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Unique weak solutions of the magnetohydrodynamic equations with fractional dissipation
Author(s) -
Dai Yichen,
Ji Ruihong,
Wu Jiahong
Publication year - 2020
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201900290
Subject(s) - uniqueness , bilinear interpolation , magnetohydrodynamic drive , mathematics , dissipation , magnetohydrodynamics , mathematical analysis , term (time) , bilinear form , physics , magnetic field , thermodynamics , statistics , quantum mechanics
This paper examines the existence and uniqueness of weak solutions to the d ‐dimensional magnetohydrodynamic (MHD) equations with fractional dissipation( − Δ ) α u and fractional magnetic diffusion( − Δ ) β b . The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting u ∈ L ∞ ( 0 , T ; B 2 , 1 d / 2 − 2 α + 1( R d ) )and b ∈ L ∞ ( 0 , T ; B 2 , 1 d / 2R d ) )when α > 1 / 2 , β ≥ 0 and α + β ≥ 1 . The case when α = 1 with ν > 0 and η = 0 has previously been studied in [7, 19]. However, their approaches can not be directly extended to the fractional case when α < 1 due to the breakdown of a bilinear estimate. By decomposing the bilinear term into different frequencies, we are able to obtain a suitable upper bound on the bilinear term for α < 1 , which allows us to close the estimates in the aforementioned Besov spaces.