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Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$

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  • The initial value problem for the cubic defocusing nonlinear Schrödinger equation $i \partial_t u + \Delta u = |u|^2 u$ on theplane is shown to be globally well-posed for initial data in $H^s (\mathbb{R}^2)$ provided $s>1/2$. The same result holds true for theanalogous focusing problem provided the mass of the initial data issmaller than the mass of the ground state. The proof relies upon analmost conserved quantity constructed using multilinear correctionterms. The main new difficulty is to control the contribution ofresonant interactions to these correction terms. The resonantinteractions are significant due to the multidimensional setting ofthe problem and some orthogonality issues which arise.
    Mathematics Subject Classification: 35Q55.

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