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GWP for the cubic wave equation in 3D

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Niveau: Supérieur, Doctorat, Bac+8
GWP for the cubic wave equation in 3D Fabrice Planchon The purpose of these notes is to give a simpler proof of KPV's result, which is GWP for H˙s data with s > 3/4. KPV's result is based on Bourgain's method : split low and high frequencies, solve the nonlinear problem for the low frequencies, deal with the pertubed equation for the high frequencies, and iterate in a clever way. Here we take a di?erent route, which can somehow be seen as dual. Such a method has previously been successfully used in the context of parabolic equations, in particular construction of infinite energy weak solutions to the Navier-Stokes equations (somehow one could trace these ideas back to nonlinear real interpolation in the 70's, with work by Peetre, Tartar...). In terms of the method, such a simpler proof (at least for the authors !) was given by I. Gallagher and the author. However, it uses at some stage improvements on the usual Strichartz estimates, and thus is not on par with KPV in term of technology. Not using this improvement leads to s > 5/6 with essentially a 1 page proof. The present note disposes with the need of improved Strichartz. We deal with ? + ?3 = 0, space dimension n = 3.

  • weak solution

  • fre- quencies land

  • strichartz estimates

  • actually like

  • navier stokes equations

  • high frequencies

  • let's start

  • result then

  • solutions up

  • kpv's result


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s˙H s> 3/4
s > 5/6
3+ = 0,
Jn = 3 =u +v || 20 0 0
J
3 J( s) ˙v + v = 0 2 H
(0 ) s
2 kv k kv k0 ˙ 0 1/2H ˙H
2 2 3u+3v u+3vu +u = 0
1˙H T(ku k )˙ 10 H
1˙v H
v
1/2+2+∞ 1/2+˙ ˙v v L (H )∩L (W )t t ∞
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1˙H
2E =k∂uk2
4H =E+kuk sup E HT T4 0tT
Z ZT T
2 2 H H + v u∂ u+ vu ∂ u .T 0 t t
0 0
4L
2J(1 s)H 20