Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

摘要

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L~2(), using "local smoothing" estimates. L~2() is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353-383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L~2(T). Our results are in line with previous work on the cubic nonlinear Schr?dinger equation, where Goubet and Molinet (Nonlinear Anal. 71, 317-320, 2009) showed weak continuity in L~2() and Molinet (Am. J. Math. 130, 635-683, 2008) showed lack of weak continuity in L~2(T).