CC BY
15
Серия «Математика» 2013. Т. 6, № 4. С. 85—106
Онлайн-доступ к журналу: http://isu.ru/izvestia
УДК 518.517
Bifurcation points of nonlinear operators: existence theorems, asymptotics and application to the Vlasov - Maxwell system
N.A. Sidorov
Irkutsk State University
Abstract. The review of existence theorems of bifurcation points of solutions for nonlinear operator equation in Banach spaces is presented. The sufficient conditions of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell system are considered. The analytical method of Lyapunov-Schmidt-Trenogin is employed.
Keywords: plasma; bifurcation points; Conley index; nonlinear analysis; Vlasov - Maxwell system; Lyapunov - Schmidt - Trenogin method.
One of the current problems in natural sciences is study of kinetic Vlasov-Maxwell (VM) system [20] describing a behaviour of many-component plasma. A large literature on the existence of solutions for the VM system is available, for example, see under references [1, 3, 11, 21, 35] and the references given there. Nevertheless, the problem of bifurcation analysis of VM system, which was first formulated by A. A. Vlasov [20], has appeared very complicated on the background of progress of bifurcation theory in other fields and it remains open up to the present time. There are only some isolated results. In [9, 10] the VM system is reduced to the system of semilinear elliptical equations for special classes of distribution functions introduced in [12]. The relativistic version of VM system for such distributions was considered in [1]. One simple existence theorem of a point of bifurcation is announced in [13], and another one is proved for this system in [14].
In memory of Professor Vladilen A. Trenogin
Introduction
Vladilen A. Trenogin laid out the fundamentals of the modern analytical branching theory of nonlinear equations. Here readers may refer to his monograph [19], chapters 7-10. The bifurcation theory have been developed by various authors [5, 6, 15, 16, 17, 18, 21, 22, 25, 30], etc. The approximate methods of construction branching solutions were constructed in [21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34], [36, 37]. The readers may refer to the pioneering research contributions presented in original paper [13, 14, 23, 29], in the monograph [21] as well as in recently published monograph [35] in the field of bifurcation analysis of the Vlasov-Maxwell systems.
The objective of the present paper is to give the survey of a general existence theorems of bifurcation points of VM system with the given boundary conditions on potentials of an electromagnetic field both the densities of charge and current. Here we apply our results of bifurcation theory from [15, 17, 21, 22, 23] and we use the index theory [2, 7, 22] for the study of bifurcation points of the VM system.
We consider the many-component plasma consisting of electrons and positively charged ions of various species, which described by the many-particle distribution function fi = /¿(r, v), i = 1, N. The plasma is confined to a domain D C R3 with smooth boundary. The particles are to interact only by self-consistent force fields, collisions among particles being neglected.
The behaviour of plasma is governed by the following version of the stationary VM system [20]
v ■ drfi + (¡i/rrii(E + -U x B) ■ dvfi = 0, (1)
r e D c R3, i = 1,...,N, curlE = 0, divB = 0
N
" (2)
divE = 4n V qk fk(r, v)dv = p,
4n N f A
curlB = — VV / vfk(r, v)dv = j.
c rri Jr3
k=i
Here p(r), j(r) are the densities of charge and current, and E(r), B(r) are the electrical and the magnetic fields.
We seek the solution E, B, f of VM system (1)-(2) for r G D C R3 with boundary conditions on the potentials and the densities
U \dD= uoi, (A, d) \dD= U02; (3)
p \dD=o, j \dD= 0, (4)
where E = —drU, B = curlA, and U, A be scalar and vector potentials.
We call a solution E0, B0, f0 for which p0 = 0 and j0 = 0 in domain D, trivial.
In the present paper we investigate the case of distribution functions of the special form [9]
f,(r, v) = Afl(-a,v2 + p,(r), v • di + ^(r)) = Af,(R, G) (5) p, : R3 ^ R; & : R3 ^ R; r e D C R3; v e R3;
A e R+; ai e R+ = [0, to); di e R3, i = 1,...,N,
where functions p,, , generating the appropriate electromagnetic field (E, B), has to be defined.
We are interested in the dependence of unknown functions p, , upon parameter A in distribution (5). Here we study the case, when A in (5) does not depend on physical parameters a, and d,. The general case of a bifurcation problem with a, = a,(A), d, = d,(A), p, = p,(A, r), = (A, r) will be considered in the following paper.
Definition 1. The point A0 is called a bifurcation point of the solution of VM system with conditions (3), (4), if in any neighbourhood of vector (A0,E0,B0,f0), corresponding to the trivial solution with p0 = 0, j0 = 0 in domain D, there is a vector (A,E,B,f) satisfying to the system (1)-(2) with (3), (4) and for which
|| E - E0 || + || B - B0 || + || f - f0 \\> 0.
