Inhomogeneous Strichartz Estimates

Foschi, Damiano

doi:10.1142/s0219891605000361

Cited by 198 publications

(198 citation statements)

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“…Finally, we mention that Foschi's estimates [6] also hold for the wave equation. One can then prove the analogue of the Perturbation Theorem for (NLS), for (NLW) and all its corollaries.…”

Section: The Hypothesis Is Verified On Imentioning

confidence: 80%

Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations

Kenig¹

2010

Journées Équations Aux Dérivées Partielles

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“…Finally, we mention that Foschi's estimates [6] also hold for the wave equation. One can then prove the analogue of the Perturbation Theorem for (NLS), for (NLW) and all its corollaries.…”

Section: The Hypothesis Is Verified On Imentioning

confidence: 80%

Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations

Kenig¹

2010

Journées Équations Aux Dérivées Partielles

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“…On the other hand, suppose that t 1 n → +∞. Then we can similarly argue that for n large, e it∆ u c (t n ) S(Ḣ 1/2 ;(−∞,0]) ≤ 1 2 δ sd , refined Strichartz estimates by Foschi [8] are sufficient to complete the long term perturbation argument 15 .…”

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confidence: 82%

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

Holmer

Roudenko

2008

Commun. Math. Phys.

287

349

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Abstract. We consider the problem of identifying sharp criteria under which radial H 1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂ t u + ∆u + |u| 2 u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the, where u 0 denotes the initial data, and M [u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution e it Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if, then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e it Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if, then the solution blows-up in finite time. The technique employed is parallel to that employed by in their study of the energy-critical NLS.

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“…It was shown in [3] that n − 2 r − 2 q ≤ n r and n − 2 r − 2 q ≤ n r (3.1) are the necessary conditions for which the inhomogeneous estimate…”

Section: Necessary Conditionsmentioning

confidence: 99%

“…(We refer the reader to [3,13,8] for other necessary conditions.) Compared with (3.1), we give here the following new necessary condition:…”

Section: Necessary Conditionsmentioning

confidence: 99%

Inhomogeneous Strichartz estimates for Schrödinger's equation

Koh

Seo

2016

Journal of Mathematical Analysis and Applications

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Abstract. Foschi and Vilela in their independent works ([3], [13]) showed that the range of (1/r, 1/ r) for which the inhomogeneous Strichartz estimateholds for some q, q is contained in the closed pentagon with vertices A, B, B ′ , P, P ′ except the points P, P ′ (see Figure 1). We obtain the estimate for the corner points P, P ′ . IntroductionIn this paper we consider the following Cauchy problem for the Schrödinger equation:where (x, t) ∈ R n × R, n ≥ 1. By Duhamel's principle, we have the solutionwhere e it∆ is the free Schrödinger propagator defined byThe Strichartz estimates for the solution play important roles in the study of well-posedness for nonlinear Schrödinger equations (cf. [1,12]). They actually consist of two parts, homogeneous (F = 0) and inhomogeneous (f = 0) part. The homogeneous Strichartz estimateholds if and only if (r, q) is admissible pair, that is, r, q ≥ 2, (n, r, q) = (2, ∞, 2) and n/r + 2/q = n/2 (see [11,4,9,6] and references therein). But determining the optimal range of (r, q) and ( r, q) for which the inhomogeneous Strichartz estimateholds is not completed yet when n ≥ 3. It was observed that this estimate is valid on a wider range than what is given by admissible pairs (r, q), ( r, q) (see [2], [5]).2010 Mathematics Subject Classification. Primary 35B45, 35Q40.

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Inhomogeneous Strichartz Estimates

Cited by 198 publications

References 12 publications

Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations

Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

Inhomogeneous Strichartz estimates for Schrödinger's equation

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