Skip to main content
SpringerLink
Log in

Dispersion Phenomena in Partial Differential Equations

  • Reference work entry
Encyclopedia of Complexity and Systems Science

  • 301 Accesses

Definition of the Subject

In a very broad sense, dispersion can be defined as the spreading of a fixed amount of matter, or energy, over a volume whichincreases with time. This intuitive picture suggests immediately the most prominent feature of dispersive phenomena: as matter spreads, its size, definedin a suitable sense, decreases at a certain rate. This effect should be contrasted with dissipation, which might be defined as an actual loss ofenergy, transferred to an external system (heat dissipation being the typical example).

This rough idea has been made very precise during the last 30 years for most evolution equations of mathematical physics. For the classical,linear, constant coefficient equations like the wave, Schrödinger, Klein–Gordon and Dirac equations, the decay of solutions can be measured in theL pnorms, and sharp estimates are available. In addition,detailed information on the profile of the solutions can be obtained, producing an accurate description of the...

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 3,499.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
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

Abbreviations

Notations:

Partial derivatives are written as u t or \( { \partial_{t}u } \), \( { \partial^{\alpha}= \partial_{x_{1}}^{\alpha_{1}}\cdot\partial_{x_{n}}^{\alpha_{n}} } \), the Fourier transform of a function is defined as

$$ \mathcal{F}f(\xi)=\widehat f(\xi)=\int \text{e}^{-ix \cdot\xi}f(x)\text{d} x $$

and we frequently use the mute constant notation \( { A \lesssim B } \) to mean \( { A\le C B } \) for some constant C (but only when the precise dependence of C from the other quantities involved is clear from the context).

Evolution equations:

Partial differential equations describing physical systems which evolve in time. Thus, the variable representing time is distinguished from the others and is usually denoted by t.

Cauchy problem:

A system of evolution equations, combined with a set of initial conditions at an initial time \( { t=t_{0} } \). The problem is well posed if a solution exists, is unique and depends continuously on the data in suitable norms adapted to the problem.

Blow up:

In general, the solutions to a nonlinear evolution equation are not defined for all times but they break down after some time has elapsed; usually the \( { L^{\infty} } \) norm of the solution or some of its derivatives becomes unbounded. This phenomenon is called blow up of the solution.

Sobolev spaces:

we shall use two instances of Sobolev space: the space \( { W^{k,1} } \) with norm

$$ \|u\|_{W^{k,1}}=\sum_{|\alpha|\le k}\|\partial^{\alpha}u\|_{L^{1}} $$

and the space \( { H^{s}_{q} } \) with norm

$$ \|u\|_{H^{s}}=\left\|(1-\Delta)^{s/2}u\right\|_{L^{q}}\:. $$

Recall that this definition does not reduce to the preceding one when \( { q=1 } \). We shall also use the homogeneous space \( { \dot H^{s}_{q} } \), with norm

$$ \left\|u\right\|_{\dot H^{s}}=\left\|(-\Delta)^{s/2}u\right\|_{L^{q}}\:. $$
Dispersive estimate:

a pointwise decay estimate of the form \( { |u(t,x)|\le Ct^{-\alpha} } \), usually for the solution of a partial differential equation.

Bibliography

  1. Artbazar G, Yajima K (2000) The \( { L\sp p } \)-continuity of wave operators for one dimensional Schrödinger operators. J MathSci Univ Tokyo 7(2):221–240

    MathSciNet  MATH  Google Scholar 

  2. Beals M (1994) Optimal \( { L\sp \infty } \) decay for solutions to the wave equation with a potential. CommPartial Diff Eq 19(7/8):1319–1369

    MathSciNet  MATH  Google Scholar 

  3. Ben-Artzi M, Trèves F (1994) Uniform estimates for a class of evolutionequations. J Funct Anal 120(2):264–299

    Google Scholar 

  4. Bergh J, Löfström J (1976) Interpolation spaces. An introduction. Grundlehrender mathematischen Wissenschaften, No 223. Springer, Berlin

    Google Scholar 

  5. Bouclet J-M, Tzvetkov N (2006) On global strichartz estimates for non trappingmetrics.

    Google Scholar 

  6. Bourgain J (1993) Fourier transform restriction phenomena for certain latticesubsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom Funct Anal 3(2):107–156

    MathSciNet  MATH  Google Scholar 

  7. Bourgain J (1995) Some new estimates on oscillatory integrals. In: (1991) Essayson Fourier analysis in honor of Elias M. Stein. Princeton Math Ser vol. 42. Princeton Univ Press, Princeton,pp 83–112

    Google Scholar 

  8. Bourgain J (1995) Estimates for cone multipliers. In: Geometric aspects offunctional analysis. Oper Theory Adv Appl, vol 77. Birkhäuser, Basel, pp 41–60