Let p0, are such constants that the corresponding p0 and j0, induced by distributions fi in the medium for p0, are equal to zero in domain D. Then VM system has the trivial solution
fi = Afi(-a,v2 + p0, v • di+&0), E0 = 0, B0 = d, p - const for
The organization of the present work is as follows. In Section 2 two theorems of existence of bifurcation points for the nonlinear operator equation in Banach space generalizing known results on a bifurcation point are proved. The method of proof of these theorems uses the index theory of vector fields [2, 7] and allows to investigate not only the point, but also the bifurcation surfaces with minimum restrictions on equation.
In Section 3 we reduce the problem on a bifurcation point of VM system to the problem on bifurcation point of semilinear elliptic system. Last one is treated as the operator equation in Banach space. We derive the branching equation (BEq) which allows to prove the principal theorem of existence of bifurcation points of VM system because of results of the Section 2. An essential moment here is that the semilinear system of elliptic equations is potential that reduces to potentiality of BEq.
It follows from our results that for the original problem (1) - (4) the bifurcation is possible only in the case, when number of species of particles N > 3.
1. Bifurcation of solutions of nonlinear equations in Banach
spaces
Let Ei, E2 are real Banach spaces; Y be normalized space. Consider the equation
Bx = R(x, e). (6)
Here B : D C E1 — E2 be closed linear operator with a dense range of definition in E1. The operator R(x,e) with values in E2 is defined, is continuous and continuously differentiable by Frechet with respect to x in a neighbourhood
Q = {x e E1, e e Y :|| x ||< r, || e ||< g}.
Thus, R(0,e) = 0, Rx(0,0) = 0. Let operator B be Fredholm. Let us introduce the basis }n in a subspace N(B), the basis {^i}™ in N(B*), and also the systems {y}n e E*, {zi}n e E2 which are biorthogonal to these basises.
Definition 2. The point eo is called a bifurcation point of the equation (6), if in any neighbourhood of point x = 0, e0 there is a pair (x,e) with x = 0 satisfying to the equation (6).
It is well known [19] that the problem on a bifurcation point of (6) is equivalent to the problem on bifurcation point of finite-dimensional system
L(C,e)=0, (7)
where C e Rn, L : Rn x Y — Rn. We call equation (7) the branching equation (BEq). We wright (6) as the system
n
B x = R(x,e) + J2 Cs Zs (8)
s=1
Cs =<x,Ys >, s = 1,...,n, (9)
~ def
where 13 = B + J^n=1 < ',Ys > zs has inverse bounded. The equation (8) has the unique small solution
n
x = £ CsPs + U (C,e) (10)
s=1
at £ — 0, e — 0. Substitution (10) into (9) yields formulas for the coordinates of vector-function L : Rn x Y — Rn
Lk(£, e) =< r( jr £sps + U(£, e),e\^k >. (11)
Here derivatives
dLk , „ „ ^-r, . _ def
d£i
|ç=c=< Rx(0,e)(I - TRX(0,e))-1>= alk(e)
are continuous in a neighbourhood of point e = 0, || rRx(0,e) ||< 1.
Let us introduce a set Q = {e | det[aik(e)] = 0}, containing point e = 0 and the following condition:
A) Suppose that in a neighbourhood of point eo £ Q there is a set S, being Jordan continuum, representable as S = S+ (JS-, e0 £ dS+ P|dS-. Moreover, there is a continuous map e(t), t £ [—1,1] such that e : [—1,0) — S-, e : (0,1] — S+, e(0) = eo, detja^(e(t))]nk= = a(t), where a(t) : [—1,1] — R1 be continuous function vanishes only at t = 0.
Theorem 1. Assume condition A, and a(t) is monotone increasing function. Then e0 be a bifurcation point of (6).
Proof. We take arbitrarily small r > 0 and 5 > 0. Consider the continuous vector field
H(£, 8) d=f L(£, e((28 — 1)5)) : Rn x R1 — Rn,
defined at £, 8 £ M, where M{£, 8 ||| £ || = r, 0 < 8 < 1}.
Case 1. If there is a pair (£*, 8*) £ M for which H(£*, 8*) = 0, then by definition 2, e0 will be a bifurcation point.
Case 2. We assume that H(£, 8) = 0 at V(£, 8) £ M and, hence, e0 is not a bifurcation point. Then vector fields H(£, 0) and H(£, 1) are homotopic on the sphere || £ ||= r. Consequently, their rotations [6] are coincided
J(H(£, 0), || £ ||= r) = J(H(£, 1), (|| £ || = r) (12)
Since vector fields H(£, 0), H(£, 1) and their linearizations
n
L-(£) =fEaik(e(—5))£k in=1,
k=1
n
def
L+(ù) = E aik(e(+ô))£k |?=i
k=l
are nondegenerated on the sphere || £ ||= r, then by smallness of r > 0, fields (H(£, 0), H(£, 1) are homotopic to the linear parts L-(£) and L+(£).
