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Singular Solutions and Pattern Forma...

Sun, Hui.

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Singular Solutions and Pattern Formation in Aggregation Equations.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Singular Solutions and Pattern Formation in Aggregation Equations./
Author:	Sun, Hui.
Description:	106 p.
Notes:	Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.
Contained By:	Dissertation Abstracts International74-09B(E).
Subject:	Applied Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3562367
ISBN:	9781303099687

Singular Solutions and Pattern Formation in Aggregation Equations.
Sun, Hui.

Singular Solutions and Pattern Formation in Aggregation Equations. - 106 p.

Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.

Thesis (Ph.D.)--University of California, Los Angeles, 2013.

In this work, we study singular solutions and pattern formation in aggregation equations and more general active scalar problems.

ISBN: 9781303099687Subjects--Topical Terms:

1669109
Applied Mathematics.

Singular Solutions and Pattern Formation in Aggregation Equations.
LDR:02729nam a2200313 4500 001 1960756
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008 150210s2013 ||||||||||||||||| ||eng d
020 $a 9781303099687
035 $a (MiAaPQ)AAI3562367
035 $a AAI3562367
040 $a MiAaPQ $c MiAaPQ
100 1 $a Sun, Hui. $3 2096467
245 1 0 $a Singular Solutions and Pattern Formation in Aggregation Equations.
300 $a 106 p.
500 $a Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.
500 $a Adviser: Andrea L. Bertozzi.
502 $a Thesis (Ph.D.)--University of California, Los Angeles, 2013.
520 $a In this work, we study singular solutions and pattern formation in aggregation equations and more general active scalar problems.
520 $a We derive a generalization of the Birkhoff-Rott equation to the case of active scalar problems with both gradient and divergence free structures. We present numerical simulations of this model demonstrating how the gradient part and the divergence free part of K influence each other and cause some nonlinear effects. Examples include superfluids, classical fluids and swarming models.
520 $a The rest of this thesis focuses on aggregation models with gradient flow structure. The discrete version of the continuum aggregation equation is the kinematic equation x&dot;i = -mi sumj≠i &dtri; U(|xi - x j|), ∀ 1 ≤ i ≤ N. For both discrete and continuum versions, we use linear stability analysis of a ring equilibrium to classify the morphology of patterns in two dimensions. Conditions are identified that assure the linear well-posedness of the ring. In addition, weakly nonlinear theory and numerical simulations demonstrate how a ring can bifurcate to more complex equilibria. Moreover, linear stability analysis of clusters equilibrium patterns are also investigated in both two-dimensional and higher-dimensional cases.
520 $a We then apply our stability results of ring patterns and clusters patterns to a family of exact collapsing similarity solutions to the aggregation equation with pairwise potential U(r) = rgamma/gamma. It was previously observed that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for gamma > 2 in all dimensions. The stability analysis for ring patterns and clusters patterns shows that the collapsing shell solution is stable for 2 < gamma 4. This holds in all spatial dimensions.
590 $a School code: 0031.
650 4 $a Applied Mathematics. $3 1669109
650 4 $a Physics, General. $3 1018488
690 $a 0364
690 $a 0605
710 2 $a University of California, Los Angeles. $b Mathematics 0540. $3 2096468
773 0 $t Dissertation Abstracts International $g 74-09B(E).
790 $a 0031
791 $a Ph.D.
792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3562367