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The partial differential equation ut +uux = μuxx -
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © Fondation Carl-Herz and Springer Nature Switzerland AG 2021
- ISSN
- 2195-4755
- eISSN
- 2195-4763
- DOI
- 10.1007/s40316-021-00175-5
- Publisher site
- See Article on Publisher Site
Abstract
We consider the equation qt+qqx=qxx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q_t+qq_x=q_{xx}$$\end{document} for q:R×(0,∞)→H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q:{{\mathbf {R}}}\times (0,\infty )\rightarrow {\mathbf {H}}$$\end{document} (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. Similar results are also established for the same equation in S1×(0,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbf{S}^1\times (0,\infty )$$\end{document}, where S1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbf{S}^1$$\end{document} is the standard one-dimensional circle.
Journal
Annales mathématiques du Québec – Springer Journals
Published: Apr 1, 2022
Keywords: Quaternionic Burgers equation; Generic well-posedness; Singularities; 35A20; 35K45; 35Q58