TY - JOUR
T1 - Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity
AU - Kato, Keiichi
AU - Ogawa, Takayoshi
PY - 2000/3
Y1 - 2000/3
N2 - We show that a solution of the Cauchy problem for the KdV equation, {∂tυ + ∂3xυ + ∂x(υ2) = 0, t ∈ (-T, T), x ∈ ℝ, υ(0, x) = φ(x). has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for Hs(ℝ) (s > -3/4) data satisfying the condition (equation presented) the 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.
AB - We show that a solution of the Cauchy problem for the KdV equation, {∂tυ + ∂3xυ + ∂x(υ2) = 0, t ∈ (-T, T), x ∈ ℝ, υ(0, x) = φ(x). has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for Hs(ℝ) (s > -3/4) data satisfying the condition (equation presented) the 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.
UR - http://www.scopus.com/inward/record.url?scp=0034384756&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0034384756&partnerID=8YFLogxK
U2 - 10.1007/s002080050345
DO - 10.1007/s002080050345
M3 - Article
AN - SCOPUS:0034384756
VL - 316
SP - 577
EP - 608
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 3
ER -