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UDC 517.9:533.7
MATHEMATICS
V. I. YUDOVICH
ON THE STABILITY OF FORCED OSCILLATIONS OF A FLUID
(Presented by Academician A. N. Kolmogorov on 27 VIII 1970)
This note gives a justification of the linearization method in the problem of stability of periodic motions of a viscous incompressible fluid. At the same time, a new method of proof is obtained for the case of stationary flows \((^{1})\).
1. Problem with initial data. Let the Navier—Stokes equations in a bounded three-dimensional domain \(\Omega\) with boundary \(S\) of class \(C^2\), under a given external body force and boundary velocity that are \(T\)-periodic in time, have a \(T\)-periodic-in-time solution* with velocity vector \(v_0(x,t)\). Seeking an arbitrary flow \(v\) in the form \(v=v_0+u\), we arrive at the nonlinear equation of perturbations
\[ \frac{du}{dt}+\nu A_0u+B(t)u=Ku, \tag{1} \]
where the following notation is used:
\[ A_0u=-\Pi\Delta u;\qquad Ku=-K_0(u,u);\quad K_0(u,v)=\Pi(u,\nabla)v, \]
\[ K_0^{0}(u,v)=K_0(u,v)+K_0(v,u);\qquad B(t)u=K_0^{0}(u,v_0). \]
Here \(\Pi\) is the operator of orthogonal projection of the vector space \(L_2\) onto the space \(H\subset L_2\), obtained by closing the set of finite smooth solenoidal vectors in \(\Omega\).
We shall regard the operators \(A_0, B, K\) as defined on the set \(D_{A_0}\) of solenoidal vectors of class \(W_2^{(2)}\) that vanish on the boundary \(S\). The operator \(A_0\) is self-adjoint and positive definite in \(H\). We denote its energy space by \(H_1\):
\[ (u,v)_{H_1}=(A^{1/2}u,A^{1/2}v)_H. \tag{2} \]
Introduce the set \(M_T\) of vector-functions \(u(t)\) of time \(t\in[0,T]\) with values in \(H\) such that \(u(t)\in D_{A_0}\) for all \(t\in[0,T]\) and the vector-functions \(u(t), A_0u(t)\) have strong derivatives of all orders with respect to \(t\) in \(H\). Define the Hilbert space \(H_2^T\) as the closure of the set \(M_T\) in the metric generated by the scalar product
\[ (u,v)_{H_2^T}=\int_0^T\left[(du/dt,dv/dt)_H+(A_0u,A_0v)_H\right]\,dt+(u(T),v(T))_{H_1}. \tag{3} \]
Lemma 1. The vector-function \(u\in H_2^T\) is strongly continuous in \(H_1\) for \(t\in[0,T]\). Moreover, the estimates
\[ \|u(t)\|_{H_1}\le c\|u\|_{H_2^T};\qquad 0\le t\le T; \tag{4} \]
\[ \int_0^T \|u(t)\|_{L_p}^{4p/(p-6)}\,dt \le c\|u\|_{H_2^T}^{4p/(p-6)}; \qquad \int_0^T \|D_xu(t)\|_{L_q}^{4q/3(q-2)}\,dt \le c\|u\|_{H_2^T}^{4q/3(q-2)}, \tag{5} \]
hold, where \(c\) depends only on the domain \(\Omega\), \(6\le p<\infty\), \(2<q\le6\).
* The existence theorem for a periodic motion is formulated in \((^{2})\).
These inequalities are easily derived with the aid of S. L. Sobolev’s embedding theorems.
Consider the Cauchy problem for equation (1) with the initial condition
\[ u(0)=a,\qquad a\in H_1. \tag{6} \]
By its solution on the time interval \([0,T]\) we shall mean a vector function \(u\in H_2^T\) satisfying equation (1) for almost all \(t\in[0,T]\) and the initial condition (6)—in the sense that
\[ \|u(t)-a\|_{H_1}\to 0\qquad (t\to 0). \tag{7} \]
Similarly, we define the solution of the Cauchy problem for the linearized equation
\[ du/dt+A_0u+B(t)u=0. \tag{8} \]
Let \(\widetilde H_2^T\) denote the closure of the set of smooth solenoidal vectors \(v(x,t)\) \((x\in\Omega,\ t\in[0,T])\) with respect to the norm
\[ \|v\|_{\widetilde H_2^T}^{2} = \int_0^T \left[ \|\partial v/\partial t\|_{L_2(\Omega)}^2 + \|v\|_{W_2^{(2)}(\Omega)}^2 \right]dt. \tag{9} \]
Lemma 2. Let \(v_0\in\widetilde H_2^T\). Then, for any \(a\in H_1\), the problems (1, 6) and (8, 6) are uniquely solvable.
