TY - JOUR
T1 - Global solutions for the gravity water waves system in 2d
AU - Ionescu, Alexandru D.
AU - Pusateri, Fabio
N1 - Funding Information:
The authors are grateful to the anonymous referees for carefully reading the manuscript and for their many useful comments. After submission of this manuscript a different proof of a similar result was given by Alazard and Delort []. The first author was partially supported by a Packard Fellowship and NSF Grant DMS 1265818. The second author was partially supported by a Simons Postdoctoral Fellowship and NSF Grant DMS 1265875.
Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.
PY - 2015/3
Y1 - 2015/3
N2 - We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu (Invent Math 177(1):45–135, 2009). We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case (Germain et al., Ann Math 175(2):691–754, 2012; Wu, Invent Math 184(1):125–220, 2011). In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2D.
AB - We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu (Invent Math 177(1):45–135, 2009). We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case (Germain et al., Ann Math 175(2):691–754, 2012; Wu, Invent Math 184(1):125–220, 2011). In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2D.
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U2 - 10.1007/s00222-014-0521-4
DO - 10.1007/s00222-014-0521-4
M3 - Article
AN - SCOPUS:84939873998
SN - 0020-9910
VL - 199
SP - 653
EP - 804
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -