Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 2 of 2
Full-Text Articles in Physics
Stability And Clustering Of Self-Similar Solutions Of Aggregation Equations, Hui Sun, David Uminsky, Andrea L. Bertozzi
Stability And Clustering Of Self-Similar Solutions Of Aggregation Equations, Hui Sun, David Uminsky, Andrea L. Bertozzi
Mathematics
In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρ t = ∇ · (ρ∇K * ρ) in Rd , d ⩾ 2, where K(r) = r γ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math.70, 2582–2603 (Year: 2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the …
Generalized Helmholtz-Kirchhoff Model For Two-Dimensional Distributed Vortex Motion, Raymond J. Nagem, Guido Sandri, David Uminsky, C. Eugene Wayne
Generalized Helmholtz-Kirchhoff Model For Two-Dimensional Distributed Vortex Motion, Raymond J. Nagem, Guido Sandri, David Uminsky, C. Eugene Wayne
Mathematics
The two-dimensional Navier-Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz-Kirchhoff model for the evolution of point vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the …