Stable and Unstable Shock Formation of the Burgers-Hilbert Equation
Name
yang-rxyang-phd-math-2022-thesis.pdf
Description
Thesis PDF
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1.42 MB
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Adobe PDF
Checksum (MD5)
eca98cb33d58e9ac837a4e714d5d5364
Author(s)
Yang, Ruoxuan
Advisor(s)
Staffilani, Gigliola
Date Issued
May 2022
Publisher
Massachusetts Institute of Technology
Abstract
The study of singularities has been an important part in the analysis of PDEs. One key type of singularities is shock. In many cases the shock has a self-similarity structure. Recently, the modulated self-similarity technique has achieved success in fluid dynamic equations. In this thesis, we apply this technique to establish finite time shock formation of the Burgers-Hilbert equation. The shocks are asymptotic selfsimilar at one single point. The shocks can be stable or unstable, both of which have an explicitly computable singularity profile, and the shock formation time and location are described by explicit ODEs. For the stable shock, the initial data are in an open set in the đ»â” -norm, and the shock profile is a cusp with Hölder 1/3 continuity. For the unstable shock, the initial data are in a co-dimension 2 subset of the đ»âč space, and the shock profile is of Hölder 1/5 continuity. Both cases utilize a transformation to appropriated self-similar coordinates, the quantitative properties of the corresponding self-similar solution to the inviscid Burgersâ equation, and transport estimates. In the case of unstable shock, we, in addition, control the two unstable directions by Newtonâs iteration.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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