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$$C^ \infty$$-solutions in time, $$n$$- $$\log$$ Lipschitz bounded in space and Euler equation. (Solutions $$C^ \infty$$ en temps, $$n$$-$$\log$$ Lipschitz bornées en espace et équation d’Euler.) (French) Zbl 0835.76012
Summary: On all $$[0,T]$$, we solve the two-dimensional incompressible Euler equations in $$L^\infty$$ within Lipschitz frame $$[u, \text{rot} (u)$$ are bounded without integrability conditions], and we show that the $$n$$- th derivative of the solution has the estimate $$C |x |\log |x |^n$$ for module of continuity; each derivation in $$t$$ is then followed by a loss of regularity in $$x$$.

##### MSC:
 76B47 Vortex flows for incompressible inviscid fluids 35Q35 PDEs in connection with fluid mechanics