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Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces

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Fefferman, Charles L., McCormick, David S., Robinson, James C. and Rodrigo, Jose L. (2017) Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Archive for Rational Mechanics and Analysis, 223 (2). pp. 677-691. doi:10.1007/s00205-016-1042-7

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Official URL: http://dx.doi.org/10.1007/s00205-016-1042-7

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Abstract

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in RdRd , where d = 2, 3, with initial data B0∈Hs(Rd)B0∈Hs(Rd) and u0∈Hs−1+ϵ(Rd)u0∈Hs−1+ϵ(Rd) for s>d/2s>d/2 and any 0<ϵ<10<ϵ<1 . The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ϵ=0ϵ=0 is explained by the failure of solutions of the heat equation with initial data u0∈Hs−1u0∈Hs−1 to satisfy u∈L1(0,T;Hs+1)u∈L1(0,T;Hs+1) ; we provide an explicit example of this phenomenon.

Item Type: Journal Article
Subjects: Q Science > QC Physics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Magnetohydrodynamics, Equations -- Numerical solutions
Journal or Publication Title: Archive for Rational Mechanics and Analysis
Publisher: Springer
ISSN: 0003-9527
Official Date: February 2017
Dates:
DateEvent
February 2017Published
1 September 2016Available
15 August 2016Accepted
Volume: 223
Number: 2
Page Range: pp. 677-691
DOI: 10.1007/s00205-016-1042-7
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access
Funder: Leverhulme Trust (LT), National Science Foundation (U.S.) (NSF), European Research Council (ERC)
Grant number: Grant RPG-2015-69 (LT), Grant DMS-09-01040 (NSF), Grant 616797 (ERC)

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