Bifurcation and Nonlinear Eigenvalue Problems: Proceedings, Université de Paris XIII, Villetaneuse, France, October 2–4, 1978

302 Pages·1980·5.336 MB·English-French

by  M. F. Barnsley (auth.), C. Bardos, J. M. Lasry, M. Schatzman (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
782
noitacrufiB dna
Nonlinear Eigenvalue
smelborP
Proceedings, Universite de Villetaneuse, Paris XIlI, France,
October 2-4, 1978
detidE yb
C. Bardos, .J .M Lasry, and .M Schatzman
galreV-regnirpS
Berlin Heidelberg New York 1980
Editors
C. Bardos
D6partement de Math6matiques
Universite Paris-Nord
Avenue .J B. Clement
93430 Villetaneuse
France
,J M. Lasry
CEREMADE
Universit~ de Paris 9 - Dauphine
75775 Paris Cedex 16
France
M, Schatzman
Analyse Numerique
Tour 55-65
5eme ~tage
Universite Pierre et Marie Curie
?5230 Paris Cedex 05
France
AMS Subject Classifications (1980): 35 G 20
ISBN 3-540-09?58-9 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-09758-9 Springer-Verlag New York Heidelberg Berlin
yrarbiL of Publication in Cataloging Congress ,ataD title: Bifurcation under entry Main
dna eigenvalue nonlinear ,smelborp in notes (Lecture ;scitamehtam 782) Bibliography: .p
index. Includes .1 equations, Differential .sessergnoC--laitraP .2 Differential ,snoitauqe
.sessergnoC--raenilnoN .3 .sessergnoC--seulavnegiE .4 Bifurcation .sessergnoC--yroeht
.I ,sodraB 1940- Claude, .II ,yrsaL .J .M III. ,namztahcS .M .VI in notes Lecture Series:
scitamehtam (Berlin; 782, QA3.L28 .on [515.3'53] 782 [QA374] 510s 85311-08
NBSI 0-387-09758-9
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PREFACE
+
This volume contains the notes of a session organized on October 2,3 and 4,1978
at the Departement of Mathematics of the University Paris 13 ,Centre Scientifique et
Polytechnique ,Villetaneuse.
The aim of this session was to gather mathematicians and scientists of other
fields : chemistry,biology ,physics and astrophysics, and to let them exchange information
and methods.
The common points to all the lectures are partial differential equations,non
linear phenc~9/~a,study of the dependence with respect to a parameter,and the methods
used are very diverse.
The lectures can be classified into three groups accordingt o their relation
to applied science : papers belonging to the first cluster deal with a phenomenolo-
gical approach;in this case,a conplete system of equations describing the experimen-
tal phenomenon is either too oonplicated or not entirely known and understood;there-
fore,a simpler system is studied which mimicks the behavior of the complete system,
and one expects qualitative results.Here belong the talks of J.Heyvaerts,J.M.Lasry,
M.Schatzman & P.Witomski,of G.Iooss,of J.P.Kernevez,G.Joly,D.Thomas & B.Bunow,and
of P. Ortoleva.
The second group is made of mathematical and nt~rerical studies of more cor~01ete
modelizations:here,them odel is better understood,and the study is more precise,so
that it may give quantitative results;this group contains the contributions of
C.M. Brauner & B.Nicolaenko,of C.Guillop~,of G.Iooss & R.Lozi,and of J.Mossino.
Though the papers of the third group are not directly concerned with natural
phenorrena,they develop theoretical tools and an understanding of non-linear pheno-
mena,which are intended to meet the needs and preoccupations of the applied scientists.
We include here the papers of H.Berestycki & P.L.Lions,of C.Bolley,M.Barnsley,
F.Mignot,F.Murat & J.P.Puel,of J.C.Saut,and of D.Serre.
Striking observational data were brought by M.D~peyrat who showed a beautiful
dynamic periodic chemical phenomenon with a movie film.
We added a paper of J.P.Chollet and M.Lesieur.They show how the nonlinearity
of the Navier-Stokes equation can create turbulence and give a phenomenological
model that fits with the Kolmogorov law.
We thank all the participants for their active presence and interesting
contributions
C. Bardos, M. J. Lasry,M. Schatzman.
+Supported by the University Paris 13,and the Centre National de la Recherche
Scientifique,A.T.P. Math~matiques pour les Sciences de l'Ing~nieur.
CONTENTS
Parameter Dependence of Solutions of Classes of Quasilinear Elliptic and
Parabolic Differential Equations, M.F. Barnsley.
