# The 2023 Abel Prize is awarded to Louis Caffarelli

Studying nature sometimes leads to seemingly very simple questions: What shape does an ice cube take when it melts, and how does the liquid flow? But the equations describing these systems can become an especially tricky playground for mathematicians. Many researchers focus specifically on ensuring that these so-called “partial derivative” equations “behave wisely”. It is one of the leading specialists in this field, Louis Caffarelli, of the University of Texas, at Austin, whom the Norwegian Academy of Sciences has decided to reward for his fundamental contribution to the regularity theory of equations for nonlinear partial derivatives, such as the Monge-Ampère equation.

Many physical processes in nature, but also in biology or economics, often lead to dealing with partial differential equations. These equations take into account the fact that the studied phenomena are often dynamic and dependent on several variables such as time and location. This is for example the case of the Fourier equation, which characterizes the propagation of heat in a substance, Maxwell’s equations, which describe the propagation of electromagnetic waves, the Schrödinger equation, in quantum mechanics, or the more Navier-Stokes equations, which explain the flow of a liquid.

These equations are often written very simply…but their solutions can be very difficult to find and understand. Therefore, mathematicians set out to study the existence, uniqueness, regularity, and stability of these solutions. Indeed, at the macroscopic scale, physical phenomena have behaviors as the sizes vary progressively. A solution that introduces singularities (“peaks”) or that tends toward infinity will not be satisfactory. This inequality function indicates, for example, that the temperature will become infinite!

In 1977, Luis Caffarelli became interested in the equations describing the melting of ice. This is an example of the so-called “free surface” problem, the latter here being the interface between ice and air. This kind of problem is difficult to deal with because the surface is not defined *probably* It is not constant throughout the process. On the contrary, it evolves over time and is therefore part of the desired solution. Thanks to the work of Luis Caffarelli, who was the first to discover how to deal with situations that have more than one dimension, and that of Alessio Figalli later, we understand why the melting of an ice cube, even if it was initially in the form of a cube with protruding edges, is Smooth its surface gradually. The tools Luis Caffarelli developed in this field are still widely used to deal with various problems.

Au début des années 1980, Luis Caffarelli a travaillé with Louis Nirenberg (lauréat du prize Abel en 2015) et Robert Kohn sur l’équation de Navier-Stokes, which decrit des écoulements de fluides, as l’eau à travers un tuyau ou dans oceans. Engineers and climatologists alike use this equation all the time, and it works pretty well. But its regularity has not been demonstrated and it is the subject of one of the seven Clay Institute Millennium Prizes (with a $1 million key reward). So the equation can contain a solution that explodes under certain conditions: the velocity of the liquid tends toward infinity! The three researchers achieved a partial result, and their solution is considered one of the most successful to date. If regions are formed where the velocity becomes infinite, then it is infinitely small. This means that these points are infinitely more rare than the points where the fluid behaves regularly.

Recently, the Argentine-born mathematician became interested in the Monge-Ampère equations. These equations appear in the field of differential geometry and are associated with optimal transport problems. Again, Luis Caffarelli has developed tools to show the regularity of some solutions.

“By combining brilliant engineering intuition with innovative analytical tools and methods, [Luis Caffarelli] He was and continues to be a major influence in this field, ”concludes Helge Holden, mathematician at the head of the Abel Commission.

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