Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces
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Abstract
This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in \({\mathbb{R}^d}\), where d = 2, 3, with initial data \({B_0\in H^s(\mathbb{R}^d)}\) and \({u_0\in H^{s-1+\epsilon}(\mathbb{R}^d)}\) for \({s > d/2}\) and any \({0 < \epsilon < 1}\). The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking \({\epsilon=0}\) is explained by the failure of solutions of the heat equation with initial data \({u_0\in H^{s-1}}\) to satisfy \({u\in L^1(0,T; H^{s+1})}\); we provide an explicit example of this phenomenon.
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