On global attractors and radiation damping for nonrelativistic particle coupled to scalar field

Komech, A.; Kopylova, E.; Spohn, H.

doi:10.1090/spmj/1492

On global attractors and radiation damping for nonrelativistic particle coupled to scalar field
HTML articles powered by AMS MathViewer

by A. Komech, E. Kopylova and H. Spohn

St. Petersburg Math. J. 29 (2018), 249-266

DOI: https://doi.org/10.1090/spmj/1492

Published electronically: March 12, 2018

PDF | Request permission

Abstract:

The Hamiltonian system of a scalar wave field and a single nonrelativistic particle coupled in a translation invariant manner is considered. The particle is also subject to a confining external potential. The stationary solutions of the system are Coulomb type wave fields centered at those particle positions for which the external force vanishes. It is proved that the solutions of finite energy converge, in suitable local energy seminorms, to the set ${\mathcal S}$ of all stationary states in the long time limit $t\to \pm \infty$. Next it is shown that the rate of relaxation to a stable stationary state is determined by the spatial decay of initial data. The convergence is followed by the radiation of the dispersion wave that is a solution of the free wave equation.

Similar relaxation has been proved previously for the case of a relativistic particle when the speed of the particle is less than the wave speed. Now these results are extended to a nonrelativistic particle with velocity, including that greater than the wave speed. However, the research is restricted to the plane particle trajectories in $\mathbb {R}^3$. Extension to the general case remains an open problem.

References

M. Abraham, Prinzipien der Dynamik des Elektrons, Ann. Phys. 10 (1903), 105–179.
V. S. Buslaev and G. S. Perel′man, Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, Algebra i Analiz 4 (1992), no. 6, 63–102 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 6, 1111–1142. MR 1199635
V. S. Buslaev and G. S. Perel′man, On the stability of solitary waves for nonlinear Schrödinger equations, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75–98. MR 1334139, DOI 10.1090/trans2/164/04
Vladimir S. Buslaev and Catherine Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 20 (2003), no. 3, 419–475 (English, with English and French summaries). MR 1972870, DOI 10.1016/S0294-1449(02)00018-5
Jürg Fröhlich and Zhou Gang, Emission of Cherenkov radiation as a mechanism for Hamiltonian friction, Adv. Math. 264 (2014), 183–235. MR 3250283, DOI 10.1016/j.aim.2014.07.013
John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
Valery Imaikin, Alexander Komech, and Herbert Spohn, Scattering theory for a particle coupled to a scalar field, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 387–396. Partial differential equations and applications. MR 2026201, DOI 10.3934/dcds.2004.10.387
Valery Imaikin, Alexander Komech, and Peter A. Markowich, Scattering of solitons of the Klein-Gordon equation coupled to a classical particle, J. Math. Phys. 44 (2003), no. 3, 1202–1217. MR 1958263, DOI 10.1063/1.1539900
V. Imaikin, A. Komech, and H. Spohn, Soliton-type asymptotics and scattering for a charge coupled to the Maxwell field, Russ. J. Math. Phys. 9 (2002), no. 4, 428–436. MR 1966018
Valery Imaikin, Alexander Komech, and Herbert Spohn, Rotating charge coupled to the Maxwell field: scattering theory and adiabatic limit, Monatsh. Math. 142 (2004), no. 1-2, 143–156. MR 2065026, DOI 10.1007/s00605-004-0232-9
Valery Imaikin, Alexander Komech, and Norbert Mauser, Soliton-type asymptotics for the coupled Maxwell-Lorentz equations, Ann. Henri Poincaré 5 (2004), no. 6, 1117–1135. MR 2105320, DOI 10.1007/s00023-004-0193-5
A. I. Komech, On attractor of a singular nonlinear $\textrm {U}(1)$-invariant Klein-Gordon equation, Progress in analysis, Vol. I, II (Berlin, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 599–611. MR 2032730
Alexander Komech and Andrew Komech, Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Ration. Mech. Anal. 185 (2007), no. 1, 105–142. MR 2308860, DOI 10.1007/s00205-006-0039-z
Alexander Komech and Andrew Komech, On global attraction to solitary waves for the Klein-Gordon field coupled to several nonlinear oscillators, J. Math. Pures Appl. (9) 93 (2010), no. 1, 91–111 (English, with English and French summaries). MR 2579377, DOI 10.1016/j.matpur.2009.08.011
Alexander Komech and Andrew Komech, Global attraction to solitary waves for Klein-Gordon equation with mean field interaction, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 3, 855–868. MR 2526405, DOI 10.1016/j.anihpc.2008.03.005
Alexander Komech and Andrew Komech, Global attraction to solitary waves for a nonlinear Dirac equation with mean field interaction, SIAM J. Math. Anal. 42 (2010), no. 6, 2944–2964. MR 2745798, DOI 10.1137/090772125
Alexander Komech, Quantum mechanics: genesis and achievements, Springer, Dordrecht, 2013. MR 3012343, DOI 10.1007/978-94-007-5542-0
Alexander Komech, Attractors of Hamilton nonlinear PDEs, Discrete Contin. Dyn. Syst. 36 (2016), no. 11, 6201–6256. MR 3543587, DOI 10.3934/dcds.2016071
Alexander Komech and Elena Kopylova, Dispersion decay and scattering theory, John Wiley & Sons, Inc., Hoboken, NJ, 2012. MR 3015024, DOI 10.1002/9781118382868
Alexander Komech and Herbert Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field, Nonlinear Anal. 33 (1998), no. 1, 13–24. MR 1623033, DOI 10.1016/S0362-546X(97)00538-5
E. A. Kopylova and A. I. Komech, Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle, Russ. J. Math. Phys. 23 (2016), no. 1, 93–100. MR 3482768, DOI 10.1134/S1061920816010076
Alexander Komech, Herbert Spohn, and Markus Kunze, Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Differential Equations 22 (1997), no. 1-2, 307–335. MR 1434147, DOI 10.1080/03605309708821264
Alexander Komech and Herbert Spohn, Long-time asymptotics for the coupled Maxwell-Lorentz equations, Comm. Partial Differential Equations 25 (2000), no. 3-4, 559–584. MR 1748357, DOI 10.1080/03605300008821524
E. A. Kopylova, Weighted energy decay for 3D wave equation, Asymptot. Anal. 65 (2009), no. 1-2, 1–16. MR 2571709
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
I. M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993), no. 2, 297–320. MR 1218303
A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), no. 1, 9–74. MR 1681113, DOI 10.1007/s002220050303
Herbert Spohn, Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge, 2004. MR 2097788, DOI 10.1017/CBO9780511535178
A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), no. 1, 119–146. MR 1071238
A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Differential Equations 98 (1992), no. 2, 376–390. MR 1170476, DOI 10.1016/0022-0396(92)90098-8