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Title: Existence and uniqueness theory for MHD systems
Author: McCormick, David S.
ISNI:       0000 0004 5350 0041
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2014
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This thesis establishes the existence and uniqueness of solutions to certain systems of equations connected to magnetohydrodynamics (MHD). The models have potential applications to the method of magnetic relaxation introduced by Moffatt (J. Fluid. Mech. 159, 359-378, 1985) to construct stationary Euler ows with non-trivial topology. Firstly, we prove existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in 2D, and existence of weak solutions in 3D; we consider the standard equations of MHD with the advective terms removed from the velocity equation. Despite the apparent simplicity of the model, the proof in 2D requires results that are at the limit of what is available, including elliptic regularity in L1 and a strengthened form of the Ladyzhenskaya inequality ||f||L3≤c||f||1/2 L2,∞||∇f||1/2 L2 Secondly, we establish the local-in-time existence and uniqueness of strong solutions in Hs for s>n=2 to the viscous, non-resistive MHD equations in Rn, n = 2; 3, as well as for a related model where the advection terms are removed from the velocity equation (the above parabolic-elliptic system with zero resistivity). The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891-907, 1988). Finally, we generalise the results of the previous chapter to prove the localin-time existence of strong solutions in the Besov space Bn/2 2;1 (ℝn) to the viscous, non-resistive MHD equations in ℝn.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EPSRC) (EP/H023364/1)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics