Bilinear estimates and applications to 2d NLS

Colliander, J.; Delort, J.-M.; Kenig, C.; Staffilani, G.

doi:10.1090/S0002-9947-01-02760-X

Bilinear estimates and applications to 2d NLS
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by J. E. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani PDF

Trans. Amer. Math. Soc. 353 (2001), 3307-3325 Request permission

Abstract:

The three bilinearities $u v, \overline {uv},\overline {u}v$ for functions $u, v : \mathbb {R}^2 \times [0,T] \longmapsto \mathbb {C}$ are sharply estimated in function spaces $X_{s,b}$ associated to the Schrödinger operator $i \partial _t + \Delta$. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.

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