## Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Jacob Bedrossian
A brief summary of nonlinear echoes and Landau damping
Journées équations aux dérivées partielles (2017), Exp. No. 2, 14 p., doi: 10.5802/jedp.652
Article PDF

Résumé - Abstract

In this expository note we review some recent results on Landau damping in the nonlinear Vlasov equations, focusing specifically on the recent construction of nonlinear echo solutions by the author [arXiv:1605.06841] and the associated background. These solutions show that a straightforward extension of Mouhot and Villani’s theorem on Landau damping to Sobolev spaces on $\mathbb{T}^n_x \times \mathbb{R}^n_v$ is impossible and hence emphasize the subtle dependence on regularity of phase mixing problems. This expository note is specifically aimed at mathematicians who study the analysis of PDEs, but not necessarily those who work specifically on kinetic theory. However, for the sake of brevity, this review is certainly not comprehensive.

Bibliographie

[1] Robert A. Adams & John J. F. Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam) 140, Elsevier/Academic Press, Amsterdam, 2003
[2] J. Bedrossian, “Nonlinear echoes and Landau damping with insufficient regularity”, arXiv:1605.06841 (2016)
[3] J. Bedrossian, “Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation”, To appear in Annals of PDE. arXiv:1704.00425 (2017)
[4] J. Bedrossian, P. Germain & N. Masmoudi, “Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold”, To appear in Mem. Amer. Math. Soc., arXiv:1506.03720 (2015)
[5] J. Bedrossian, P. Germain & N. Masmoudi, “Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold”, arXiv:1506.03721 (2015)
[6] J. Bedrossian, P. Germain & N. Masmoudi, “On the stability threshold for the 3D Couette flow in Sobolev regularity”, Ann. of Math. 157 (2017) no. 1
[7] J. Bedrossian, P. Germain & N. Masmoudi, “Stability of the Couette flow at high Reynolds number in 2D and 3D”, arXiv:1712.02855 (2017)
[8] J. Bedrossian, N. Masmoudi & C. Mouhot, “Landau damping in finite regularity for unconfined systems with screened interactions”, To appear in Comm. Pure Appl. Math. (2016)
[9] J. Bedrossian, N. Masmoudi & V. Vicol, “Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow”, Arch. Rat. Mech. Anal. 216 (2016) no. 3, p. 1087-1159
[10] J. Bedrossian, V. Vicol & F. Wang, “The Sobolev stability threshold for 2D shear flows near Couette”, To appear in J. Nonlin. Sci.. Preprint: arXiv:1604.01831 (2016)
[11] Jacob Bedrossian, Michele Coti Zelati & Vlad Vicol, “Vortex axisymmetrization, inviscid damping, and vorticity depletion in the linearized 2D Euler equations”, arXiv preprint arXiv:1711.03668 (2017)
[12] Jacob Bedrossian & Nader Masmoudi, “Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations”, Publ. math. de l’IHÉS (2013), p. 1-106
[13] Jacob Bedrossian, Nader Masmoudi & Clement Mouhot, “Landau damping: paraproducts and Gevrey regularity”, Annals of PDE 2 (2016) no. 1, p. 1-71
[14] J.M. Bony, “Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non lináires”, Ann.Sc.E.N.S. 14 (1981), p. 209-246
[15] T. J. M. Boyd & J. J. Sanderson, The physics of plasmas, Cambridge University Press, Cambridge, 2003 Article
[16] E. Caglioti & C. Maffei, “Time asymptotics for solutions of Vlasov-Poisson equation in a circle”, J. Stat. Phys. 92 (1998) no. 1/2
[17] James Colliander, Markus Keel, Gigiola Staffilani, Hideo Takaoka & Terence Tao, “Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation”, Inventiones mathematicae 181 (2010) no. 1, p. 39-113
[18] P. Degond, “Spectral theory of the linearized Vlasov-Poisson equation”, Trans. Amer. Math. Soc. 294 (1986) no. 2, p. 435-453
[19] Erwan Faou & Frédéric Rousset, “Landau damping in Sobolev spaces for the Vlasov-HMF model”, Arch. Ration. Mech. Anal. 219 (2016) no. 2, p. 887-902 Article
[20] Bastien Fernandez, David Gérard-Varet & Giambattista Giacomin, “Landau damping in the Kuramoto model”, Preprint arXiv:1410.6006, to appear in Ann. Institut Poincaré - Analysis nonlinéaire
[21] Robert Glassey & Jack Schaeffer, “Time decay for solutions to the linearized Vlasov equation”, Transport Theory Statist. Phys. 23 (1994) no. 4, p. 411-453 Article
[22] Robert Glassey & Jack Schaeffer, “On time decay rates in Landau damping”, Comm. Part. Diff. Eqns. 20 (1995) no. 3-4, p. 647-676 Article
[23] François Golse, Pierre-Louis Lions, Benoît Perthame & Rémi Sentis, “Regularity of the moments of the solution of a transport equation”, Journal of functional analysis 76 (1988) no. 1, p. 110-125
