Justin Holmer

Brown Univ. Math. Dept.
Box 1917
151 Thayer Street
Providence, RI 02912
Email:
Office: 210 Kassar-Gould House
Phone: (401) 863-1123
Fax: (401) 863-9013

I am an associate professor in the Department of Mathematics at Brown University. I am currently Director of Graduate Studies for the department.

Current Courses

  • Fall 2014, MATH 237, PDE I, MWF 10:00-10:50 in Kassar 105
  • Spring 2015, MATH 226, Complex Analysis II, MWF 10:00-10:50.

Research Papers and Preprints

  • (with Xuwen Chen) Correlation structures, many-body scattering processes and the derivation of the Gross-Pitaevskii hierarchy, preprint, pdf.
  • (with Xuwen Chen) Focusing quantum many-body dynamics II: the rigorous derivation of the 1D focusing cubic nonlinear Schroedinger equation from 3D, preprint, pdf.
  • (with Xuwen Chen) Focusing quantum many-body dynamics: the rigorous derivation of the 1D focusing cubic nonlinear Schroedinger equation, preprint pdf.
  • (with Xuwen Chen) On the Klainerman-Machedon conjecture for quantum BBGKY hierarchy with self-interaction, to appear in J. Eur. Math. Soc. (JEMS) pdf.
  • (with Galina Perelman and Svetlana Roudenko) A solution to the focusing 3d NLS that blows up on a contracting sphere, to appear in Transactions of the AMS, preprint pdf.
  • (with Xuwen Chen) On the rigorous derivation of the 2D cubic nonlinear Schroedinger equation from 3D quantum many-body dynamics, to appear in Archive for Rational Mechanics and Analysis, preprint pdf.
  • (with M. Christ and D. Tataru) Low regularity a priori bounds for the modified Korteweg-de Vries equation, to appear in Libertas Mathematica (new series), preprint pdf.
  • (with Quanhui Lin) Phase-driven interaction of widely separated nonlinear Schroedinger solitons, to appear in Journal of Hyperbolic Differential Equations, pdf.
  • (with Svetlana Roudenko) Blow-up solutions on a sphere for the 3d quintic NLS in the energy space, Analysis & PDE 5-3 (2012), pp. 475--512, pdf.
  • (with Svetlana Roudenko) A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle, AMRX Appl Math Res Express (2011) 2011 (1): 23-94, doi: 10.1093/amrx/abq016, pdf.
  • Dynamics of KdV solitons in the presence of a slowly varying potential, to appear in IMRN Internat. Math. Res. Notices, pdf.
  • (with Galina Perelman and Maciej Zworski, and an appendix by Bernd Sturmfels) Effective dynamics of double solitons for perturbed mKdV, Communications in Mathematical Physics 305 (2011) pp. 363--425, pdf.
    The numerical illustrations of the results and MATLAB codes can be found here.
  • (with Rodrigo Platte and Svetlana Roudenko) Blow-up criteria for the 3d cubic nonlinear Schroedinger equation , Nonlinearity 23 (2010) pp. 977-1030. pdf
  • (with Svetlana Roudenko) Divergence of infinite-variance nonradial solutions to the 3d NLS equation , Commun. Partial Differential Equations 35 (2010) pp. 878--905. pdf
  • (with I. Bejenaru, S. Herr, and D. Tataru) On the 2d Zakharov system with L^2 Schroedinger data, Nonlinearity 22 (2009) pp. 1063--1089. pdf
  • (with Maciej Zworski) Breathing patterns in nonlinear relaxation, Nonlinearity 22 (2009) pp. 1259--1301. pdf
  • (with Maciej Zworski) Geometric structure of NLS evolution, unpublished. pdf
  • (with Nikos Tzirakis) Asymptotically linear solutions in H^1 of the 2-d defocusing nonlinear Schroedinger and Hartree equations, Journal of Hyperbolic Differential Equations 7 (2010) pp. 117-138. pdf
  • (with Kiril Datchev) Fast soliton scattering by attractive delta impurities, Commun. Partial Differential Equations 34 (2009) pp. 1074--1113 pdf
  • (with Thomas Duyckaerts and Svetlana Roudenko) Scattering for the non-radial 3d cubic nonlinear Schroedinger equation, Math. Res. Lett. 15 (2008) pp. 1233--1250. pdf
  • (with Maciej Zworski) Soliton interaction with slowly varying potentials, IMRN Internat. Math. Res. Notices 2008 (2008), Art. ID runn026, 36 pp. pdf
  • (with Svetlana Roudenko) A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation, Comm. Math. Phys. 282 (2008) pp. 435--467. pdf.
    We have an unpublished erratum that corrects an improper use of the Strichartz estimates in S 2.
  • (with Svetlana Roudenko) On blow-up solutions to the 3d cubic nonlinear Schroedinger equation, AMRX Appl. Math. Res. Express 2007 (2007), Art. ID abm004, 29 pp. pdf
  • (with Maciej Zworski) Slow soliton interaction with delta impurities, J. Modern Dynamics 1 (2007), pp. 689--718. pdf.
  • (with J. Colliander, M. Visan, and X. Zhang) Global existence and scattering for rough solutions to generalized nonlinear Schroedinger equations on R, Commun. Pure Appl. Anal. 7 (2008), no. 3, pp. 467--489. pdf.
  • (with Jeremy Marzuola and Maciej Zworski) Soliton splitting by external delta potentials, J. Nonlinear Science 17 (2007), pp. 349-367. pdf.
    This is a numerical companion to our paper, Fast soliton scattering by delta impurities, posted below.
  • (with James Colliander and Nikolaos Tzirakis) Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schroedinger systems, Trans. Amer. Math. Soc. 360 (2008), pp. 4619-4638. pdf.
  • Local ill-posedness of the 1d Zakharov system, Electron. J. Differential Equations, Vol. 2007(2007), No. 24, pp. 1-22. pdf
  • (with Jeremy Marzuola and Maciej Zworski) Fast soliton scattering by delta impurities, Comm. Math. Phys. 274 (2007), pp. 187-216. pdf
  • The initial-boundary value problem for the Korteweg-de Vries equation, Comm. Partial Differential Equations 31 (2006), pp. 1151-1190. pdf
  • The initial-boundary value problem for the 1d nonlinear Schroedinger equation on the half-line, Differential Integral Equations 18 (2005), pp. 647-668. pdf
  • Ph.D. Thesis, University of Chicago, 2004. Uniform estimates for the Zakharov system and the initial-boundary value problem for the Korteweg-de Vries and nonlinear Schroedinger equations. 210 pages. pdf. Ph.D. Thesis summary. 10 pages. pdf.

Expository

  • Lecture notes for the Park City Math Institute graduate summer school course "The Cauchy problem for the Quasilinear Schrodinger equation," delivered by Carlos Kenig, July 13-18, 2003. These notes are authored by Carlos Kenig. Revised June 2005, 125 pages. pdf.