106 Pages

Cedric Villani Ecole Normale Superieure de Lyon Institut Henri


Gain access to the library to view online
Learn more


Niveau: Secondaire
Landau damping Cedric Villani Ecole Normale Superieure de Lyon & Institut Henri Poincare, 11 rue Pierre et Marie Curie, F-75231 Paris Cedex 05, FRANCE E-mail address :

  • landau damping

  • asymptotic behavior

  • equation near

  • qualitative recap

  • nonlinear stability

  • linearized vlasov

  • ens de lyon

  • vlasov equation



Published by
Reads 58
Language English

Landau damping
C´edric Villani
´ ´Ecole Normale Superieure de Lyon & Institut Henri
E-mail address: cvillani@umpa.ens-lyon.fr`A MarseilIe, en juillet 2010Contents
Foreword 7
Chapter 1. Mean field approximation 9
1. The Newton equations 9
2. Mean field limit 10
3. Precised results 14
4. Singular potentials 15
Bibliographical notes 17
Chapter 2. Qualitative behavior of the Vlasov equation 19
1. Boundary conditions 19
2. Structure 20
3. Invariants and identities 21
4. Equilibria 23
5. Speculations 24
Bibliographical notes 25
Chapter 3. Linearized Vlasov equation near homogeneity 29
0. Free transport 29
1. Linearization 32
2. Separation of modes 33
3. Mode-by-mode study 35
4. The Landau–Penrose stability criterion 38
5. Asymptotic behavior of the kinetic distribution 42
6. Qualitative recap 43
Bibliographical notes 45
Chapter 4. Nonlinear Landau damping 47
1. Nonlinear stability? 47
2. Elusive bounds 48
3. Backus’s objection 48
4. Numerical simulations 49
5. Theorem 49
6. The information cascade 53
Bibliographical notes 55
Chapter 5. Gliding analytic regularity 57
1. Preliminary analysis 57
2. Algebra norms 58
3. Gliding regularity 61
4. Functional analysis 62
Bibliographical notes 63
Chapter 6. Characteristics in damped forcing 65
1. Damped forcing 65
2. Scattering 66
Bibliographical notes 68
Chapter 7. Reaction against an oscillating background 69
1. Regularity extortion 69
2. Solving the reaction equation 70
3. Analysis of the kernel K 72
4. Analysis of the integral equation 74
5. Effect of singular interactions 76
6. Large time estimates via exponential moments 78
Bibliographical notes 80
Chapter 8. Newton’s scheme 81
0. The classical Newton scheme 81
1. Newton scheme for the nonlinear Vlasov equation 83
2. Short time estimates 84
3. Large time estimates 86
4. Main result 93
Bibliographical notes 93
Chapter 9. Conclusions 95
Bibliographical notes 97
Bibliography 99Abstract. This course was taught in the summer of 2010 in
the Centre International des Rencontres Math´ematiques as part
of a programon mathematical plasmaphysics related to the ITER
project; itconstitutesanintroductiontotheLandaudampingphe-
nomenon in the linearized and perturbative nonlinear regimes, fol-
lowing the recent work [74] by Mouhot & Villani.
In 1936, Lev Landau devised the basic collisional kinetic model
for plasma physics, now commonly called the Landau–Fokker–Planck
equation. With this model he was introducing the notion of relaxation
in plasma physics: relaxation `a la Boltzmann, by increase of entropy,
or equivalently loss of information.
relaxation without entropy increase, with preservation of information.
The revolutionary idea that conservative phenomena may exhibit ir-
reversible features has been extremely influential, and later led to the
concept of violent relaxation.
Thisideahasalsobeencontroversial andintriguing,triggeringhun-
dredsofpapersandmanydiscussions. ThebasicmodelusedbyLandau
was the linearized Vlasov–Poisson equation, which is only a formal ap-
proximation of the Vlasov–Poisson equation. In the present notes I
shall present the recent work by Cl´ement Mouhot and myself, extend-
ing Landau’s results to the nonlinear Vlasov–Poisson equation in the
perturbative regime. Although this extension is still far from handling
the mysterious fully nonlinear regime, it already turned out to be rich
These notes start with basic reminders about classical particle sys-
tems and Vlasov equations, assuming no prerequisite from modeling
nor physics. Standard notation is used throughout the text, except
maybe for the Fourier transforms: if h = h(x,v) is a function on the
bposition-velocity phase space, then h stands for the Fourier transform
ein thex variable only, whileh stands for the Fourier transform in both
x and v variables. Precise conventions will be given later on.
A preliminary version of this course was taught in the summer of
2010 in Cotonou, Benin, on the invitation of Wilfrid Gangbo; it is
a pleasure to thank the audience for their interest and enthusiasm.
The first version of the notes was mostly typed during the nights of a
meeting on wave turbulence organized by Christophe Josserand, in the
welcoming library of the gorgeous Domaine des Treilles of the Fonda-
tion Schlumberger. Then the notes were polished as I was teaching the
´course, on the invitation of Eric Sonnendru¨cker, as part of the Cem-
racs 2010 program on plasma physics and mathematics of ITER, in
the Centre International des Rencontres Math´ematiques (CIRM), Lu-
miny, near Marseille, France. I hope this text has retained a bit of the
magical atmosphere of work and play which was in the air during that
summer in Provence.
This is also an opportunity to honor the memory of Naoufel Ben
Abdallah, who tragically passed away, only days before this course
was held. Naoufel was a talented researcher, an energetic colleague, a
reliable leader as well as a joyful fellow. I cherish the memory of an
astonishing hike which we did together, also with his wife Najla and
ourcommon friend Jean Dolbeault, in the Haleakalacrater onHawai‘i,
back in 1998. These memories of good times will not fade, and neither
will the beauty of Naoufel’s contribution to science.