[24] François Golse, Benoıt Perthame & Rémi Sentis, “Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport”, CR Acad. Sci. Paris Sér. I Math 301 (1985) no. 7, p. 341-344
[25] Marcel Guardia & Vadim Kaloshin, “Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation”, Journal of the European Mathematical Society 17 (2015) no. 1, p. 71-149
[26] D. Han-Kwan, “Quasineutral limit of the Vlasov-Poisson system with massless electrons”, Comm. Part. Diff. Eqns. 36 (2011) no. 8, p. 1385-1425
[27] D. Han-Kwan & M. Iacobelli, “The quasineutral limit of the Vlasov-Poisson equation in Wasserstein metric”, arXiv preprint arXiv:1412.4023 (2014)
[28] D. Han-Kwan & F. Rousset, “Quasineutral limit for Vlasov-Poisson with Penrose stable data”, arXiv preprint arXiv:1508.07600 (2015)
[29] Zaher Hani, Benoit Pausader, Nikolay Tzvetkov & Nicola Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, in Forum of Mathematics, Pi, Cambridge University Press, 2015
[30] L. Hörmander, “The Nash-Moser theorem and paradifferential operators”, Analysis, et cetera (1990), p. 429-449
[31] H. J. Hwang & J. J. L. Velaźquez, “On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem”, Indiana Univ. Math. J (2009), p. 2623-2660
[32] Pierre-Emmanuel Jabin & Luis Vega, “A real space method for averaging lemmas”, Journal de mathématiques pures et appliquées 83 (2004) no. 11, p. 1309-1351
[33] Herbert Koch, Daniel Tataru & Monica Visan, Dispersive equations and nonlinear waves, in Oberwolfach Seminars, Springer, 2014
[34] N. Krall & A. Trivelpiece, Principles of plasma physics, San Francisco Press, 1986
[35] Lev Landau, “On the vibration of the electronic plasma”, J. Phys. USSR 10 (1946) no. 25
[36] D. Levermore & M. Oliver, “Analyticity of solutions for a generalized Euler equation”, J. Diff. Eqns. 133 (1997), p. 321-339
[37] Zhiwu Lin & Chongchun Zeng, “Small BGK waves and nonlinear Landau damping”, Comm. Math. Phys. 306 (2011) no. 2, p. 291-331 Article
[38] J. Malmberg, C. Wharton, C. Gould & T. O’Neil, “Plasma wave echo”, Phys. Rev. Lett. 20 (1968) no. 3, p. 95-97
[39] Clément Mouhot & Cédric Villani, “On Landau damping”, Acta Math. 207 (2011), p. 29-201
[40] Thomas M O’Neil, “Effect of Coulomb collisions and microturbulence on the plasma wave echo”, The Physics of Fluids 11 (1968) no. 11, p. 2420-2425
[41] W. Orr, “The stability or instability of steady motions of a perfect liquid and of a viscous liquid, Part I: a perfect liquid”, Proc. Royal Irish Acad. Sec. A: Math. Phys. Sci. 27 (1907), p. 9-68
[42] Raymond E. A. C. Paley & Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications 19, American Mathematical Society, Providence, RI, 1987, Reprint of the 1934 original
[43] O. Penrose, “Electrostatic instability of a uniform non-Maxwellian plasma”, Phys. Fluids 3 (1960), p. 258-265
[44] AA Schekochihin, JT Parker, EG Highcock, PJ Dellar, W Dorland & GW Hammett, “Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence”, Journal of Plasma Physics 82 (2016) no. 2
[45] T. Stix, Waves in plasmas, Springer, 1992
[46] CH Su & C Oberman, “Collisional damping of a plasma echo”, Physical Review Letters 20 (1968) no. 9
[47] T. Tao, “Nonlinear dispersive equations”, CBMS Regional Conference Series in Mathematics 106 (2006)
[48] Lloyd Nicholas Trefethen & Mark Embree, Spectra and pseudospectra: the behavior of nonnormal matrices and operators, Princeton University Press, 2005
[49] I. Tristani, “Landau damping for the linearized Vlasov Poisson equation in a weakly collisional regime”, arXiv:1603.07219 (2016)
[50] N.G. van Kampen, “On the theory of stationary waves in plasmas”, Physica 21 (1955), p. 949-963
[51] A. A. Vlasov, “The vibrational properties of an electron gas”, Zh. Eksp. Teor. Fiz. 291 (1938) no. 8, In russian, translation in english in Soviet Physics Uspekhi, vol. 93 Nos. 3 and 4, 1968
[52] D. Wei, Z. Zhang & W. Zhao, “Linear inviscid damping for a class of monotone shear flow in Sobolev spaces”, Communications on Pure and Applied Mathematics (2015)
[53] Dongyi Wei, Zhifei Zhang & Weiren Zhao, “Linear inviscid damping and vorticity depletion for shear flows”, arXiv preprint arXiv:1704.00428 (2017)
[54] J.H. Yu & C.F. Driscoll, “Diocotron wave echoes in a pure electron plasma”, IEEE Trans. Plasma Sci. 30 (2002) no. 1
[55] J.H. Yu, C.F. Driscoll & T.M. O‘Neil, “Phase mixing and echoes in a pure electron plasma”, Phys. of Plasmas 12 (2005) no. 055701
[56] Christian Zillinger, “Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity”, Archive for Rational Mechanics and Analysis 221 (2016) no. 3, p. 1449-1509
Copyright Cellule MathDoc 2018 | Crédit | Plan du site