Higher order commutator estimates and local existence for the non-resistive MHD equations and related models

By Charles Fefferman, D. McCormick, James C. Robinson, José L. Rodrigo. 2014

246 citations recorded by the source corpus.

This paper establishes the local-in-time existence and uniqueness of strong solutions in Hs for s > n/2 to the viscous, non-resistive magnetohydrodynamics (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 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).

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6fccfe4d8e912c153bd5d676f85ccb8429aafb74

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

On Kato–Ponce and fractional Leibniz (2016, background): “In recent [9], Fefferman, McCormick, Robinson and Rodrigo (FMRR) considered a class of non-resistive MHD equations and proved a new Kato-Ponce type inequality ‖D((u · ∇)B) − (u · ∇)(DB)‖L2(Rd) .”
Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces (2016, unknown): “Jiu & Niu (2006) established the local existence of solutions in 2D for initial data in H for integer s ≥ 3, and in a previous paper (Fefferman et al., 2014) we proved a local existence result for these equations when u0, B0 ∈ H(R) with s > d/2 for d = 2, 3.”
Local existence for the non-resistive MHD equations in Besov spaces (2015, background): “A detailed discussion of previous work on the subject can be found in the introduction to Fefferman et al. (2014).”
Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces (2016, unknown): “1) in R , d = 2, 3 with the initial data (u0, B0) ∈ H (R) × H(R), s > d 2 in [7] and (u0, B0) ∈ H (R) × H(R), s > d 2 , 0 < ε < 1 in [8].”
Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium (2019, background): “GLOBAL SOLUTIONS OF 3D MHD EQUATIONS WITH MIXED DISSIPATION 3 some of the other exciting results in [9, 13, 14, 16–18, 23–25, 35, 38–43, 45] and the references therein.”