UDC 513.882

MATHEMATICS

I. A. BAKHTIN

APPLICATION OF TOPOLOGICAL METHODS TO THE STUDY OF THE CRITICAL REGIME OF A REACTOR

(Presented by Academician I. N. Vekua on 24 V 1965)

In the present work, by the topological methods of the monograph of M. A. Krasnosel’skii \((^1)\), a special system of equations of the form

\[ \begin{aligned} x&=\mu A(x,y),\\ y&=B(x,y). \end{aligned} \tag{1} \]

is investigated.

The results obtained are applied to the study of the critical regime of a reactor.

1. We introduce some notation which we shall use. Let \(E_1\) and \(E_2\) be two real Banach spaces with cones \(K_1\) and \(K_2\) \((^2)\). Denote by \(E=E_1\times E_2\) the Banach space of elements \(z=(x,y)\) \((x\in E_1,\ y\in E_2)\) with norm \(\|z\|=\|x\|+\|y\|\). Obviously, \(K=K_1\times K_2\) is a cone in the space \(E\).

Let \(A\) and \(B\) be operators acting respectively from the space \(E\) into \(E_1\) and \(E_2\). Then the operator \(C=(A,B)\) will act in the space \(E\). Denote by \(C(\mu)\) the operator \((\mu A,B)\). It is easy to see that system (1) can be written in the form of a single equation

\[ z=C(\mu)z. \tag{2} \]

By \(P\) and \(Q\) we denote the Fréchet derivatives at zero \(\theta\) (if they exist) of the operators \(A\) and \(B\). By \(D\) and \(D(\mu)\) we denote respectively the operators \((P,Q)\) and \((\mu P,Q)\). Obviously, the operator \(D(\mu)\) is the Fréchet derivative at zero \(\theta\) of the operator \(C(\mu)\). By \(Q_1\) and \(Q_2\) we denote the operator \(Q\) respectively on the subspaces \(E_1\) and \(E_2\).

2. In this item we shall formulate two theorems on the computation of rotations of vector fields, analogous to the theorems of Leray and Schauder \((^1)\).

A solution \(z=(x,y)\) of equation (2) will be called a semiproper vector of the operator \(C\), if \(x\ne\theta\). The corresponding number \(\mu\) \((\lambda=1/\mu)\) will be called a semicharacteristic (semiproper) number of the operator \(C\). The totality of all semiproper numbers of the operator \(C\) will be called the semispectrum of the operator \(C\). The multiplicity of the characteristic number \(\mu\) of the operator

\[ P_{0}x=P\bigl[x,(I-Q_{2})^{-1}Q_{1}x\bigr] \]

will be called the \(\pi\)-multiplicity of the semicharacteristic number \(\mu\) of the linear operator \(D\).

Let \((T_1)_{r_1}\subset E_1\) be the ball \(\|x\|\le r_1\), and \((T_2)_{r_2}\subset E_2\) the ball \(\|y\|\le r_2\). Denote by \((S_1)_{r_1}\) and \((S_2)_{r_2}\), respectively, the spheres \(\|x\|=r_1\) and \(\|y\|=r_2\).

Theorem 1. Let the linear operator \(D=(P,Q)\) be completely continuous. Let unity not be a point of the spectrum of the operator \(Q_2\). Then the rotation \(\gamma\) of the vector field \(z-D(\mu_0)z\) \((\mu_0\ne0)\) \((^1)\) on the boundary \(\Gamma\) of the set \((T_1)_{r_1}\times(T_2)_{r_2}\) is equal to the product of the rotations \(\gamma_1\) and \(\gamma_2\) of the vector fields \(x-\mu_0 P[x,(I-Q_2)^{-1}Q_1x]\) and \(y-Q_2y\), respectively, on the spheres

\((S_1)_{r_1}\) and \((S_2)_{r_2}\), i.e. \(\gamma = (-1)^{\beta}\gamma_2\), where \(\beta\) is the sum of the \(\pi\)-multiplicities of all semi-characteristic numbers \(\mu \in (0,\mu_0)\) of the operator \(D\).

