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Singular Solutions and Pattern Formation in Aggregation Equations


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

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.

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. 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.

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) = rν/ν. It was previously observed that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for ν > 2 in all dimensions. The stability analysis for ring patterns and clusters patterns shows that the collapsing shell solution is stable for 2 < ν < 4, while always unstable and destabilizes into clusters that form a simplex for ν > 4. This holds in all spatial dimensions.

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