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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
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 | ||||||||
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| 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: |
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| 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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