Prof. Lia Bronsard

424 Hamilton Hall

905-525-9140, ext. 23418

FAX: 905-522-0935

bronsard@mcmaster.ca

Bienvenue!!

TEACHING

Courses for 2010:

Math 2X03 - Vector Calculus I

RESEARCH INTERESTS

The analysis of the structure and evolution of point or line singularities in partial differential equations (PDE) problems has led to a wealth of new mathematical tools, results and questions over the past several years. One important class of PDEs arises from Ginzburg--Landau models: the state of a physical system is described by an ``order parameter'' (a vector-valued function) and a free energy functional with a small parameter epsilon.

I am interested in gradient flows associated to this energy and its stationary equilibria, in the limit as epsilon tends to zero, especially the qualitative information on the singularities (vortices or interfaces separating regions of different phase.) Often these problems lead to challenging analytical questions as well as nice geometrical results for the singularities thus obtained.

DOWNLOAD: CV | Research statement

PREPRINTS

  • “Vortices for a rotating toroidal Bose–Einstein Condensate.” with S. Alama, J. A. Montero. Preprint (2005).
  • "On the shape of interlayer vortices in the Lawrence--Doniach model,'' with S. Alama and Etienne Sandier (2003). To appear in Transactions AMS. PDF

PUBLICATIONS

  • ``Vortices and pinning effects for the Ginzburg--Landau model in multiply connected domains,'' with S. Alama. Comm. Pure Appl. Math. 59 (2006), no. 1, 36–70.
  • ``On the Ginzburg--Landau model of a superconducting ball in a uniform field,'' with S. Alama, J.A. Montero. Preprint (2004), Ann. Inst. H. Poincar Anal. Non Linaire 23 (2006), no. 2, 237–267.
  • "Fractional degree vortices for a spinor Ginzburg--Landau model,'' with S. Alama. Commun. Contemp. Math. 8 (2006), no. 3, 355–380.
  • ``Giant vortex and the breakdown of strong pinning in a rotating Bose-Einstein Condensate,'' with A. Aftalion, S. Alama. Arch. Ration. Mech. Anal. 178 (2005), no. 2, 247–286.
  • "Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions," with S. Alama, J. Math. Phys. 46 (2005), no. 9, 095102, 39 pp.
  • ``Vortices and the lower critical field for a Ginzburg--Landau model with ferromagnetic interactions,'' with S. Alama. Proc. Roy. Soc. Edinburgh Sect. A, vol. 135 (2005), no. 2, pp. 223-252.
  • "Long-time behavior for Competition-Diffusion systems via viscosity comparison'', with S.-A Shim, preprint, 2003. Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 561–581.
  • "On the second critical field for a Ginzburg--Landau model with ferromagnetic interactions,'' with S. Alama. Rev. Math. Phys., vol. 16, No. 2 (2004), 147-174.
  • "Des vortex fractionnaires pour un modele Ginzburg--Landau spineur / Half-integer Vortices for a Spin-coupled Ginzburg--Landau model,'' with S. Alama. C. R. Acad. Sci. Paris, Serie 1, vol. 337 (2003), 243--247.
  • "Minimizers of the Lawrence--Doniach energy in the small-coupling limit: finite width samples in a parallel field'', with S. Alama and J. Berlinsky. Annales IHP-Analyse nonlineaire, vol. 19 (2002), 281--312.
  • "Symmetric Vortex solutions in the U(1) and SO(5) Ginzburg--Landau Models of Superconductivity,'' with S. Alama. In Nonlinear PDE's in Condensed Matter and Reactive Flows, H. Berestycki et Y. Pomeau (eds.),pp. 323--337, Kluwer Academic Publishers, 2002.