Some Applications of the Method of Super and Subsolutions, H. Berestycki and
P.L. Lions. 16
Multiple Solutions of a Bifurcation Problem, C. Bolley. 42
On non linear eigenvalue problems which extend into free boundary problems,
C.M. Brauner and .B Nicolaenko. 16
Introduction aux Theories statistiques de la Turbulence pleinement d~velopp~e,
M. Lesieur and J.P. Chollet. 10I
Experimental Study of the Mechanism of a new hydrodynamical Instability
observed at some Interfaces between immiscible Liquids, M. Dupeyrat. 122
Remarques sur un Probl~me de Valeurs Propres non lin~aires faisant intervenir
des Fonctions non diff~rentiables, .C Guillop~. 130
Solar Flares: a non linear eigenvalue Problem in an unbounded Domain,
J. Heyvaerts, J.M. Lasry, M. Schatzman, and P. Witomski. 160
Bifurcation of invariant Tori in]R ,3 G. looss. 192
Pattern Formation and Wave Propagation in the s-a System, J.P. Kernevez,
.G Joly, D. Thomas and B. Bunow. 102
Variation d'un point de Retournement par rapport au Domaine, F. Mignot,
.F Murat et J.P. Puel. 222
Dynamic Padg Approximants and Behavior Singularities in non linear Physico-
chemical Systems, °P Ortoleva. 255
Remarks on a non linear Equation arising in Population Genetics, J.C.Saut. 265
Triplets de Solutions d'une Equation aux D~riv~es partielles elliptique
non lin~aire, D. Serre. 283
The contributions of J.Mgssino,A non Linear Eigenvalue Problem and a Variational
Problem connected with In--sing Rearrangement in Plasma Physics,and of G.Iooss and
R.Lozi,Convection between two ~Otating Plates and Dynamo Effect occuring f~m a
Secondary Bifurcation will ai~pear elsewhere.
LIST OF PARTICIPANTS
ORGANIZERS
.C Bardos,D~partement de Math~matiques,Universit~ Paris XIII,Av. J.B. Cl~ment,
93430 Villetaneuse.
J.M. Lasry,CEREMADE,Universit~ Paris IX,Place De Lattre de Tassigny,
75775 Paris CEDEX .61
.M Sehatzman,C.N.R.S.,L.A. 981 Analyse Num~rique,Universit~ Pierre et Marie
Curie,2 Place Jussieu,75230 Paris CEDEX 05.
CONTRIBUTORS
M.F. Barnsley,C.E.A.,Division de la Physique,B.P.2,91190 Gif sur Yvette.
H.Berestycki,C.N.R.S.,Laboratoire d'Analyse Num~rique,Universit~ Pierre
et Marie Curie,2 Place Jussieu;75230 Paris CEDEX .50
C.Bolley,Laboratoire d'Analyse Num~rique,l.N.S.A. de Rennes,B.P. 41 A,
35031Rennes CEDEX.
C.M. Brauner,Laboratoire de Math~matiques-lnformatique-Syst~mes,Ecole
Centrale de Lyon,69130 Ecully.
B.Bunow,National Institute of Health,Bethesda,Maryland 20014,U.S.A.
J.P. Chollet,lnstitut de M~canique de Grenoble,B.P. 53 X,38041 Grenoble
CEDEX.
M.Dupeyrat,Laboratoire de Chimie Physique,Universit~ Pierre et Marie Curie,
II Rue Pierre et Marie Curie,75230 Paris CEDEX 05.
C.Guillop~,C.N.R.S.,Analyse num~rique et Fonctionnelle,B~t. 425,Universit~
Paris-Sud,91405 Orsay.
J.Heyvaerts,Observatoire de Meudon,92|90 Meudon.
.G looss,lnstitut de Math~matiques et de Sciences Physiques,Pare Valrose,
06034 Nice.
G.Joly,Universit~ de Technologie de Compi~gne,B.P. 233,60206 Compi~gne.
J.P. Kernevez,Universit~ de Technologie de Compi~gne,B.P. 233,
60206 Compi~gne.
P.L. Lions,C.N.R.S.,Laboratoire d'Analyse Num~rique,Universit~ Pierre et
Marie Curie,2 Place Jussieu, 75230 Paris CEDEX 05.
M.Lesieur,lnstitut de M~canique de Grenoble,B.P. 53 X,3804! Grenoble CEDEX.
.R Lozi, Institut de Math~matiques et de Sciences Physiques,06034 Nice.
