On the defect of compactness for the Strichartz estimates of the Schrödinger equations. (English) Zbl 1038.35119

The author studies the Cauchy problem for non-linear Schrödinger equation \[ i v_t + \tfrac12 \triangle v= F(v) \qquad v(0,x)=\phi_0(x). \tag{1} \] To this purpose, he uses a family of space-time estimates for the solutions of the associated free problem: \[ v_{tt} - \tfrac12 \triangle v =0, \qquad v(0,x)=\phi_0(x), \] classically called Strichartz inequalities, which play an important role in the study of nonlinear Schrödinger equations. For example it was the fundamental tool used by T. Kato [Ann. Inst. Henri Poincaré, Phys. Théor. 46,113–129 (1987; Zbl 0632.35038)] to establish some results of wellposedness for the subcritical Schrödinger equation.
First, the author proves that every sequence of solutions to the linear Schrödinger equation with bounded data can be written, up to a subsequence, as an almost orthogonal sequence with a small remainder term in Strichartz norms. Second, for the equation (1) for \(F(v)=v^4v\) he proves a similar one, with the initial data belonging to a ball in the energy space where the equation is solvable. Third, he proves the existence of an a priori estimate of the Strichartz norms in terms of the energy.


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
35B45 A priori estimates in context of PDEs


Zbl 0632.35038
Full Text: DOI


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