Nonlocal variational problems arising in long wave propagatioN

Orlando Lopes

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 5, page 501-528
  • ISSN: 1292-8119

Abstract

top
In this paper we study the existence of minimizer for certain constrained variational problems given by functionals with nonlocal terms. This type of functionals are first integrals of evolution equations describing long wave propagation and the existence of minimizer gives the existence and the stability of traveling waves for these equations. Due to loss of compactness, the major problem is to prevent dichotomy of minimizing sequences. Our approach is an alternative to the concentration-compactness method and it allows us to deal with some functionals for which the verification of the strict subadditivity seems to be difficult.

How to cite

top

Lopes, Orlando. "Nonlocal variational problems arising in long wave propagatioN." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 501-528. <http://eudml.org/doc/197302>.

@article{Lopes2010,
abstract = { In this paper we study the existence of minimizer for certain constrained variational problems given by functionals with nonlocal terms. This type of functionals are first integrals of evolution equations describing long wave propagation and the existence of minimizer gives the existence and the stability of traveling waves for these equations. Due to loss of compactness, the major problem is to prevent dichotomy of minimizing sequences. Our approach is an alternative to the concentration-compactness method and it allows us to deal with some functionals for which the verification of the strict subadditivity seems to be difficult. },
author = {Lopes, Orlando},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlocal variational problems; stability of traveling waves.; stability of traveling waves; constrained variational problems},
language = {eng},
month = {3},
pages = {501-528},
publisher = {EDP Sciences},
title = {Nonlocal variational problems arising in long wave propagatioN},
url = {http://eudml.org/doc/197302},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Lopes, Orlando
TI - Nonlocal variational problems arising in long wave propagatioN
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 501
EP - 528
AB - In this paper we study the existence of minimizer for certain constrained variational problems given by functionals with nonlocal terms. This type of functionals are first integrals of evolution equations describing long wave propagation and the existence of minimizer gives the existence and the stability of traveling waves for these equations. Due to loss of compactness, the major problem is to prevent dichotomy of minimizing sequences. Our approach is an alternative to the concentration-compactness method and it allows us to deal with some functionals for which the verification of the strict subadditivity seems to be difficult.
LA - eng
KW - Nonlocal variational problems; stability of traveling waves.; stability of traveling waves; constrained variational problems
UR - http://eudml.org/doc/197302
ER -

References

top
  1. R. Adams, Sobolev Spaces. Academic Press (1975).  
  2. J. Albert, Concentration-Compactness and stability-wave solutions to nonlocal equations. Contemp. Math.221, AMS (1999) 1-30.  Zbl0936.35159
  3. J. Albert, J. Bona and D. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Phys. D24 (1987) 343-366.  Zbl0634.35079
  4. J. Albert, J. Bona and J.C. Saut, Model equations for waves in stratified fluids. Proc. Roy. Soc. London Ser. A453 (1997) 1233-1260.  Zbl0886.35111
  5. J. Bergh and J. Lofstrom, Interpolation Spaces. Springer-Verlag, New-York/Berlin (1976).  Zbl0344.46071
  6. P. Blanchard and E. Bruning, Variational Methods in Mathematical Physics. Springer-Verlag (1992).  Zbl0756.49023
  7. H. Brezis and E. Lieb, Minimum Action Solutions of Some Vector Field Equations. Comm. Math. Phys.96 (1984) 97-113.  Zbl0579.35025
  8. A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension. Proc. Roy. Soc. Edinburgh Sect. A126 (1996) 89-112.  Zbl0861.35094
  9. I. Catto and P.L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Part I. Comm. Partial Differential Equations17 (1992) 1051-1110.  Zbl0767.35065
  10. T. Cazenave and P.L. Lions, Orbital Stability of Standing waves for Some Nonlinear Schrödinger Equations. Comm. Math. Phys.85 (1982) 549-561.  Zbl0513.35007
  11. S. Coleman, V. Glazer and A. Martin, Action Minima among to a class of Euclidean Scalar Field Equations. Comm. Math. Phys.58 (1978) 211-221.  
  12. T. Colin and M. Weinstein, On the ground states of vector nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor.65 (1996) 57-79.  Zbl0863.35101
  13. G.H. Derrick, Comments on Nonlinear Wave Equations as Models for Elementary Particles. J. Math. Phys.5, 9 (1964) 1252-1254.  
  14. M. Grillakis, J. Shatah and W. Strauss, Stability of Solitary Waves in the Presence of Symmetry I. J. Funct. Anal.74 (1987) 160-197.  Zbl0656.35122
  15. L. Hormander, Estimates for translation invariant operators in Lp spaces. Acta Math.104 (1960) 93-140.  Zbl0093.11402
  16. O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Springer, Heidelberg (1993).  Zbl0797.58005
  17. P. Lax, Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math.21 (1968) 467-490.  Zbl0162.41103
  18. S. Levandosky, Stability and instability of fourth-order solitary waves. J. Dynam. Differential Equations10 (1998) 151-188.  
  19. E. Lieb, Existence and uniqueness of minimizing solutions of Choquard's nonlinear equation. Stud. Appl. Math.57 (1977) 93-105.  Zbl0369.35022
  20. P.L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. Ann. Inst. H. Poincaré Anal. Non Linéaire1 (1984) Part I 109-145, Part II 223-283.  Zbl0541.49009
  21. P.L. Lions, Solutions of Hartree-Fock Equations for Coulomb Systems. Comm. Math. Phys.109 (1987) 33-97.  
  22. O. Lopes, Radial symmetry of minimizers for some translation and rotation invariant functionals. J. Differential Equations124 (1996) 378-388.  Zbl0842.49004
  23. O. Lopes, Sufficient conditions for minima of some translation invariant functionals. Differential Integral Equations10 (1997) 231-244.  Zbl0891.49001
  24. O. Lopes, A Constrained Minimization Problem with Integrals on the Entire Space. Bol. Soc. Brasil Mat. (N.S.)25 (1994) 77-92.  Zbl0805.49005
  25. O. Lopes, Variational Systems Defined by Improper Integrals, edited by L. Magalhaes et al., International Conference on Differential Equations. World Scientific (1998) 137-153.  Zbl0961.35034
  26. O. Lopes, Variational problems defined by integrals on the entire space and periodic coefficients. Comm. Appl. Nonlinear Anal.5 (1998) 87-120.  Zbl1108.49300
  27. J. Maddocks and R. Sachs, On the stability of KdV multi-solitons. Comm. Pure. Appl. Math.46 (1993) 867-902.  Zbl0795.35107
  28. J.C. Saut, Sur quelques généralizations de l'équation de Korteweg-de Vries. J. Math. Pure Appl. (9)58 (1979) 21-61.  Zbl0449.35083
  29. H. Triebel, Interpolation Theory, Functions Spaces, Differential Operators. North-Holland, Amsterdam (1978).  
  30. M. Weinstein, Liapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations. Comm. Pure Appl. Math.39 (1986) 51-68.  Zbl0594.35005
  31. M. Weinstein, Existence and dynamic stability of solitary wave solution of equations arising in long wave propagation. Comm. Partial Differential Equations12 (1987) 1133-1173.  Zbl0657.73040

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.