Inertial Manifolds and Spectral Gap Properties for Wave Equations with Weak and Strong Dissipation

Abstract

Sufficient conditions for the existence of an inertial manifold for the equation $u_{tt}-2\gamma _{s} \varDelta u_t +2\gamma _{w} u_t - \varDelta u = f(u)$, $\gamma _{s} > 0$, $\gamma _{w} \ge 0$ are found. The nonlinear function $f$ is supposed to satisfy Lipschitz property. The proof is based on construction of a new inner product in the phase space in which the conditions of a general theorem on the existence of inertial manifolds for an abstract differential equation in a Hilbert space are satisfied.

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1 Introduction

In the theory of nonlinear evolution partial differential equations, great attention is paid to long-time behavior of dynamic systems. Some way of such description relates with notion of an inertial manifold (see [5, 6, 9]).

Let us consider an initial-value problem for an abstract differential equation in a Hilbert space,

$$\begin{aligned} \frac{d}{dt}y+&\mathbf {A}y=F(y), \qquad y\in \fancyscript{H}, \end{aligned}$$

(14.1)

$$\begin{aligned}&y\big |_{t=0} = y_0 \in \fancyscript{H}. \end{aligned}$$

(14.2)

Here $\mathbf {A}$ is a linear operator and ${F}$ is a nonlinear operator. Suppose problem (14.1), (14.2) has a unique solution $y$ for any $y_0\in \fancyscript{H}$. Hence, this problem generates a continuous semigroup $\{S(t) \mid t\ge 0\}$, acting in the space $\fancyscript{H}$ by the formula $S(t)y_0=y(t)\in \fancyscript{H}$.

Definition 14.1.

A Lipschitz finite dimensional manifold $\fancyscript{M} \subset \fancyscript{H}$ is an inertial manifold for the semigroup $S(t)$ if it is invariant (i.e., $S(t)\fancyscript{M} = \fancyscript{M}$, $\forall t\ge 0$) and it satisfies the following asymptotic completeness property:

$$\begin{aligned} \forall y_0\in \fancyscript{H} \exists \tilde{y}_0\in \fancyscript{M} \text { such that } \Vert S(t)y_0 - S(t)\tilde{y}_0\Vert _\fancyscript{H} \le q(\Vert y_0\Vert _\fancyscript{H}) e^{-ct}, t\ge 0, \end{aligned}$$

where the positive constant $c$ and the monotonic function $q$ are independent of $y_0$.

Inertial manifolds enable one to reduce the study of the behavior of an infinite-dimensional dynamical system to the investigation of this problem for some finite-dimensional dynamical system generated by original system on an inertial manifold.

For the abstract equation of the form (14.1), there are known sufficient conditions under which there is an inertial manifold in the Hilbert space $\fancyscript{H}$ (see [3]). Let us present these conditions. Let $\mathbf {A}$ be a linear closed (possibly unbounded) operator with dense domain $\fancyscript{D}(\mathbf {A})$ in $\fancyscript{H}$ and let the spectrum $\sigma (\mathbf {A})$ of $\mathbf {A}$ be disjoint from the strip $\{m<\mathfrak {R}\zeta < M\}$, where $M\ge 0$, $M>m$. Denote by $P$ the orthogonal projection to the invariant subspace of $\mathbf {A}$ corresponding to the part of the spectrum $\sigma \cap \{\mathfrak {R}\zeta \le m\}$ and write $Q=\mathrm{Id}-P$. Assume that the space $P(\fancyscript{H})$ is finite-dimensional.

Theorem 14.1.

Let the space $\fancyscript{H}$ be equiped with an inner product in such a way that the space $P(\fancyscript{H})$ and $Q(\fancyscript{H})$ are orthogonal and the following relations hold:

$$\begin{aligned} \begin{array}{l l} (\mathbf {A}y,y)\le m |y|^2 &{} \quad \forall y\in P(\fancyscript{H}),\\ (\mathbf {A}y,y)\ge M |y|^2 &{} \quad \forall y\in Q(\fancyscript{H})\cap \fancyscript{D}(\mathbf {A}). \end{array} \end{aligned}$$

(14.3)

Moreover, let $F(y)$ be a nonlinear function such that $F(0) = 0$ and let $F$ satisfy the Lipschitz condition with the constant $L$, where

$$\begin{aligned} 2L < M-m.\end{aligned}$$

(14.4)

In this case, there is an inertial manifold $\fancyscript{M}$ in the Hilbert space $\fancyscript{H}$, and this manifold is the graph of a Lipschitz continuous function $\varPhi$: $P(H)\rightarrow Q(H)$.

