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On the local well-posedness of the Benjamin-Ono equation in H S (). (English) Zbl 1039.35106

The authors prove the following theorem for the Benjamin-Ono equation

u t +𝐇u xx +uu x =0,u(0,x)=u 0 (x),(8)

where 𝐇 denotes the Hilbert transform.

Fix s>5 4. Then for every u 0 𝐇 s (), there exist T||u 0 || 𝐇 s -4 and a unique solution of (8) on the time interval [0,T] satisfying

uC([0,T],L 2 ()),u x L 1 ([0,T],L ())·

Moreover, for any R>0, there exists TR -4 such that the nonlinear map u 0 u is continuous from the ball of radius R of 𝐇 s () to C([0,T],𝐇 s ()).

Conditions for an improvement of the theorem are given.

35Q53KdV-like (Korteweg-de Vries) equations
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
42B25Maximal functions, Littlewood-Paley theory