A mathematical model of solar flares
Authors:
Jean Heyvaerts, Jean-Michel Lasry, Michelle Schatzman and Patrick Witomski
Journal:
Quart. Appl. Math. 41 (1983), 1-30
MSC:
Primary 85A30; Secondary 58E99
DOI:
https://doi.org/10.1090/qam/700658
MathSciNet review:
700658
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Abstract: The phenomenon of solar flares is modeled assuming that the magnetic field is force-free and that its evolution is quasi-static. This model is simplified so as to be tractable and yields a semi-linear elliptic equation in a halfplane depending on a parameter $\lambda$ which describes the time evolution. It is proved that there are (at least) two branches of solutions which have distinct asymptotic behaviors at infinity. The upper branch exists for all $\lambda > 0$, but the lower branch exists only on a finite interval $\left [ {0,{\lambda ^c}} \right ]$. As stable solutions must have the same asymptotic behavior as the lower branch of solutions, and as this is impossible after ${\lambda ^c}$, we contend that no stable solution exists after ${\lambda ^c}$ and that a solar flare is thus triggered.
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI https://doi.org/10.1002/cpa.3160120405
A. Bahri, thèse d’état, Université Pierre et Marie Curie, Paris, 1981
H. Berestycki, thèse d’état, Université Pierre et Marie Curie, Paris, 1981
H. Berestycki and P. L. Lions, Existence of solitary waves in nonlinear Klein-Gordon equations, Part I: the ground state; Part II: existence of infinitely many bound states, Arch. Rat. Mech. Anal., 1981
R. Chiapinelli and C. A. Stuart, Bifurcations when the linearization has no eigenvalues, J. Diff. Eqs. 30, 296–307 (1978)
- Michael G. Crandall and Paul H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topological degree., J. Math. Mech. 19 (1969/1970), 1083–1102. MR 0259232
R. Courant and D. Hilbert, Methods of mathematical physics, II, Interscience, New York, 1963
M. J. Esteban, thèse de troisième cycle, Université Pierre et Marie Curie, Paris, 1981
G. Geymonat and P. Grisvard, Problèmes aux limites elliptiques dans L$^{p}$, Secrétariat mathématique d’Orsay, Orsay, 1964
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
- B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. MR 619749, DOI https://doi.org/10.1080/03605308108820196
- Herbert B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, Applications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976) Academic Press, New York, 1977, pp. 359–384. Publ. Math. Res. Center, No. 38. MR 0455353
F. Kikuchi, Finite-element approximation to bifurcaton problems of turning-point type, in Computing methods in applied sciences and engineering, ed. R. Glowinski and J. L. Lions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1979
- Klaus Kirchgässner and Jürgen Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Differential Equations 32 (1979), no. 1, 119–148. MR 532767, DOI https://doi.org/10.1016/0022-0396%2879%2990055-X
T. Küpper, The lowest point of the continuous spectrum as a bifurcation point, Report 78.12 of the Mathematics Department of the University of Köln, 1978
T. Küpper, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Report 78.25 of the Mathematics Department of the University of Köln, 1978
T. Küpper and D. Reimer, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonlinear Anal. 3, 555–561 (1979)
P. L. Lions, Thèse d’état, Université Pierre et Marie Curie, Paris, 1979
- G. Moore and A. Spence, The calculation of turning points of nonlinear equations, SIAM J. Numer. Anal. 17 (1980), no. 4, 567–576. MR 584731, DOI https://doi.org/10.1137/0717048
J. C. Paumier, Calcul numérique des points de retournement, séminaire, Université Pierre et Marie Curie, Laboratoire d’analyse numérique, 1980
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
L. Reinhardt, Sur la solution numérique de problèmes aux limites non linéaires par des méthodes de continuation, thèse de troisième cycle, Université Pierre et Marie Curie, Paris, 1980
C. A. Stuart, Global properties of components of solutions of nonlinear second-order ordinary differential equations on the half-line, Ann. Sc. Norm. Sup. Pisa 11, 265–286 (1975)
- C. A. Stuart, Bifurcation for Neumann problems without eigenvalues, J. Differential Equations 36 (1980), no. 3, 391–407. MR 576158, DOI https://doi.org/10.1016/0022-0396%2880%2990057-1
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12, 623–727 (1959)
A. Bahri, thèse d’état, Université Pierre et Marie Curie, Paris, 1981
H. Berestycki, thèse d’état, Université Pierre et Marie Curie, Paris, 1981
H. Berestycki and P. L. Lions, Existence of solitary waves in nonlinear Klein-Gordon equations, Part I: the ground state; Part II: existence of infinitely many bound states, Arch. Rat. Mech. Anal., 1981
R. Chiapinelli and C. A. Stuart, Bifurcations when the linearization has no eigenvalues, J. Diff. Eqs. 30, 296–307 (1978)
M. G. Crandall and P. H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topologival degree, J. Math. Mech. 19, 161–180 (1969–70)
R. Courant and D. Hilbert, Methods of mathematical physics, II, Interscience, New York, 1963
M. J. Esteban, thèse de troisième cycle, Université Pierre et Marie Curie, Paris, 1981
G. Geymonat and P. Grisvard, Problèmes aux limites elliptiques dans L$^{p}$, Secrétariat mathématique d’Orsay, Orsay, 1964
B. Gidas, Y. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68, 209–243 (1979)
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, preprint
H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of bifurcation theory, ed. P. H. Rabinowitz, Academic Press, New York, 1977
F. Kikuchi, Finite-element approximation to bifurcaton problems of turning-point type, in Computing methods in applied sciences and engineering, ed. R. Glowinski and J. L. Lions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1979
K. Kirchgässner and J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Diff. Eqs. 32, 119–148 (1979)
T. Küpper, The lowest point of the continuous spectrum as a bifurcation point, Report 78.12 of the Mathematics Department of the University of Köln, 1978
T. Küpper, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Report 78.25 of the Mathematics Department of the University of Köln, 1978
T. Küpper and D. Reimer, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonlinear Anal. 3, 555–561 (1979)
P. L. Lions, Thèse d’état, Université Pierre et Marie Curie, Paris, 1979
G. Moore and A. Spence, The calculation of turning points of nonlinear equations, SIAM J. Numer. Anal. 17, (1980)
J. C. Paumier, Calcul numérique des points de retournement, séminaire, Université Pierre et Marie Curie, Laboratoire d’analyse numérique, 1980
M. Protter and H. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, N. J., 1971
L. Reinhardt, Sur la solution numérique de problèmes aux limites non linéaires par des méthodes de continuation, thèse de troisième cycle, Université Pierre et Marie Curie, Paris, 1980
C. A. Stuart, Global properties of components of solutions of nonlinear second-order ordinary differential equations on the half-line, Ann. Sc. Norm. Sup. Pisa 11, 265–286 (1975)
C. A. Stuart, Bifurcation without eigenvalues, J. Diff. Eqs. 36, 391–407 (1980)
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© Copyright 1983
American Mathematical Society