Therefore
J(H(C, 0), || C ||= r) = J(l-(C), || C ||= r) (13)
J(H(C, 1), || C ||= r) = J(L+ (C), || C ||= r). (14)
Because of nondegeneracy of linear fields L^(C), by the theorem about Kronecker index, the following equalities hold
J(L-(C), || C ||= r) = signa(-S),
J(L+(C), || C ||= r) = signa(+5).
Since a(S) < 0, a(+5) > 0, then the equality (12) is impossible by (13), (14). Hence, we find a pair (C*, ©*) e M for which H(C*, ©*) = 0 and e0 be a bifurcation point.
Remark 1. If the conditions of the theorem 1 are satisfied for Ve e Q0 C Q, then Q0 be a bifurcation set of (6). If moreover, Q0 is connected set and its every point is contained in a neighbourhood, which is homeomorphic to some domain of Rn, then Q0 is called n-dimensional manifold of bifurcation.
For example, it is true, if Y = Rn+1, n > 1, Q0 be a bifurcation set of (6) containing point e = 0 and V£ detfa^(e)] |£=0= 0. It follows from the theorem 1 at Y = R1 the generalization [17], and also other known strengthenings of M.A. Krasnoselskii theorem on a bifurcation point of odd multiplicity [6]. An important results in the theory of bifurcation points were obtained for (6) with potential BEq to C, when
L(C,e) = grads U (C,e). (15)
This condition is valid, if a matrix 1 is symmetric. By differentiation
of superposition, one finds from (11) that
dL ( n \ ( dU \
-QjT =< Rx + 17£^£J + gfj ' ^k
(16)
where according to (8), (10)
dU _ „ „ ,_1
<pi + 1rF = (I-TRx)-l<pi. (17)
The operator I — TRx is continuously invertible because || TRx ||< 1 for sufficiently small by norm C and e. Substituting (17) into (16) we obtain equalities
dL
—— =< RX{I - TRxi)~lLpi,^k >, i, k = 1,..., n.
Oqi
It follows the following claim:
Lemma 1. In order BEq (7) to be potential it is sufficient that a matrix
- = [< Rx (r Rx)mpi >}'n,k=1 to be symmetric at V(x,e) in a neighbourhood of point (0,0). Corollary 1. Let all matrices
[< Rx(rRx)mPi, ^k >]£k=1, m = 0,1,2,...
are symmetric in some neighbourhood of point (0,0). Then BEq (7) be potential.
Corollary 2. Let E1 = E2 = H, H be Hilbert space. If operator B is symmetric in D, and operator Rx(x,e) is symmetric for V(x,e) in a neighbourhood of point (0,0) in D, then BEq be potential.
In the paper [16] more delicate sufficient conditions of BEq potentiality have been proposed.
Suppose that BEq (7) is potential. Then it follows from the proof of lemma 1 that the corresponding potential U in (15) has the form
1n i,k=1
where || u(£,e) ||= 0(| £ |2) at £ — 0.
Theorem 2. Let BEq (7) be potential. Assume condition A). Moreover, let the symmetrical matrix [a^(e(t))] possesses at least v1 positive eigenvalues at t > 0 and at least v2 positive eigenvalues at t < 0, v1 = v2. Then e0 will be a bifurcation point of (6).
Proof. We take the arbitrary small 5 > 0 and we consider the function U(£,e((2 8 — 1)5)), defined at 8 £ [0,1] in a neighbourhood of the critical point £ = 0.
Case 1. If there is 8* £ [0,1] such that £ = 0 is the nonisolated critical point of the function U(£,e((28* — 1)5), then by definition 2, e0 will be a bifurcation point.
Case 2. Assume that point £ = 0 will be the isolated critical point of the function U(£,e((28 —1)5)) at V8 £ [0,1], where e(t) be continuous function from condition A). Then at V8 £ [0,1], the Conley index [2] K© of the critical point £ = 0 of this function is defined. Let us remind that
det || -—- ||c=0= «((28 - 1)5).
Since a((28 — 1)5) / 0 at 0 / then the critical point £ = 0 at 8 / \ is nonsingular. Therefore, index K@ for any 8 / A by the definition
(here readers may refer to p.6 [2]), is necessary equal to number of positive eigenvalues of the corresponding Hessian. Thus, K© = v1, K1 = v2, where v1 = v2 by the condition of theorem 2. Hence, K© = K1. Suppose that e0 is not a bifurcation point. Then VgU(£,e((20 — 1)a) = 0 at 0 <|| £ ||< r, where r > 0 is small enough, 0 E [0,1]. Because of homotopic invariancy of Conley index (see theorem 4, p.52 in [2]), K© is constant at 0 E [0,1] and K0 = K1. Hence, in the second case we find a pair (£*, 0*) for arbitrary small r > 0, 5 > 0, where 0 <1 £* ||< r, 0* E [0,1], satisfying to the equation VgU(£,e((20 — 1)5) = 0 and e0 is a bifurcation point.
