The TURBO method for well-posedness of the incompressible Euler equations in Sobolev spaces in any domain
I. Kukavica, W. S. Ożański
TL;DR
The paper addresses local‑in‑time well‑posedness of the incompressible Euler equations in Sobolev spaces $H^r$ on bounded domains with Sobolev regularity. It introduces the TURBO method, which builds an analytic problem on an analytically approximated domain $Q$ and solves it in analytic spaces $X(\tau)$, $Y(\tau)$ with a radius of analyticity $\tau(t)$, using analytic persistence to propagate Sobolev bounds back to the original domain. The proof proceeds through a sequence of steps: establishing commutator and pressure estimates in analytic spaces, constructing an analytic solution on $Q$ over $[0,T_0]$, obtaining a Sobolev limit via analytic approximation and Aubin–Lions, and finally transferring the result to the original Sobolev domain by a careful limit process. The approach preserves the original equations, does not rely on modifications or regularizations, and is presented as adaptable to other PDEs in incompressible fluid mechanics, including Boussinesq, Navier–Stokes, free‑boundary Euler, and ideal MHD.
Abstract
We introduce a new method for constructing local-in-time solutions the incompressible Euler equations in Sobolev spaces on an arbitrary Sobolev bounded domain. The method is based on construction of an analytic solution in an analytically approximated domain, after which we apply analytic persistence to extend the analytic solution given a priori bounds in Sobolev spaces. The method does not introduce any modification or regularization of the equations themselves and appears applicable to many other PDEs.