Luc TARTAR Compensated Compactness with more ... - ICMS

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I chose Jacques-Louis LIONS as thesis advisor (since he looked more interested in applications than Laurent SCHWARTZ), but when (in the mid 1970s) I decided to understand more about Continuum Mechanics for developing some of the new mathematical tools needed, it appeared that he was not interested. Although his Theory of Distributions helped understand formal results by HEAVISIDE (1850–1925) and by DIRAC (1902–1984), Laurent SCHWARTZ had no interest in Physics. I think that Sergei SOBOLEV (1908–1989) introduced his H 1 space in relation to a physical question; that he did not publish more about his theory seems related to USSR politics. It was in relation to (simplified) Navier–Stokes equation that Jean LERAY (1906–1998) used weak solutions, but he stopped working on PDE in Continuum Mechanics when he was a prisoner of war during World War II. The approach which I was taught for nonlinear PDE, of introducing adapted Sobolev spaces and proving various estimates for showing existence and uniqueness of solutions used either a compactness argument, according to the original ideas of Jean LERAY, or a monotonicity argument, introduced by Eduardo ZARANTONELLO (1918–2010) for a problem in Fluid Dynamics, and by George MINTY (1930–1986) for a problem in Electricity; while I was a student, monotonicity then developed into a question of Functional Analysis, mostly studied by Haïm BREZIS, Felix BROWDER, Jacques-Louis LIONS, and Terry ROCKAFELLAR (in alphabetic order). I unified these various aspects in my Compensated Compactness Method, based on my joint work in Compensated Compactness with François MURAT. 2
In the late 1960s, a student at École Polytechnique from the year after mine told me that I should read Foundations of Mechanics by ABRAHAM: I bought the book and it was useful since it contained plenty of results about manifolds, which I had only heard about in a vague way, but I ended up quite puzzled, because the book contained no Mechanics at all! I realized afterward that it is common for differential geometers to be stuck at the 18th century point of view of Mechanics (Classical Mechanics) which uses ODE (ordinary differential equations), as if they were not bright enough to learn the 19th century point of view of Mechanics (Continuum Mechanics) which uses PDE. Of course, I observed a similar limitation when I later explained that the 20th century point of view of Mechanics/Physics uses PDE with small scales, so that one needs to understand questions of Homogenization for identifying limiting effective equations, which often force to consider a larger class of equations (still a little vague) which I call beyond PDE. A little later (still in the late 1960s) a friend told me about the ideas of René THOM (1923–2002), whose book I only glanced at since he seemed to believe that the laws governing Nature are ODE, again the 18th century point of view, as if he decided to ignore the 19th century point of view of PDE. I also was puzzled with his choice of the term “catastrophes” for “singularities of differentiable mappings”. In 1970, I was shown an article by David EBIN and Jerry MARSDEN (1942–2010), on Euler equation for an incompressible fluid on a compact Riemannian manifold, based on an idea of Vladimir ARNOL’D (1937–2010), of considering a flow of volume preserving diffeomorphisms. 3
Page 1: Differential Geometry and Continuum
Page 5 and 6: Has their framework been useful for
Page 7 and 8: However, for a scalar wave equation
Page 9 and 10: If one applies to the Maxwell-Heavi
Page 11 and 12: My physics courses did not mention
Page 13 and 14: For a weakly converging sequence u
Page 15 and 16: The equation which one calls Navier
Page 17 and 18: If a n only take k particular value
Page 19 and 20: I thought that the mass of a “par
Page 21 and 22: If one considers two (or more) link
Page 23 and 24: He then took some paste out of a ja
Page 25 and 26: In 1807, POISSON (1781-1840) analyz
Page 27 and 28: My Compensated Compactness Method (
Page 29 and 30: I used σ = f(u), but without using
Page 31: A few years ago, after Amit ACHARYA

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