Applied Fluid Mechanics Colloquium
Data: Ar, Aza 10 2026
Ordua: 12:30 - 15:00
Lekua: Maryam Mirzakhani Seminar Room
Hizlariak: Frank Merle (Cergy Paris Université & IHÉS)
Title: On soliton resolution for the radial critical wave equation
Abstract:
We consider mostly the energy-critical focusing wave equation in dimension d=3,5, then briefly over dimension and other models. The equation has a nonzero radial stationary solution W, which is unique up to scaling and sign change. Together with T. Duyckaerts, C. Kenig, and later C. Collot, we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated W, decoupled by the scaling, and a radiation term.
The proof in dimension 3 follows from a exceptional property of W (true only in dimension 3,4), that any none zero radial initial data different from W will produce at the initial time in some sense a radiation.
The proof in dimension 5 essentially boils down to the fact that the equation does not have purely nonradiative multisoliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case by reducing the study of a multisoliton solution to a finite dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multisolitons.
Zoom: https://us06web.zoom.us/j/89636021614?pwd=E5othQOK4Zju4gave0B88id2DXQxTi.1
Meeting ID: 896 3602 1614
Passcode: 099217
Hizlari baieztatuak:
Frank Merle is full professor at Cergy Paris Université. He also holds the Analysis Chair at the Institut des Hautes Études Scientifiques.
After graduation from the École Normale Supérieure (ENS), he received his Ph.D. in 1987 from the Université Paris VI under H. Berestycki direction. He then became a CNRS researcher at École Normale Supérieure. In 1989/90 he was assistant professor at the Courant Institute of Mathematical Sciences at NYU. Since 1991 Merle has been a professor at the University of Cergy-Pontoise (now Cergy Paris Université). From 1998 to 2003 he was a member of the Institut Universitaire de France. In the fall of 1996, the fall of 2001, and the academic year 2003–2004 he was a visiting scholar at the Institute for Advanced Study. He was a visiting professor at Stanford University, Rutgers University, the University of Chicago, the Mathematical Sciences Research Institute (MSRI) at Berkeley, Leiden University, and the University of Tokyo.
He was invited twice to the International Congress of Mathematicians (1998 invited speaker, 2014 invited plenary speaker).
Frank Merle is a worldwide recognised specialist in the analysis of partial differential equations and mathematical physics.
His researches contributed to revolutionise the study of nonlinear dispersive PDEs, such as the nonlinear Schrödinger equation and the Korteweg–De Vries equation, and of the qualitative properties of their solutions, notably finite time blow-up phenomena. Such researches earned him the "Prix de l'Institut Poincaré en Physique théorique" (1997), "Prix Charles-Louis de Saulse de Freycinet" of the Académie des Sciences de Paris (2000), the "Médaille d'argent du CNRS" and "Bôcher Prize" (2005), "Prix Ampère" awarded by the French Academy of Sciences (2018), the "Bôcher Memorial Prize" and the "Clay Research Award" (2023). He also received the 2026 Breakthrough Prize in Mathematics for significantly advancing the modern understanding of nonlinear evolution equations.
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