This lemma is not difficult to prove by the method of successive approximations, regarding the terms \(B(t)u\) and \(Ku\) as perturbations. The necessary estimates follow from Lemma 1 and the identity
\[ \|u(t)\|_{H_1}^{2} + \int_0^t \left( \left\|\frac{du}{dt}\right\|_H^2 + \|A_0u\|_H^2 \right)d\tau = \|a\|_{H_1}^{2} + \int_0^T \|f(\tau)\|_H^2\,d\tau; \]
\[ f=\frac{du}{dt}+A_0u;\qquad a=u(0). \tag{10} \]
Introduce the operators \(N_t,\ U_t\) of displacement along the trajectories of equations (1) and (8), putting, for any \(a\in H_1\),
\[ N_ta=u(t);\qquad U_ta=\widetilde u(t), \tag{11} \]
where \(u,\widetilde u\) are the solutions of the Cauchy problem (1, 6) and the problem (8, 6).
Lemma 3. The operator \(N_t:H_1\to H_1\) \((0\le t\le T)\) is defined in a neighborhood of zero in the space \(H_1\), is completely continuous and continuously differentiable. Its Fréchet differential at the point \(0\) is \(U_t\). The operator \(U_t\) is also completely continuous in \(H_1\).
2. Stability condition. We shall say that the solution \(v_0\) is Lyapunov stable in the space \(H_1\) if problem (1, 6), for every \(a\) from some neighborhood of zero in the space \(H_1\), has a solution on the time interval \([0,\infty)\), and for every \(\varepsilon>0\) one can indicate a \(\delta>0\) such that from the condition \(\|a\|_{H_1}<\delta\) it follows that \(\|N_ta\|_{H_1}<\varepsilon\) for \(t\ge0\). If, in addition, \(\|N_ta\|_{H_1}\to0\) \((t\to\infty)\), then we shall say that the flow \(v_0\) is asymptotically stable in \(H_1\).
Seeking a solution of equation (6) in the form \(u(t)=e^{\sigma t}w(t)\), where \(w(t)\) is a \(T\)-periodic vector function, we arrive at the problem
\[ \frac{dw}{dt}+\sigma w+A_0w+B(t)w=0;\qquad w(t+T)\equiv w(t). \tag{12} \]
The set of complex numbers \(\sigma\) for which problem (12) has a nonzero solution is called the stability spectrum of the flow \(v_0\) and is denoted by \(\Sigma(v_0)\).
If \(\sigma\in\Sigma(v_0)\), then \(\rho=e^{T\sigma}\) is an eigenvalue of the monodromy operator \(U_T\); conversely, if \(\rho\) is an eigenvalue of the operator \(U_T\), then
\[ \sigma_k=\frac{1}{T}\ln\rho+\frac{2\pi}{T}K_i\in\Sigma(v_0),\qquad k=0,\mp 1,\ldots \]
From the complete continuity of the operator \(U_T\) it follows that it can have no more than a finite number of eigenvalues with modulus greater than 1. Therefore the stability spectrum contains no more than a finite number of eigenvalues \(\sigma\) with positive and distinct real parts.
Theorem 1. Let the stability spectrum of the periodic flow \(v_0\) lie in the left half-plane
\[ \operatorname{Re}\sigma<-\sigma_0<0;\qquad \sigma\in\Sigma(v_0). \tag{13} \]
Then the flow \(v_0\) is asymptotically stable in \(H_1\). Moreover, for every solution of the Cauchy problem (1,6), for sufficiently small \(\|a\|_{H_1}\), the estimates
\[ \|u(t)\|\leq Ce^{-\sigma_0 t}\|a\|_{H_1};\qquad \int_0^t e^{2\sigma_0\tau}\left(\left\|\frac{du}{dt}\right\|_H^2+\|A_0u\|_H^2\right)d\tau \leq C^2\|a\|_{H_1}^2. \tag{14} \]
hold.
The proof of this theorem is obtained by applying to the operator \(N_T=N\) the following lemma.