    Google Scholar 

  9. Bourgain J (1998) Refinements of Strichartz' inequality and applications to2D-NLS with critical nonlinearity. Int Math Res Not 5(5):253–283

    MathSciNet  Google Scholar 

  10. Brenner P (1975) On \( { L\sb{p}-L\sb{p\sp{\prime}} } \) estimates for the wave‐equation. Math Z145(3):251–254

    MathSciNet  MATH  Google Scholar 

  11. Brenner P (1977) \( { L\sb{p}-L\sb{p^{\prime}} } \)-estimates for Fourier integral operators related tohyperbolic equations. Math Z 152(3):273–286

    MathSciNet  MATH  Google Scholar 

  12. Burq N, Gérard P, Tzvetkov N (2003) The Cauchy problem for the nonlinearSchrödinger equation on a compact manifold. J Nonlinear Math Phys 10(1):12–27

    Google Scholar 

  13. Cazenave T (2003) Semilinear Schrödinger equations. Courant Lecture Notes inMathematics, vol 10. New York University Courant Institute of Mathematical Sciences, New York

    Google Scholar 

  14. Choquet‐Bruhat Y (1950) Théorème d'existence pour les équations de lagravitation einsteinienne dans le cas non analytique. CR Acad Sci Paris 230:618–620

    Google Scholar 

  15. Christodoulou D, Klainerman S (1989) The nonlinear stability of the Minkowskimetric in general relativity. In: Bordeaux (1988) Nonlinear hyperbolic problems. Lecture Notes in Math, vol 1402. Springer, Berlin,pp 128–145

    Google Scholar 

  16. D'Ancona P, Fanelli L (2006) Decay estimates for the wave and dirac equationswith a magnetic potential. Comm Pure Appl Anal 29:309–323

    MATH  Google Scholar 

  17. D'Ancona P, Fanelli L (2006) \( { {L}^p } \) – boundedness of the wave operator for the one dimensionalSchrödinger operator. Comm Math Phys 268:415–438

    MathSciNet  MATH  ADS  Google Scholar 

  18. D'Ancona P, Fanelli L (2008) Strichartz and smoothing estimates for dispersiveequations with magnetic potentials. Comm Partial Diff Eq 33(6):1082–1112

    MathSciNet  MATH  Google Scholar 

  19. D'Ancona P, Pierfelice V (2005) On the wave equation with a large roughpotential. J Funct Anal 227(1):30–77

    MathSciNet  MATH  Google Scholar 

  20. D'Ancona P, Georgiev V, Kubo H (2001) Weighted decay estimates for the waveequation. J Diff Eq 177(1):146–208

    MathSciNet  MATH  Google Scholar 

  21. Foschi D (2005) Inhomogeneous Strichartz estimates. J Hyperbolic Diff Eq2(1):1–24

    MathSciNet  MATH  Google Scholar 

  22. Ginibre J, Velo G (1985) The global Cauchy problem for the nonlinearSchrödinger equation revisited. Ann Inst Poincaré H Anal Non Linéaire 2(4):309–327

    Google Scholar 

  23. Ginibre J, Velo G (1995) Generalized Strichartz inequalities for the waveequation. J Funct Anal 133(1):50–68

    MathSciNet  MATH  Google Scholar 

  24. Hassell A, Tao T, Wunsch J (2006) Sharp Strichartz estimates on nontrappingasymptotically conic manifolds. Am J Math 128(4):963–1024

    MathSciNet  MATH  Google Scholar 

  25. Hörmander L (1997) Lectures on nonlinear hyperbolic differential equations.Mathématiques & Applications (Mathematics & Applications), vol 26. Springer, Berlin

    Google Scholar 

  26. John F (1979) Blow-up of solutions of nonlinear wave equations in three spacedimensions. Manuscr Math 28(1–3):235–268

    MATH  Google Scholar 

  27. John F, Klainerman S (1984) Almost global existence to nonlinear waveequations in three space dimensions. Comm Pure Appl Math 37(4):443–455

    MathSciNet  MATH  Google Scholar 

  28. Journé J-L, Soffer A, Sogge CD (1991) Decay estimates for Schrödingeroperators. Comm Pure Appl Math 44(5):573–604

    Google Scholar 

  29. Kato T (1965/1966) Wave operators and similarity for somenon‐selfadjoint operators. Math Ann 162:258–279

    Google Scholar 

  30. Keel M, Tao T (1998) Endpoint Strichartz estimates. Am J Math120(5):955–980

    MathSciNet  MATH  Google Scholar 

  31. Klainerman S (1980) Global existence for nonlinear wave equations. Comm PureAppl Math 33(1):43–101

    MathSciNet  MATH  Google Scholar 

  32. Klainerman S (1981) Classical solutions to nonlinear wave equations andnonlinear scattering. In: Trends in applications of pure mathematics to mechanics, vol III. Monographs Stud Math, vol 11. Pitman, Boston,pp 155–162

