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

Abstract

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

Lower Bounds on Blowing-Up Solutions of the Three-Dimensional Navier--Stokes Equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$

“…A proof of the first can be found in Fefferman et al (2014); the second is an immediate consequence of Bernstein's inequality (see McCormick, Robinson, and Rodrigo (2013), for example).…”

“…In fact we prove a somewhat more general result in Corollary 5.4, which in turn is a consequence of the following commutator estimate (cf. Kato and Ponce (1988), Fefferman et al (2014)). …”

“…The counterexample in the appendix to Fefferman et al (2014) shows that one cannot remove the second term on the right-hand side, at least in the case n = 2. We will use this estimate in the form of the following corollary, which provides a partial generalisation of Lemma 1.1 from Chemin (1992).…”

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

“…In a previous paper [6] we proved a local existence result for these equations taking arbitrary initial data in u 0 , B 0 ∈ H s (R d ) with s > d/2 for d = 2, 3. However, given the presence of the diffusive term in the equation for u, it is natural DSMcC is supported by Leverhulme Trust research project grant RPG-2015-69.…”

On the uniqueness for the 2D MHD equations without magnetic diffusion

“…Fefferman et al [4] obtained the local existence and uniqueness for (1.1) and related models with the initial data (u 0 , B 0 ) ∈ H s (R d ), s > Very recently, Chemin et al in [2] obtain the local existence for (1.1) in 2D and 3D.…”