Global well-posedness of critical nonlinear Schrödinger equations below $L^2$

Cho, Yonggeun; Hwang, Gyeongha; Ozawa, Tohru

doi:10.3934/dcds.2013.33.1389

Cited by 16 publications

(10 citation statements)

References 23 publications

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“…The main tool for proving the theorem is the weighted Strichartz estimates in [7]. We adapt the idea in [8] in which the Hartree equation was considered. To apply the weighted Strichartz estimate, we shall accordingly construct some weighted spaces.…”

Section: )mentioning

confidence: 99%

“…Most of these works concern the mass-critical and the energy-critical cases. Recently, there is a development obtained by Cho and Hwang [8] on the line of well-posedness theory for the Hartree equation with the critical Sobolev space of negative order. They used the weighted Strichartz estimate in symmetric form.…”

Section: )mentioning

confidence: 99%

“…The usual Strichartz estimates do not work. Cho and Hwang [8] proved for dimensions n 3 the small data global well-posedness for the radial data and 3 2 < < 2 and for the general data with angular regularity and 2 3 2nC2 < < 2 by using the weighted Strichartz estimate. In the following, we shall take full advantage of the weighted Strichartz estimate with nonsymmetric spatial-time norms.…”

Section: An Improvement For the Hartree Equation With Negative Criticmentioning

confidence: 99%

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Small data global well‐posedness for the nonlinear wave equation with nonlocal nonlinearity

Cheng

Gao

2012

Math Methods in App Sciences

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In this paper, we investigate the Cauchy problem of the nonlinear wave equation @ 2 t u C u D V.u/u, .t, x/ 2 R R n , where V.u/ D j j juj 2 , 2 R n f0g, 0 < < min.4, n/ and n 3. We prove small data global well-posedness for the radial data and for the general data with angular regularity. We also give an improved result of the Hartree equation with negative critical regularity.The well-posedness theory for the wave equation with Hartree-type nonlinearity was first studied by Menzala and Strauss [3]. They proved the local well-posedness for the H 1 .R n / L 2 .R n / solution of (HNLW) when 0 < < 3 for n D 3 and 0 < Ä 3 for n 4. Mochizuki [4] showed by using the mixed norm estimates of Pecher that the small P H 1 .R n / L 2 .R n / solution of (HNLW) with D 4 < n scatters. Later, Mochizuki and Motai [5] extended the range of to 2 C 2 3.n 1/ < < n. In [6], Hidano proved that small data scattering holds for > 2 in some weighted space of P H 1 .R n / L 2 .R n / when n 3. In this paper, we shall establish the small data global well-posedness for (HNLW) in P H s c P H s c 1 with 0 < s c < 1. More precisely, the main result is the following. 99-112Thus, there exists a unique solution to (HNLW). This completes the proof of Theorem 1(b).

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Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: An Improvement For the Hartree Equation With Negative Criticmentioning

confidence: 99%

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Small data global well‐posedness for the nonlinear wave equation with nonlocal nonlinearity

Cheng

Gao

2012

Math Methods in App Sciences

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“…When it comes to the critical case below L 2 (i.e., the case s < 0), the small data global well-posedness is known in [8] only for radial (at best angularly regular) data. The main contribution of this paper is to develop the well-posedness theory for general data in this critical case.…”

Section: Introductionmentioning

confidence: 99%

Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations

Kim¹,

Lee²,

Seo³

2021

Preprint

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In this paper we study the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = λ|x| −α |u| β u below L 2 . The wellposedness theory for this equation in the critical case has been intensively studied in recent years, but much less is understood below L 2 . The only known result is the small data global well-posedness for radial (at best angularly regular) data. The main contribution of this paper is to develop the well-posedness theory for general data. To this end, we significantly improve the previously known L p Strichartz estimates with singular weights and indeed sharpen them.

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“…Well-posedness issues of the Hartree equation which corresponds to the particular case p = 2 in (1.1) were widely investigated, see for instance [5,12,26,27].…”

Section: Introductionmentioning

confidence: 99%