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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.

MSC:

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

Citations:

Zbl 0632.35038
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References:

[1] Bourgain, J., Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. amer. math. soc., 12, 145-171, (January)
[2] Bahouri, H.; Gérard, P., High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. math., 121, 131-175, (1999) · Zbl 0919.35089
[3] Cazenave, T.; Weissler, F., The Cauchy problem for the critical nonlinear Schrödinger equation in Hs, Nonlinear anal., 14, 807-863, (1990)
[4] Constantin, P.; Saut, J.C., Local smoothing properties of dispersive equations, Amer. math. soc., 1, (1988) · Zbl 0667.35061
[5] Gérard, P., Description du défaut de compacité de l’injection de Sobolev, Esaim.cocv, 3, 213-233, (1998)
[6] Gérard, P.; Meyer, Y.; Oru, F., Inégalités de Sobolev précisées, Séminaire équations aux Dérivées partielles, École polytechnique, palaiseau, (1996) · Zbl 1066.46501
[7] Ginibre, J.; Velo, G., Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. math. pures appl., 64, 363-401, (1984) · Zbl 0535.35069
[8] Kato, T., On nonlinear Schrödinger equations, Ann. inst. H. Poincaré phys. théor., 46, 113-129, (1987) · Zbl 0632.35038
[9] Merle, F.; Vega, L., Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equations in 2D, Internat. math. res. notices, 8, 399-425, (1998) · Zbl 0913.35126
[10] Moyua, A.; Vargas, A.; Vega, L., Restriction theorems and maximal operators related to oscillatory integrals in \(R\)^{3}, Duke math. J., 96, 547-574, (1999) · Zbl 0946.42011
[11] Segal, I., Space-time decay for solutions of wave equations, Adv. math., 22, 305-311, (1976) · Zbl 0344.35058
[12] Sjölin, P., Regularity of solutions to the Schrödinger equations, Duke math. J., 55, 699-715, (1987) · Zbl 0631.42010
[13] Strichartz, R., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equation, Duke math. J., 44, 705-714, (1977) · Zbl 0372.35001
[14] Tomas, P., A restriction theorem for the Fourier transform, Bull. amer. math. soc., 81, 477-478, (1975) · Zbl 0298.42011
[15] Vega, L., Schrödinger equation: pointwise convergence to the initial data, Proc. amer. math. soc., 102, 874-878, (1988) · Zbl 0654.42014
[16] Yajima, K., Existence of solutions for Schrödinger evolution equations, Comm. math. phys., 110, 415-426, (1987) · Zbl 0638.35036
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