Remark 2. Other proof of the theorem 2 with application of the Roll theorem is given in [18] for the case Y = R1, v+ = n, v— =0.
Remark 3. The theorems 1, 2 (see remark 1) allow to construct not only the bifurcation points, but also the bifurcation sets, surfaces and curves of bifurcation.
Corollary 3. Let Y = R1 and BEq be potential. Moreover, let [aik(e)]nk=1 be positively definite matrix at e E (0, r) and negatively defined at e E (—r, 0). Then e = 0 is a bifurcation point of (6).
Consider the connection of eigenvalues of matrix [aik(e)] with eigenvalues of operator B — Rx(0,e).
Lemma 2. Let E1 = E2 = E, e E R1; v = 0 be isolated Fredholm point of operator-function B — vI. Then
k n
signA(e) = (—1)k sign\\vi (e) = signW^i(e),
i i
where k be a root number of operator B; }n are eigenvalues of matrix [aik(e)], A(e) = det[aik(e)].
Proof. Since {^i}n are eigenvalues of matrix [aik(e)], then l^i(e) = A(e). Thus, it is sufficient to prove the equality A(e) = (—1)krL vi(e). Since zero is the isolated Fredholm point of operator-function B — vI, then operators B and B* have the corresponding complete Jordan systems [19]
= (rrV(1), 4s) = (r*rV(1), i = 1,...,n; s = 1,...,Pi. (18)
Here
<Vl >= 5ij; <<fi j >= 5ij, i,j = 1,...,n;^ P = k.
i=1
Let us remind that
(n \ — 1
b+r < ,4pi) >4P)) ,
n
where k = l1 + ... + ln we call a root number of operator B — Rx(0,e). The small eigenvalues v(e) of operator B — Rx(0,e) satisfy to the following branching equation [19]
L(v,e)= det |< Rx(0,e)+vI)(I—rRx(0,e)—vr)-1 p>|nj=1= 0. (20)
Because of preliminary Weierstrass theorem [19], p. 66, by the equalities (18), (19), equation (20) in a neighbourhood of zero will be transformed to the form
L(v,e) = (vk + Hk-1(e)vk-1 + ... + H(e))Q(e,v) = 0,
where Hk-1(e),..., H0(e) = A(e) are continuous functions of e, Q(0,0) = 0, H0(0) = 0. Consequently, operator B — Rx(0,e) has k > n small eigenvalues Vi(e), i = 1,...,n, which we may define from the equation
vk + Hk-1(e)vk-1 + ... + A(e) = 0.
Then nk vi(e) = A(e)(—1)k.
Assume now e £ R1. Consider the calculation of asymptotics of eigenvalues n(e) and v(e). Let us introduce the block representation of matrix [aik}nk=1, satisfying the following condition:
B) Let [aik(e)]nk=1 = [Aik(e)]i,k=1 ~ [erifc^0k]i,k=1 at e — 0, where [Aik] are blocks of dimensionality [ni x nk], n1 + ... + ni = n, min(ri1,..., rn) =
rii== ri E rik > ri at k > i (or at k < i), i = 1,..., l. Let H1 det[A0i] = 0. The condition B) means that matrix [aik(e)]nk=1 admits the block representation being "asymptotically trianglar"at e — 0.
Lemma 3. Assume B). Then
det[aik(e)]nk=1 = eniri+-+niri (]] det | ^0i | +0(1)) ,
formulas
Hi = eri(Ci + 0(1)), i = 1,...,l (21)
define the principal terms of all n eigenvalues of matrix | aik(e) \ik=1, where Hi, Ci £ Rni; Ci be vector of eigenvalues of matrix Al0i.
Proof. By B) and the property of linearity of determinant, we have
' A0n +0(1), 0(1)......, ......0(1) ■
+ + A21 +0(1), A02 + 0(1), 0(1) ...0(1) det[aik(e)] = eniri+-+nir det
41 + 0(1), ......, AO + 0(1)
eni[] det | A° | +0(1)) .
i
Substituting i = eric(e), i = 1,... ,l into equation det | aik(e) — ¡i5ik !nk=r 0 and using the property of linearity of determinant we obtain equation
i-1
£ni ri+...+Ui-iri-l + (ni + ...+Ul )ri det | A°-
j=1
det(A° - (e)E)(e)ni+1+...+ni + al(e)} = 0, i = l,...,l, (22)
where ai(e) ^ 0 at e ^ 0. Hence, the coordinates of unknown principal terms Ci in asymptoticses (21) satisfy to the equations det | A0 — E |= 0, i = l,...,l.
