On uniqueness for the critical wave equation Nader Masmoudi

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On uniqueness for the critical wave equation Nader Masmoudi Courant Institute of Mathematical Sciences, 251 Mercer Street, New York NY 10012 and Fabrice Planchon Laboratoire Analyse, Geometrie & Applications UMR 7539, Institut Galilee Universite Paris 13, 99 avenue J.B. Clement 93430 Villetaneuse FRANCE Abstract We prove the uniqueness of weak solutions to the critical defocusing wave equation in 3D under a local energy inequality condition. More precisely, we prove the uniqueness of u ? L∞t (H˙1)? W˙ 1,∞t (L2), under the condition that u verifies some local energy inequalities. 1 Introduction and statement of result We consider the defocusing quintic wave equation in 3D, (1) { u+ u5 = 0, u(t = 0) = u0, ut(t = 0) = u1. Existence of global weak solutions goes back to Segal ([9], under milder as- sumptions on the nonlinearity). Existence of global smooth solutions was proved by Grillakis ([3]), while global solutions in the energy space C(R;H1)? C1(R;L2) were constructed by Shatah and Struwe [11]. Uniqueness was proved only under an additional space-time integrability of Strichartz type, 1

weak solution

among all weak

local existence

let ?

propagation

quintic wave

equation

energy density

existence result

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Publié par

profil-vieg-2012

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On uniqueness for the critical wave equation

Nader Masmoudi
Courant Institute of Mathematical Sciences,
251 Mercer Street, New York NY 10012
masmoudi@cims.nyu.edu
and
Fabrice Planchon
Laboratoire Analyse, G´om´trie & Applications
UMR 7539, Institut Galil´e
Universit´ Paris 13, 99 avenue J.B. Cl´ment
93430 Villetaneuse FRANCE
fab@math.univ-paris13.fr

Abstract
We prove the uniqueness of weak solutions to the critical defocusing
wave equation in 3D under a local energy inequality condition.More
1,∞
∞1 2
˙ ˙
precisely, we prove the uniqueness ofu∈L(H)∩W(L), under
t t
the condition thatuveriﬁes some local energy inequalities.

Introduction and statement of result

We consider the defocusing quintic wave equation in 3D,

5
u+u= 0,
(1)
u(t= 0) =u0, ut(t= 0) =u1.

Existence of global weak solutions goes back to Segal ([9], under milder
assumptions on the nonlinearity).Existence of global smooth solutions was
1
proved by Grillakis ([3]), while global solutions in the energy spaceC(R;H)∩
1 2
C(R;L) were constructed by Shatah and Struwe [11].Uniqueness was
proved only under an additional space-time integrability of Strichartz type,

Voir