Luc Tartar Livres

Homogenization is not about periodicity, or Gamma-convergence, but about understanding which effective equations to use at macroscopic level, knowing which partial differential equations govern mesoscopic levels, without using probabilities (which destroy physical reality); instead, one uses various topologies of weak type, the G-convergence of Sergio Spagnolo, the H-convergence of François Murat and the author, and some responsible for the appearance of nonlocal effects, which many theories in continuum mechanics or physics guessed wrongly. For a better understanding of 20th century science, new mathematical tools must be introduced, like the author’s H-measures, variants by Patrick Gérard, and others yet to be discovered.

Equations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the „mean free path between collisions“ tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e. g. from quantum mechanics: there are no „particles“, so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity!

After publishing an introduction to the Navier–Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with another set of lecture notes based on a graduate course in two parts, as indicated by the title. A draft has been available on the internet for a few years. The author has now revised and polished it into a text accessible to a larger audience.

In the spring of 1999, I taught a graduate course on Partial Differential Equations Models in Oceanography at Carnegie Mellon University and created lecture notes for my students. These notes were later shared online and distributed at a Summer School in Lisbon in July 1999. After some time, I decided to publish a revised version to reach a broader audience. While there seems to be a growing interest in the Navier–Stokes equation, many who express this interest lack a fundamental understanding of continuum mechanics, raising questions about the depth of their engagement. The renewed focus may be linked to the Clay Millennium Prizes; however, the conjectures associated with this prize appear to have been selected by individuals not deeply versed in continuum mechanics, as they lack significant physical relevance. Although invariance by translation or scaling is mentioned, the omission of rotational invariance and Galilean invariance is notable. The latter is crucial for understanding that, in ordinary fluid dynamics where velocities are much smaller than the speed of light, relativistic corrections are unnecessary, and thus Galilean invariance should be applied. However, it's important to recognize that the mathematical formulation does not guarantee that solutions will adhere to velocities limited by the speed of light.