If k = n, then operator B — Rx(0,e), as well as the matrix [a^(e)]nk=1 has n small eigenvalues. In this case we state a result:
Corollary 4. Let operator B has not I - joined elements and let the condition B) holds. Then the formula
Vi = —eri(Ci + 0(1)), i = l,...,l, (23)
defines all n small eigenvalues of operator B — Rx(0,e), where Ci G Rni be vector of eigenvalues of matrix A0, i = 1,...,l, n1 + ... + ni = n.
Proof. By lemma 2 in this case ^n P = n (root number k = n) and operator B — Rx(0, e) possesses at least n small eigenvalues. Since ^ 1 ni = n, A0 is quadratic matrix, then formula (23) yields n eigenvalues, where the principal terms coincide to within a sign with principal terms in (21). For calculation of eigenvalues v of operator B — Rx(0,e) we transform (20) to the form
L(v, e) = det[aifc(e) + £ ]^=i = 0, (24)
j=1
where
b(k =< [(I — rRx(0,e))-1 r]j-1(I — rRx(0,e))-1 Vi,Yk > .
Substituting v = —eric(e) into (24) and taking into account the property of linearity of determinant we shall receive the equation, which differs from (22) by error term ai(e) only. Then in conditions of corollary 4 the principal terms of all small eigenvalues of operator B — Rx(0, e) and matrix —[aik(e)] are defined from the same equations and therefore, are equal.
Conclusions. 1) By lemma 3 we can replace condition A) in the theorem 1 with the following one:
A*). Let E1 = E2 = E; v = 0 be isolated Fredholm point of operator-function B — vl. Let in a neighbourhood of point e0 £ Q there is a set S, containing point e0 and be continuum represented as S = S+ (JS-. Moreover, assume
e0 £ dS+p| dS-; [] vi(e) |£eS+ •[] vi(e) |eeS_ < 0, ii
where {vi(e)} are small eigenvalues of operator B — Rx(0,e).
2) If the principal terms of asymptotics of small eigenvalues of operator B—Rx(0, e) and matrix [aik(e)]nk=1 coincide, then we may use eigenvalues of such operator in the theorem 2. By corollary 4 it is possible, if E1 = E2 = H, operators B and Rx(0,e) are symmetric and condition B) is valid. Let us note that condition B) is valid in papers [15, 16, 17, 18] about bifurcation point with potential BEq, thus r1 = ... = rn = 1.
2. Statement of boundary-value problem and problem on a bifurcation point for the system (32) [9]
We begin with a one preliminary result on reduction of VM system (1)-(2) with conditions (3) to the quasilinear system of elliptical equations for distribution (5), was first investigated in [10]. Assume the following condition:
C). f(R, G) are fixed, differentiable functions in distribution (5); ai, d are free parameters; | di |= 0; pi = cu + lip(r), = C2i + ki4(r); C1i, C2i-const; the parameters li, ki are connected by relations
m-1 aiQi q1 qi
k =--, h—di = —di, ki = h = 1, (25)
a1 q1 mi m1 mi
and the integrals fR3 fdv, fR3 fivdv converge at Vpi,
= = =
Let us introduce notations m1 = m, a1 = a, q1 = q.
Theorem 3. Let fi are defined as weJl as in (5) and the condition C) is valid. Let the vector-function (p, 4) is a solution, of the system of equations
= /x V qk i fkdv, h = ^^
k=1 jr3 m
(26)
N
A4 = ^y^qW (v,d)fkdv, v = — k=1 Jr3
4nq
mc2
i 2aq q
\dD=--«01, W \dD= -U02 (27)
m mc
on a subspace
(drw,di) = 0, (drii,di) = 0, i = 1,...,N. (28)
Then the VM system (1), (2) with conditions (5) possesses a solution m d f1 mc
E = B = ^fj + i {d X J(ir)'r)di) " x (29)'
where
N
J A 47T y-v f p _ const_
c ti JR3
The potentials
m mc
= A = ^d + A1{r), (A1,d)= 0 (30)
isatisfying to condition (3) are defined through this solution.
The proof of theorem 3 follows from theorem 1 of paper [14]. Introduce notations
ji = vfidv, pi = fidv, i = 1,...,N
J R3 JR3
and the following condition:
D). There are vectors /3i <E R3 such that ji = ¡3ipi, i = 1,...,N. For example, the condition D holds for distribution
fi = fi(a(—aiv2 + + b((di, v) + ii)) (31)
for Pi = i^di, a,b-const.
Suppose that condition D is valid. Then the system (26) will be transformed to the following
NN
Ap = X^qiAi, Ai = \vY,qi(fr,d)Ai, (32)
i=l i=l
where
Ai(li<p,ki i,ai ,di ) = fidv.
R3
Further, we shall suppose that the auxiliary vector d in (5) is directed along axes Z. Because of conditions (28) we put in system (32) p = p(x,y), ip = ip(x,y), x,y e D c R2. Moreover, let N > 3 and ¥■ / const.
