UDC 517.9:533.7

MATHEMATICS

V. I. YUDOVICH

ON THE STABILITY OF SELF-OSCILLATIONS OF A FLUID

(Presented by Academician A. N. Kolmogorov, 27 VIII 1970)

A linearization law is established in the stability problem for a periodic self-oscillation of a viscous incompressible fluid. An infinite-dimensional analogue of the Andronov—Witt theorem \((^{1,2})\) is proved.

1. The general stability problem for a cycle. Let a dynamical system \((Q_t, X)\) be given, i.e., a nonlinear partial semigroup of operators \(Q_t : X \to X\) \((0 \leq t < \infty)\), acting in a Banach space \(X\). This means that for each \(x \in X\) an element \(Q_t x\) is defined for \(0 \leq t < t_0(x)\), with \(Q_0x=x\); \(Q_{t+\tau}x=Q_tQ_\tau x\) \((t,\tau \geq 0;\ t+\tau < t_0(x))\).

Suppose that the operator \(Q_T\) for some \(T>0\) has a fixed point \(q_0\). Then \(q_\tau=Q_\tau q_0\) is also a fixed point of the operator \(Q_T\) for all \(\tau>0\). If \(C=\bigcup_{0\leq \tau\leq T} q_\tau\) contains more than one point, we shall say that \(C\) is a cycle (or self-oscillation) of the dynamical system \((Q_t,X)\). We shall call it (asymptotically) stable if, for every \(\varepsilon>0\), one can indicate such a \(\delta>0\) that from \(\rho(x_0,C)<\delta\) it follows that \(t_0(x_0)=\infty\) and \(\rho(Q_t x_0,C)<\varepsilon\) for \(t\geq 0\) (and \(\rho(Q_t x_0,C)\to 0\) as \(t\to+\infty\)). If \(\|Q_t x_0-q_{t+h}\|\to 0\) as \(t\to+\infty\), then we shall say that the trajectory \(Q_t x_0\) has asymptotic phase \(h\).

Next, we shall call the cycle \(C\) smooth if in some neighborhood of it
\(\omega_\eta=\{x\in X:\rho(x,C)<\eta\}\) the operators \(Q_t\) are continuously Fréchet differentiable (with respect to \(x\)), the derivative \(d/d\tau\, q_\tau=\dot q_\tau\) exists \((0\leq \tau\leq T)\), and, moreover, for any \(\varepsilon_1,\varepsilon_2>0\) there exist \(\delta_1,\delta_2>0\) such that from \(\|a\|<\delta_1\), \(|s|<\delta_2\) follow estimates, uniform in \(\tau\in[0,T]\),

\[ \|\Delta_1(\tau,a)\|\equiv\|Q_T(q_\tau+a)-q_\tau-Q_T'(q_\tau)a\|\leq \varepsilon_1\|a\|; \tag{1} \]

\[ \|\Delta_2(\tau,s)\|\equiv\|q_{\tau+s}-q_\tau-\dot q_\tau s\|\leq \varepsilon_2|s|. \tag{2} \]

The operator \(U_{T,\tau}=Q_T'(q_\tau)\) will be called the monodromy operator (corresponding to the initial instant \(\tau\)). Differentiating the equality \(Q_Tq_\tau=q_\tau\) with respect to \(\tau\), we infer that \(\varphi_\tau=\dot q_\tau\) is an eigenvector of the operator \(U_{T,\tau}\) corresponding to the eigenvalue \(1\): \(U_{T,\tau}\varphi_\tau=\varphi_\tau\).

Lemma 1. The spectrum of the monodromy operator \(U_{T,\tau}\) does not depend on \(\tau\).

This lemma is easily derived by taking into account the equalities

\[ U_{T,\tau}=Q_\tau'(q_0)Q_{T-\tau}'(q_\tau),\qquad U_{T,0}=Q_{T-\tau}'(q_\tau)Q_\tau'(q_0) \tag{3} \]

and using the following simple assertion.

Lemma 2. Let \(U,V:X\to X\) be linear bounded operators. Then the operators \(UV\) and \(VU\) have one and the same nonzero spectrum:
\(\sigma(UV)-\{0\}=\sigma(VU)-\{0\}\). Moreover, if \(0\) is excluded, the point, continuous, and residual spectra, respectively, coincide. The multiplicities of the nonzero eigenvalues are then the same.

