Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

Abstract

We show that a solution of the Cauchy problem for the KdV equation,has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity atthe solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac δ measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator 3t∂ t + x∂ x .

Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity

“…This result generalized the result given by W. Craig et al [7]. On the other hand, because (1.1)-(1.3) is a coupled system of Korteweg-de Vries equations, it is natural to ask whether it has a smoothing effect up to real analyticity if the initial data only has a single point singularity at x = 0 similar to those results obtained by K. Kato and T. Ogawa [10] for the Korteweg-de Vries equation. In this paper, we give an answer to this question which will improve those results obtained in O. Vera [19] (see also [1]).…”

“…In 2004, F. Linares and M. Panthee [18] In 2001, O. Vera [19], following the idea of W. Craig et al [7] shown that C ∞ solutions (u(·, t), v(·, t)) to (1.1)-(1.3) are obtained for t > 0 if the initial data (u 0 (x), v 0 (x)) belong to a suitable Sobolev space satisfying reasonable conditions as |x| → ∞. In 2000, K. Kato and T. Ogawa [10] shown that a solution of the Korteweg-de Vries equation has a smoothing effect up to real analyticity if the initial data only has a single point singularity at x = 0. It is shown that for H s (R) (s > −3/4) data satisfying the condition the solution is analytic in both space and time variables.…”

Nonanalytic solutions of the KdV equation

“…For example, G. Łysik in a private communication mentioned that in the nonperiodic case, he can show that the Cauchy problem for the KdV with initial data ϕ(x) = 1/(1 + x 2 ) is not analytic (like in the heat equation). For more results about the analyticity and smoothing effects of the KdV, we refer the reader to the paper of Kato and Ogawa [6] and the references therein.…”

Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity

“…[6,7,19,27,29,34,41] for linear equations with constant coefficients and [16,17,23,31] for nonlinear equations. For Schro¨dinger equations, these results have been extended to the cases with decaying potentials [2,8,20] or with potentials which grow at most quadratically at infinity [43].…”