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

In this paper we investigate the pointwise convergence problem for the Korteweg–de Vries–Benjamin–Ono equation u t + γ H ( ∂ x 2 u ) − β ∂ x 3 u = 0 , ( x , t ) ∈ R × R , u ( x , 0 ) = f ( x ) ∈ H s ( R ) , where γ β > 0 . We prove that the solution u ( x , t ) = U t f ( x ) converges pointwisely to the initial data f ( x ) for a.e. x ∈ R when f ∈ H s ( R ) with s ≥ 1 4 , and that the Hausdorff dimension of the divergence set of points of the solution is α U ( s ) = 1 − 2 s when 1 4 ≤ s ≤ 1 2 . 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 L 2 ( R ) , via the randomization technique.