Lemma 4. Let \(N\) be a continuously differentiable operator mapping a neighborhood of zero of the \(B\)-space \(X\) into \(X\). Let \(N(0)=0\), \(N'(0)=U\), and let the spectrum of the operator \(U\) be contained inside the unit disk
\[ |\sigma(U)|<\rho<1. \tag{15} \]
Then \(\|N^n x_0\|\to 0\) \((n\to\infty)\), if \(\|x_0\|\) is sufficiently small. Moreover, the estimate
\[ \|N^n x_0\|\leq C\rho^n\|x_0\| \tag{16} \]
holds.
3. Condition of instability. Theorem 2. Let the stability spectrum \(\Sigma(v_0)\) contain at least one eigenvalue \(\sigma_0\) with positive real part. Then the flow \(v_0\) is unstable in \(H_1\).
This theorem is derived from the following lemma.
Lemma 5. Let \(N,U\) be the same as in Lemma 4. Let the spectrum of the operator \(U\) be represented as the union of nonintersecting closed sets \(\sigma_1(U)\) and \(\sigma_2(U)\), where
\[ |\sigma_1(U)|>1;\qquad |\sigma_2(U)|\leq 1. \tag{17} \]
Then there exists \(\varepsilon_0>0\) such that, for any \(\delta>0\), one can specify a vector \(a\in X\) and a natural number \(n\) for which
\[ \|a\|<\delta;\qquad \|N^n a\|\geq \varepsilon_0. \tag{18} \]
4. Conditional stability. Theorem 3. Let \(\Sigma(v_0)\) contain no points of the imaginary axis. Then, in a neighborhood of zero of the space \(H_1\), there are defined a finite-dimensional manifold \(Y_1\) and a manifold of finite codimension \(Y_2\), possessing the following properties. 1) If \(a\in Y_2\), then \(N_t a\to 0\) in \(H_1\) as \(t\to+\infty\). 2) If \(\|a\|_{H_1}\) is small and \(a\notin Y_2\), then there exists \(t>0\) such that \(\|N_t a\|_{H_1}\geq\varepsilon_0\); \(\varepsilon_0>0\) does not depend on \(a\). 3) If \(a\in Y_1\), then the Cauchy problem (1,6) has a solution \(N_t a\), defined for all \(t<0\), and \(\|N_t a\|_{H_1}\to 0\) as \(t\to-\infty\). 4) If \(a\notin Y_1\) and \(\|a\|_{H_1}\) is small, then either \(\|N_t a\|_{H_1}\geq 0\) for some \(t<0\), or \(N_t a\) is not defined for some \(t<t_0\leq 0\).
This theorem follows from the following lemma.
Lemma 6. Let \(N, U\) be the same as in Lemma 4, and suppose the spectrum of the operator \(U\) can be represented as the union of closed sets \(\sigma_1(U)\) and \(\sigma_2(U)\), with
\[ |\sigma_1(U)|>1;\qquad |\sigma_2(U)|<1. \tag{19} \]
Then, in some neighborhood \(D_r=\{x\in X:\|x\|<r\};\ r>0\) of zero in the space \(X\), there are manifolds \(Y_1\) and \(Y_2\), invariant with respect to the operator \(N\), which at zero are tangent respectively to the invariant subspaces \(X_1, X_2\) of the operator \(U\) corresponding to the spectral sets \(\sigma_1(U)\) and \(\sigma_2(U)\). Moreover: 1) \(\|N^n x\|\to0\) if \(x\in Y_2\); 2) for every \(x\in D_r-Y_2\) there exists a natural number \(n\) such that \(\|N^n x\|\ge r\); 3) for every \(x\in Y_1\) the preimage \(N^{-1}x\) is uniquely determined, and \(\|N^{-n}x\|\to0\) as \(n\to+\infty\); 4) if \(x\in D_r-Y_1\), then either, starting from some \(n\), the element \(N^{-n}x\) is not defined, or \(\|N^{-n}x\|\ge r\) for some \(n\).
The proof of this lemma is close to that given in \((^3)\) for the finite-dimensional case.
- Examples. 1) If the velocity \(v_0\) is small: \(\|v_0\|_{L_3(\Omega)}<Cv\) (\(C\) is an absolute constant), then the flow \(v_0\) is stable. 2) The conditions of Theorems 1 and 2 are preserved under a small perturbation of the basic flow. Therefore, from each example of a stable or unstable stationary flow \((^{4-11})\) one obtains corresponding examples of periodic flows.
Rostov State University
Received
20 VIII 1970
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