In the present chapter, an initial-boundary value problem for a wave equation with weak and strong dissipation is considered. The nonlinear term depends on the unknown function $u$, these term is assumed to be Lipschitzian,

$$ u_{tt}- 2\gamma _s\varDelta u_t + 2\gamma _w u_t -\varDelta u = f(u). $$

For this equation, we obtain a condition on the Lipschitz constant of the function $f$ which ensures the existence of an inertial manifold. The result is stated in Theorems 14.2 and 14.3. The proof is based on construction of a new inner product in the phase space in which the conditions of Theorem 14.1 are satisfied.

2 Statement of the Problem and Spectrum of the Linear Operator

In a bounded domain $\varOmega$, we consider the inertial-boundary value problem for a wave equation with dissipation,

$$\begin{aligned}&u_{tt}-2\gamma _s\varDelta u_t+2\gamma _w u_t-\varDelta u = f(u), \qquad u|_{\partial \varOmega }=0,\end{aligned}$$

(14.5)

$$\begin{aligned}&u|_{t=0} = u_0(x)\in H_0^1(\varOmega ),\qquad u_t|_{t=0} = p_0\in L_2(\varOmega ). \end{aligned}$$

(14.6)

Here $\gamma _w$ and $\gamma _s$ are positive coefficients of the dissipation, and the nonlinear function $f$ is continuously differentiable and satisfy the global Lipschitz condition,

$$\begin{aligned} |f(v_1)-f(v_2)| \le l|v_1-v_2|\qquad \forall v_1,v_2\in \mathbb {R}, \end{aligned}$$

(14.7)

Moreover, let $f(0)=g(0)=0$.

Under these assumptions, problem (14.5), (14.6) has a unique weak solution $u\in {C}\big ([0,T];H_0^1(\varOmega )\big )$, $\partial _t u\in {C}\big ([0,T];L_2(\varOmega )\big )$ for any $T>0$ (see [7, 8, 10]). Hence, this problem generates a continuous semigroup $\{S(t)\}$, $t\ge 0$, acting in the phase space $\fancyscript{H}=H_0^1(\varOmega )\times L_2(\varOmega )$ by the formula

$$ S(t)(u_0(x),p_0(x))=y(t)\equiv (u(t,x),p(t,x))\in H, $$

where $u(t,x)$ is a solution of the problem (14.5), (14.6), $p(t,x)=\partial _t u(t,x)$ stands for the derivative of this solution w.r.t. $t$, and $y=(u,p)\in \fancyscript{H}$.

Let us represent the initial-boundary value problem in the form of an ordinary differential equation to find the unknown vector function $y=(u,p)\in \fancyscript{H}$,

$$\begin{aligned} \frac{d}{dt}y(t)+\mathbf {A}y=F(y), \quad \mathbf {A}y=\begin{pmatrix}0&{}-1 \\ -\varDelta &{}2\gamma _w-2\gamma _s\varDelta \end{pmatrix}y, \quad F(y)=\begin{pmatrix} 0\\ f(u)\end{pmatrix}. \end{aligned}$$

Let $e_k(x)$ and $\lambda _k$ be the eigenfunctions and the eigenvalues of the operator $-\varDelta$ in the domain $\varOmega$ with the Dirichlet conditions on the boundary,

$$\begin{aligned} -\varDelta e_k (x)&=\lambda _k e_k(x),\quad e_k(x)\big |_{\partial \varOmega }=0, \quad e_k(x)\not \equiv 0,\\&0<\lambda _1<\lambda _2\le \lambda _{3}\le \dots \rightarrow +\infty . \end{aligned}$$

Denote by $(\cdot ,\cdot )_\fancyscript{H}$ and $\Vert \cdot \Vert$ the standard inner product and the corresponding norm in the space $\fancyscript{H}$, namely,

$$ (y, \tilde{y})_\fancyscript{H}=(\nabla u, \nabla \tilde{u}) +(p,\tilde{p}) =\sum _{k=1}^\infty \left( \lambda _k u_k\tilde{u}_k+p_k\tilde{p}_k\right) , $$

where $u_k=(u,e_k)$, $p_k=(p,e_k)$, and $(\cdot ,\cdot )$ stands for the inner product in $L_2(\varOmega )$.

The two-dimensional subspace $\fancyscript{H}_k$ with basis $(e_k,0)$, $(0,e_k)$ is invariant under the operator $\mathbf {A}$. The restriction of the operator $\mathbf {A}$ to the subspace $\fancyscript{H}_k$ has the matrix $A_k=\left( \begin{array}{cc} ~0~ &{} -1\\ \lambda _k &{} 2(\gamma _w+\gamma _s \lambda _k) \end{array} \right)$. The eigenvalues of $A_k$ are equal to

$$\begin{aligned} \mu _k=\gamma _k - \sqrt{\gamma _k^2-\lambda _{k}}\quad \text {and} \quad \nu _k=\gamma _k + \sqrt{\gamma _k^2-\lambda _{k}} \end{aligned}$$

where we denote $\gamma _k = \gamma _w+\gamma _s\lambda _k$. In Figs. 14.1 and 14.2, we show the qualitative displacement of these eigenvalues on the complex plane in two cases, namely, $4\gamma _w\gamma _s<1$ and $4\gamma _w\gamma _s\ge 1$. In the first case, the operator $A$ has both real and nonreal eigenvalues and, in the other case, all eigenvalues are real.

