Trajectorial hypocoercivity and application to control theory
Helge Dietert1 ; Frédéric Hérau2 ; Harsha Hutridurga3 ; Clément Mouhot4
1 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG F-75006 Paris, France.
2 Laboratoire de Mathématiques Jean Leray, Nantes Université 2 rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 3, France
3 Department of Mathematics, Indian Institute of Technology Bombay Powai, Mumbai 400076, India
4 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge Wilberforce Road, CB3 0WA Cambridge, UK
Séminaire Laurent Schwartz — EDP et applications (2021-2022), Exposé no. 8, 10 p.

We present the quantitative method of the recent work [6] in a simple setting, together with a compactness argument that was not included in [6] and has interest per se. We are concerned with the exponential stabilisation (spectral gap) for linear kinetic equations with degenerate thermalisation, i.e. when the collision operator vanishes on parts of the spatial domain. The method in [6] covers both scattering and Fokker-Planck type operators, and deals with external potential and boundary conditions, but in these notes we present only its core argument and restrict ourselves to the kinetic Fokker-Planck in the periodic torus with unit velocities and a thermalisation degeneracy (this equation is not covered by the previous results [2, 9, 7]).

Publié le :
DOI : 10.5802/slsedp.156
Helge Dietert 1 ; Frédéric Hérau 2 ; Harsha Hutridurga 3 ; Clément Mouhot 4

1 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG F-75006 Paris, France.
2 Laboratoire de Mathématiques Jean Leray, Nantes Université 2 rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 3, France
3 Department of Mathematics, Indian Institute of Technology Bombay Powai, Mumbai 400076, India
4 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge Wilberforce Road, CB3 0WA Cambridge, UK 
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Helge Dietert; Frédéric Hérau; Harsha Hutridurga; Clément Mouhot. Trajectorial hypocoercivity and application to control theory. Séminaire Laurent Schwartz — EDP et applications (2021-2022), Exposé no. 8, 10 p. doi : 10.5802/slsedp.156. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.156/

[1] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), pp. 1024–1065. | DOI | MR | Zbl

[2] É. Bernard and F. Salvarani, On the exponential decay to equilibrium of the degenerate linear Boltzmann equation, Journal of Functional Analysis, 265 (2013), p. 1934–1954. | DOI | MR | Zbl

[3] M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR, 248 (1979), pp. 1037–1040.

[4] —Solutions of some problems of vector analysis, associated with the operators div and grad , in Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, vol. 1980 of Trudy Sem. S. L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 5–40, 149.

[5] J. Bourgain and H. Brezis, On the equation divy=f and application to control of phases, Journal of the American Mathematical Society, 16 (2002), p. 393–427. | DOI | MR

[6] H. Dietert, F. Hérau, H. Hutridurga, and C. Mouhot, Quantitative geometric control in linear kinetic theory, 2022. | arXiv

[7] J. Evans and I. Moyano, Quantitative rates of convergence to equilibrium for the degenerate linear boltzmann equation on the torus, 2019. | arXiv

[8] Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), pp. 1104–1135. | DOI | MR | Zbl

[9] D. Han-Kwan and M. Léautaud, Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium, Annals of PDE, 1 (2015). | DOI | MR | Zbl

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