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. [ABSTRACT FROM AUTHOR]