.F Mignot,U.E.R. de Math~matiques,Universit~ de Lille 1,59650 Villeneuve
d'Ascq.
J.Mossino,C.N.R.S.,Analyse Num~rique et Fonctionnelle,B~t. 425,Universit~
Paris-Sud,91405 Orsay.
IIIV
.F Murat,Laboratoire d'Analyse Num~rique,Universit~ Pierre et Marie Curie,
2 Place Jussieu, 75230 Paris CEDEX 05.
B.Nicolaenko,University of California,Math. Anal. Group,T7,Los Alamos
scientific Laboratory,Los Alamos,New Mexico 87545,U.S.A.
P.Ortoleva,Department of Chemistry,Indiana University,Bloomington,Indiana
47401,U.S.A.
J.P. Puel,D~partement de Math~matiques et Informatique,Universit~ de Nancy ,II
42 Avenue de la Liberation,54000 Nancy.
J.C. Saut,Analyse Num~rique et Fonctionnelle,B~t. 425,Universit~ Paris-Sud,
91405 0rsay.
.D Serre,C.N.R.S.;Analyse Num~rique et Fonctionnelle,B~t. 425,Universit~
Paris-Sud,91405 Orsay.
.D Thomas,Universit~ de Technologic de Compi~gne,B.P. 233660206 Compi~gne.
.P Witomski, Laboratoire IMAG,B.P. 53 ,X 3804l Grenoble CEDEX.
PARAMETER DEPENDENCE OF SOLUTIONS OF CLASSES OF
QUASI-LINEAR ELLIPTIC AND PARABOLIC DIFFERENTIAL EQUATIONS
by
M.F. Barnsley
Service de Physique th~orique
C.E.N. Saclay
B.P. n°2
9||90 Gif s/Yvette
ABSTRACT
Earlier work, on the dependence of solutions of certain classes
of quasi-linear elliptic and parabolic differential equations on
embedded parameters, is extended and generalized. In particular, generic
classes of linearly perturbed, and inhomogeneously perturbed, quasi-linear
elliptic and parabolic boundary values problems whose stable positive
solutions are Laplace transforms of positive measures, are identified.
For a particular class of such problems the conjecture that the solution
is a Stieltjes transform of a positive measure is explored. It si shown
that low order rational fraction Pad~ approximants provide useful bounds,
independently of whether or not the conjecture itself is true.
.I INTRODUCTION
We consider some extensions and generalizations of earlier work [1'2] concerning
the dependence of solutions of certain quasi-linear elliptic and parabolic differen-
tial equations on an external parameter. Our interest is in those cases where
the solution, as a function of the external parameters, can be expressed as a trans-
form of a positive measure. In such cases one can use moment theory to yield
convergent sequences of upper and lower bounds on the solution throughout the range
of the parameter, as described in [2]. To construct the bounds one needs to know
either an initial sequence of terms in a perturbation expansion of the solution
in the parameter (in some cases these can be obtained by solving a set of linear
equations), or a set of experimental points corresponding to different values of
the parameter. The latter possibility is attractive because the resulting bounds
[I]
are to some extent "model-independent" as described in
In II and III we describe two generic situations which, in the elliptic case,
have positive stable solutions which are Laplace transforms of positive measures.
As such, they are amenable to analysis using generalized Pad6 approximants.
In IV we consider a class of nonlinearly perturbed elliptic boundary value
problems for which it is conjectured that the positive stable solution is a Stieltjes
transform of a positive measure in the perturbation parameter. In certain cases
this conjecture has been established, and then rational fraction Pad~ approximants
provide not only convergent sequences of bounds on the solution but also they yield
bounds on the associated turning point. It is shown that, for low order Pad6
approximants, similar results pertain in general, whether o/ not the conjecture
itself is true.
II. LINEARLY PERTURBED NONLINEAR EQUATI, ,S,NO
We consider quasi-linear differential equations of the form
L~ + F(~) + %p~ = f in D~
B~ = 0 on SD J (2.1)
Here D denotes a bounded domain of real N-space ~N with boundary ~D and closure D.
