Sharp well-posedness for the free boundary MHD equations
Vortrag von Dr. Mitchell Taylor
Datum: 27.03.25 Zeit: 16.15 - 18.00 Raum: Y27H46
I will discuss a series of recent works written jointly with M. Ifrim, B. Pineau and D. Tataru where we develop a new \emph{fully Eulerian} approach to free boundary fluid dynamics, obtaining the first sharp well-posedness theorems for the free boundary Euler and MHD equations. Our well-posedness theories include (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all at optimal Sobolev regularity; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of scaling, requiring merely integrability in time of the Lipschitz norm of \((v,B)\) and the \(C^{1,\frac{1}{2}}\) norm of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In particular, for the Euler equations we show that solutions can be continued as long as the velocity is in \(L_T^1W^{1,\infty}\), the free surface is in \(L_T^1C^{1,\frac{1}{2}}\) and the surface does not self-intersect; (vi) A novel construction of regular solutions, which was not known at any regularity for MHD.