Determining nodes for the Ginzburg-Landau equations of superconductivity

Discrete and Continuous Dynamical Systems

Volumn 4, Issue 2, 1998, Pages 205-224

  Kaper, Hans G ^a   Wang, Bixiang ^b   Wang, Shouhong ^c  

^a ARGONNE NATIONAL LABORATORY   (United States)

^b TSINGHUA UNIVERSITY   (China)

^c INDIANA UNIVERSITY   (United States)

Author keywords

Determining nodes; Ginzburg Landau equations; Large time asymptotic behavior; Superconductivity

Indexed keywords

EID: 0032384865     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (24)

1. A.A. Abrikosov, "Fundamentals of the Theory of Metals," North-Holland Publ. Co., Amsterdam, 1988.
2. R.A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
3. F. Bethuel and T. Rivière, Vortices for a variational problem related to supraconductlvity, Ann. Inst. Henri Poincaré 12 (1995), 243-303.
4. P. DeGennes, "Superconductivity in Metals and Alloys," Benjamin, New York, 1966.
5. J. Fleckinger-Pellé and H.G. Kaper, Gauges for the Ginzburg-Landau equations of superconductivity, Proc. ICIAM 95. Z. Angew. Math. Mech. (1996), to appear. Preprint ANL/MCS-P527-0795, Mathematics and Computer Science Division, Argonne National Laboratory, October 1995.
6. J. Fleckinger-Pellé, H.G. Kaper, and P. Takáč, Dynamics of the Ginzburg-Landau equations of superconductivity, submitted for publication. Preprint ANL/MCS-P565-0296, Mathematics and Computer Science Division, Argonne National Laboratory, February 1996.
7. C. Foias and I. Kukavica, Determining nodes for the Kuramoto-Sivashinsky equation, J. Dynam. Diff. Eq. 7 (1995), 365-373.
8. C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comp. 43 (1984), 117-133.
9. V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds, Ann. Mat. Pura Appl. (4) 122 (1979), 159-198.
10. V.L. Ginzburg and L.D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz. (USSR) 20 (1950), 1064-1082; Engl. transl. in D. ter Haar, "L.D. Landau; Men of Physics," Vol. I, Pergamon Press, Oxford, 1965, 138-167.
11. V.L. Ginzburg and L.D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz. (USSR) 20 (1950), 1064-1082; Engl. transl. in D. ter Haar, "L.D. Landau; Men of Physics," Vol. I, Pergamon Press, Oxford, 1965, 138-167.
12. V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations," Springer-Verlag, New York, 1986.
13. L.P. Gor'kov and G.M. Eliashberg, Generalizations of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities, Zh. Eksp. Teor. Fiz. 54 (1968), 612-626; Soviet Phys. JETP 27 (1968), 328-334.
14. L.P. Gor'kov and G.M. Eliashberg, Generalizations of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities, Zh. Eksp. Teor. Fiz. 54 (1968), 612-626; Soviet Phys. JETP 27 (1968), 328-334.
15. D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, New York, 1981.
16. R.L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, preprint, Dept. of Mathematics, University of Illinois at Urbana/Champaign, June 24, 1997.
17. D.A. Jones and E.S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J. 42 (1993), 875-887.
18. H.G. Kaper and P. Takáč, Equivalence relation for the Ginzburg-Landau equations of superconductivity, Z. Angew. Math. Phys., to appear. Preprint ANL/MCS-P588-0496, Mathematics and Computer Science Division, Argonne National Laboratory, April 1996.
19. H.G. Kaper and P. Takáč, Ginzburg-Landau dynamics with a time-dependent magnetic field, submitted for publication. Preprint ANL/MCS-P620-1196, Mathematics and Computer Science Division, Argonne National Laboratory, November 1996.
20. I. Kukavica, On the number of determining nodes for the Ginzburg-Landau equation, Nonlinearity 5 (1992), 997-1006.
21. A. Schmid, A time dependent Ginzburg-Landau equation and its application to the problem of resistivity in the mixed state, Phys. Kondens. Mater. 5 (1966), 302-317.
22. F. Takens, Detecting strange attractors in turbulence. In: D.A. Raud and L.-S. Young (eds.), Lecture Notes in Math., Vol. 898, Springer-Verlag, New York, pp. 366-381.
23. Q. Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Physica D 88 (1995), 139-166.
24. M. Tinkham, "Introduction to Superconductivity," 2nd ed., McGraw-Hill, Inc., New York, 1996.