We assume that ~D belongs to the class C 2+~, where & E (0,I) is fixed. L is the
uniformly elliptic differential operator
N N ~(x)
L~ =- Z (x) ~2~(x) Z ak(x) + a(x)~(x) (2.2)
i,j=l ai'j ~ +
1 3 k=l
with real coefficients ai, j 6 C2+a(D), k 6 a CI+~(D), and a 6 Ca(D) where we assume
that,for all x = (xl,x x ..... 2 N) 6
a(x) k 0 (2.3)
The matrix (aij) is supposed to be uniformly positive definite over D. B is either
of the boundary operators
B~ m ¢(x) on SD, (2.4)
Be m ~(x)¢(x) + ~¢(x)/$w on ~D, (2.5)
where B(x) 6 CI+~(~D) and satisfies
~(x) ~ 0 for all x 6 ~D, (2.6)
~/~ denotes the outward conormal derivative. In the case where (2.5) applies we
assume that a(x) and B(x) do not both vanish identically. The functions p and f
in (2.1) belong to C~(D) and satisfy
p(x) > O, and f(x) ~ 0 for all x 6 D, (2.7)
with f # 0.
The real valued function F(~) = F(x,~) may depend explicitly on both x 6
and ~. We suppose that it has the following properties
(i) F(0) - f ! 0 for all x 6 D.
(ii) There exists a constant C > 0 such that
F(C) - f ~ 0 for all x 6 D.
(iii) F(~) is C in ~ for all ~ 6 [O,C], each of its derivatives in this range
belonging to C~(D) in x, and such that uniformly in x 6 D and n = 2,3,4 ....
-F i F(n)(~) ! 0 for all ~ 6 [0,C]
for some constant F > O.
(iv) F(1)(#) ~ 0 for all x 6 D and ~ 6 [O,C]. This means that the linear operator
in~2(D) corresponding to L + F(1)(~) together with the boundary condition in (2.1)
has strictly positive least eigenvalue for all smooth ~(x) £ [0,C].
The conditions above on (L,B) are such that are such that the Maximam Principle ]3[
and the Positivity Lena ]4[ apply. Moreover, the smoothness conditions of F(~) mean
that A~ann's Theorem ]5[ on the existence of solutions via sandwiching between
upper and lower solutions, applies to (2.1). The key condition on F(~) which ensures
the establishment of the Laplace transform property (Proposition 1.2) is (iii).
Proposition (2.1). The problem (2.1) with % k 0 possesses exactly one solution
6 C2+a(D) which satisfies 0 ~ t(x) ~ C for all x 6 .D
Proof : The existence of at least one solution in. the desired range is provided
by conditions (i) and (ii), upon application of Amann's Theorem. Zero is a lower
solution while the constant C is an upper solution, for all % ~ O.
To establish uniqueness let ~I and ~2 be two solutions. Then Taylor's Theorem
with remainder provides
F(@ )l - F(@ )2 = F(1)(@3)(~ - 1 @2 ) )8.2(
for some @3 lying between I@ and @2' so that @3 6 [0,C]. Hence
[e + %p + F(1)(@3)](@I - @2 ) = 0 in 1 D (2.9)
B(~ 1 - ~2 ) = 0 on ~D j
and condition (iv) now yields l~q = "2~c Q.E.D.
We will denote the solution referred to above by @(%). In order to examine
its analytic nature let %0 >- 0 be held fixed, let % E C be given, and set
# = % - %0 (2.10)
Then we will investigate the formal series
co 1
7~ = ~[%0,p] = ~ ~-~ ~n[%0]p n (2.11)
n=0
where the p-independent functions ~n = 0%[n#~ ] are supposed to satisfy the set of
equations obtained by equating the coefficients of the different powers of p which
occur in the formal expansion of
L~ + )7~(F + + (%0 p)p~ = f in D,I (2.12)
B~ = 0 on ,D3
and where ~0 si constrained by
0 S ~0 i C, for all x E 5 (2.13)
The equations to be satisfied by the ~n'S are found to be
L~0 + F(~0) + %0P~0 = f in D] (2.14.0)
B~0 = 0 on ,D~ ~0 E [0,C]
[L+F(1)(~O) +%0P]~l +p~0 = 0 in D~
J (2.14.1)
= ~I 0 B on ~D
[L+F(1)(~O) +%0P]~2 + F(2)(~O)~ + 2p@] = 0 in D,I
(2.14.2)
B ~2 = 0 on SD,
[L+F(1)(~O) + %oP]~n + nP~n_ 1
+ ~ F(m) (@) f E
C(n;k ..... I km_l)~(n_km_l )
m= 2 0 \O<kl<k2<...<km_l< n
(2.14.n)
~(km_ l_km_2)-..@(k2_kl ) ~kl } = 0 in D
B~n= 0 on ~D, = n 2,3,4,...
/'n-1 ~k m 1 -1~ .,/k 2- ~1
where C(n,k I ..... km_l) = kkm_l/\km- 2 j.. \kl
7 I