On the existence and stability of self-similar blowup in nonlinear wave equations
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2018
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ndltd-OhioLink-oai-etd.ohiolink.edu-osu15236929383167942021-08-03T07:06:04Z On the existence and stability of self-similar blowup in nonlinear wave equations Glogic, Irfan Mathematics We study the existence and stability of singularity formation in nonlinear wave equations.First, we formulate the problem of mode stability of self-similar blowup solutions to a class of nonlinear wave equations. We then develop a method of proving mode stability in this context and illustrate it on two important examples, the spherically symmetric Yang-Mills equation and equivariant wave maps into a sphere, for which the problem was open for almost a decade. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations. This part of the thesis is based on two research papers that already appeared in print, see [13,16].Next, we study the singularity formation in wave maps into a negatively curved target manifold. More precisely, we consider wave maps on (1+d)-dimensional Minkowski space. For each dimension d≤8 we construct a negatively curved, d-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem. In particular, we illustrate how the problem of mode stability plays a decisive role in the proof of the (nonlinear) stability of self-similar blowup. This part is based on a joint work with Roland Donninger [19]. 2018-09-18 English text The Ohio State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=osu1523692938316794 http://rave.ohiolink.edu/etdc/view?acc_num=osu1523692938316794 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
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NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics |
| spellingShingle |
Mathematics Glogic, Irfan On the existence and stability of self-similar blowup in nonlinear wave equations |
| author |
Glogic, Irfan |
| author_facet |
Glogic, Irfan |
| author_sort |
Glogic, Irfan |
| title |
On the existence and stability of self-similar blowup in nonlinear wave equations |
| title_short |
On the existence and stability of self-similar blowup in nonlinear wave equations |
| title_full |
On the existence and stability of self-similar blowup in nonlinear wave equations |
| title_fullStr |
On the existence and stability of self-similar blowup in nonlinear wave equations |
| title_full_unstemmed |
On the existence and stability of self-similar blowup in nonlinear wave equations |
| title_sort |
on the existence and stability of self-similar blowup in nonlinear wave equations |
| publisher |
The Ohio State University / OhioLINK |
| publishDate |
2018 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=osu1523692938316794 |
| work_keys_str_mv |
AT glogicirfan ontheexistenceandstabilityofselfsimilarblowupinnonlinearwaveequations |
| _version_ |
1719453667531161600 |