Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations

Fabrice Planchon

Journées équations aux dérivées partielles (1999)

  • page 1-11
  • ISSN: 0752-0360

Abstract

top
We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space B ˙ 2 n 2 - 2 p , ∞ ( 𝐑 n ) , when the nonlinearity is of type u p , for p ∈ 𝐍 . This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.

How to cite

top

Planchon, Fabrice. "Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations." Journées équations aux dérivées partielles (1999): 1-11. <http://eudml.org/doc/93386>.

@article{Planchon1999,
abstract = {We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space $\dot\{B\}^\{\{\frac\{n\}\{2\}-\frac\{2\}\{p\},\infty \}\}_2(\mathbf \{R\}^n)$, when the nonlinearity is of type $u^\{p\}$, for $p\in \mathbf \{N\}$. This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.},
author = {Planchon, Fabrice},
journal = {Journées équations aux dérivées partielles},
keywords = {initial value problem; well-posedness; Besov space; self-similar solutions},
language = {eng},
pages = {1-11},
publisher = {Université de Nantes},
title = {Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations},
url = {http://eudml.org/doc/93386},
year = {1999},
}

TY - JOUR
AU - Planchon, Fabrice
TI - Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 11
AB - We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space $\dot{B}^{{\frac{n}{2}-\frac{2}{p},\infty }}_2(\mathbf {R}^n)$, when the nonlinearity is of type $u^{p}$, for $p\in \mathbf {N}$. This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.
LA - eng
KW - initial value problem; well-posedness; Besov space; self-similar solutions
UR - http://eudml.org/doc/93386
ER -

References

top
  1. [1] J. Bergh and J. Löfstrom. Interpolation Spaces, An Introduction. Springer-Verlag, 1976. Zbl0344.46071
  2. [2] J.-M. Bony. Calcul symbolique et propagation des singularités dans les équations aux dérivées partielles non linéaires. Ann. Sci. Ecole Norm. Sup., 14:209-246, 1981. Zbl0495.35024MR84h:35177
  3. [3] J. Bourgain. Refinements of Strichartz inequality and applications to 2-D NLS with critical nonlinearity. I.M.R.N., 5:253-283, 1998. Zbl0917.35126MR99f:35184
  4. [4] M. Cannone. Ondelettes, Paraproduits et Navier-Stokes. Diderot Editeurs, Paris, 1995. Zbl1049.35517MR2000e:35173
  5. [5] T. Cazenave and F. Weissler. The Cauchy problem for the critical nonlinear Schrödinger equation in Hs. Nonlinear Anal. T.M.A., 14:807-836, 1990. Zbl0706.35127MR91j:35252
  6. [6] T. Cazenave and F. Weissler. Asymptotically self-similar global solutions of the non linear Schrödinger and heat equations. Math. Zeit., 228:83-120, 1998. Zbl0916.35109MR99d:35149
  7. [7] T. Cazenave and F. Weissler. More self-similar solutions of the nonlinear Schrödinger equation. No D.E.A., 5:355-365, 1998. Zbl0990.35121MR99f:35185
  8. [8] J. Ginibre and Velo. The global Cauchy problem for the NLS equation revisited. Ann. IHP, An. non-linéaire, 2:309-327, 1985. Zbl0586.35042MR87b:35150
  9. [9] M. Keel and T. Tao. Enpoint Strichartz estimates. American Journal of Mathematics, 120(5):955-980, 1998. Zbl0922.35028MR2000d:35018
  10. [10] H. Lindbladt and C. D. Sogge. On existence and scattering with minimal regularity for semilinear wave equations. J. funct. Anal., 130:357-426, 1995. Zbl0846.35085MR96i:35087
  11. [11] R. O'Neil. Convolution operators and L(p,q) spaces. Duke Mathematical Journal, 30:129-142, 1963. Zbl0178.47701MR26 #4193
  12. [12] F. Oru. Rôle des oscillations dans quelques problèmes d'analyse non-linéaire. PhD thesis, ENS Cachan, 1998. 
  13. [13] H. Pecher. Self-similar and asymptotically self-similar solutions of nonlinear wave equations. preprint. Zbl0960.35067
  14. [14] J. Peetre. New thoughts on Besov Spaces. Duke Univ. Math. Series. Duke University, Durham, 1976. Zbl0356.46038MR57 #1108
  15. [15] F. Planchon. On the Cauchy problem in Besov spaces for a non-linear Schrödinger equation. preprint. Zbl0961.35148
  16. [16] F. Planchon. Self-similar solutions and semi-linear wave equations in Besov spaces. preprint. Zbl0979.35106
  17. [17] F. Planchon. Asymptotic Behavior of Global Solutions to the Navier-Stokes Equations. Rev. Mat. Iberoamericana, 14(1), 1998. Zbl0910.35096MR99k:35144
  18. [18] F. Planchon. Solutions autosimilaires et espaces de données initiales pour une équation de Schrödinger non-linéaire. C. R. Acad. Sci. Paris, 328, 1999. Zbl0931.35038MR2000d:35028
  19. [19] F. Ribaud and A. Youssfi. Self-similar solutions of nonlinear wave equation. preprint. Zbl0933.35140
  20. [20] F. Ribaud and A. Youssfi. Regular and self-similar solutions of nonlinear Schrödinger equations. J. Math. Pures Appl., 9(10):1065-1079, 1998. Zbl0928.35159MR2000b:35240
  21. [21] F. Ribaud and A. Youssfi. Solutions globales et solutions auto-similaires de l'équation des ondes non linéaire. C. R. Acad. Sci. Paris, 328, 1999. Zbl0933.35140MR2001b:35219
  22. [22] R. Strichartz. Restriction of Fourier transform to quadratic surfaces and decay of solutions of the wave equations. Duke Mathematical Journal, 44:705-714, 1977. Zbl0372.35001MR58 #23577
  23. [23] T. Tao. Low regularity semi-linear wave equations. Comm. in Partial Diff. Equations, 24(3&4):599-629, 1999. Zbl0939.35123MR2000a:35175

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.