A team of four mathematicians, including Frank Merle, holder of the CY Cergy Paris Université-IHES Chair in Analysis, Pierre Raphaël, holder of the Schlumberger Chair for mathematical sciences at IHES and Professor at the University of Cambridge, Igor Rodnianski, Professor at Princeton University, and Jérémie Szeftel, CNRS Research Director at Sorbonne Université and a regular visitor of IHES, have been awarded the 2023 Bôcher Memorial Prize of the American Mathematical Society.
The prize was announced on December 5, and it acknowledges the laureates’ “groundbreaking work establishing the existence of blow-up solutions to the defocusing non-linear Schrödinger equation in some supercritical regimes and to the compressible Euler and Navier-Stokes equations” published in [1,2,3].
The collaboration that led to these results involved long-distance conversations as well as long eye-to-eye discussions, many of which took place at IHES. The four researchers, who have known each other and collaborated on different projects for many years, started working on these problems in 2012.
“Supercritical regimes were unexplored ground, so we needed to develop a completely new theoretical framework”, explains Frank Merle, a mathematician specializing in partial differential equations who has been associated with IHES since 2006 and part time at CY Cergy Paris Université. “There were no rigorous mathematical results, and for a long time it felt like walking in the dark. This implied many trials and errors, especially because intuition based on subcritical and critical regimes revealed not to be the best guide when dealing with systems as complex as the defocusing Schrödinger equation and the compressible Navier-Stokes equation.”
It took about five years to find the key that allowed the quartet to tackle the problem. This was the first, and arguably the most important part of their work: “What is most important, Szeftel explains, is to find the good idea to take advantage of a weak point of the problem, and that takes time because it follows a process that you cannot always control and that requires a deep understanding of the problem”.
After many discussions, a lot of thinking, and some faux pas, the four mathematicians found in 2017 that a specific class of solutions of the compressible Euler equation could be transposed into the compressible Navier-Stokes. That was the key that opened up the way to obtaining singular solutions of the complete equation and of the defocusing Schrödinger equation. “The strength of our team is the diversity of our scientific culture”, says Raphaël. “This creates quite a bit of chaos, and we do get lost many times, but exploring unknown territory in good company is the best part of our job”.
The second part of the work, which led to the publication of an impressive series of three articles totaling more than 450 pages, was more systematic, albeit not easier, and consisted in applying the mathematical skills and expertise that the four mathematicians had honed over years of experience.
Numerical simulations and mathematical intuition had led mathematicians to rule out the possibility of singular solutions for the defocusing non-linear Schrödinger equation, to the point that Fields medalist and former permanent professor at IHES Jean Bourgain had formalized that in a conjecture. The solutions found by the quatuor instead are divergent, thus making the results acknowledged by the 2023 Bôcher Prize all the more groundbreaking.
Frank Merle, who already received the Bôcher Prize in 2005 for his work in the analysis of nonlinear dispersive equations, is the only researcher to have received it twice. “I am very grateful for this award. The Bôcher prize I received in 2005 meant a lot to me, as it was the first time that my work as a mathematician was recognized at a high level. I am particularly happy to be able to share this second prize with Pierre, Igor and Jérémie”.
The Bôcher Memorial Prize is awarded every three years for a notable research work in analysis published in a peer-reviewed journal during the preceding six years. The 2023 prize will be awarded on Wednesday, January 4 during the Joint Prize Session at the 2023 Joint Mathematics Meetings of the AMS in Boston.
 On the implosion of a compressible fluid II: singularity formation. Ann. of Math. (2) 196 (2022), no. 2, 779–889. arXiv:1912.11009
 On the implosion of a compressible fluid I: smooth self-similar inviscid profiles. Ann. of Math. (2) 196 (2022), no. 2, 567–778. arXiv:1912.10998
 On blow up for the energy super critical defocusing nonlinear Schrödinger equations. Invent. Math. 227 (2022), no. 1, 247–413. arXiv:1912.11005