Theorem 2. Let zero \(\theta\) be a fixed point of the completely continuous vector field \(\Phi = I - C(\mu_0)\) \((\mu_0 \ne 0)\) \((^1)\). Let \(D\) be the Fréchet derivative of the operator \(C\) at zero \(\theta\). Finally, suppose that unity is not an eigenvalue of either the operator \(D(\mu_0)\) or \(Q_2\).

Then \(\theta\) is an \((^1)\) isolated fixed point of the vector field \(\Phi\), and the index \((^1)\) of this point is equal to \((-1)^{\beta}\gamma_2\), where \(\beta\) is the sum of the \(\pi\)-multiplicities of all semi-characteristic numbers of the operator \(D\) from the interval \((0,\mu_0)\), and \(\gamma_2\) is the index of zero \(\theta\) of the field \(I-Q_2\) in \(E_2\).

1. We now present a theorem on bifurcation points of operators. We shall call the number \(\mu_0\) a point of \(\pi\)-bifurcation of the operator \(C\) if, for every \(\varepsilon>0\), there exists a semi-characteristic number \(\mu\) of the operator \(C\) such that \(|\mu-\mu_0|<\varepsilon\) and to the number \(\mu\) there corresponds at least one semi-eigenvector \(z:C(\mu)z=z\), with norm \(\|z\|<\varepsilon\).

Theorem 3. Let the completely continuous operator \(C\) \((C\theta=\theta)\) have at zero \(\theta\) the Fréchet derivative \(D\). Suppose that unity is not a point of the spectrum of the operator \(Q_2\).

Then:

1) Only semi-characteristic numbers of the operator \(D\) and the number zero can be points of \(\pi\)-bifurcation of the operator \(C\).

2) Every semi-characteristic number \(\mu_0\) of odd \(\pi\)-multiplicity of the operator \(D\) is a point of \(\pi\)-bifurcation of the operator \(C\); moreover, at this point there corresponds a continuous branch \((^1)\) of semi-eigenvectors of the operator \(C\). If it is additionally known that zero \(\theta\) is an isolated fixed point of the vector field \(I-C(\mu_0)\), then the semispectrum of the operator \(C\) includes some interval.

1. In the present section we formulate a theorem that is an analogue of the topological principle of M. A. Krasnosel’skii (see \((^1)\), pp. 244–249, Theorem 1.1).

Theorem 4. Let the positive operator \(C=(A,B)\) \((CK \subset K)\) be completely continuous. Suppose that

\[ \inf_{x\in \Gamma_1\cap K_1,\; y\in K_2\cap (T_2)_{r_2}} \|A(x,y)\| > 0, \]

where \(\Gamma_1\) is the boundary of some open set \(G_1\subset E_1\) containing zero \(\theta\). Finally, suppose that for any \(x\in \Gamma_1\cap K_1\) the operator \(B_xy=B(x,y)\) maps the convex set \(K_2\cap (T_2)_{r_2}\) into itself.

Then the operator \(C\) has a semi-eigenvector \(z=(x,y)\) in the cone \(K\), with \(x\in \Gamma_1\cap K_1,\; y\in K_2\cap (T_2)_{r_2}\).

1. We now turn to the application of the results obtained. Let the distribution of the neutron flux \(\Phi\) and of the temperature \(T\) over the volume \(V\) of the reactor be described by the \((^3)\) equations

\[ \begin{gathered} \nabla^2 \Phi + \mu a(T)\Phi = 0,\qquad \Phi|_{\Sigma}=0;\\ \nabla^2 T + b(T)\Phi = 0,\qquad T|_{\Sigma}=T_0, \end{gathered} \tag{3} \]

where \(\Sigma\) is the boundary of the reactor, assumed to be sufficiently smooth and convex. In view of the physical meaning \((^3)\), the coefficients \(a(T)\) and \(b(T)\) are positive and continuous. The system (3) of differential equations can be replaced by the equivalent system of integral equations

\[ \Phi(P)=\mu\int_V K(P,Q)a[T(Q)+T_0]\Phi(Q)\,dQ=\mu A(\Phi,T), \]

\[ T(P)=\int_V K(P,Q)b[T(Q)+T_0]\Phi(Q)\,dQ=B(\Phi,T). \tag{4} \]

The system (4) was studied by O. B. Moskalev in paper (3). We investigate it under other restrictions.