Fig. 14.1

$4\gamma _w\gamma _s<1$

Fig. 14.2

$4\gamma _w\gamma _s>1$

If the orthogonal projection $P$ satisfies the assumptions of the Theorem 14.1, then the image $P(\fancyscript{H})$ (which is finite-dimensional) must correspond to finitely many eigenvalues of $\mathbf {A}$ belonging to the domain $\{ \mathrm{Re}\zeta \le m\}$. However, $\mu _k\rightarrow 1/(2\gamma _s)$ and $\nu _k\rightarrow +\infty$ as $\lambda _k\rightarrow +\infty$, and thus the quantity $m$ must be less than $1/(2\gamma _s)$. In the case $4\gamma _w\gamma _s < 1$, to the values $\mu _k$ and $\nu _k$ lying to the left of the accumulation point $1/(2\gamma _s)$ there correspond values $\lambda _k <\frac{1-2\gamma _w\gamma _s}{2\gamma _s^2}$. If $4\gamma _w\gamma _s \ge 1$, then $\mu _k < 1/(2\gamma _s)$ for any $k$.

3 Sufficient Conditions for the Existence of Inertial Manifolds

In this section, we present conditions for the existence of a gap both in the real part (Theorem 14.2) and in the nonreal part (Theorem 14.3) of the spectrum of $\mathbf {A}$.

First let us consider a gap in the real part of the spectrum. Thus, for $4\gamma _w\gamma _s < 1$, the additional condition $m < \frac{1-\sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s}$ is imposed, which corresponds to the inequality $\lambda _k < \frac{1 - 2\gamma _w\gamma _s - \sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s^2}$.

Remark 14.1.

If Eq. (14.5) has not strongly dissipative term (i.e., $\gamma _s=0$), then the circle to which a part of eigenvalues of the operator $\mathbf {A}$ belongs (see Fig. 14.1) is transformed to the vertical line $\{\mathfrak {R}\zeta = \gamma _w\}$ (see Fig. 14.3), and the condition on $m$ becomes $m < \gamma _w$.

Fig. 14.3

Weak dissipation, $\gamma _s=0$

Write

$$\begin{aligned} \gamma _\star = {\left\{ \begin{array}{ll} \gamma _1, &{}\text { if } 1\le 2\gamma _s\gamma _1; \\ 1/(2\gamma _s), &{}\text { if } 2\gamma _s\gamma _1\le 1 \le 2\gamma _s\gamma _{N+1}; \\ \gamma _{N+1}, &{}\text { if } 2\gamma _s\gamma _{N+1}\le 1; \end{array}\right. } \quad \lambda _\star = \frac{\gamma _\star -\gamma _w}{\gamma _s}. \end{aligned}$$

Theorem 14.2.

Let $f$ satisfy condition (14.7). Moreover, suppose that there is an $N$ such that the following inequality holds:

$$\begin{aligned} 2\frac{l}{\sqrt{\gamma _\star ^2-\lambda _\star }} < \mu _{N+1}-\mu _N= \gamma _{N+1} - \sqrt{\gamma _{N+1}^2-\lambda _{N+1}}-\gamma _N + \sqrt{\gamma _N^2-\lambda _N}, \end{aligned}$$

(14.8)

and, if $4\gamma _w\gamma _s < 1$, then the following inequality also holds:

$$\begin{aligned} \lambda _{N+1}<\frac{1-2\gamma _w\gamma _s- \sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s^2}. \end{aligned}$$

In this case, there is an $N$-dimensional inertial manifold for problem (14.5), (14.6) in the space $\fancyscript{H}$.

Remark 14.2.

If $\gamma _s = 0$, then condition (14.8) coincides with the similar condition obtained in [4].

Remark 14.3.

If there is no weak dissipation, then all real point of the spectrum of the operator $\mathbf {A}$ are located to the right of the number $1/(2\gamma _s)$ (see Fig. 14.4), and Theorem 14.2 cannot be applied to this situation.

Fig. 14.4

Strong dissipation, $\gamma _w=0$

Now we consider case of spectral gap in nonreal part of spectrum. Hence we assume that $4\gamma _w\gamma _s < 1$.

Let values $m$ and $M$ be chosen in such a way that

$$\begin{aligned} \frac{1-\sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s}\le m < M \le \frac{1}{2\gamma _s}, \end{aligned}$$

(14.9)

and the spectrum $\sigma (\mathbf {A})$ of $\mathbf {A}$ be disjoint from the strip $\{m < \mathfrak {R}\zeta < M\}$, but the set $\sigma (\mathbf {A})\cap \{\mathfrak {R}\zeta \le m\}$ is not empty.

