Randomized final-data problem for systems of nonlinear Schrödinger equations and the Gross–Pitaevskii equation

Abstract

We consider the final-data problem for systems of nonlinear Schrödinger equations (NLS) with L 2 subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in L 2 (R d ), extending the result of J. Murphy [29] to powers equal to or lower than the Strauss exponent. In particular, systems with quadratic nonlinearity can be treated in three space dimensions, and by the same argument, the Gross-Pitaevskii equation in the energy space. The extension is by use of the Strichartz estimate with a time weight.

Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

“…We are now ready to provide the improvement in space-time integrability of the linear flow e it∆ f ω with data randomized in frequency space, the angular variable and in physical space which originates in the unit-scale decomposition in physical space. We obtain the same set of estimates as for the pure physical-space randomization, see [36,28]. Hence, the additional randomization in frequency space and the angular variable does not impair the improvements of the physical-space randomization.…”

“…Roughly speaking, this randomization allows to employ the dispersive estimate for solutions of the linear Schrödinger equation although the data only belongs to L 2 . These techniques were further refined in [28].…”

“…for every f ∈ L 2 (R 4 ). The physical-space randomization means the randomization with respect to this decomposition, see [27,28,36].…”

“…x , we exploit that the randomization (1.17) yields the same improved decay of the linear flow as the pure physical-space randomization in [28,36] (see Proposition 3.6). On the other hand, we can still gain regularity for the linear flow as for the randomization (1.14) (see Proposition 3.4).…”

Randomized final-state problem for the Zakharov system in dimension three

“…Roughly speaking, the physical-space randomization gives access to the dispersive estimate for linear solutions of the Schrödinger equation although the data only belongs to L 2 . In [38] the authors observed that this dispersive decay can be used to study the final-state problem in time-weighted spaces, improving on the results in [37].…”