**Title:** Beyond Partial Differential Equations

**Speaker:** Professor Luc Tartar

**Speaker Info:** Carnegie Mellon University

**Brief Description:**

**Special Note**:

**Abstract:**

Problems in Mechanics and Physics have played an important role in the past for inducing mathematicians to create new mathematical tools, but nowadays the original reasons for developing a particular mathematical tool are rarely taught and as most mathematicians have no serious knowledge in Mechanics or Physics anymore, many do not understand what the actual challenges are, and lacking a basic knowledge in Mechanics or Physics might be the reason why so many mathematicians easily fall prey to fashions, whose leaders play with terms from Mechanics or Physics but show a quite deluded level of understanding of these fields, a fact which every educated person from older generations like mine perceives easily. Learning Mechanics or Physics is then made more difficult for mathematicians as they may easily be lured in the wrong direction, but even for those with enough critical judgment for avoiding some traps, it is not an easy task to guess what one should believe from all that is said about the real world and the mathematical models which are used for describing it.I was trained in Paris in the late 60s in Functional Analysis and Partial Differential Equations, having great teachers like Laurent SCHWARTZ and Jacques-Louis LIONS} (whom I chose as advisor), but beyond the questions of Interpolation which answered the questions of my advisor, I had a personal interest in understanding more in Continuum Mechanics and Physics, interest that my teachers did not share. However, it was a purely academic question of optimization that I was studying with Francois MURAT which put me on the way to a new approach, where we first rediscovered an idea which Laurence C. YOUNG had developed in the 40s, related to what one calls now Young measures, but which I initially called parametrized measures (the name that I had heard in seminars in Control Theory), and then rediscovered an idea which Sergio SPAGNOLO had introduced in the late 60s under the guidance of Ennio DEGIORGI, which he had named G-convergence (convergence of Green kernels), but our slightly more general approach became H-convergence (in connection with the term Homogenization, a term introduced by Ivo BABUKA), but it was thanks to some earlier work of Henri ANCHEZ-PALENCIA that I understood that what we had been doing had something to do with properties of mixtures (qualified of effective by physicists). It was because I had also been taught Mechanics (Classical Mechanics, as well as Continuum Mechanics) and Physics (Clasical Physics, as well as Quantum Physics and Statistical Physics) when I was a student at Ecole Polytechnique, that I realized that from the new approach that we had developped I could discover how to give a mathematical meaning to some of the strange facts that I had been taught in these courses, and it is important that no probabilistic framework is used for linking the properties at various levels (microscopic, mesoscopic, macroscopic).

It is important to realize that if Ordinary Differential Equations is the adapted mathematical tool for Classical Mechanics which is an 18th Century point of view, and Partial Differential Equations is the adapted mathematical tool for Continuum Mechanics which is an 19th Century point of view, but the mathematical tools which existed in the mid 60s were only adapted to linear problems, and part of my research work stemmed from looking at the difficulties posed by nonlinear effects. The two theories that I partly developed with Francois MURAT, Homogenization, and then Compensated Compactness, and finally the theory of H-measures and their variants which I developed afterward form a piece of this new theory that I have started to build, which lies Beyond Partial Differential Equations, and should be adapted to 20th century problems in Continuum Mechanics, like Plasticity or Turbulence, and 20th century Physics, i.e. quantum effects.

Anyone who has learned Continuum Mechanics knows that Turbulence has nothing to do with letting time tend to infinity, and this is just one of the deluded description of Mechanics that I alluded to , but in order to understand why it is a problem of Homogenization, one must consider the term Homogenization with the generality that I had given it in the late 70s (in my Peccot lectures at Coll\`ege de France, whose title was ``Homog\'en\'eisation dans les \'equations aux d\'eriv\'ees partielles'', i.e. ``Homogenization in partial differential equations''), which was certainly not restricted to periodic questions for which Ivo B{\eightrm ABU\v{S}KA} had introduced the term; however, identifying that Turbulence falls into the family of Homogenization problems does not help to solve it because very little is known yet for what concerns hyperbolic situations like transport along a flow, but some particular cases of first order equations which have been solved suggest that there should be some apparition of nonlocal effects. Similarly, a spectroscopy experiment consists in sending a wave in an heterogeneous medium whose property vary, again an Homogenization problem for which the effective equation probably involves some nonlocal effects; if one could identify the kernel appearing in the nonlocal term one would probably find what physicists call a density of absorption and reemission, but that would be done without involving any probabilities, and there would be none of the well defined rays of absorption which had led physicists to look for a game creating a list of numbers (as rays were created by the lack of precision of the measuring apparatus), so no spectrum of an operator would be necessary and some of the rules invented by physicists might look a little silly after all that.

I had introduced H-measures in order to compute some second order corrections of effective coefficients in the case where one mixes materials with similar properties, and an analogous tool might be useful for computing a first approximation of the kernels involved in the effective equation, but I have also used H-measures for explaining properties like Geometric Optics for limits of solutions of the wave equation, and it suggests a quite different interpretation of quantum effects than that proposed by physicists, as it would not be that particles sometimes are particles and sometimes are waves, but that there are no particles at all but only waves, and in the limit of infinite frequencies some H-measure associated to the waves satisfies a first order equation, which one may interpret as describing the motion of some idealized particles. In the end, it would not be that one starts from an Hamiltonian framework and one builds a partial differential equation like a Schr\"odinger equation, because the world is not described by Ordinary Differential Equations but by Partial Differential Equations, with the particular difficulty that the solutions of these equations have small scale effects which require then the yet unfinished theory Beyond Partial Differential Equations for describing what these effects are.

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