Fig. 14.5

A spectral gap in nonreal part of the spectrum

Let numbers $k_1$, $k_2$ are such that values $\nu _{k_1}$ and $\nu _{k_2+1}$ belong to the domain $\{\mathfrak {R}\zeta \ge M\}$, and numbers $\nu _{k_1+1}$ and $\nu _{k_2}$ belong to the domain $\{\mathfrak {R}\zeta \le m\}$ (see Fig. 14.5). Thus for $\nu _1 \not \in \mathbb {R}$ or $\nu _1\in \mathbb {R}$, $\nu _1\le m$ we have $k_1 = 0$; for the converse case we get $\mathfrak {R}\nu _{k_1+1} \le m < M \le \mathfrak {R}\nu _{k_1}$.

If there are not numbers $\nu _k$ to the left of the strip, then we have $M\le \mathfrak {R}\nu _{k_2+1}$ and $M\le \mathfrak {R}\nu _{k_1}=\nu _{k_1}=\nu _{k_2}$. Otherwise number $k_2$ is such that $\mathfrak {R}\nu _{k_2} \le m < M \le \mathfrak {R}\nu _{k_2+1}$.

Denote numbers $\varkappa _I$, $\varkappa _{II}$, $\varkappa _{III}$ and $\varkappa _{IV}$. First if $k_1 = 0$ then formally write $\varkappa _I = +\infty$. In the other case write $\varkappa _I = \sqrt{\gamma _{k_1}^2-\lambda _{k_1}}$. Secondly if $k_2 = k_1$ then formally write $\varkappa _{II} = \varkappa _{III} = +\infty$. Otherwise denote $\varkappa _{II} = s_{k_1+1}$, $\varkappa _{III} = s_{k_2}$, where

$$\begin{aligned} s_k = \sqrt{m^2-2m\gamma _k+\lambda _k} + m-\gamma _k. \end{aligned}$$

(14.10)

Finally write $\lambda _M = (M-\gamma _w)/\gamma _s$ and $\varkappa _{IV} = \sqrt{\lambda _M - M^2}$.

Theorem 14.3.

Let nonlinear function $f$ satisfies condition (14.7). Moreover, suppose that the following inequality holds:

$$\begin{aligned} 2l < (M-m)\min \{\varkappa _{I}, \varkappa _{II}, \varkappa _{III}, \varkappa _{IV}\}. \end{aligned}$$

(14.11)

Then there is a $(2k_2-k_1)$-dimensional inertial manifold for problem (14.5), (14.6) in the space $\fancyscript{H}$.

Remark 14.4.

It follows from condition (14.11) that there are enough large gaps in the spectrum of operator $-\varDelta$ in domain $\varOmega$. Actually, we have

$$\begin{aligned} \varkappa _{IV}&= \sqrt{\lambda _M - M^2}= \sqrt{\frac{M -\gamma _w-\gamma _s M^2}{\gamma _s}}\,{=}\\&=\sqrt{\frac{4\gamma _s M{-}4\gamma _w\gamma _s{-}4\gamma _s^2 M^2}{4\gamma _s^2}}{=}\!\sqrt{\frac{1-4\gamma _w\gamma _s-(2\gamma _s M-1)^2}{4\gamma _s^2}}\!&\!{<}\frac{\sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s}. \end{aligned}$$

Moreover, the inequalities $\gamma _{k_2}\le m$ and $M\le \gamma _{k_2+1}$ hold by definition of the number $k_2$. Indeed if $\nu _{k_2}\in \mathbb {R}$, then we have $\nu _{k_2}< \frac{1}{2\gamma _s}$, $\gamma _{k_2} < \frac{1-\sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s}\le m$ (see (14.9)); otherwise we have $\gamma _{k_2} = \mathfrak {R}\nu _{k_2}\le m$. Similarly if $\nu _{k_2+1}\in \mathbb {R}$, then we have $\nu _{k_2}>\frac{1}{2\gamma _s}$, $\gamma _{k_2} > \frac{1+\sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s}> M$; otherwise we get $\gamma _{k_2+1} = \mathfrak {R}\nu _{k_2+1}\ge M$.

Thus, by (14.11) it follows the inequality,

$$ 2l < (\gamma _{k_2+1}-\gamma _{k_2}) \frac{\sqrt{1-4\gamma _w\gamma _s}}{2\gamma _s} = (\lambda _{k_2+1}-\lambda _{k_2}) \frac{\sqrt{1-4\gamma _w\gamma _s}}{2}. $$

This means that there are spectral gaps on the order of $l$:

$$ \lambda _{k_2+1}-\lambda _{k_2} > 4l\sqrt{1-4\gamma _w\gamma _s}. $$

The proofs of Theorems 14.2 and 14.3 are based on the construction of a new norm in the phase space $\fancyscript{H}$, in which the assumptions of Theorem 14.1 are satisfied. Note the schemes of the new inner product construction are essentially different for gaps in the real part and in the nonreal part of the spectrum. Then this two cases are considered separately. In the present chapter we prove Theorem 14.3. The proof of Theorem 14.2 presented in [1].

