WELL-POSEDNESS FOR EQUATIONS OF BENJAMIN-ONO TYPE

摘要

The Cauchy problem u_t - |D|~αu_x+uu_x = 0 in (-T,T) × R, u(0) = u_0, is studied for 1 ＜ α ＜ 2. Using suitable spaces of Bourgain type, local well-posedness for initial data u_0 ∈ H~s(R) ∩ H~(-ω)(R) for any s ＞ - 3/4(α - 1), ω: = 1/α - 1/2 is shown. This includes existence, uniqueness, persistence, and analytic dependence on the initial data. These results are sharp with respect to the low frequency condition in the sense that if ω ＜ 1/α - 1/2, then the flow map is not C~2 due to the counterexamples previously known. By using a conservation law, these results are extended to global well-posedness in H~s(R) ∩ H~(-ω)(R) for s ≥ 0, ω = 1/α - 1/2, and real valued initial data.