Let D be bounded domain in R2 with the boundary dD of class C2,a, a £ (0,1]. The boundary conditions (4) on the densities of local charge and current induce the equalities: I.
N N
E QkAk(lkkk4°, ai, di) = 0; ^ qk(fik, d)Ak(lk, k4°, ai, di) = 0 k=i k=i
(33)
for ye £ i, where i is a neighbourhood of point e = 0 and
° 2aq ° q
<p =--«01, ip = —U02• (34)
m mc
Remark 4. If N = 2 and A = then by condition I and equalities
(/3i, d) = j: we have alternative: or in condition I: A\ = A2 = 0 or hi = k,
i = 1,2. In this case, and also at |i=const the system (32) is reduced to one equation and bifurcation of solutions in such approach, as it is considered in this paper is impossible.
By (33), (34) system (32) with boundary conditions
p \dD= 4 \dD= 4° (35)
has a trivial solution p = , 4 = 4° at £ R+.
Then because of theorem 3 the VM system with boundary conditions
(3), (4) has a trivial solution at m
E° =-drtp° = 0, B° = pdi, r £ D c R2,
2aq
f ° = fi(-aiv2 + cii + hp0, (v, di) + C2i + ki4°).
Thus, the densities p and j vanish at domain D.
Now our purpose is to find X° in neighbourhood of which system (32), (35) has a nontrivial solution. Then the corresponding densities p and j will be identically vanish at domain D, and the point X° is a bifurcation point of the VM system with conditions (4), (5).
Let functions fi are analytical in (5). Using the expansion in Taylor series
i>° ' y and selecting linear terms, we transform (32) to operator form
(L° - XLi)u - Xr(u) = 0. (36)
Here
Lo
N
Li = ^ qs
s=i
A 0 0 A
u = (p - p0, ^ -
(37)
^s °dx ^s 9dy
T ¡J,T2 VT3 m
œ n
r(u) = Y^ CJis(u)bs,
i>l s=i
where
Qs d d
Qis{u) = -7(^1— + ksu2—)% As{lsLp° ,ksiP) i! dx dy
A
(38)
(39)
are i homogeneous forms by u;
gii +i2
dxi1 dyi2
As(x,y) \x=lsV°,y=ksi>° =0 at
2 < ¿1 + ¿2 < l - 1, s = 1,...,N; l > 2; bs = (p,v(&,d))'.
We study the problem of existence of a bifurcation point A0 for (32), (34) as the problem on bifurcation point for operator equation (36). Let us introduce Banach spaces C2,a(D) and C0,a(D) with norms || • ||2,a, || • Ho,« and W2'2(D), which is usual L2 Sobolev space in D. Let us introduce Banach
space E of vectors u = (u1,u2)', where ui E L2(D), L2 be real Hilbert space with internal product ( , ) and the corresponding norm || • Hl2 (D).
As a range of definition D(L0) we take set of vectors u = (u1,u2) with ◦ 2,2 ◦ 2,2 ui eW (D). Here W (D) denotes W2,2 functions with trace 0 on dD. Hence, L0 : D C E ^ E is linear self-adjoint operator. By virtue of embedding
W2'2(D) C C 'a(D), 0 < a < 1
(40)
the operator r : W2,2 C E — E be analytical in neighbourhood of zero. The operator Li E L(E — E) is linear bounded. For matrix corresponding to operator Li we shall keep same notations. By embedding (40) any solution of the equation (36) will be Holder in D(L0). Moreover, because the coefficients of (36) are constant, then vector r(u) will be analytical, dD E C2,a and thanks to well-known results of the regularity theory of weak solutions [8],
o 2,2
the being searching generalized solutions of (36) in W (D) belong to C2,a.
By theorem 3 on reduction of VM system, the bifurcation points of problem (32), (34) are the bifurcation points of solutions of VM system (1), (2) with boundary conditions (3), (4). Thanks to given conditions on L0 and L\,
all singular points of operator- function L(X) = L0 — XL1 be Fredholm. The bifurcation points of nonlinear equation (36) we can found only among points of a spectrum for linearized system
(L° - XL\)u = 0.
(41)
For study of spectrum problem (41) we preliminary find the eigenvalues and the eigenfunctions of matrix Li in (41) for physically admissible parameters. With this purpose, we introduce the following condition: II: (T1T4 — T2T3) > 0, Ti < 0.
Lemma 4. Let ^ = ^ > 0, i = 1,..., N at x = Up0, y
Assume
dx dy
N i-1
(ljki - kjli)(ßi - ßj,d) > 0,
i=2 j=1
where ai = Çî^j^, then condition II is valid.
