the density patch problem, the main novelty is an $L^1$-global-in-time Lipschitz estimate on the velocity field. This enables us to fully describe the long-time behavior of the patch, which is the rigid motion of an emerging Lipschitz domain. This is based on joint work with Alessandro Violini. | |
