Congratulations to Didier Bresch and Pierre-Emmanuel Jabin, the 2021 recipients of the SIAM Activity Group on Analysis of Partial Differential Equations Best Paper Prize, and Giacomo Canevari, the 2021 recipient of the SIAM Activity Group on Analysis of Partial Differential Equations Early Career Prize. Both prizes will be presented at the 2022 SIAM Conference on Analysis of Partial Differential Equations (PD22), to be held in a virtual format March 14–18, 2022.
Didier Bresch and Pierre-Emmanuel Jabin
Didier Bresch and Pierre-Emmanuel Jabin are the recipients of the 2021 SIAM Activity Group on Analysis of Partial Differential Equations Best Paper Prize. The award will be presented at the 2022 SIAM Conference on Analysis of Partial Differential Equations (PD22). Bresch and Jabin will give a talk at the conference titled “Global Weak Solutions for Heat-Conducting Compressible Navier-Stokes equations with a Truncated Virial Pressure Law” on Thursday, March 17, 2022 at 8:00 a.m. ET.
The SIAM Activity Group on Analysis of Partial Differential Equations awards this prize every two years to the author(s) of the most outstanding paper, as determined by the prize committee, on a topic in partial differential equations published in the four calendar years preceding the award year.
Didier Bresch and Pierre-Emmanuel Jabin
Didier Bresch is currently Research Director at CNRS assigned to the Laboratoire de Mathématiques UMR5127 CNRS in the Université Savoie Mont-Blanc (France). He received his Ph.D. from Clermont Auvergne University in 1997 before being recruited as a Junior CNRS researcher in 1998 (Clermont Auvergne University 1998-2002 and LJK Grenoble University 2002-2006), and his HDR in 2002 before obtaining his promotion to Research Director in 2006 (second class from 2006 to 2010 and then promoted first class in 2010). He was a recipient with collaborators of the Maurice Audin Prize in 2007 and La Recherche Prizes in 2009 and 2011. He was Director of LAMA UMR5127 CNRS (Laboratoire de Mathématiques), University Savoie Mont-Blanc from 2010 to 2014. From September 2016 to August 2021, he was President of the CNRS hiring, evaluation and promotion national committee for mathematics and interactions of mathematics (section 41) in France. Motivated by climate issues and by a desire to show the important role that mathematics can play in interaction with other disciplines, he was also the head of the national “MathsInTerre” Research Think Tank in 2012-2013.
Pierre-Emmanuel Jabin has been professor at the Pennsylvania State University since August 2020. He was a student of École Normale Supérieure from 1995 to 1999; he earned his Ph.D. in 2000 and his HRD in 2003 both at Université Pierre et Marie Curie (Paris VI). Jabin was an assistant professor (agrégé-préparateur) at the École Normale Supérieure in Paris from 2000 to 2004. He then became professor in the J.-A. Dieudonné laboratory at the Université Nice-Sophia Antipolis in France from 2004 to 2011, and held a visiting faculty position at the Institute for Research in Informatics and Automatics in Sophia-Antipolis, France, from 2007 to 2011. He was more recently a professor at the University of Maryland from 2011 to 2020, where he was also Director of the Center for Scientific Computation and Mathematical Modeling from 2016 to 2020. Jabin was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro in 2018.
Q: Why are you excited to receive the SIAG/APDE Best Paper Prize?
A: We are grateful and honored to be awarded this prize, especially given how many breakthrough papers have been published in the field of PDE in the four calendar years preceding 2021. By such recognition, we really hope to attract young researchers to work on applied mathematics and of course on applications to compressible flows, for example to describe wet granular flows (more general state laws), solar events (law of virial pressure), geophysical flows (turbulent viscosity), or situations in biology (anisotropy in viscosity and pressure which presents zones of attraction or repulsion with thresholds evolving in time and space).
Q: Could you tell us a bit about the research that won you the prize?
A: The Navier-Stokes equations provide a basic mathematical model for describing the motion of a fluid. In a famous paper published in Acta Mathematica in 1934, Jean Leray introduced the concept of weak solutions (and to do this he also defined what is now called a Sobolev space) by giving a precise definition of what a so-called weak solution is and he proved that such a weak solution exists globally in time for the homogeneous incompressible Navier-Stokes equations. We now call these solutions of minimal regularity (finite energy): solution à la Leray.
While existence of weak solutions does not imply any well-posedness character of the system, such analysis has an immediate physical signification because assumptions on the initial data are minimal and strongly related to physical quantities. Existence of weak solutions can be seen as some stability property of the system and can also contribute to the design stable numerical schemes that often do not preserve strong regularity estimates.
Existence of solutions à la Leray on the compressible Navier-Stokes equation with barotopic pressure laws has been obtained for the first time by Pierre-Louis Lions in 1998 and E. Feireisl, A. Novotny, and H. Petzeltova in 2001, who considered the case of isotropic and constant shear and bulk viscosities as well as a monotone pressure state law or at least monotone out of a fixed compact interval.
