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Holomorphic structures with weak spatial regularity in fluid mechanics. (Structures holomorphes à faible régularité spatiale en mécanique des fluides.) (French) Zbl 0849.35111
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).
Reviewer: Ph.Serfati (Paris)

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76B47 Vortex flows for incompressible inviscid fluids 35C05 Solutions to PDEs in closed form