Abstract
We consider an initial boundary value problem for a quantum version of the Zakharov system arising in plasma physics. We prove the global well-posedness of this problem in some Sobolev type classes and study properties of solutions. This means that the quantum Zakharov model is coherent with the observation of an absence of collapse of (quantum) Langmuir waves, hence might be a valid model for the description of electronic plasma waves. In the dissipative case the existence of a finite-dimensional global attractor is established, and regularity properties of this attractor are studied. For this we use the recently developed method of quasi-stability estimates. In the case when external forces are C ∞ functions, we show that every trajectory in the attractor is C ∞ in both time and spatial variables. This can be interpreted as the absence of sharp coherent structures in the limiting dynamics.
Similar content being viewed by others
References
Babin A., Vishik M.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)
Bourgain J.: On the Cauchy and invariant measure problem for the periodic Zakharov system. Duke Math. J. 76, 175–202 (1994)
Bourgain J., Colliander J.: On well-posedness of the Zakharov system. Int. Math. Res. Not. 11, 515–546 (1996)
Chueshov, I.: Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/
Chueshov I., Lasiecka I.: Attractors for second-order evolution equations with a nonlinear damping. J. Dyn. Differ. Eqs. 16, 469–512 (2004)
Chueshov, I., Lasiecka, I.: Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS 912. AMS, Providence (2008)
Chueshov I., Lasiecka I.: Von Karman Evolution Equations. Springer, New York (2010)
Chueshov, I., Lasiecka, I.: Well-Posedness and Long Time Behavior in Nonlinear Dissipative Hyperbolic-Like Evolutions with Critical Exponents. preprint ArXiv:1204.5864v1 (2012)
Chueshov I., Shcherbina A.: On 2D Zakharov system in a bounded domain. Differ. Int. Eqs. 18, 781–812 (2005)
Flahaut I.: Attractors for the dissipative Zakharov system. Nonlinear Anal. 16, 599–633 (1991)
Garcia L.G., Haas F., Goedert J., Oliveira L.P.L.: Modified Zakharov equations for plasmas with a quantum correction. Phys. Plasmas 12, 012302 (2005)
Ghidaglia J.M., Temam R.: Regularity of the solutions of second order evolution equations and their attractors. Ann. della Scuola Norm. Sup. Pisa 14, 485–511 (1987)
Ginibre J., Tsutsumi Y., Velo G.: On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151, 384–436 (1997)
Glangetas L., Merle F.: Existence and self-similar blow up solutions for Zakharov equation in dimension two, Part I. Commun. Math. Phys. 160, 173–215 (1994)
Goubet O., Moise I.: Attractor for dissipative Zakharov system. Nonlinear Anal. 7, 823–847 (1998)
Haas F., Shukla P.K.: Quantum and classical dynamics of Langmuir wave packets. Phys. Rev. E 79, 066402 (2009)
Ladyzhenskaya O.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)
Lions J.L., Magenes E.: Problemes aux Limites Non Homogenes et Applications. Dunod, Paris (1968)
Málek J., Nečas J.: A finite dimensional attractor for three dimensional flow of incompressible fluids. J. Differ. Eqs 127, 498–518 (1996)
Málek J., Pražak D.: Large time behavior via the method of l-trajectories. J. Differ. Eqs 181, 243–279 (2002)
Shcherbina A.S.: Gevrey regularity of the global attractor for the dissipative Zakharov system. Dyn. Syst. 18, 201–225 (2003)
Simon J.: Compact sets in the space L p(0,T;B). Ann. di Mat. Pura Appl., Ser.4 148, 65–96 (1987)
Simpson G., Sulem C., Sulem P.L.: Arrest of Langmuir wave collapse by quantum effects. Phys. Rev. 80, 056405 (2009)
Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)
Triebel H.: Interpolation Theory, Functional Spaces and Differential Operators. North Holland, Amsterdam (1978)
Zakharov V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908–912 (1972)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chueshov, I. Quantum Zakharov model in a bounded domain. Z. Angew. Math. Phys. 64, 967–989 (2013). https://doi.org/10.1007/s00033-012-0278-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-012-0278-9