Global smoothing for the periodic Benjamin equation in low-regularity spaces

This paper is intended as an attempt to set up the global smoothing for the periodic Benjamin equation.It is shown that for Hs(T) initial data with s >-1/2 and for any s < s1< min{s+1,3s+1},the diference of the evolution with the linear evolution is in Hs1(T) for all times,with at most polynomial growing Hs1 norm.Unlike Korteweg-de Vries(KdV) equation,there are less symmetries of the Benjamin system,especially for the resonant function.The new ingredient is that we need to deal with some new difculties that are caused by the lack of symmetries.

This paper is intended as an attempt to set up the global smoothing for the periodic Benjamin equation. It is shown that for Hs（T） initial data with 8 〉 -1/2 and for any s 〈 s1〈 min{s ＋ 1,3s ＋ 1}, the difference of the evolution with the linear evolution is in Hs1 （T） for all times, with at most polynomial growing HS1 norm. Unlike Korteweg-de Vries （KdV） equation, there are less symmetries of the Benjamin system, especially for the resonant function. The new ingredient is that we need to deal with some new difficulties that are caused by the lack of symmetries.