Remark 14.5.

The case of the gap in the nonreal part of the spectrum was partially studied in [2], where a strongly dissipative wave equation (i.e., $\gamma _w = 0$) was considered.

4 Proof of Theorem 14.3

Let us decompose the entire phase space $\fancyscript{H}$ in direct sum of spaces that are pairwise orthogonal, $\fancyscript{H}=\fancyscript{H}_1\oplus \fancyscript{H}_2\oplus \ldots \oplus \fancyscript{H}_{k_2}\oplus \fancyscript{H}_\infty$, where every subspace $\fancyscript{H}_k$, $k=1,\dots ,k_2$, is two-dimensional and corresponds to the eigenvector $e_k$ with respect to $u$ and $p$, and $\fancyscript{H}_\infty =(\fancyscript{H}_1\oplus \fancyscript{H}_2\oplus \ldots \oplus \fancyscript{H}_{k_2})^{\bot }$ is the subspace of codimension $2{k_2}$ which corresponds to the eigenvectors $e_{{k_2}+1},e_{{k_2}+2},\dots$ of the Laplace operator. Note that the spaces $\fancyscript{H}_k$, $k = 1, \ldots , {k_2}$, and $\fancyscript{H}_\infty$ are invariant with respect to the action of the linear operator $\mathbf {A}$.

The new inner product $[\cdot ,\cdot ]$ introduced below preserves the condition that the spaces $\fancyscript{H}_k$, $k=1,\dots , {k_2},\infty$, are pairwise orthogonal and modifies the inner product in each of these subspaces. Thus, if $y=(u,p)\in \fancyscript{H}$ and the orthogonal projections of $y$ to $\fancyscript{H}_k$ are denoted by $y_k=(u_ke_k,p_ke_k)\in \fancyscript{H}_k$, $k=1, \dots , {k_2}, \infty$, then the new norm in $\fancyscript{H}$ is defined by the formula

$$ |||y|||^2=\sum _{k=1}^{k_2}|||y_k|||_k^2+|||y_\infty |||_\infty ^2. $$

4.1 New Norm in the Spaces $\fancyscript{H}_k, k = 1, \ldots ,k_1$

By definition the number $k_1$, for $k = 1, \ldots , k_1$ the eigenvalues $\mu _k$ and $\nu _k$ are real and lie to the different sides of the strip $\{m < \mathfrak {R}\zeta < M\}$. We introduce the new inner product in such a way that the eigenvectors $\xi _k$ and $\eta _k$, which correspond to the eigenvalues $\mu _k$ and $\nu _k$, are orthogonal with respect to this inner product.

Define a new inner product $[\cdot ,\cdot ]_k$ of vectors $y=(u,p)$, $\tilde{y}=(\tilde{u},\tilde{p})$, $y, \tilde{y}\in \fancyscript{H}_k$ by the rule

$$\begin{aligned}{}[y,\tilde{y}]_k = (2\gamma _k^2 -\lambda _k)(u,\tilde{u})+ \gamma _k(u,\tilde{p})+ \gamma _k(p,\tilde{u})+(p,\tilde{p}). \end{aligned}$$

The following assertions hold.

Lemma 14.1.

The eigenvectors $\xi _k$ and $\eta _k$ corresponding to the eigenvalues $\mu _k$ and $\nu _k$, are orthogonal with respect to the new inner product.

Proof.

The eigenvectors of the matrix $A_k$ in the space $\fancyscript{H}_k$ are the vectors $\xi _k = (1, -\mu _k)$ and $\eta _k = (1, -\nu _k)$. It follows from $\mu _{k}+\nu _{k}=2\gamma _{k}$ and $\mu _{k}\nu _{k}=\lambda _{k}$ that

$$\begin{aligned} \qquad \quad \quad [\xi _k, \eta _k]_{k} = 2\gamma _k^2 -\lambda _k - \gamma _k(\mu _k+\nu _k)+\mu _k\nu _k = 0.\qquad \end{aligned}$$

Since $\gamma _k^2 > \lambda _k$ for $k \le k_1$, it follows that the new inner product defines the norm

$$ |||y|||_k^2 =[y,y]_k= (\gamma _k^2 - \lambda _k)\Vert u\Vert ^2+\Vert \gamma _k u+p\Vert ^2. $$

Let us prove that

Lemma 14.2.

The minimum of the function $\kappa _1(\gamma ) =\gamma ^2-\lambda (\gamma )$, where $\lambda (\gamma ) = \frac{\gamma -\gamma _w}{\gamma _s}$, on the interval $\gamma \in [\gamma _1, \gamma _{k_1}]$ is achieved at the point $\gamma = \gamma _{k_1}$.

