Long time behavior of solutions of nonlinear dispersive equations

• Project/Area Number: 15K17570
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Nagoya University
• Principal Investigator: Roy Tristan 名古屋大学, 高等研究院(多元), 特任助教 (80751073)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Discontinued (Fiscal Year 2018)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: dispersive equations / global well-posedness / scattering / local well-posednes / loglog supercritical / local well-posedness / third-order nonlinearity / Dispersive equations / Global existence / Scattering / Schrodinger / Wave, Klein-Gordon / Local well-posedness / Fully nonlinear problems / Asymptotic behavior / Solilton conjecture / Local behavior / Fully nonlinear PDEs

Outline of Annual Research Achievements

I worked on several problems. I proved global existence and scattering of nonsmooth solutions of some barely supercritical Schrodinger equations with a defocusing (i.e repulsive) nonlinearity: this lead to a paper recently published in International Mathematics Research Notices. I also proved global existence and scattering of smooth and nonsmooth solutions below ground state of some size-dependent barely supercritical Schrodinger equations with a focusing (i.e attractive) nonlinearity: this lead to a preprint (arXiv:1701.02689). I also worked on a problem with Prof Kotaro Tsugawa (Nagoya University and Chuo University) that involves the construction of local smooth solutions of fully nonlinear equations that satisfy
a dispersive property in a weak sense. This collaboration is about to lead to a preprint.

Research Products

• [Journal Article] On Jensen-type Inequalities for Nonsmooth Radial Scattering Solutions of a Loglog Energy-Supercritical Schrodinger Equation (2018)
  • Author(s): Tristan Roy
  • Journal Title: International Mathematics Research Notices Volume: - Issue: 8 Pages: 2501-2541
  • DOI: 10.1093/imrn/rny045
• [Journal Article] Blow-up Of The Critical Sobolev Norm For Nonscattering Radial Solutions Of Supercritical Wave Equations In Space Dimension 3 (2017)
  • Author(s): Thomas Duyckaerts, Tristan Roy
  • Journal Title: Bulletin de la SMF Volume: 145 Pages: 507-573
  • Open Access / Int'l Joint Research
• [Journal Article] A weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrodinger equations (2017)
  • Author(s): Tristan Roy
  • Journal Title: Journal of Hyperbolic Differential Equations Volume: Vol. 14, no. 2 Issue: 02 Pages: 249-300
  • DOI: 10.1142/s0219891617500072
• [Journal Article] On The Interpolation With The Potential Bound For Global Solutions Of The Defocusing Cubic Wave Equation on T2 (2016)
  • Author(s): Tristan Roy
  • Journal Title: Journal Of Functional Analysis Volume: 9 Issue: 9 Pages: 3280-3306
  • DOI: 10.1016/j.jfa.2016.02.018
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Blow-up Of The Critical Sobolev Norm For Nonscattering Radial Solutions Of Supercritical Wave Equations In Space Dimension 3 (2016)
  • Author(s): Thomas Duckaerts, Tristan Roy
  • Journal Title: Bulletin De La Societe Mathematique De France (to appear) Volume: 0000 Pages: 0000-0000
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Jensen-type inequalities and nonsmooth radial solutions of loglog supercritical Schrodinger equations (2018)
  • Author(s): Tristan Roy
  • Organizer: Nagoya University (Japan)
• [Presentation] Jensen-type inequalities and nonsmooth solutions of a loglog supercritical Schr (2018)
  • Author(s): Tristan Roy
  • Organizer: Paris 13 (France)
• [Presentation] Jensen-type inequalities and nonsmooth solutions of a loglog supercritical Schrodinger equation (2018)
  • Author(s): Tristan Roy
  • Organizer: Ecole Polytechnique (France)
• [Presentation] Jensen-type inequalities an nonsmooth solutions of a loglog supercritical Schrodinger equation (2018)
  • Author(s): Tristan Roy
  • Organizer: Universite Paris 13 (France)
• [Presentation] Jensen-type inequalities an nonsmooth solutions of a loglog supercritical Schrodinger equation (2018)
  • Author(s): Tristan Roy
  • Organizer: Ecole Polytechnique (France)
• [Presentation] On Jensen-type inequalities for unbounded radial scattering solutions of barely supercritical Schrodinger equations (2017)
  • Author(s): Tristan Roy
  • Organizer: Tokyo University of Science
• [Presentation] On Jensen-type inequalities for unbounded radial scattering solutions of barely supercritical Schrodinger equations (2017)
  • Author(s): Tristan Roy
  • Organizer: Tohoku University.
• [Presentation] On Jensen-type inequalities for unbounded radial scattering solutions of barely supercritical Schrodinger equations (2017)
  • Author(s): Tristan Roy
  • Organizer: Osaka University
• [Presentation] On the asymptotic behavior of solutions of barely supercritical Schrodinger equations with radial data (2016)
  • Author(s): Tristan Roy
  • Organizer: Kyoto University (PDEs)
  • Place of Presentation: Kyoto University
  • Year and Date: 2016-11-18
• [Presentation] On The Interpolation With The Potential Bound For Global Solutions Of The Defocusing Cubic Wave Equation On T2 (2016)
  • Author(s): Tristan Roy
  • Organizer: Osaka University (PDEs seminar)
  • Place of Presentation: Osaka University
  • Year and Date: 2016-01-07
  • Invited
• [Presentation] On The Interpolation With The Potential Bound For Global Solutions Of The Defocusing Cubic Wave Equation On T2 (2015)
  • Author(s): Tristan Roy
  • Organizer: Shizuoka University (PDEs seminar)
  • Place of Presentation: Shizuoka University
  • Year and Date: 2015-12-10
  • Invited
• [Presentation] Solutions Radiales Des Equations Supercritiques Des Ondes En Dimension 3 (2015)
  • Author(s): Tristan Roy
  • Organizer: Universite de Cergy-Pontoise (PDEs seminar)
  • Place of Presentation: Universite de Cergy-Pontoise
  • Year and Date: 2015-06-15
  • Invited
• [Remarks] Tristan Roy's homepage
  • URL: http://www.math.nagoya-u.ac.jp/~tristanroy/
• [Remarks] Tristan Roy's webpage
  • URL: http://www.math.nagoya-u.ac.jp/~tristanroy/