Global versions of the Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations
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- by Leonardo Enrique Abbrescia and Willie Wai Yeung Wong PDF
- Trans. Amer. Math. Soc. 374 (2021), 773-802 Request permission
Abstract:
We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $L^p$ ($p < \infty$) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $L^\infty$ endpoint was first introduced by Klainerman and is the basis of the classical vector field method, which is now one of the standard techniques for studying long-time behavior of nonlinear evolution equations. We were motivated in our pursuit by settings where the vector field method is applied to an energy hierarchy with growing higher order energies. In these settings the use of the $L^p$ endpoint versions of Sobolev inequalities can allow one to gain essentially one derivative in the estimates, which would then give a corresponding gain of decay rate. The paper closes with the analysis of one such model problem, where our new estimates provide an improvement.References
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Additional Information
- Leonardo Enrique Abbrescia
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 1190580
- ORCID: 0000-0001-5188-5265
- Email: leonardo.abbrescia@vanderbilt.edu
- Willie Wai Yeung Wong
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- MR Author ID: 877992
- ORCID: 0000-0002-5050-2119
- Email: wongwwy@math.msu.edu
- Received by editor(s): September 26, 2019
- Received by editor(s) in revised form: August 11, 2020
- Published electronically: November 25, 2020
- Additional Notes: The first author was supported by an NSF Graduate Research Fellowship (DGE-1424871).
The second author was supported by a Collaboration Grant from the Simons Foundation, #585199. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 773-802
- MSC (2010): Primary 35A23, 35L71, 35B45
- DOI: https://doi.org/10.1090/tran/8277
- MathSciNet review: 4196377