Mathematics and the Earth: a long story
For several centuries, great mathematicians have been interested in our planet. P. Fermat studied the weight of the Earth, C.F. Gauss contributed to the development of geomagnetism and A. Tikhonov developed regularization techniques commonly used in geophysics. The work of H. Poincaré in celestial mechanics should also not be forgotten or that of L. Euler, J. R. D’Alembert, H. Navier and G. Coriolis which culminated in the Navier-Stokes equations with the Coriolis term, providing us with central elements of simulations in meteorology. The history of this equation is recounted with humour and a degree of irreverence by the comic strip L’équation du millénaire, which the FMSP distributes free on the internet. In this section, we describe three other examples. It should be noted that the initiative Un jour, une brève is a much more exhaustive treatment (see Further reading).
Living world: The stripes of the zebra or the polygons of the giraffe are among the most spectacular morphogenetic manifestations in the natural world. In 1952, the founder of information technology, A. Turing, proposed a system of two reaction-diffusion equations that mimed these structures. Then R. Thom was the first to mathematically define morphogenesis. More recently, problems of invasion in a heterogeneous milieu (periodic, random) and non-local interactions (competition between individuals of different genetic traits) have led to the study of integro-differential equations that sometimes exhibit very complex behaviour with multiple stationary states.
Boundary layers: The boundary layer methods, initiated by L. Prandtl in 1904, divide the flow velocity field into an inner part and a part close to the edges. This approach was taken up by the oceanographers V.W. Ekman and W. Munk on very simplified geophysical models. Next came the development of the mathematical methods of singular perturbations after 1950, the date of the publication by K.O. Friedrichs. The literature in this field is already rich: P.A. Lagerstrom, J.D. Cole, M. Van Dyke, W. Eckhaus and J-L Lions. Important results on the degeneration of boundary layers were only published recently.
Optimal transport: The theory of optimal transport was developed by G. Monge in the 18th century in the context of moving a pile of sand into a hole in the most economic way. This theory was then taken up again in 1940 by a mathematician, L. Kantorovich, for problems relating to optimal resource allocation. Other mathematicians finally succeeded in solving the problem, in the case of an infinite number of grains of sand, by formulating a continuous model. A little later, optimal transport was used as a powerful demonstration tool in EDPs, geometry and probability. The work of Y. Brenier, C. Villani et al. provides a perfect example.