Theorem 1. Let \(C=\bigcup_{0\leq \tau\leq T}Q_\tau q_0\) be a smooth cycle of the dynamical system \((Q_t,X)\), and let the spectrum of the monodromy operator \(U_{T,\tau}\) have the form

\[ \sigma(U_{T,\tau})=\{1\}\cup\sigma_0(U_{T,\tau}),\qquad |\sigma_0(U_{T,\tau})|<\alpha<1. \tag{4} \]

Let 1 be a simple proper number. Then the cycle \(C\) is asymptotically stable, and every trajectory \(\{Q_t x^0\}\), \(t \ge 0\), has an asymptotic phase, provided only that the quantity \(\rho(x_0,C)\) is sufficiently small.

Proof. Let \(\psi_\tau\) be a fixed vector of the adjoint operator \(U_{T,\tau}^{*}\), normalized by the condition \((\varphi_\tau,\psi_\tau)=1\). Define the operator \(V_\tau\) by setting

\[ V_\tau x=U_{T,\tau}x-(x,\psi_\tau)\varphi_\tau . \tag{5} \]

It is clear that \(\sigma(V_\tau)=\sigma_0(U_{T,\tau})\). Therefore, for sufficiently large natural \(m\), the operator \(V_\tau^m\) is a contraction (uniformly in \(\tau\)): \(\|V_\tau^m\|<\theta<1\). We shall assume that this already holds for \(m=1\); this case can be attained by introducing the substitution \(T\to mT\).

Let \(\rho(x_0,C)<\delta\). Then for some \(\tau_0\), \(0\le \tau_0\le T\), we have

\[ \|x_0-q_{\tau_0}\|<\delta . \tag{6} \]

Define sequences of time instants \(\tau_n\) and elements \(a_n\) of the space \(X\) by setting

\[ \tau_{n+1}=\tau_n+s_n,\qquad s_n=(a_n,\psi_{\tau_n}),\qquad a_n=x_n-q_{\tau_n},\quad n=0,1,\ldots \tag{7} \]

We shall show that, if \(\delta\) is sufficiently small, then the estimates

\[ \rho(Q_T^n x_0,C)\le \|a_n\|\le \theta^n\delta,\qquad |s_n|\le l\theta^n\delta,\qquad l=\max\|\psi_\tau\| \tag{8} \]

hold.

For \(n=0\), the estimates (8) follow immediately from (6). If the estimates (8) have already been proved for \(n=k\), then for \(n=k+1\) we derive them, using conditions (1) and (2), from the relation

\[ \rho(x_{k+1},C)\le \|a_{k+1}\| =\|V_{\tau_k}a_k+\Delta_1(\tau_k,a_k)+\Delta_2(\tau_k,s_k)\|. \]

Here it is sufficient to choose \(\varepsilon_1,\varepsilon_2,\delta\) so small that the inequalities

\[ \max_\tau\|V_\tau\|+\varepsilon_1+l\varepsilon_2<\theta,\qquad \delta<\delta_1,\qquad l\delta<\delta_2 \tag{9} \]

are satisfied.

It is now not difficult to establish that

\[ \|Q_t x_0-q_{h+t}\|\to 0\quad (t\to+\infty),\qquad h=\tau_0+\sum_{n=0}^{\infty}s_n, \tag{10} \]

which completes the proof.

Theorem 2. Let \(C\) be a smooth cycle of the dynamical system \((Q_t,X)\). Let the mapping \(Q_t\) be differentiable with respect to \(t\), and let the derivative \(\dfrac{d}{dt}Q_t=\dot Q_t\) be continuous in \((x,t)\): \(x\in\omega_n\), \(0\le t<t_0(x)\). Let the spectrum of the monodromy operator have the form

\[ \sigma(U_{T,\tau})=\sigma_1(U_{T,\tau})\cup\sigma_2(U_{T,\tau});\quad |\sigma_1(U_{T,\tau})|>\beta>1,\quad |\sigma_2(U_{T,\tau})|\le 1 . \tag{11} \]

Then the cycle \(C\) is unstable.

Proof. Let \(\psi\in X^{*}\) and \((\varphi_0,\psi)=1\) (it is easy to prove that \(\varphi_0=\dot q_0\ne 0\)). Consider in the space \(X\) the plane \(X_0=\{a\in X:(a,\psi)=0\}\) and the hyperplane \(\Gamma=\{x:x=q_0+a;\ a\in X_0\}\). Define a mapping \(K\) of a neighborhood of zero in the space \(X_0\) into \(X_0\) by setting

\[ Ka=Q_{t_*}(q_0+a)-q_0. \tag{12} \]

Here \(t_*=t_*(x)\) is the instant of the first return of the trajectory \(Q_t x\), \(x\in\Gamma\), to the hyperplane \(\Gamma\). To prove the existence of \(t_*\), it is sufficient to apply the implicit-function theorem to the equation

\[ F(t,a)\equiv (Q_t(q_0+a)-q_0,\psi)=0. \tag{13} \]

Indeed, \(F(T,0)=0\), \(F_t(T,0)=(\dot q_0,\psi)=1\), and the function \(F\) is continuously differentiable in a neighborhood of the point \((T,0)\in R\times X_0\).

