Research Statement
Th´ eophile Dolmaire
∗June 5, 2019
1 Context and global introduction
Back to 1900, David Hilbert presented his famous problems[4] in the Sorbonne, during the International Congress of Mathematicians. Among the two aspects which compose the sixth problem, concerning generally speaking the mathe- matical treatment of the axioms of Physics, the second proposes to set a theory able to rigorously study the processes ”which lead from the atomistic view to the laws of motion of continua”1.
In a more specific context, one can try to address the very difficult Cauchy problem, or at least to better understand the structure of the solutions of the continuous fluid equations of hydrodynamics such as the Euler or the Navier- Stokes equations, starting from a microscopic description of a gas.
A program was suggested [Par qui ?] to obtain results, using mesoscopic de- scription as an interim level of description. To be more precise, one starts with a numberN of particles (in the most basic setting, they are spheres) of radius ε(N is meant to be very large, while εis meant to be very small), evolving in a domain Ω of the euclidian spaceRd (dbeing 2 or 3 in the physically relevant settings), and interacting between each other in a way that will be described below.
The first step of the program, called the low-density limit, consists in taking a positive numberαfixed, and setting the condition
N εd−1=α,
which, physically speaking, means that each particle travels on a given distance before interacting with an other particle. Letting N going to infinity, one de- scribes in the limit a gas which is very diluted, and one can formally obtain, as it was shown by Boltzmann[1], the following famous kinetic equation which now bears his name :
∂tf+v· ∇xf =Q(f, f), (1)
∗Laboratoire IMJ-PRG, Universit Paris 7 - Paris Diderot - theophile.dolmaire@imj-prg.fr
1see the page 454.
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wheref is the 1-particle density, bringing a statistical description of the fluid.
In other words, the unkown f of the Boltzmann equation, which depends on the three variablest (the time), x(the position) andv (the velocity) provides an information that can be understood as follows. The positive number
f(t, x, v) dxdv
represents the quantity of particles that lie in the elementary volumex+dx⊂Ω, with a velocity belonging tov+ dv⊂Rd, at time t.
Of course, since the Boltzmann equation is itself a fascinating and very active subject of research, this step of the program presents an important interest in itself, beside the crucial role it plays to study hydrodynamics equations. My work is inscribed in the movement which aims a better comprehension of this first step.
Finally, the last step consists in letting the parameterαgoing to infinity, which corresponds to increasing again and again the density of the gas. This last step is known as the hydrodynamic limit, carefully studied in [7].
2 The work during the thesis
When Boltzmann introduced the equation (1) that he obtained formally, he no- ticed that a gas which verifies this evolution equation follows a dynamics which is irreversible. However, Loschmidt[6] and Zermelo later[8] noticed apparent paradoxes which are direct consequences of this irreversibility.
From this, the rigorous study of the low-density limit turned out to be a mean- ingful topic, but it was left as an open question until the decisive result, known as the Lanford theorem, first stated in [5], while the proof was completed re- cently in [3]. This theorem states that this low-density limit holds on a non trivial time interval, and more precisely :
Theorem 1. Let f0 : R2d 7→ R+ be a continuous density of probability such that :
f0(x, v) exp β 2|v|2
L∞(Rd)
<+∞
for someβ >0. Consider the system ofN hard spheres of diameterε, initially distributed according tof0and independent. Then, in the Boltzmann-Grad limit :
ß N →+∞, N εd−1= 1,
its ditribution function converges to the solution of the Boltzmann equation (1), with initial dataf0 in the sense of observables.
Here, the collision term of the Boltzmann equation (1), that is the right-hand- side of the equation is not detailed, but depends crucially on the interaction between the particles chosen for the study.
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The main theorems stated in [3] takes the setting of particles which evolve in the whole euclidian spaceRd. My work consists in changing the domain in which the particles move. To be explicit, I adapted the proof in the case of a convex obstacle removed from the whole euclidian spaceRd. Physically speak- ing, it means that one considers a large number of particles evolving around an obstacle, and studies the global behaviour of those particles when their number becomes larger and larger, in the very specific case of the hard-sphere, which is the setting of the theorem 4 in [3].
3 Possible developments and new axis of research
The first theorems stated in [3] takes the setting of the hard-spheres, that is par- ticles move in straight line and at constant velocity until two of them collide, that is when there centers are at distance ε. At the moment of the collision, their velocities are changed according to laws of dynamics which preserve the kinetic energy and the momentum of the system composed by all the particles.
The collision is therefore elastic.
A challenging problem consists in modifying those laws, defining new ones such that one enables a loss of kinetic energy during collisions (which are then in- elastic). The Cauchy problem for the system of particles, when their number is fixed, which is the very first step of the proof, is not even solved.
An other way to extend the result lies in the modification of the interaction between the particles and the obstacle. In the work done during the thesis, the study was made under the hypothesis of specular reflexion. A difficult question, which would imply a radical change in the core of the proof, is to consider in- stead diffusive boundary condition, or even a mix between specular and diffusive reflexion, as in the article [2].
References
[1] L. Boltzmann, A. Gallotti, H. B´enard, and M. Brillouin. Le¸cons sur la th´eorie des gaz. Number vol. 1 `a 2 in Lecons sur la th´eorie des gaz. Gauthier- Villars, 1902.
[2] Marc Briant and Yan Guo. Asymptotic stability of the Boltzmann equa- tion with Maxwell boundary conditions. Journal of Differential Equations, 261(12):7000–7079, 2016.
[3] Isabelle Gallagher, Laure Saint-Raymond, and Benjamin Texier.From New- ton to Boltzmann: hard spheres and short-range potentials. European math- ematical society, 2013.
[4] David Hilbert. Mathematical problems.Bull. Amer. Math. Soc., 8(10):437–
479, 1902.
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[5] Oscar E Lanford. Time evolution of large classical systems. InDynamical systems, theory and applications, pages 1–111. Springer, 1975.
[6] J Loschmidt. Uber das Warmegleichgewicht eines Sys-tems von Korpern mit Riicksicht auf die Schwere. Sitzungsberichte der Akademie der Wis- senschaften, 1876.
[7] Laure Saint-Raymond. Hydrodynamic limits of the Boltzmann equation.
Number 1971. Springer Science & Business Media, 2009.
[8] Ernst Zermelo. Uber einen Satz der Dynamik und die mechanische¨ W¨armetheorie. Annalen der Physik, 293(3):485–494, 1896.
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