TY - JOUR
T1 - Global smoothing for the periodic KdV evolution
AU - Erdoǧan, Mehmet Burak
AU - Tzirakis, Nikolaos
N1 - Funding Information:
This work was partially supported by NSF grants DMS-0900865 (to B.E.) and DMS-0901222
PY - 2013/1/1
Y1 - 2013/1/1
N2 - The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for Hs initial data, and for any, the difference of the nonlinear and linear evolutions is in Hs1 for all times, with at most polynomially growing Hs1 norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case s≥0. Our result and a theorem of Oskolkov for the Airy evolution imply that if one starts with continuous and bounded variation initial data, then the solution of KdV (given by the L2 theory of Bourgain) is a continuous function of space and time. In addition, we demonstrate smoothing for the defocusing modified KdV equation on the torus for.
AB - The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for Hs initial data, and for any, the difference of the nonlinear and linear evolutions is in Hs1 for all times, with at most polynomially growing Hs1 norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case s≥0. Our result and a theorem of Oskolkov for the Airy evolution imply that if one starts with continuous and bounded variation initial data, then the solution of KdV (given by the L2 theory of Bourgain) is a continuous function of space and time. In addition, we demonstrate smoothing for the defocusing modified KdV equation on the torus for.
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U2 - 10.1093/imrn/rns189
DO - 10.1093/imrn/rns189
M3 - Article
AN - SCOPUS:84885936398
SN - 1073-7928
VL - 2013
SP - 4589
EP - 4614
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 20
ER -