Solutions globales d’énergie infinie pour l’équation des ondes critique

Germain, Pierre

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2006-2007), Exposé no. 11, 31 p.

Nous considérons dans cet article l’équation des ondes semilinéaire critique

$${\left(NLW\right)}_{2^{*}-1}\left\{\begin{array}{c}\square u+{\left|u\right|}^{2^{*}-2}u=0\hfill \\ u_{|t=0}=u_0\hfill \\ {\partial }_t{u}_{|t=0}=u_1,\hfill \end{array}\right.$$

posée dans tout l’espace ${ℝ}^d$, avec $2^{*}=\frac{2d}{d-2}·$ Shatah et Struwe [31] ont prouvé que si les données initiales sont d’énergie finie, c’est à dire si $(u_0,u_1)\in {\dot{H}}^1\times L^2$, alors il existe une solution globale. Planchon [22] a montré que c’est aussi le cas pour certaines données initiales d’énergie infinie : il suffit que les données initiales soient de norme petite dans ${\dot{B}}_{2,\infty }^1\times {\dot{B}}_{2,\infty }^0$. Nous construisons ici des solutions globales de ${\left(NLW\right)}_{2^{*}-1}$ pour des données initiales d’énergie infinie arbitrairement grandes, en utilisant deux méthodes qui reviennent à interpoler entre solutions d’énergie finie et solutions d’énergie infinie : la méthode de Bourgain et la méthode de Calderón. Ces deux méthodes donnent des résultats complémentaires.

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