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Nonlocal variational problems arising in long wave propagatioN

  • Orlando Lopes (a1)


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.



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[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.
[3] Albert, J., Bona, J. and Henry, D., Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Phys. D 24 (1987) 343-366.
[4] Albert, J., Bona, J. and Saut, J.C., Model equations for waves in stratified fluids. Proc. Roy. Soc. London Ser. A 453 (1997) 1233-1260.
[5] J. Bergh and J. Lofstrom, Interpolation Spaces. Springer-Verlag, New-York/Berlin (1976).
[6] P. Blanchard and E. Bruning, Variational Methods in Mathematical Physics. Springer-Verlag (1992).
[7] Brezis, H. and Lieb, E., Minimum Action Solutions of Some Vector Field Equations. Comm. Math. Phys. 96 (1984) 97-113.
[8] de Bouard, A., Stability and instability of some nonlinear dispersive solitary waves in higher dimension. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 89-112.
[9] Catto, I. and Lions, P.L., Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Part I. Comm. Partial Differential Equations 17 (1992) 1051-1110.
[10] Cazenave, T. and Lions, P.L., Orbital Stability of Standing waves for Some Nonlinear Schrödinger Equations. Comm. Math. Phys. 85 (1982) 549-561.
[11] Coleman, S., Glazer, V. and Martin, A., Action Minima among to a class of Euclidean Scalar Field Equations. Comm. Math. Phys. 58 (1978) 211-221.
[12] Colin, T. and Weinstein, M., On the ground states of vector nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 65 (1996) 57-79.
[13] G.H. Derrick, Comments on Nonlinear Wave Equations as Models for Elementary Particles. J. Math. Phys. 5, 9 (1964) 1252-1254.
[14] Grillakis, M., Shatah, J. and Strauss, W., Stability of Solitary Waves in the Presence of Symmetry I. J. Funct. Anal. 74 (1987) 160-197.
[15] Hormander, L., Estimates for translation invariant operators in L p spaces. Acta Math. 104 (1960) 93-140.
[16] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Springer, Heidelberg (1993).
[17] Lax, P., Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968) 467-490.
[18] Levandosky, S., Stability and instability of fourth-order solitary waves. J. Dynam. Differential Equations 10 (1998) 151-188.
[19] Lieb, E., Existence and uniqueness of minimizing solutions of Choquard's nonlinear equation. Stud. Appl. Math. 57 (1977) 93-105.
[20] P.L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) Part I 109-145, Part II 223-283.
[21] Lions, P.L., Solutions of Hartree-Fock Equations for Coulomb Systems. Comm. Math. Phys. 109 (1987) 33-97.
[22] Lopes, O., Radial symmetry of minimizers for some translation and rotation invariant functionals. J. Differential Equations 124 (1996) 378-388.
[23] Lopes, O., Sufficient conditions for minima of some translation invariant functionals. Differential Integral Equations 10 (1997) 231-244.
[24] Lopes, O., Constrained Minimization Problem, A with Integrals on the Entire Space. Bol. Soc. Brasil Mat. (N.S.) 25 (1994) 77-92.
[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.
[26] Lopes, O., Variational problems defined by integrals on the entire space and periodic coefficients. Comm. Appl. Nonlinear Anal. 5 (1998) 87-120.
[27] Maddocks, J. and Sachs, R., On the stability of KdV multi-solitons. Comm. Pure. Appl. Math. 46 (1993) 867-902.
[28] Saut, J.C., Sur quelques généralizations de l'équation de Korteweg-de Vries. J. Math. Pure Appl. (9) 58 (1979) 21-61.
[29] H. Triebel, Interpolation Theory, Functions Spaces, Differential Operators. North-Holland, Amsterdam (1978).
[30] Weinstein, M., Liapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations. Comm. Pure Appl. Math. 39 (1986) 51-68.
[31] Weinstein, M., Existence and dynamic stability of solitary wave solution of equations arising in long wave propagation. Comm. Partial Differential Equations 12 (1987) 1133-1173.


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Nonlocal variational problems arising in long wave propagatioN

  • Orlando Lopes (a1)


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