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- Publisher
- Springer Journals
- Copyright
- Copyright © Springer Nature B.V. 2019
- Subject
- Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-019-00274-0
- Publisher site
- See Article on Publisher Site
Abstract
In this paper we study the decay characterization in the space HσK+1(R3)×HσK(R3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$H^{K+1}_{\sigma }(\mathbb{R}^{3})\times H^{K}_{\sigma }(\mathbb{R} ^{3})$\end{document} of solutions to a 3D magnetohydrodynamics-α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha $\end{document} model (MHD-α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha $\end{document} model for short) in the whole space R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathbb{R}^{3}$\end{document}, namely,∥∇mu(t)∥2+α2∥∇m+1u(t)∥2+∥∇mB(t)∥2≤C(1+t)−min(r∗+m+32,m+52),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \bigl\| \nabla ^{m} u(t)\bigr\| ^{2}+\alpha ^{2}\bigl\| \nabla ^{m+1} u(t)\bigr\| ^{2}+ \bigl\| \nabla ^{m} \mathbf{B}(t)\bigr\| ^{2}\leq C(1+t)^{-\min (r^{*}+m+\frac{3}{2}, m+ \frac{5}{2} )}, $$\end{document} where m≤K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$m\leq K$\end{document}, r∗=min(r∗(u0),r∗(B0))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$r^{*}=\min (r^{*}(u_{0}), r^{*}(\mathbf{B}_{0}))$\end{document} is the decay character of the initial datum (u0,B0)∈HσK+1(R3)×HσK(R3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(u_{0}, \mathbf{B}_{0}) \in H^{K+1}_{\sigma }(\mathbb{R}^{3})\times H^{K}_{\sigma }( \mathbb{R}^{3})$\end{document}. We also get the optimal lower bounds for decay rates of solutions to the MHD-α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha $\end{document} model when −3/2<r∗≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$-{3}/{2}< r^{*}\leq 1$\end{document}. The results obtained are extensions/improvements of previous results on decay rates of solutions to this 3D MHD-α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha $\end{document} model, the classical 3D MHD model, and the 3D viscous Camassa-Holm equations.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Jun 24, 2020