Proof.

Let us show that the derivative of $\kappa _1(\gamma )$ is negative on the interval $\gamma \in [\gamma _1, \gamma _{k_1}]$. Indeed, by definition of the number $k_1$ we get $\gamma <\gamma _{k_1} < 1/(2\gamma _s)$. Hence for $\gamma < \gamma _{k_1}$ we have

$$\begin{aligned} \gamma _s{\kappa _1}_\gamma ' = 2\gamma \gamma _s-1 < 0. \end{aligned}$$

Thus, the function $\kappa _1(\gamma )$ decreases on the interval $\gamma \in [\gamma _1, \gamma _{k_1}]$, and its minimum is attained at $\gamma = \gamma _{k_1}$.

Since Lemma 14.2 the following estimate of the norm of the vector $y = y_1+ \cdots +y_{k_1}$, $y_k=(u_k, p_k)\in \fancyscript{H}_k$, holds

$$\begin{aligned} |||y|||^2&= \sum _{k=1}^{k_1} |||y_k|||_k^2\ge \sum _{k=1}^{k_1} (\gamma _k^2-\lambda _k) \Vert u_k\Vert ^2 \ge \min _{1\le k\le k_1}&\left\{ \gamma _k^2-\lambda _k\right\} \cdot \sum _{k=1}^{k_1}\Vert u_k\Vert ^2\,{=}\nonumber \\&= (\gamma _{k_1}^2-\lambda _{k_1}) \Vert u\Vert ^2 = \varkappa _I^2 \Vert u\Vert ^2. \end{aligned}$$

(14.12)

4.2 New Norm in the Spaces $\fancyscript{H}_k$, $k = k_1+1, \ldots , k_2$

By definition the numbers $k_1$, $k_2$ for $k = k_1+1,\ldots , k_2$ the eigenvalues $\mu _k$ and $\nu _k$ belong to the domain $\{\mathfrak {R}\zeta < m\}$. In this section, we introduce the new inner product $[\cdot , \cdot ]_k$ in the spaces $\fancyscript{H}_k$, $k = k_1+1, \ldots , k_2$, in such a way that $[\mathbf {A}y,y]_k \le m [y,y]_k$ for any vector $y\in \fancyscript{H}_k$.

Define the new inner product $[\cdot ,\cdot ]_k$ of the vectors $y=(u,p)$, $\tilde{y}=(\tilde{u},\tilde{p})$, $y, \tilde{y}\in \fancyscript{H}_k$ by the rule

$$\begin{aligned}{}[y,\tilde{y}]_k = b_k(u,\tilde{u})+\gamma _k(u,\tilde{p})+ \gamma _k(p,\tilde{u})+(p,\tilde{p}), \end{aligned}$$

where $b_k = \gamma _k^2+s_k^2$ and the numbers $s_k$ are defined in (14.10).

Define the auxiliary function

$$\begin{aligned} s(\gamma ) = \sqrt{m^2-2\gamma m+\lambda (\gamma )}+m-\gamma , \end{aligned}$$

where $\lambda (\gamma ) = (\gamma -\gamma _w)/\gamma _s$. Then $s(\gamma _k) = s_k$. For $\gamma \in [\gamma _{k_1+1}, \gamma _{k_2}]$ the value $s(\gamma )$ is real. Actually, by the choice of $k_1$, $k_2$ we have $m \ge \mathfrak {R}\nu = \mathfrak {R}\left( \gamma +\sqrt{\gamma ^2-\lambda (\gamma )}\right)$ for $\gamma \in [\gamma _{k_1+1}, \gamma _{k_2}]$. Hence $m\ge \gamma$, $m^2-2\gamma m+\lambda (\gamma )\ge 0$.

Since the numbers $s_k$ are real, we see that the inner product defines the norm

$$\begin{aligned} |||y|||_k^2 =[y,y]_k=s_k^2\Vert u\Vert ^2+\Vert \gamma _k u+p\Vert ^2. \end{aligned}$$

The following assertions hold.

Lemma 14.3.

For any vector $y=(u,p)\in \fancyscript{H}_k$, $[\mathbf {A} y, y]_k \le m[y, y]$.

Proof.