The operator \(K\) is continuously differentiable in a neighborhood of zero, and

\[ K'(0)a=U_{\tau,0}a-\varphi_0(U_{\tau,0}a,\psi),\qquad a\in X_0. \tag{14} \]

The spectrum of the operator \(K'(0)\) obviously contains the set \(\sigma_1(U_{\tau,\tau})\). Now Theorem 2 is easily derived from Lemma 5 of paper \((^3)\), if one further notes that for points \(x=q_0+a,\ a\in X_0;\ \|a\|<\varepsilon\), for sufficiently small \(\varepsilon>0\), there exists a constant \(\gamma>0\) such that \(\rho(x,C)\geq \gamma\|a\|\).

2. Application to the Navier—Stokes equations. Let a viscous incompressible homogeneous fluid fill a three-dimensional bounded domain \(\Omega\) with boundary \(S\) of class \(C^2\). Let the body forces and the boundary value of the velocity be prescribed and independent of time. Then the Navier—Stokes equations and boundary conditions have the form

\[ v_t+(v,\nabla)v-\nu\Delta v=-\nabla P+F(x),\qquad x\in\Omega; \tag{15} \]

\[ \operatorname{div}v=0; \tag{16} \]

\[ v|_S=a(x). \tag{17} \]

Suppose that there exists a (sufficiently smooth) \(T\)-periodic in time \(t\) self-oscillatory solution of system (15)—(17), with velocity vector \(v_0(x,t)\) and pressure \(P_0(x,t)\). We shall be interested in its stability with respect to perturbations from the Hilbert space \(H_1\). The latter is the closure of the set of smooth solenoidal vector fields vanishing on the boundary in the metric

\[ (u,v)_{H_1}=\int_{\Omega}\sum_{k=1}^3 \frac{\partial u}{\partial x_k}\frac{\partial v}{\partial x_k}\,dx. \tag{18} \]

We shall call the cycle \(C=\bigcup_t v_0(\cdot,t)\) (asymptotically) stable in \(H_1\), if for every \(\varepsilon>0\) there corresponds a \(\delta>0\) such that

\[ \rho(v(\cdot,t),C)=\inf_\tau\|v(\cdot,t)-v_0(\cdot,\tau)\|_{H_1}<\varepsilon \]

for any solution of system (15)—(17) with velocity vector \(v\), provided that \(\rho(v(\cdot,0),C)<\delta\).

The stability spectrum \(\Sigma(v_0)\) \((^3)\) is the set of those \(\sigma\) for which there exists a nonzero \(T\)-periodic solution of the linearized system

\[ u_t+\sigma u+(v_0,\nabla)u+(u,\nabla)v_0-\nu\Delta u=-\nabla q,\qquad \operatorname{div}u=0,\qquad u|_S=0. \tag{19} \]

Obviously, \(\sigma_k=-2k\pi i/T\in\Sigma(v_0)\) \((k=0,\mp1,\ldots)\): the corresponding solution of system (19) is \(v_{0t}\exp\sigma_k t,\ P_{0t}\exp\sigma_k t\). Applying Theorems 1 and 2, we arrive at the following conclusions.

Theorem 3. Let \(\sigma_0=0\) be a simple eigenvalue of system (19), and let all points of the stability spectrum \(\Sigma(v_0)\), except \(\sigma_k:\ k=0,\mp1,\ldots\), lie inside the left half-plane. Then the cycle \(C\) is asymptotically stable in \(H_1\).

Theorem 4. If the stability spectrum \(\Sigma(v_0)\) contains at least one point \(\sigma\) with \(\operatorname{Re}\sigma>0\), then the cycle \(C\) is unstable.

For examples of self-oscillatory regimes, see \((^{4-7})\).

Rostov State University

Received
20 VIII 1970

REFERENCES

\(^1\) A. Andronov, A. Witt, ZhETF, 3, issue 3 (1933).
\(^2\) B. P. Demidovich, Lectures on the Mathematical Theory of Stability, “Nauka,” 1967.
\(^3\) V. I. Yudovich, DAN, 195, No. 2 (1970).
\(^4\) N. N. Brushlinskaya, DAN, 157, No. 5 (1964).
\(^5\) N. N. Brushlinskaya, DAN, 162, No. 4 (1965).
\(^6\) V. I. Yudkovich, PMM, 29, No. 3 (1965).
\(^7\) V. I. Yudovich, Abstracts of brief scientific communications, Section 12, International Mathematical Congress, Moscow, 1966.