= ki 4°
A
Proof. Without loss of generality we put q = qi < 0, qi > 0, i = 2,...,N. Then via (25) signqili = signq. Further, because of definition of Ti (see.(38)), we verify that Ti < 0. The positiveness of T1T4 - T2T3 follows from equality
T1T4 - T2T3 = £ liai^] ki(ßi, d)ai-E kia^ kß, d)
)ai =
N i- 1
E E ai aj (lj ki - kj li )(ßi - ßj, d). i=2 j=1
Example. If ft = then (ft, d) = and
N i- 1
2 d2
^J ^J = aiüjiljh - likj)2 ■ -¡-j- > 0.
i=2 j=1 ~ i j
Lemma 5. Let distribution function has a form (31) and f'i > 0. Then conditions D and II hold for ft = and the system (32) will be
transformed to the potential form
a1 0 - dV - dip
A .4. = X 0 a2 dV - dip -
100
where
N
k=l
Ok
Ik J0
alk p+bkkf ud?
Ak(s)ds, ai=n/a, a2 = —. (43)
The proof is conducted by direct substitution (43) into the system (42).
Lemma 6. Let r = x e Rl, v e R2, d = d2. Then the system (32) with potential (43) can be written as Hamiltonian system
Pv = -dvH, p = dPv H
P^ = -d^ H, 4 = dp^ H with Hamiltonian function of the form
22
Here
N m talkV f N m rbkkf
V(<p, 4) = Aai E f y I , 4)ds + Xa2 ^ I I s)ds.
k=l
R2
k=l
' R2
The proof follows from lemma 2.2 (p.1152) of work [4].
Lemma 7. Assume II. Then matrix Li in (38) has one positive eigenvalue
X+ = T + 0(1)
and one negative
X-=nTlT*-T2T3e + 0(e), = 0 (44)
Ti
m
+ A 1
at e = -T
2
0.
Eigenvalue x- induces the eigenvactors of matrices Ll and L\ respectively
+ O(e).
The readers may refer to [14] for the proof.
Let us now consider the calculation of bifurcation points Ao of equation (36). Setting in (36) A = Aq + e, we consider the equation
" Cl " — r T2 1 Ti + O(e), 'Cl' — 0
. C2 . 0 _C2. 1
(Lo - (Ao + e)LlU - (Ao + e)r(u) = 0
(45)
in neighbourhood of point A0. Let T2 = 0 and T3 = 0, or T2 = T3 = 0. With the purpose of symmetrization of system at T2 = 0 and T3 = 0 having multiplicated both parts of (45) on matrix
M = f V where a = ^ / 0, V 0 a) vT
we write (45) as
Bu = eBiu + (A0 + e)K(u). (46)
Here B = M(L0 - A0Li); K(u) = Mr(u) = (n(u),r2(u)); Bi E L(E — E) be Fredholm self- adjoint operator. If As = As(alsp + bkkthen
dA^ .» dAs w _ i ,, d , n b ds
dy s dx s 2aa a 2as
In expansion (39)
Qis = jA<i\alsp0 + bk8i)°){al8u 1 + bksu2)\
Thus, in this case = matrix №u(u) will be symmetric for Vu and operator : E — E is self-adjoint for Vu.
Remark 5. If T2 = T3 = 0, then we put a = 1. If T2 = 0, T3 = 0 or T3 = 0, T2 = 0, then the problem (36) has not the property of symmetrization and we should work with (45). In this case for study of the problem on bifurcation point we may use our results from [13].
Let n be eigenvalue of the Dirichlet problem
-Ae = ¡ie e \dD= 0 (47)
and {e1,..., en} be orthonormalized basis in a subspace of eigenfunctions. Denote by c- = (c1,c2)/ the eigenvector of matrix L1, which corresponds to eigenvalue x- < 0.
Lemma 8. Let A0 = -^/x-- Then A0 > 0, dimN(B) = n and the isystem ^}'n=1, where ei = c-ei forms basis in a subspace N(B).
Proof. Let us introduce matrix of columns A, which are the eigenvectors of matrix L1 corresponding to eigenvalues x-, X+. Moreover,
¡X- 0 \ A-1L1A = , L0A = AL0
V 0 x+)
and equation Bu = 0 by change u = AU will be transformed to the form M[L0AU - A0L1AU] = M[A(L0U - A0A-1 L1AU)] = 0.
Hence, from here follows that the linear system (41) is decomposed onto two linear elliptical equations
AUi - X0X-U1 = 0, Ui \du= 0, AU2 — XoX+ U2 = 0, U2 \dD= 0, (48)
where AoX- = ^oX+ > 0. From (47) follows that ¡i £ a(—A). Hence, U1 = Y^i=1 aiei, ai — const, U2 = 0 and
U1 = AU = C1_ c1+ — U1 — C1-
U2 C2- c2+ 0 C2-
E
i=1
a. ei
Let us construct Lyapunov-Schmidt BEq for equation (46).
