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

By Charles Fefferman, D. McCormick, James C. Robinson, José L. Rodrigo. Archive for Rational Mechanics and Analysis, 2016

148 citations recorded by the source corpus.

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^d}$$\end{document}, where d = 2, 3, with initial data B0∈Hs(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B_0\in H^s(\mathbb{R}^d)}$$\end{document} and u0∈Hs-1+ϵ(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_0\in H^{s-1+\epsilon}(\mathbb{R}^d)}$$\end{document} for s>d/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s > d/2}$$\end{document} and any 0<ϵ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0 < \epsilon < 1}$$\end{document}. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ϵ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\epsilon=0}$$\end{document} is explained by the failure of solutions of the heat equation with initial data u0∈Hs-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_0\in H^{s-1}}$$\end{document} to satisfy u∈L1(0,T;Hs+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u\in L^1(0,T; H^{s+1})}$$\end{document}; we provide an explicit example of this phenomenon.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/24a68b159e8c3b92bcd601d16d98de7b6054ea67

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Citations in the corpus

Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces (2016, background): “al-in-time existence of strong solutions to (1.1) in Rd, d = 2,3 with the initial data (u 0,B0) ∈ Hs(Rd) × Hs(Rd), s > d 2 in [7] and (u0,B0) ∈ Hs−1−ε(Rd) × Hs(Rd), s > d 2 , 0 < ε < 1 in [8]. Chemin et al. in [2] proved the local existence of solutions to (1.1) in Rd, d = 2,3 with the i”
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.”