Sharp well-posedness results of the Benjamin-Ono equation in H^s(T,R) and qualitative properties of its solution · arXivDesk
2004.04857Apr 9, 2020The paper is a greatly extended version of arXiv:1909.07314. In particular, we have included the result on ill-posedness of the Benjamin-Ono equation in $H^{-1/2}$ and added an appendix on the restriction of the Birkhoff map to the scale of Sobolev space $H^s$, $s>0$, and another one on a sharper form of ill-posedness of the Benjamin-Ono equation in $H^{s}_{r}$ with $s<-1/2$
Sharp well-posedness results of the Benjamin-Ono equation in Hs(T,R) and qualitative properties of its solution
We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space Hs(T,R) for any s>−1/2 and ill-posed for s≤−1/2
Nearby in the stack
. Hence the critical Sobolev exponent
sc=−1/2
of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in
Hs(T,R)
for any
s>−1/2
. Novel conservation laws and a nonlinear Fourier transform on
Hs(T,R)
with
s>−1/2
are key ingredients into the proofs of these results.