{"title":"线性波或Schrödinger方程能量的微局部分配","authors":"Jean-Marc Delort","doi":"10.2140/tunis.2022.4.329","DOIUrl":null,"url":null,"abstract":"We prove a microlocal partition of energy for solutions to linear half-wave or Schrödinger equations in any space dimension. This extends well-known (local) results valid for the wave equation outside the wave cone, and allows us in particular, in the case of even dimension, to generalize the radial estimates due to Côte, Kenig and Schlag to non radial initial data. 0 Introduction The goal of this paper is to revisit the property of space partition of energy when time goes to infinity for solutions of linear wave equations that has been uncovered by Duyckaerts, Kenig and Merle [6, 7] in odd dimensions and by Côte, Kenig and Schlag [4] in even dimensions, and to extend it to other dispersive equations. Recall that if w solves the linear wave equation on R× Rd (∂ t −∆x)w = 0 w|t=0 = w0 ∂tw|t=0 = w1 and if one defines the energy at time t outside the wave cone by (1) E(w0, w1, t) = ∫ |x|>|t| [ |∂tw(t, x)| + |∇xw(t, x)| ] dx, then it has been proved in [6, 7] that, if d is odd, either ∀t ≥ 0, E(w0, w1, t) ≥ 1 2 [ ‖w1‖L2 + ‖∇xw0‖ 2 L2 ] or ∀t ≤ 0, E(w0, w1, t) ≥ 1 2 [ ‖w1‖L2 + ‖∇xw0‖ 2 L2 ] . (2) 2020 Mathematics Subject Classification: 35L05, 35Q41.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Microlocal partition of energy for linear wave or\\nSchrödinger equations\",\"authors\":\"Jean-Marc Delort\",\"doi\":\"10.2140/tunis.2022.4.329\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove a microlocal partition of energy for solutions to linear half-wave or Schrödinger equations in any space dimension. This extends well-known (local) results valid for the wave equation outside the wave cone, and allows us in particular, in the case of even dimension, to generalize the radial estimates due to Côte, Kenig and Schlag to non radial initial data. 0 Introduction The goal of this paper is to revisit the property of space partition of energy when time goes to infinity for solutions of linear wave equations that has been uncovered by Duyckaerts, Kenig and Merle [6, 7] in odd dimensions and by Côte, Kenig and Schlag [4] in even dimensions, and to extend it to other dispersive equations. Recall that if w solves the linear wave equation on R× Rd (∂ t −∆x)w = 0 w|t=0 = w0 ∂tw|t=0 = w1 and if one defines the energy at time t outside the wave cone by (1) E(w0, w1, t) = ∫ |x|>|t| [ |∂tw(t, x)| + |∇xw(t, x)| ] dx, then it has been proved in [6, 7] that, if d is odd, either ∀t ≥ 0, E(w0, w1, t) ≥ 1 2 [ ‖w1‖L2 + ‖∇xw0‖ 2 L2 ] or ∀t ≤ 0, E(w0, w1, t) ≥ 1 2 [ ‖w1‖L2 + ‖∇xw0‖ 2 L2 ] . (2) 2020 Mathematics Subject Classification: 35L05, 35Q41.\",\"PeriodicalId\":36030,\"journal\":{\"name\":\"Tunisian Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2022-08-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Tunisian Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/tunis.2022.4.329\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2022.4.329","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
摘要
我们证明了在任何空间维度上线性半波或薛定谔方程解的能量的微局部分配。这扩展了对波锥外波动方程有效的已知(局部)结果,并特别允许我们在偶数维的情况下,将Côte、Kenig和Schlag的径向估计推广到非径向初始数据。0引言本文的目标是重新审视Duyckaerts、Kenig和Merle[6,7]在奇维和Côte、Kenig、Schlag[4]在偶维中发现的线性波动方程解在时间无穷大时能量的空间分配性质,并将其扩展到其他色散方程。回想一下,如果w求解R×Rd上的线性波动方程(⏴t-∆x)w=0 w | t=0=w0⏴[|w1 |L2+|xw0 |L2]或∀t≤0,E(w0,w1,t。(2) 2020数学学科分类:35L05、35Q41。
Microlocal partition of energy for linear wave or
Schrödinger equations
We prove a microlocal partition of energy for solutions to linear half-wave or Schrödinger equations in any space dimension. This extends well-known (local) results valid for the wave equation outside the wave cone, and allows us in particular, in the case of even dimension, to generalize the radial estimates due to Côte, Kenig and Schlag to non radial initial data. 0 Introduction The goal of this paper is to revisit the property of space partition of energy when time goes to infinity for solutions of linear wave equations that has been uncovered by Duyckaerts, Kenig and Merle [6, 7] in odd dimensions and by Côte, Kenig and Schlag [4] in even dimensions, and to extend it to other dispersive equations. Recall that if w solves the linear wave equation on R× Rd (∂ t −∆x)w = 0 w|t=0 = w0 ∂tw|t=0 = w1 and if one defines the energy at time t outside the wave cone by (1) E(w0, w1, t) = ∫ |x|>|t| [ |∂tw(t, x)| + |∇xw(t, x)| ] dx, then it has been proved in [6, 7] that, if d is odd, either ∀t ≥ 0, E(w0, w1, t) ≥ 1 2 [ ‖w1‖L2 + ‖∇xw0‖ 2 L2 ] or ∀t ≤ 0, E(w0, w1, t) ≥ 1 2 [ ‖w1‖L2 + ‖∇xw0‖ 2 L2 ] . (2) 2020 Mathematics Subject Classification: 35L05, 35Q41.