WK Differential Equations - Student Member:
Gonca Aki

Faculty of Mathematics
Vienna University
Nordbergstr. 15, C, 4. floor (room 420)
A 1090 Wien, Austria

Telephone: (++43)(1)4277-50674
Fax: (++43)(1)4277-50665

Advisor: Peter Markowich

My main research interest has recently developed on the semi-relativistic Hartree systems.

Semi-relativistic Hartree systems have interesting applications in the quantum theory as a mean-field limit of large systems of self-interacting, relativistic bosons. Depending on the sign of the self-interacting potential it might model either repulsive or attractive interacting particles. While the repulsive case can be regarded as a relativistic correction to the usual Hartree systems which models electron-electron interactions, the attractive case commonly used to model self-gravitating boson stars.

The (Global) well-posedness of this problem has been understood very well with a smallness assumption on the initial mass in the case of self-gravitating particles. Under the supervision of Peter Markowich and Christof Sparber, we have recently established the classical limit (which is relativistic Vlasov-Poisson equation) of this system with Hartree type nonlinearity using Wigner measure approach whose efficiency has been proved in numerous applications.


Another interesting approach could be proving the existence of ground states of the self-gravitating system. This turns out to be a challenging problem for the mixed states since behavior of the energy functional (which is supposed to be minimized by the ground states) along the minimizing sequences is not easy to be understood.


And lately, with collaboration of Federica Pezzotti, I have been introduced to the idea of understanding the mean-field limits at the same time with either semi-classical or non-relativistic limits. Roughly speaking, we want to understand if the mean-field limit can commute with semi-classical (or non-relativistic) limit.


Work in progress:

[1] “Existence of steady states of gravitating Hartree systems”

[2] (With F. Pezzotti) “Commutator behaviour of mean field and non-relativistic limits”

Papers in journals:

[3] (With P. A. Markowich, C. Sparber) “Classical limit for semi-relativistic Hartree systems”, J. Math. Phys. 49 (2008), 102110 - 102120.

Papers in refereed conference proceedings:

[4] (With H. A. Erbay) "Variational Formulation of Dynamical Equations of Hiperelastic Membrane Tubes and Wave Propagation", Proceedings of 14th National Mechanics Congress, pp. 42-49, Hatay, Turkey, 2005 (in Turkish).




  • 2007, Feb 21-23: Mini course on "Computational High Frequency Waves" by Shi Jin, WPI, Vienna, Austria
  • 2007, Jul 9-14: DEASE Summer-school and Annual Meeting, WPI, Vienna, Austria
  • 2007, Jul 22 - 27: Summer School "Topics in Nonlinear PDEs", CIM/UC Coimbra, Portugal
  • 2008, April 7-11: Summer School "Topics and PDE's and applications 2008", FisyMat Granada, Spain
  • 2008, May 5-9: Short course "Multi-scale problems and models in traffic flow", WPI, University of Vienna
  • 2008, June 9-14: Summer School "Methods and Models of kinetic theory", Porto Ercole, Italy
  • 2008, Sept 18-20: DEASE Summer-school and Annual Meeting, Hamburg Uni, Hamburg, Germany
  • 2008, Sept 24-27: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices", WPI, Vienna, Austria

Talk: “Classical limit for semi-relativistic Hartree systems

  • 2009, March 9- June 12: Supported (by IPAM) research stay at scientific program “Quantum and kinetic transport: Analysis, computations, and new applications”, IPAM, UCLA, USA