Project Details
Projekt
Vortex dynamics and blow-up phenomena in two dimensions
Subject Area
Mathematics
Term
since 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 444615381
The dynamics of N point-vortices in a domain or on a Riemann surface is described by a first order Hamiltonian system that goes back to Kirchhoff and Routh (1880s). There are many fascinating relations between this Hamiltonian system and partial differential equations. It has been derived as a singular limit from the 2D Euler equations for an ideal fluid assuming that the vorticity is concentrated in N different points, analogous to the concept of point-masses in celestial mechanics. The Kirchhoff-Routh Hamiltonian depends on the Green function of the Dirichlet Laplace operator and is singular when vortices come close to each other or approach the boundary of the domain. The same type of Hamilton function appears in other problems from mathematical physics, for instance as renormalized energy for the dynamics of vortices in the Ginzburg-Landau theory.Statistical mechanics for large numbers of vortices leads to partial differential equations of mean field type, going back to work of Onsager on two-dimensional turbulence. Surprisingly, mean field type equations arise also in differential geometry. Solutions can be obtained with variational methods, the associated Euler-Lagrange functionals, however, may not be compact for certain critical parameter values: there may be orbits of the gradient flow that converge in the sense of measures to sums of Dirac deltas which are located at critical points of a vortex Hamiltonian that is similar to but more complicated than the Kirchhoff-Routh function. Therefore information about critical points of vortex Hamiltonians is crucial for understanding the loss of compactness of the functionals associated to mean field type equations. This information is also important when one wants to construct blow-up solutions, i.e. families of solutions that converge in the sense of measures towards sums of Dirac deltas as a parameter approaches a critical value. As a last example where vortex type Hamilton functions appear we mention the study of blow-up phenomena for Toda systems.The principal goals of this project are:1. to investigate point-vortex dynamics in a domain or on a Riemann surface, in particular to find equilibria as critical points of the Kirchhoff-Routh Hamiltonian, and to find periodic solutions.2. to develop Morse theory for mean field type equations, in particular to find critical points of vortex Hamiltonians, the ultimate goal being the proof of existence and multiplicity of solutions of mean field equations as well as the construction of blow-up solutions.3. to develop Morse theory for the SU(3) Toda system with Neumann boundary conditions or on surfaces in order to establish existence and multiplicity of solutions and to construct blow-up solutions.
DFG Programme
Research Grants
International Connection
Italy, Tunisia
Cooperation Partners
Professor Dr. Mohamed Ben Ayed; Professorin Dr. Angela Pistoia
