On the boundary Strichartz estimates for wave and Schrödinger equations

Zihua Guo; Ji Li; Kenji Nakanishi; Lixin Yan

doi:10.1016/j.jde.2018.07.010

On the boundary Strichartz estimates for wave and Schrödinger equations

Zihua Guo^*, Ji Li, Kenji Nakanishi, Lixin Yan

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

13 Citations (Scopus)

Abstract

We consider the L_t ²L_x ^r estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show L_t ²L_x ^∞ estimates fail at the critical regularity in high dimensions by using stable Lévy process in R^d. Moreover, we show that some spherically averaged L_t ²L_x ^∞ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double L_t ²-type estimates.

Original language	English
Pages (from-to)	5656-5675
Number of pages	20
Journal	Journal of Differential Equations
Volume	265
Issue number	11
DOIs	https://doi.org/10.1016/j.jde.2018.07.010
Publication status	Published - 5 Dec 2018

Keywords

Nonlinear Schrödinger equation
Nonlinear wave equation
Strichartz estimates

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10.1016/j.jde.2018.07.010

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@article{af5b49bbe2ec4c3ba2a49773c6e5f63d,

title = "On the boundary Strichartz estimates for wave and Schr{\"o}dinger equations",

abstract = "We consider the Lt 2Lx r estimates for the solutions to the wave and Schr{\"o}dinger equations in high dimensions. For the homogeneous estimates, we show Lt 2Lx ∞ estimates fail at the critical regularity in high dimensions by using stable L{\'e}vy process in Rd. Moreover, we show that some spherically averaged Lt 2Lx ∞ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double Lt 2-type estimates.",

keywords = "Nonlinear Schr{\"o}dinger equation, Nonlinear wave equation, Strichartz estimates",

author = "Zihua Guo and Ji Li and Kenji Nakanishi and Lixin Yan",

year = "2018",

month = dec,

day = "5",

doi = "10.1016/j.jde.2018.07.010",

language = "English",

volume = "265",

pages = "5656--5675",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Academic Press Inc.",

number = "11",

}

TY - JOUR

T1 - On the boundary Strichartz estimates for wave and Schrödinger equations

AU - Guo, Zihua

AU - Li, Ji

AU - Nakanishi, Kenji

AU - Yan, Lixin

PY - 2018/12/5

Y1 - 2018/12/5

N2 - We consider the Lt 2Lx r estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show Lt 2Lx ∞ estimates fail at the critical regularity in high dimensions by using stable Lévy process in Rd. Moreover, we show that some spherically averaged Lt 2Lx ∞ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double Lt 2-type estimates.

AB - We consider the Lt 2Lx r estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show Lt 2Lx ∞ estimates fail at the critical regularity in high dimensions by using stable Lévy process in Rd. Moreover, we show that some spherically averaged Lt 2Lx ∞ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double Lt 2-type estimates.

KW - Nonlinear Schrödinger equation

KW - Nonlinear wave equation

KW - Strichartz estimates

UR - https://www.scopus.com/pages/publications/85049643313

U2 - 10.1016/j.jde.2018.07.010

DO - 10.1016/j.jde.2018.07.010

M3 - Article

AN - SCOPUS:85049643313

SN - 0022-0396

VL - 265

SP - 5656

EP - 5675

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 11

ER -

On the boundary Strichartz estimates for wave and Schrödinger equations

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