Skip to main content
SpringerLink
Log in

The Nash-Moser Iteration Technique with Application to Characteristic Free-Boundary Problems

  • Conference paper
  • First Online:
Hyperbolic Conservation Laws and Related Analysis with Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 49))

  • 1514 Accesses

Abstract

These notes are an overview of the Nash-Moser iteration technique for solving PDEs (or other non-linear problems) via linearisation, where the linearised equations admit estimates with a loss of regularity with respect to the source term, coefficients and/or boundary/initial data. We first introduce the abstract setting along with a version of the iteration scheme due to Hörmander (Arch Ration Mech Anal 62(1):1–52, 1976). We then introduce some modifications which allow the scheme to be applied to some characteristic free-boundary problems for hyperbolic conservation laws. We focus on the case of supersonic vortex sheets in 2D as considered by Coulombel and Secchi in Ann Sci Éc Norm Supér (4) 41(1):85–139, 2008.

2010 Mathematics Subject Classification 76N10 (35L65 35L67 35Q35 35R35)

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. R.A. Adams, J.J.F. Fournier, Sobolev Spaces. Volume 140 of Pure and Applied Mathematics (Amsterdam), 2nd edn. (Elsevier/Academic, Amsterdam, 2003)

    Google Scholar 

  2. S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Commun. Partial Differ. Equ. 14(2), 173–230 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Alinhac, P. Gérard, Pseudo-differential Operators and the Nash-Moser Theorem (American Mathematical Society, Providence, 2007)

    MATH  Google Scholar 

  4. W.I. Axford, The stability of plane current-vortex sheets. Q. J. Mech. Appl. Math. 13(3), 314–324 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  5. G.-Q. Chen, Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics. Arch. Ration. Mech. Anal. 187(3), 369–408 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. G.-Q. Chen, Y.-G. Wang, Characteristic discontinuities and free boundary problems for hyperbolic conservation laws, in Nonlinear Partial Differential Equations – The Abel Symposium 2010, Oslo, ed. by H. Holden, K. Karlsen. Volume 7 of Abel Symposia (Springer, 2012)

    Google Scholar 

  7. J.-F. Coulombel, P. Secchi, The stability of compressible vortex sheets in two space dimensions. Indiana Univ. Math. J. 53(4), 941–1012 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. J.-F. Coulombel, P. Secchi, Nonlinear compressible vortex sheets in two space dimensions. Ann. Sci. Éc. Norm. Supér. (4) 41(1), 85–139 (2008)

    Google Scholar 

  9. J.-F. Coulombel, A. Morando, P. Secchi, P. Trebeschi, A priori estimates for 3D incompressible current-vortex sheets. Commun. Math. Phys. 311(1), 247–275 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. I. Ekeland, An inverse function theorem in Fréchet spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(1), 91–105 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. L.C. Evans, Partial Differential Equations. Volume 19 of Graduate Studies in Mathematics, 2nd edn. (American Mathematical Society, Providence, 2010)

    Google Scholar 

  12. J.A. Fejer, Hydromagnetic stability at a fluid velocity discontinuity between compressible fluids. Phys. Fluids 7, 499–503 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  13. J.A. Fejer, J.W. Miles, On the stability of a plane vortex sheet with respect to three-dimensional disturbances. J. Fluid Mech. 15, 335–336 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  14. R.S. Hamilton, The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)

    Google Scholar 

  15. L. Hörmander, The boundary problems of physical geodesy. Arch. Ration. Mech. Anal. 62(1), 1–52 (1976)

    Article  MATH  Google Scholar 

  16. A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer, New York, 1984)

    Book  MATH  Google Scholar 

  17. J.W. Miles, On the disturbed motion of a plane vortex sheet. J. Fluid Mech. 4, 538–552 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Morando, P. Trebeschi, Two-dimensional vortex sheets for the nonisentropic Euler equations: linear stability. J. Hyperbolic Differ. Equ. 5(3), 487–518 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Morando, Y. Trakhinin, P. Trebeschi, Stability of incompressible current-vortex sheets. J. Math. Anal. Appl. 347(2), 502–520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Moser, A new technique for the construction of solutions of nonlinear differential equations. Proc. Natl. Acad. Sci. USA 47, 1824–1831 (1961)

    Article  MATH  Google Scholar 

  21. J. Nash, The imbedding problem for Riemannian manifolds. Ann. Math. (2) 63, 20–63 (1956)

    Google Scholar 

  22. L. Nirenberg, Topics in Nonlinear Functional Analysis. Volume 6 of Courant Lecture Notes in Mathematics (New York University Courant Institute of Mathematical Sciences, New York, 2001). Chapter 6 by E. Zehnder, Notes by R. A. Artino, Revised reprint of the 1974 original.

  23. J. Schwartz, Nonlinear Functional Analysis (Gordon and Breach, New York, 1969)

    MATH  Google Scholar 

  24. Y. Trakhinin, Existence of compressible current-vortex sheets: variable coefficients linear analysis. Arch. Ration. Mech. Anal. 177(3), 331–366 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics. Arch. Ration. Mech. Anal. 191(2), 245–310 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Y.-G. Wang, F. Yu, Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets. Arch. Ration. Mech. Anal. (2013). doi:10.1007/s00205-012-0601-9

    Google Scholar 

Download references

Acknowledgements

My research is supported by a UK EPSRC grant to the Department of Mathematics at Oxford University. I would like to thank my supervisor, Gui-Qiang G. Chen, for helpful discussions on this problem.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK

    Ben Stevens

Authors
  1. Ben Stevens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben Stevens .

Editor information

Editors and Affiliations

  1. University of Oxford Mathematical Institute, Oxford, United Kingdom

    Gui-Qiang G. Chen

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  3. University of Oslo Centre of Mathematics for Applications, Oslo, Norway

    Kenneth H. Karlsen

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Stevens, B. (2014). The Nash-Moser Iteration Technique with Application to Characteristic Free-Boundary Problems. In: Chen, GQ., Holden, H., Karlsen, K. (eds) Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proceedings in Mathematics & Statistics, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39007-4_13

Download citation

Publish with us

Policies and ethics

Search

Navigation