Sur une équation mi onde dégénérée Workshop Handdy île de Berder septembre

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The equation Compared to the Szego solution Integrable effective dynamics for a non linear wave equation Sandrine Grellier Universite d'Orleans- Federation Denis Poisson WORSHOP HANDDY SEPT. 2011 jointwork with P. Gerard (Universite Paris sud) Sandrine Grellier Integrable effective dynamics for a non linear wave equation

  • half wave

  • szego solution

  • universite de paris sud

  • universite d'orleans- federation

  • toy model

  • integrable effective

  • ck eikx


Publié le : lundi 18 juin 2012
Lecture(s) : 28
Source : univ-orleans.fr
Nombre de pages : 61
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Integrable effective dynamics for a non linear wave equation
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Universite´dOrle´ans-Fe´d´erationDenisPoisson WORSHOPHANDDYSEPT. 2011
jointwork with rardP. Ge´is´tvire(nUd)issuePar
Sandrine Grellier
ofscimanydevitceeeavrweainnlnoraerllniGenardSeeffrablntegierI
Consider the half wave equation
itu− |D|u=|u|2u,tR,xT
where
=X |D|kXZckeikx!:kZck|k|eikx.
Toy model for NLS on degenerate geometries leading to lack of dispersion. Admits the same conservation laws as NLS : H(u) =12(|D|u,u)L2+14kukL44, Q(u) =kukL22, M(u) = (Du,u)L2.
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itu− |D|u=|u|2u,tR,xT
auitevqeraawlineanonsforamicaSdnir
Toy model for NLS on degenerate geometries leading to lack of dispersion. Admits the same conservation laws as NLS : H(u) =12(|D|u,u)L2+41kukL44, 2 Q(u) =kukL2, M(u) = (Du,u)L2.
where
|D|kXZckeikx!:=kXZck|k|eikx.
Consider the half wave equation
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itu− |D|u=|u|2u,tR,xT
ehnqoe
Toy model for NLS on degenerate geometries leading to lack of dispersion. Admits the same conservation laws as NLS : 4 H(u) =12(|D|u,u)L2+14kukL4, Q(u) =kuk2L2, M(u) = (Du,u)L2.
a
where
u
D|Xc |kZkeikx!:=kXZck|k|eikx.
i
Consider the half wave equation
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whereu±:= Π±uand Π+(Xckeikx) :=Xckeikx, kZk0 Π(Xceikx) :=Xckeikx k. nZk<0
Compared to the ”usual” 1D-NLS
itu+x2u=|u|2u,
this is anon dispersiveequation : (i(t+x)u+= Π+(|u|2u), i(tx)u= Π(|u|2u),
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itu+2u|u|2u, x=
this is anon dispersiveequation : (i(t+x))uu+=ΠΠ=+((||uu||22uu)),, i(tx
whereu±:= Π±uand Π+(Xckeikx) :=Xcke, ikx kZk0 Π(Xckeikx) :=Xckeikx. nZk<0
Compared to the ”usual” 1D-NLS
uqtaoin
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t 0Z0|(|u(τ)|2u(τ))dτ u(t) =eit|D|u iei(tτ)|D
isnot bounded inHsfors<12! Indeed, boundedness would require Z1 0keit|D|fk4L4(T)dt.kfk4Hs/2.
n
This fails ifs<1 2.
The scaling isL2critical,but the first iteration of the Duhamel formula
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u(t) =eit|D|u0iZ0tei(tτ)|D|(|u(τ)|2u(τ))dτ
isnot bounded inHsfors<12! Indeed, boundedness would require 1 Z0keit|D|fkL44(T)dt.kfk4Hs/2.
This fails ifs<21.
The scaling isL2critical, but the first iteration of the Duhamel formula
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t u(t) =eit|D|u0iZ0ei(tτ)|D|(|u(τ)|2u(τ))dτ
The scaling isL2critical, but the first iteration of the Duhamel formula
This fails ifs<21.
on
isnot bounded inHsfors<21! Indeed, boundedness would require 4 Z10keit|D|fk4L4(T)dt.kfkHs/2.
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csminadyvetiecffeelbargetnIreill
The scaling isL2critical, but the first iteration of the Duhamel formula
This fails ifs<21.
n
isnot bounded inHsfors<21! Indeed, boundedness would require Z10keit|D|fk4L4(T)dt.kfk4Hs/2.
u(t) =eit|D|u0iZ0tei(tτ)|D|(|u(τ)|2u(τ))dτ
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Optimality ?
ku(t)kHs.eeCs t
Notice : provides ,bad large time estimates
Proposition Givenu0H12(T), there existsuC(R,H21(T))unique such that itu− |D|u=|u|2u,u(0,x) =u0(x). Moreover ifu0Hs(T)for somes>12, thenuC(R,Hs(T)).
One can only prove, using the energy conservation law and logarithmic inequalities,
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