Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations

Article


Robinson, J. and Sharples, N. 2013. Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations. Journal of Differential Equations. 254, pp. 4144-4167. https://doi.org/10.1016/j.jde.2013.02.012
TypeArticle
TitleDimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations
AuthorsRobinson, J. and Sharples, N.
Abstract

We provide a criterion for a generalised flow solution of a non-autonomous ordinary differential equation to avoid a subset of the phase space. This improves on that established by Aizenman for the autonomous case, where avoidance is guaranteed if the underlying vector field is sufficiently regular and the subset has sufficiently small box-counting dimension. We define the r-codimension print of a subset $S\subset \R^{n}\times [0,T]$, which is a subset of $(0,\infty]^{2}$ that encodes the dimension of S in a way that distinguishes spatial and temporal detail. We prove that the subset S is avoided by a generalised flow solution with underlying vector field in $L^{p}(0, T; L^{q}(R^{n}))$ if the Holder conjugates (q^{*}; p^{*}) are in the r-codimension print of S.

LanguageEnglish
PublisherElsevier
JournalJournal of Differential Equations
ISSN0022-0396
Publication dates
Online05 Mar 2013
Print15 May 2013
Publication process dates
Deposited15 Oct 2015
Accepted04 Dec 2012
Output statusPublished
Accepted author manuscript
Web address (URL)http://www.sciencedirect.com/science/article/pii/S0022039613000879
Digital Object Identifier (DOI)https://doi.org/10.1016/j.jde.2013.02.012
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