Pointwise convergence problem of the Korteweg–de Vries–Benjamin–Ono equation

Abstract

In this paper we investigate the pointwise convergence problem for the Korteweg–de Vries–Benjamin–Ono equation ut+γH(∂x2u)−β∂x3u=0,(x,t)∈R×R,u(x,0)=f(x)∈Hs(R),where γβ>0. We prove that the solution u(x,t)=Utf(x) converges pointwisely to the initial data f(x) for a.e. x∈R when f∈Hs(R) with s≥14, and that the Hausdorff dimension of the divergence set of points of the solution is αU(s)=1−2s when 14≤s≤12. We also obtain the stochastic continuity for the initial data with much less regularity, i.e. for a large class of the initial data in L2(R), via the randomization technique.