Our own research has focused on whether it is possible to consider more realistic state laws: Can we consider viscous anisotropy depending on the directions? Can we cover the case of non-monotone pressure laws? These state laws are related to practical situations that may be encountered in geophysical flows or solar events.
To give an answer to these two open problems, we analyze the fine structure of the equations combined with a new approach of compactness to the transport equations with non-smooth velocity by the introduction of appropriate weights and a precise description of their mathematical properties. This leads to a novel method which quantifies and propagates the critical regularity of the system. This new approach allowed us to build global solutions in the case of pressure state laws, which can be thermodynamically unstable (hence possibly non-monotone pressure) and in the case of some anisotropy in the stress tensor law (different shear viscosity coefficients compared to the size of the bulk viscosity). This requires a fine analysis of the structure of the equations combined with a new approach of compactness to the transport equations with non-smooth velocity by the introduction of appropriate weights and a precise description of their mathematical properties.
Q: What does your work mean to the public?
A: Our regularity estimates have direct implications on the amplitude of oscillations in fluid mechanics models. In particular we can apply for the first time such a control to several complex models in engineering and physics.
As mentioned above, the stability properties of weak solutions could also help to design a new class of numerical simulation algorithms adapted, for instance, to fluid mechanics, geophysics, or biology with a focus on environmental purposes.
Q: What does being a member of SIAM mean to you?
A: We really appreciate the scientific diversity represented in SIAM which allows you to connect the many different branches of mathematics with applications. We also want to emphasize the key role played by the various activity groups for organizing and strengthening the applied math community. The many activities organized by SIAM also help publicize and highlight new, original fields of research and applications of mathematics.
Giacomo Canevari
Giacomo Canevari is the recipient of the 2021 SIAM Activity Group on Analysis of Partial Differential Equations Early Career Prize; this is the first time that the prize will be awarded. The award will be presented at the 2022 SIAM Conference on Analysis of Partial Differential Equations (PD22). Canevari will give a talk at the conference titled “A Few Variational Approaches to the Description of Defects in Liquid Crystals” on Thursday, March 17, 2022 at 11:10 a.m. ET.
The SIAM Activity Group on Analysis of Partial Differential Equations awards this prize every two years to an early career researcher who has made outstanding, influential, and potentially long-lasting contributions within six years of receiving a Ph.D. or equivalent degree as of January 1st of the award year. At least one of the papers containing this work must be published in English in a peer-reviewed journal or conference proceedings.
Giacomo Canevari is a researcher at the University of Verona (Italy). After graduating in Mathematics at the University of Pavia (Italy), he received his Ph.D. at the UPMC-Paris 6, under the supervision of Professor Fabrice Bethuel. He was a postdoc at the University of Oxford, in the group led by Professor Sir John M. Ball, and at the Basque Centre for Applied Mathematics (Bilbao, Spain), with Professor Arghir D. Zarnescu. His research interests are in the mathematical analysis of variational models for materials science, with a focus on liquid crystals.
Q: Why are you excited to receive the SIAG/APDE Early Career Prize?
A: I am very honored to be selected for this prize. I am grateful to the people whom I had the opportunity to meet – exceptional mentors, collaborators, and colleagues. I am glad to see that our efforts have been recognized by the SIAM community.
Q: Could you tell us a bit about the research that won you the prize?
A: Nematic liquid crystals are particular phases of matter, where the constituent molecules have local orientational order. Oversimplifying things a little, you may think your material is composed by rigid, rod-shaped particles that tend to align parallel to each other, locally. However, the orientational order is not uniform in space; at some places, the order may be broken, and we have an optical defect. At a defect, the preferred direction of molecular alignment is not well-defined. Defects typically occur at isolated points or along lines. In my work, I considered a variational, continuum model for three-dimensional nematics – the Landau-de Gennes model – and studied the asymptotic behavior of minimizers in the large-domain limit. In this approach, both line and point defects appear as singularities of the limit maps and they can be rigorously studied using variational and PDE-based techniques. Since then, other groups of researchers in the PDE community have made further progress in the analysis of defects in nematics, but many questions remain open, for instance, the behavior of defects in other classes of materials, such as biaxial nematics.
Q: What does your work mean to the public?
A: Liquid crystals are very important to technological applications – they are widely used in the display industry, for instance – and they are still the object of a lot of experimental research. For example, if we are able to understand and control the behavior of defects in a liquid crystal shell, then we may be able to use this information to our advantage and, ideally, tune the properties of the material by defect engineering. I think mathematicians could contribute to this process. Even though mathematical results are often obtained in idealized settings, which are not nearly as complex as real-world applications, they may still grasp a few aspects of the physical phenomena and serve as basis for more detailed studies later on.
Q: What does being a member of SIAM mean to you?
A: I appreciate that SIAM promotes connections between different branches of mathematics, favoring the collaboration between scientists whose interests are oriented towards the applications and those who are more theory oriented. SIAM conferences, for instance, are a great opportunity to keep up to date with the latest advances in your field, but also to broaden your interests and meet researchers from other areas.