    Google Scholar 

  33. Klainerman S (1982) Long-time behavior of solutions to nonlinear evolutionequations. Arch Rat Mech Anal 78(1):73–98

    MathSciNet  MATH  Google Scholar 

  34. Klainerman S (1985) Long time behaviour of solutions to nonlinear waveequations. In: Nonlinear variational problems. Res Notes in Math, vol 127. Pitman, Boston, pp 65–72

    Google Scholar 

  35. Klainerman S, Nicolò F (1999) On local and global aspects of the Cauchyproblem in general relativity. Class Quantum Gravity 16(8):R73–R157

    Google Scholar 

  36. Marzuola J, Metcalfe J, Tataru D. Strichartz estimates and local smoothingestimates for asymptotically flat Schrödinger equations. To appear on J Funct Anal

    Google Scholar 

  37. Metcalfe J, Tataru D. Global parametrices and dispersive estimates forvariable coefficient wave equation. Preprint

    Google Scholar 

  38. Pecher H (1974) Die Existenz regulärer Lösungen für Cauchy- undAnfangs‐Randwert‐probleme nichtlinearer Wellengleichungen. Math Z 140:263–279 (in German)

    MathSciNet  Google Scholar 

  39. Reed M, Simon B (1978) Methods of modern mathematical physics IV. In: Analysisof operators. Academic Press (Harcourt Brace Jovanovich), New York

    Google Scholar 

  40. Segal I (1968) Dispersion for non‐linear relativistic equations. II. AnnSci École Norm Sup 4(1):459–497

    Google Scholar 

  41. Shatah J (1982) Global existence of small solutions to nonlinear evolutionequations. J Diff Eq 46(3):409–425

    MathSciNet  MATH  Google Scholar 

  42. Shatah J (1985) Normal forms and quadratic nonlinear Klein–Gordonequations. Comm Pure Appl Math 38(5):685–696

    MathSciNet  MATH  Google Scholar 

  43. Shatah J, Struwe M (1998) Geometric wave equations. In: Courant Lecture Notesin Mathematics, vol 2. New York University Courant Institute of Mathematical Sciences, New York

    Google Scholar 

  44. Strichartz RS (1970) Convolutions with kernels having singularities ona sphere. Trans Am Math Soc 148:461–471

    MathSciNet  MATH  Google Scholar 

  45. Strichartz RS (1970) A priori estimates for the wave equation and someapplications. J Funct Anal 5:218–235

    MathSciNet  MATH  Google Scholar 

  46. Strichartz RS (1977) Restrictions of Fourier transforms to quadratic surfacesand decay of solutions of wave equations. J Duke Math 44(3):705–714

    MathSciNet  MATH  Google Scholar 

  47. Tao T (2000) Spherically averaged endpoint Strichartz estimates for thetwo‐dimensional Schrödinger equation. Comm Part Diff Eq 25(7–8):1471–1485

    MATH  Google Scholar 

  48. Tao T (2003) Local well‐posedness of the Yang–Mills equation inthe temporal gauge below the energy norm. J Diff Eq 189(2):366–382

    MATH  Google Scholar 

  49. Taylor ME (1991) Pseudodifferential operators and nonlinear PDE. Progr Math100. Birkhäuser, Boston

    Google Scholar 

  50. von Wahl W (1970) Über die klassische Lösbarkeit des Cauchy‐Problems fürnichtlineare Wellengleichungen bei kleinen Anfangswerten und das asymptotische Verhalten der Lösungen. Math Z114:281–299 (in German)

    MathSciNet  MATH  Google Scholar 

  51. Wolff T (2001) A sharp bilinear cone restriction estimate. Ann Math (2)153(3):661–698

    MathSciNet  Google Scholar 

  52. Yajima K (1987) Existence of solutions for Schrödinger evolutionequations. Comm Math Phys 110(3):415–426

    MathSciNet  MATH  ADS  Google Scholar 

  53. Yajima K (1995) The \( { W\sp {k,p} } \)-continuity of wave operators for Schrödinger operators. J Math Soc Japan47(3):551–581

    MathSciNet  MATH  Google Scholar 

  54. Yajima K (1995) The \( { W\sp {k,p} } \)-continuity of wave operators for schrödinger operators. III.even‐dimensional cases \( { m\geq 4 } \). J Math Sci Univ Tokyo 2(2):311–346

    Google Scholar 

  55. Yajima K (1999) \( { L\sp p } \)-boundedness of wave operators for two‐dimensional schrödingeroperators. Comm Math Phys 208(1):125–152

    MathSciNet  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Unversità di Roma “La Sapienza”, Roma, Italy

    Piero D'Ancona

Authors
  1. Piero D'Ancona
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

D'Ancona, P. (2009). Dispersion Phenomena in Partial Differential Equations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_128

Download citation

Publish with us

Policies and ethics

Search

Navigation