Since $\gamma _k = \gamma _w+\gamma _s\lambda _k$, we see that $\mathbf {A} y = (-p, \lambda _k u+2\gamma _k p)$ and

$$\begin{aligned}{}[\mathbf {A} y, y]_k&= -b_k(p, u)-\gamma _k(p, p)+\gamma _k (\lambda _k u+2\gamma _k p, u) + (\lambda _k u+2\gamma _k p, p)\,{=} \\&=\gamma _k\lambda _k \Vert u\Vert ^2 +(2\gamma _k^2-b_k+\lambda _k)(u, p)+\gamma _k\Vert p\Vert ^2. \end{aligned}$$

Then

$$\begin{aligned}{}[\mathbf {A} y, y]_k - m[y,y]_k&= (\gamma _k\lambda _k - m b_k) \Vert u\Vert ^2\,{+} \\&\qquad +(2\gamma _k^2 -b_k+\lambda _k - 2m \gamma _k)(u, p)+(\gamma _k - m)\Vert p\Vert ^2. \end{aligned}$$

Simple monomorphisms can show that the determinant of the last quadratic form is equal to

$$\begin{aligned} D&=(2\gamma _k^2-b_k+\lambda _k - 2m \gamma _k)^2-4(\gamma _k\lambda _k -m b_k)(\gamma _k - m)\,{=}\\&=(b_k-\lambda _k-2(m-\gamma _k)^2)^2 -4(\gamma _k-m)^2(m^2-2\gamma _k m+\lambda _k). \end{aligned}$$

The reader will easily prove that

$$\begin{aligned} b_k-2m^2+2\gamma _k(2m-\gamma _k)-\lambda _k = 2(m-\gamma _k)\sqrt{m^2 - 2\gamma _k m+\lambda _k}. \end{aligned}$$

Thus $D=0$. Moreover, since $\gamma _k-m \le 0$ then the quadratic form $[\mathbf {A} y, y]_k - m[y,y]_k$ is confluent and nonpositive. This completes the proof of the lemma.

Let us show that $\min _{k_1+1\le k\le k_2}\{s_k\} = \min \{s_{k_1+1}, s_{k_2}\}$.

Lemma 14.4.

The minimum of the function $s(\gamma )$ on the closed interval $I=[\gamma _{k_1+1}, \gamma _{k_2}]$ is attained at the ends of the closed interval.

Proof.

The derivative of $s(\gamma )$ is given by

$$\begin{aligned} s'_\gamma =\frac{-2\gamma _s m+1}{2\gamma _s\sqrt{m^2-2\gamma m+\lambda (\gamma )}}-1. \end{aligned}$$

Since $2\gamma _s m < 1$ then $s'_\gamma$ has the same sign as the following expression

$$\begin{aligned} (1-2\gamma _s m)^2&-4\gamma _s^2(m^2-2\gamma m+\lambda (\gamma )) =1-4\gamma _s m+4\gamma _s^2 m^2\,{-}\\ -4\gamma _s^2(m^2-2\gamma m)&-4\gamma _s(\gamma -\gamma _w) =1-4\gamma _s m+4\gamma _s\gamma _w+4\gamma _s(2\gamma _s m-1)\gamma . \end{aligned}$$

The last expression is linear with respect to $\gamma$ and the leading coefficient is negative. Hence, $s'_\gamma$ may have only one root on the interval $I$ and this root corresponds to the maximum of $s(\gamma )$. We get that the minimum of $s$ is attained at the ends of the closed interval.

By Lemma 14.4 the minimum of $s_k$ for $k_1+1\le k\le k_2$ is achieved either at $k = k_1+1$ or at $k=k_2$. This implies the following estimate of the norm of vector $y=y_{k_1+1}+\ldots +y_{k_2}$, $y_k \in \fancyscript{H}_k$,

$$\begin{aligned} |\!|\!|y|\!|\!|^2&= \sum _{k=k_1+1}^{k_2} |\!|\!|y_k|\!|\!|_k^2\ge \sum _{k=k_1+1}^{k_2} s_k^2 \Vert u_k\Vert \ge \min _{k_1+1\le k\le k_2}\left\{ s_k^2\right\} \sum _{k=k_1+1}^{k_2} \Vert u_k\Vert ^2 \ge \nonumber \\&\quad \ge \min \{s_{k_1+1}^2, s_{k_2}^2\} \Vert u\Vert ^2 = \min \{\varkappa _{II}^2, \varkappa _{III}^2\} \Vert u\Vert ^2. \end{aligned}$$

(14.13)

4.3 New Norm in the Space $\fancyscript{H}_\infty$

The space $\fancyscript{H}_\infty$ is infinitely-dimensional. We introduce the new inner product $[\cdot , \cdot ]_\infty$, which is equivalent to the standard one, in such a way that for any vector $y\in \fancyscript{H}_\infty$, $[\mathbf {A} y, y]_\infty \ge M[y, y]_\infty$.