Below, the roles of \(E_1, E_2\) and \(K_1, K_2\) are played respectively by the space \(C(V)\) of functions continuous on \(V\) and the cone \(K(V)\) of nonnegative functions of the space \(C(V)\).

In paper (3) it is shown that the operator \(C=(A,B)\) is positive and completely continuous.

Theorem 5. The points of \(\pi\)-bifurcation of the system (4) can only be the characteristic numbers of the operator

\[ A_0\Phi=a(T_0)\int_V K(P,Q)\Phi(Q)\,dQ . \]

Each characteristic number of odd multiplicity of the operator \(A_0\) is a point of \(\pi\)-bifurcation of the system (4). In particular, the characteristic number \(\mu_0\) of the operator \(A_0\), to which there corresponds the eigenvector \(\Phi_0\in K_1\), is a point of \(\pi\)-bifurcation of the system (4).

If, in addition, it is known that the kernel \(K(P,Q)\) of the system (4) is symmetric and that in some neighborhood of the point \(T=0\) the functions \(a(T+T_0)\) and \(b(T+T_0)\) have continuous derivatives respectively up to orders \(n\) and \(m\) inclusive, where
\[ a'(T_0)=\ldots=a^{(n-1)}(T_0)=0,\quad a^{(n)}(T_0)\ne 0 \]
and
\[ b'(T_0)=\ldots=b^{(m-1)}(T_0)=0,\quad b^{(m)}(T_0)\ne 0, \]
then for any characteristic number \(\mu_*\) of odd multiplicity of the operator \(A_0\):

a) if \(n\) is an even number and \(a^{(n)}(T_0)<0\), then the half-spectrum of the system (4) completely fills some interval \((1/\mu_1,1/\mu_*)\), where \(\mu_1<\mu_*\) if \(\mu_*<0\), and \(\mu_2<\mu_*\) if \(\mu_*>0\).

b) if \(n\) is an even number and \(a^{(n)}(T_0)>0\), then the half-spectrum of the system (4) completely fills some interval \((1/\mu_2,1/\mu_*)\), where \(\mu_2>\mu_*\) if \(\mu_*<0\), and \(\mu_2<\mu_*\) if \(\mu_*>0\).

Definition. We shall say that the semi-eigenvectors \(z\) of the operator \(C\) form a conditionally continuous branch of infinite length if, for the boundary \(\Gamma_1\) of any bounded open set \(G_1\subset E_1\) containing zero \(\theta\), there exists such a semi-eigenvector \(z=(x,y)\) of the operator \(C\) that \(x\in\Gamma_1\).

Theorem 6. Suppose that for all \(T\ge 0\)

\[ b(T+T_0)\le \alpha+\beta T^q, \]

where \(\alpha,\beta>0,\ q\in(0,1)\).

Then the semi-eigenvectors \(z=(\Phi,T)\in K\) of the system (4) form a conditionally continuous branch of infinite length.

If, in addition, it is known that:

a) there exists the limit
\[ \lim_{T\to\infty} a(T+T_0)=a_0<\infty; \]

b) for all \(T\ge 0\)
\[ b(T+T_0)\ge b_0>0, \]

then the half-spectrum of the system (4) includes the interval
\[ \left(\frac{1}{\mu_0},\ \frac{1}{\mu_0}\frac{a_0}{a(T_0)}\right). \]

In conclusion, the author expresses sincere gratitude to M. A. Krasnosel’skii for his attention to the work.

Voronezh State
Pedagogical Institute

Received
15 V 1965

REFERENCES

1. M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Moscow, 1956.
2. M. G. Krein, M. A. Rutman, UMN, 3, No. 1 (23) (1948).
3. O. B. Moskalev, Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki, 3, No. 2 (1963).