Holomorphic structures with weak spatial regularity in fluid mechanics (Q1914527)

We construct a new frame of study for the Euler equation on \(\mathbb{R}^n\) and we show that in Lagrangian frame and for initial speeds \(u_0\in C^{m+ s}\), \(m\geq 1\), this equation is a holomorphic ODE and its group of resolution \((t, u_0)\to S(u_0)(t)\in C^{m+ s}\), is holomorphic in \((t, u_0)\) if \(\text{Im}(t)\), \(\text{Re}(t)\) small (\(\text{Re}(t)\) large if \(n= 2\), \(D^{\underline m} u_0\), \(\underline m\leq m\), not being then necessarily in one \(L^p\), \(p< \infty\)). The power series in \(t\) of all the Lagrangian variables have coefficients in \(u_0\) deduced from multilinear applications and linked by a recurrence involving iterative commutators of \(D^p \Delta^{- 1}\), \(p\in \mathbb{N}\). We present finally global structures in \(t\), the speeds being at gradient nilpotent (and we give explicit solutions).