Gevrey regularity of the periodic gKdV equation

Abstract

Using the multilinear estimates, which were derived for proving well-posedness of the generalized Korteweg–de Vries (gKdV) equation, it is shown that if the initial data belongs to Gevrey space Gσ, σ⩾1, in the space variable then the solution to the corresponding Cauchy problem for gKdV belongs also to Gσ in the space variable. Moreover, the solution is not necessarily Gσ in the time variable. However, it belongs to G3σ near 0. When σ=1 these are analytic regularity results for gKdV.