Strichartz estimates and local regularity for the elastic wave equation with singular potentials

Seongyeon Kim; Yehyun Kwon; Ihyeok Seo

doi:10.3934/dcds.2020344

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2021, Volume 41, Issue 4: 1897-1911. Doi: 10.3934/dcds.2020344

This issue Previous Article Reversible perturbations of conservative Hénon-like maps Next Article Diffeomorphisms with a generalized Lipschitz shadowing property

Strichartz estimates and local regularity for the elastic wave equation with singular potentials

1.
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
2.
School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

^* Corresponding author: Ihyeok Seo
^* Corresponding author: Ihyeok Seo

Received: June 30, 2020

Revised: July 31, 2020

Early access: October 2020

Published: April 2021

The second author is supported by a KIAS Individual Grant (MG073701) at Korea Institute for Advanced Study and NRF-2020R1F1A1A01073520. The thrid author is supported by NRF-2019R1F1A1061316

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Abstract

We obtain weighted $ L^2 $ estimates for the elastic wave equation perturbed by singular potentials including the inverse-square potential. We then deduce the Strichartz estimates under the sole ellipticity condition for the Lamé operator $ -\Delta^\ast $. This improves upon the previous result in [1] which relies on a stronger condition to guarantee the self-adjointness of $ -\Delta^\ast $. Furthermore, by establishing local energy estimates for the elastic wave equation we also prove that the solution has local regularity.

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Mathematics Subject Classification: Primary: 35B45, 35B65; Secondary: 35L05.

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