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Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations
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Robinson, James C. and Sharples, Nicholas (2013) Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations. Journal of Differential Equations, Volume 254 (Number 10). pp. 4144-4167. doi:10.1016/j.jde.2013.02.012 ISSN 0022-0396.
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Official URL: http://dx.doi.org/10.1016/j.jde.2013.02.012
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⊂Rn×[0,T], which is a subset of (0,∞]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 Lp(0,T;Lq(Rn)) with 1⩽p,q⩽∞ if the Hölder conjugates (q⁎,p⁎) are in the r-codimension print of S.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal of Differential Equations | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-0396 | ||||
Official Date: | 2013 | ||||
Dates: |
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Volume: | Volume 254 | ||||
Number: | Number 10 | ||||
Page Range: | pp. 4144-4167 | ||||
DOI: | 10.1016/j.jde.2013.02.012 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Grant number: | EP/G007470/1 (EPSRC) |
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