  • "Periodic vortex lattices for the Lawrence--Doniach model of layered superconductors in
    a parallel field'', with S. Alama and J. Berlinsky. Commun. Contemp. Math., vol. 3 (2001), no. 3, 457--494.
  • "Vortices with antiferromagnetic cores in the $SO(5)$ theory of superconductivity'', with S. Alama, J. Berlinsky, and T. Giorgi. Phys. Rev. B. vol. 60, no. 9, pp. 6901--6906, 1999.
  • "Analysis of some macroscopic models of high--$T_c$ superconductivity,'' with S. Alama. CRM Proceedings and Lecture Notes, AMS, vol. 27, pp.1--16, 2001.
  • "Vortex Structures for an SO(5) Model of High-T_C Superconductivity and Antiferromagnetism'', with S. Alama and T. Giorgi. Proc. Roy. Soc. Edin., ser. A, vol. 130 (2000), no. 6, 1183--1215.
  • "Uniqueness of Symmetric Vortex Solutions in the Ginzburg--Landau Model of Superconductivity,'' with S. Alama and T. Giorgi. Journal of Functional Analysis, vol. 167, pp. 399--424, 1999.
  • "A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretization for the geometric evolution problem'', with H. Garcke and B. Stoth, Proc. of the Royal Soc. of Edinborough, Vol 128A, pp. 481--506, 1998.
  • "The Singular Limit of a Vector-Valued Reaction-Diffusion Process'', with B. Stoth, Trans. AMS , Vol 350, no. 12, pp. 4931--4953, 1998.
  • "A Singular Limit of the Ginzburg-Landau Equations for Superconductivity and the one-phase Stefan problem'', with B. Stoth, Annales IHP-Analyse nonlin\'eaire, Vol 15,no. 3, pp. 371--397, 1998.
  • Slow motion in the gradient theory of phase transitions via energy and spectrum,'', with N. Alikakos and G. Fusco, Calc. of Variation and PDE, Vol 6, pp. 39--66, 1998.
  • "Volume Preserving Mean Curvature Flow as a Limit of a Nonlocal Ginzburg-Landau Equation'', with B. Stoth, SIAM J. Math. Anal., Vol 28, no 4, pp.769-807, July 1997.
  • "Stationary layered solutions in $\bold R^2$ for an Allen-Cahn system with multiple well potential'' with S. Alama and C. Gui, Calc. of Variation and PDE, vol. 5, pp 359-390, 1997.
  • "A Three Layered Minimizer in $\bold R^2$ for a Variational Problem with a Symmetric Three Well Potential'', with C. Gui and M. Schatzman, Comm. Pure Appld. Math., vol 49, pp 677-715, 1996.
  • "On the Existence of High Multiplicity Interfaces'', with B. Stoth, Math. Res. Lett., vol 3, pp 41-50, 1996.
  • "A Numerical Method for Tracking Curve Networks Moving with Curvature Motion'', with B. Wetton, Jour. Comp. Phys., vol 120, pp 66-87, 1995.
  • "On Three-Phase Boundary Motion and the Singular Limit of a Vector-Valued Ginzburg-Landau Equation,'' with F. Reitich, Arch Rat. Mech. and Analysis, vol 124, no 4, pp 355-379, 1993.
  • "Front Propagation for Reaction-Diffusion Equations of Bistable Type'', with G. Barles and P. E. Souganidis, Ann. I. H. P.-non lineaire, vol 9 no 5, pp 479-496, 1992.
  • "On the Slow Dynamics for the Cahn-Hilliard Equation in One Space Dimension'', with D. Hilhorst, Proc. Roy. Soc. Lon.-series A (math. phys. Sci.}, vol 439 no 1907, pp 669-682, 1992.
  • "Motion by Mean Curvature as the Singular Limit of Ginzburg-Landau Dynamics'', with R. V. Kohn, Jour. of Diff. Eq., vol. 90, pp. 211-237, 1991.
  • "On the Slowness of Phase Boundary Motion in One Space Dimension'', with R. V. Kohn, Comm. Pure Appl. Math., vol. XLIII, pp. 983-997, 1990.