scholarly journals Dispersive estimates, blow-up and failure of Strichartz estimates for the Schrödinger equation with slowly decaying initial data

2020 ◽  
Vol 2 (2) ◽  
pp. 519-532
Author(s):  
Rainer Mandel
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Strichartz Estimates

scholarly journals Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

2021 ◽  
Vol 9 ◽  
Author(s):  
Neal Bez ◽  
Sanghyuk Lee ◽  
Shohei Nakamura
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Oscillatory Integral ◽  
Oscillatory Integrals ◽  
Strichartz Estimates ◽  
Optimal Decay Rates ◽  
Fractional Schrödinger Equations ◽  
Optimal Decay ◽  
Klein Gordon

Abstract We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form $|\xi |^{-\lambda }$ or $(1 + |\xi |^2)^{-\lambda /2}$ , where $\lambda \in \mathbb {R}$ . We achieve optimal decay rates by considering such weights with appropriate $\lambda \in \mathbb {C}$ . For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein. Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.

scholarly journals The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation

1991 ◽  
Vol 117 (3-4) ◽  
pp. 251-273 ◽  
Author(s):  
Thierry Cazenave ◽  
Fred B. Weissler
Keyword(s):  
Schrödinger Equation ◽  
Global Solution ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Structure Of Solutions ◽  
Nonlinear Schrödinger ◽  
Conformally Invariant

SynopsisWe study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

scholarly journals COUNTEREXAMPLES TO STRICHARTZ ESTIMATES FOR THE MAGNETIC SCHRÖDINGER EQUATION

2011 ◽  
Vol 13 (02) ◽  
pp. 213-234 ◽  
Author(s):  
LUCA FANELLI ◽  
ANDONI GARCIA
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Strichartz Estimates

In space dimension n ≥ 3, we consider the magnetic Schrödinger Hamiltonian H = -(∇ - iA(x))2and the corresponding Schrödinger equation [Formula: see text] We show some explicit examples of potentials A, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schrödinger admissibility.

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