Without loss of generality we assume that the eigenvector c1- of matrix L\ is chosen such that %_(cf_ + Fc\_) = 1, where F = Then the system of vectors {B1ei}'n=1 is biorthogonal to ^}™=1. Thus, operator
n
B = B + ^ < -,Yi >Yi
A
with y = B1ei has inverse bounded r £ L(E ^ E), r = r*, = e^ Rewrite (46) as the system
(B - eBi)u = (Ac + e)K(u) + ^ Ça.
Ç.i =<u,Yi >, i = 1,...,n. By the theorem on inverse operator we have from (49)
u = (Ac + e)(I - erBi)-1m(u) +
1
1 - e
Ec.
i=1
e..
(49)
(50)
(51)
From (50) we have
1e
1e
(52)
where K(u) = ^(u) + !Rl+1(u) + .... Because of the theorem on implicit operator, equation (51) has unique solution for sufficiently small e, | £ |.
u = ui(£e, e) + (Aq + e)(I - eTBi)-1T{ui(£e, e) + ui+i(£e, e) + ...}. (53)
Here
Ui(C e,e) =
1
1e
EC
i=1
e
ui(£e, e) = (£e,e)), ui+i(£e,e) = Ki+i(ui (£e,e)) +
C 0, l > 2
1 m2M£e, e))(Ao + e)(I — erBi)-1ru2(£e, e), l = 2 and etc. Substituting the solution (53) into (52) we obtain desired BEq
L(£,e)=0 (BEq)
with L = (Li, ...,Ln),
v = zr^Zi + hX°\i+1[< m*,*i) > < Ml+ii&ei) >] +
1 — e (1 — e)l+i 1 — e
0, l> 2
+ n (£,e),
^ K^eil-eTB^T^e)^^, 1 = 2
ri = o(| £ |l+i), i = 1,...,n. If L(£,e) = gradU(£,e), then we call BEq potential. In potential case matrix Lg(£,e) is symmetric.
Let in (46) fi = fi(alp+bki4), i = 1,...,N. Then from explained above matrix Ku(u) will be symmetric at Vu and we have the following statement:
Lemma 8. Let conditions C), D), I-II and A0 = —//X- hold. Then equation (46) possesses so much small solutions u ^ 0 at X ^ X0, as small solutions £ ^ 0 possesses BEq at e ^ 0. If in system (32) Ai = Ai(alip + bki4), i = 1,..., N; a, b—-const, then BEq will be potential.
Theorem 4 (Principal theorem). Let N > 3. Let conditions C, D, I-II and X0 = —//x are valid, where / is n multiple eigenvalue of Dirichlet problem (47). Number x- see in (44). If n is odd, or distribution function has the form fi = fi(a(—aiv2 + p,,) + b((di, v) + 4)), i = 1,---,N, then X0 be a bifurcation point of VM system (1)-(2) with conditions (3)-(4).
Proof. Case 1. Let n is odd. Then in BEq
dLki
A(e) = det
d£
-(0,e)
ikk=i v1 — e
Since n is odd, then A(e) > 0 for e e (0,1), and A(e) < 0 for e e (—1,0) and the statement of theorem follows from theorem 1.
Case 2. Let fi = fi(a(—aiv2+pi)+b((di,v)+4i)). Then BEq is potential, moreover
dLk (0, e) e
d£i 1 — e
&ik, i,k = 1,...,n.
Hence, all eigenvalues of matrix || aZ/fc(°'£) || are positive at e > 0 and are negative at e < 0. Thus, the validity of the theorem in case 2 follows from theorem 2.
3. Conclusion
The distributions functions fi in VM system depend not only upon A, but also on parameters ai, di, ki, li. It seems interest to investigate a behaviour of solutions of (1)-(2) with conditions (3), (4) depending from these parameters. Applying theorems 1, 2 and their corollaries in the present paper, we can prove the existence theorems of points and surfaces of bifurcation for this more complicated case.
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Н. А. Сидоров
Точки бифуркации нелинейных операторов: теоремы существования и приложения к исследованию систем Власова-Максвелла
Аннотация. Дан обзор теорем существования точек бифуркации решений нелинейных операторных уравнений в банаховых пространствах. Получены достаточные условия ветвления решений граничных задач для систем Власова-Максвелла. При построении асимптотики решений граничной задачи используется аналитический метод Ляпунова-Шмидта-Треногина.
Ключевые слова: точка бифуркации, нелинейный анализ, система Власова-Максвелла, плазма, индекс Конли, метод Ляпунова-Шмидта-Треногина.
Сидоров Николай Александрович, доктор физико-математических наук, профессор, Институт математики, экономики и информатики, Иркутский государственный университет, 664033, Иркутск, ул. К.Маркса, 1 ([email protected])
Sidorov Nikolay, Irkutsk State University, 1, K. Marks St., Irkutsk, 664003, professor, Phone: (3952)242210 ([email protected])
