Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2190))

  • 494 Accesses

Abstract

The concept of Two-Scale Convergence was introduced in two papers of Nguetseng

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Ailliot, E. Frénod, V. Monbet, Long term object drift forecast in the ocean with tide and wind. Multiscale Model. Simul. 5(2), 514–531 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Ailliot, E. Frénod, V. Monbet, Modeling the coastal ocean over a time period of several weeks. J. Differ. Equ. 248, 639–659 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Bensoussan, J.L. Lions, G. Papanicolaou, in Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, vol. 5 (North Holland, Amsterdam/New York, 1978)

    Google Scholar 

  5. S. Bochner, Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind. Fundam. Math. 20, 262–276 (1933)

    Article  MATH  Google Scholar 

  6. M. Bostan, Periodic solutions for evolution equations. Electron. J. Differ. Equ., Monograph 03 (2002), pp. 41

    Google Scholar 

  7. M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. second order approximation. Multiscale Model. Simul. 8(5), 1923–1957 (2010)

    Google Scholar 

  8. A. Braides, \(\Gamma\) -convergence for Beginners (Oxford University Press, Oxford, 2002)

    Book  MATH  Google Scholar 

  9. N. Crouseilles, E. Frénod, S.A. Hirstoaga, A. Mouton, Two-scale macro-micro decomposition of the Vlasov equation with a strong magnetic field. Math. Models Methods Appl. Sci. 23(8), 1527–1559 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Dal Maso, An Introduction to \(\Gamma\) -Convergence (Birkhäuser Boston, Inc., Boston, MA, 1993)

    Book  MATH  Google Scholar 

  11. J. Diestel, J.J. Uhl, Vector Measures. Mathematical Surveys and Monographs, vol. 15 (American Mathematical Society, Providence, 1977), pp. 322

    Google Scholar 

  12. B. Engquist, Computation of oscillatory solutions to partial differential equations, in Nonlinear Hyperbolic Problems, (St. Etienne, 1986). Lecture Notes in Mathematics, vol. 1270 (Springer, Berlin, 1987), pp. 10–22

    Google Scholar 

  13. I. Faye, E. Frénod, D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment. Discrete Contin. Dyn. Syst. 29(3), 1001–1030 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Faye, E. Frénod, D. Seck, Long term behaviour of singularly perturbed parabolic degenerated equation. J. Nonlinear Anal. Appl. 2016(2), 82–105 (2016)

    Google Scholar 

  15. E. Frénod, Homogénéisation d’équations cinétiques avec potentiels oscillants, PhD thesis, 1994

    Google Scholar 

  16. E. Frénod, Application of the averaging method to the gyrokinetic plasma. Asymptot. Anal. 46(1), 1–28 (2006)

    MathSciNet  MATH  Google Scholar 

  17. E. Frénod, K. Hamdache, Homogenisation of kinetic equations with oscillating potentials. Proc. Roy. Soc. Edinburgh Sect. A 126(6), 1247–1275 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Frénod, E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field. Asymptot. Anal. 18(3–4), 193–214 (1998)

    MathSciNet  MATH  Google Scholar 

  19. E. Frénod, E. Sonnendrücker, The finite Larmor radius approximation. SIAM J. Math. Anal. 32(6), 1227–1247 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Frénod, F. Watbled, The Vlasov equation with strong magnetic field and oscillating electric field as a model for isotop resonant separation. Electron. J. Differ. Equ. 2002(6), 1–20 (2002)

    MathSciNet  MATH  Google Scholar 

  21. E. Frénod, M. Gutnic, S.A. Hirstoaga, First order two-scale particle-in-cell numerical method for the Vlasov equation. Esaim: Proc. 38, 348–360 (2012). (Cemracs 2011Project Proceeding)

    Google Scholar 

  22. E. Frénod, P.A. Raviart, E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field. J. Math. Pures Appl. 80(8), 815–843 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Frénod, A. Mouton, E. Sonnendrücker, Two scale numerical simulation of the weakly compressible 1d isentropic Euler equations. Numer. Math. 108(2), 263–293 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Frénod, F. Salvarani, E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method. Math. Models Methods Applied Sci. 19(2), 175–197 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. E. Frenod, S.A. Hirstoaga, M. Lutz, Long time simulation of a highly oscillatory Vlasov equation with an exponential integrator. Comptes Rendus de Mécanique 342(10–11), 595–609 (2014)

    Article  Google Scholar 

  26. E. Frenod, S. Hirstoaga, M. Lutz, E. Sonnendrücker, Long time behaviour of an exponential integrator for a Vlasov-Poisson system with strong magnetic field. Commun. Comput. Phys. 18(2), 263–296 (2015)

    Article  MathSciNet  Google Scholar 

  27. P. Gérard, Microlocal defect measures. Commun. Partial Differ. Equ. 16(11), 1761–1794 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Ghendrih, M. Hauray, A. Nouri, Derivation of a gyrokinetic model. existence and uniqueness of specific stationary solution. Kinet. Relat. Models. AIMS 2, 707–725 (2009)

    Google Scholar 

  29. F. Murat, L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications (Birkhäuser Boston, Boston, MA, 1997), pp. 21–43

    Google Scholar 

  30. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics. SIAM J. Math. Anal. 21(6), 1394–1414 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. A.A. Pankov, G-convergence and Homogenization of Nonlinear Partial Differential Operators (Kluwer Academic Publishers, Dordrecht, 1997)

    Book  MATH  Google Scholar 

  33. E. Sanchez-Palencia, in Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127 (Springer, Berlin, 1980)

    Google Scholar 

  34. L. Schwartz, Analyse III: Calcul Intégral (Hermann, Paris, 1993)

    MATH  Google Scholar 

  35. L. Tartar, Cours Peccot (Collège de France, Paris, 1977)

    Google Scholar 

  36. L. Tartar, Compensated compactness and applications to partial differential equations: Heriot-watt symposium, in Nonlinear Analysis and Mechanics. Research Notes in Mathematics, vol. 4 (Pitman, Boston, MA/London, 1979), pp. 136–211

    Google Scholar 

  37. L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115(3–4), 193–230 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. L. Tartar, The General Theory of Homogenization. A Personalized Introduction(Springer, Berlin, 2009)

    Google Scholar 

  39. K. Yosida, Functional Analysis, 6th edn. (Springer, Berlin/New York, 1980), Grundlehren der Mathematischen Wissenschaften 123

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMBA, Université Bretagne Sud, Vannes, France

    Emmanuel Frénod

Authors
  1. Emmanuel Frénod
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Frénod, E. (2017). Introduction. In: Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations. Lecture Notes in Mathematics, vol 2190. Springer, Cham. https://doi.org/10.1007/978-3-319-64668-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation