# Space-time L 2 estimates, regularity and almost global existence for elastic waves

Space-time L 2 estimates, regularity and almost global existence for elastic waves AbstractIn this paper, we first establish a kind of weighted space-time L2{L^{2}}estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author[D. Zha, Space-time L2L^{2}estimates for elastic waves and applications,J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama,On almost global existence and local well posedness for some 3-D quasi-linear wave equations,Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John,Almost global existence of elastic waves of finite amplitude arising from small initial disturbances,Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris,On almost global existence for nonrelativistic wave equations in 3D,Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularityof initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case,we establish the almost global existence of a low regularity solution for every smallinitial data in H3×H2{H^{3}\times H^{2}}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

# Space-time L 2 estimates, regularity and almost global existence for elastic waves

Forum Mathematicum, Volume 30 (5): 17 – Sep 1, 2018
17 pages

/lp/de-gruyter/space-time-l-2-estimates-regularity-and-almost-global-existence-for-2EuV0bivG6
Publisher
de Gruyter
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5337
eISSN
1435-5337
DOI
10.1515/forum-2018-0050
Publisher site
See Article on Publisher Site

### Abstract

AbstractIn this paper, we first establish a kind of weighted space-time L2{L^{2}}estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author[D. Zha, Space-time L2L^{2}estimates for elastic waves and applications,J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama,On almost global existence and local well posedness for some 3-D quasi-linear wave equations,Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John,Almost global existence of elastic waves of finite amplitude arising from small initial disturbances,Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris,On almost global existence for nonrelativistic wave equations in 3D,Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularityof initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case,we establish the almost global existence of a low regularity solution for every smallinitial data in H3×H2{H^{3}\times H^{2}}.

### Journal

Forum Mathematicumde Gruyter

Published: Sep 1, 2018

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