SHARP WELL-POSEDNESS RESULTS FORTHE GENERALIZED BENJAMIN-ONO EQUATIONWITH HIGH NONLINEARITY

摘要

We establish the local well posedness of the generalizedBenjamin-Ono equation _tu+H_x~2u±u~k_xu = 0 inH~s(R), s> 1/2-kfor k ≥ 12 and without smallness assumption on the initial data. Thecondition s > 1/2 — 1/k is known to be sharp since the solution mapu_0→u is not of class C~(k+1)onH~s(R)fors <1/2 — k.On the otherhand, in the particular case of the cubic Benjamin-Ono equation, weprove the ill posedness in H~s(R), s.