Define the inner product of vectors $y=(u,p)\in \fancyscript{H}_\infty$, $\tilde{y}=(\tilde{u},\tilde{p})\in \fancyscript{H}_\infty$, by the rule

$$\begin{aligned}{}[y,\tilde{y}]_\infty =(1-2M\gamma _s)(\nabla u,\nabla \tilde{u}) +2M\gamma _s\lambda _M(u,\tilde{u})+ M(u,\tilde{p})+ M(p,\tilde{u}) +(p,\tilde{p}), \end{aligned}$$

where $\lambda _M= \frac{M-\gamma _w}{\gamma _s}$. By (14.9) we have $\lambda _M > M^2$. Moreover, for any vector $y=(u,p)\in \fancyscript{H}_\infty$,

$$\begin{aligned} \Vert \nabla u\Vert ^2\ge \lambda _{k_2+1}\Vert u\Vert ^2 = \frac{\gamma _{k_2+1}-\gamma _w}{\gamma _s}\Vert u\Vert ^2\ge \lambda _M \Vert u\Vert ^2. \end{aligned}$$

(14.14)

Corresponding norm is defined by the formula

$$\begin{aligned} |\!|\!|y|\!|\!|_\infty ^2=(1-2M\gamma _s)\Vert \nabla u\Vert ^2+ M(2\gamma _s\lambda _M-M)\Vert u\Vert ^2 +\Vert M u+p\Vert ^2. \end{aligned}$$

Lemma 14.5.

The norms $|\!|\!|y|\!|\!|_\infty$ and $\Vert y\Vert _H$ are equivalent on the space $H_\infty$.

Proof.

Since $2\gamma _s M < 1$ and

$$ |\!|\!|y|\!|\!|_\infty ^2 \le (1-2M\gamma _s)\Vert \nabla u\Vert ^2+ M(2\gamma _s\lambda _M-M)\Vert u\Vert ^2 +(M \Vert u\Vert + \Vert p\Vert )^2, $$

it follows that the quantity $|\!|\!|y|\!|\!|_\infty ^2$ is bounded above by a quantity depending on $\Vert \nabla u\Vert ^2$ and $\Vert p\Vert ^2$.

Let us find a lower bound for $|\!|\!|y|\!|\!|_\infty ^2$. For some $\varepsilon >0$, we have $\lambda _M(1-\varepsilon ) > M^2$. With regard to (14.14), we have

$$\begin{aligned} |\!|y|\!|\!|_\infty ^2&=(1-2M\gamma _s)\Vert \nabla u\Vert ^2+(2M\gamma _s\lambda _M-M^2)\Vert u\Vert ^2 +\Vert M u+p\Vert ^2\ge \\&\quad \ge \varepsilon \Vert \nabla u\Vert ^2 +(1-2M\gamma _s-\varepsilon )\Vert \nabla u\Vert ^2 +\lambda _M(2M\gamma _s-1+\varepsilon )\Vert u\Vert ^2\,{+}\\&\quad +\Vert M u+p\Vert ^2\ge \varepsilon \Vert \nabla u\Vert ^2+\Vert M u+p\Vert ^2 \ge \frac{\varepsilon }{2}\Vert \nabla \,{+}\, u\Vert ^2 +\frac{\varepsilon \lambda _M}{2}\Vert u\Vert ^2\,{+}\\&\quad +(M\Vert u\Vert -\Vert p\Vert )^2. \end{aligned}$$

The expression on the right-hand side is a positive-defined quadratic form in $\Vert \nabla u\Vert$, $\Vert u\Vert$ and $\Vert p\Vert$, which can be estimated below by multiple of $\Vert \nabla u\Vert ^2+\Vert p\Vert ^2$.

Lemma 14.6.

For any vector $y=(u,p)\in \fancyscript{H}_\infty$,

$$\begin{aligned} |\!|\!|y|\!|\!|_\infty \ge \sqrt{\lambda _M-M^2}\Vert u\Vert = \varkappa _{IV}\Vert u\Vert . \end{aligned}$$

(14.15)

Proof.

By (14.14) we have

$$\begin{aligned} |\!|\!|y|\!|\!|_\infty ^2&\ge ((1-2M\gamma _s)\lambda _M + 2M\gamma _s\lambda _M)\Vert u\Vert ^2-2M\Vert u\Vert \Vert p\Vert +\Vert p\Vert ^2\,{=}\, \\&=(\lambda _M-M^2) \Vert u\Vert ^2 + (M\Vert u\Vert -\Vert p\Vert )^2 \ge (\lambda _M-M^2) \Vert u\Vert ^2. \end{aligned}$$

Lemma 14.7.

For any vector $y=(u,p)\in \fancyscript{H}_\infty \cap \mathcal{D}(\mathbf {A})$, $[\mathbf {A} y,y]_\infty \ge M [y,y]_\infty$.

Proof.

With regard to $M = \gamma _w+\gamma _s\lambda _M$, we have

$$\begin{aligned}{}[y,y]_\infty&=(1-2M\gamma _s)\Vert \nabla u\Vert ^2 +2M(M-\gamma _w)\Vert u\Vert ^2 +2M(u,p)+\Vert p\Vert ^2;\\ \mathbf {A} y&=(-p,-\varDelta u+2\gamma _w p-2\gamma _s \varDelta p); \end{aligned}$$

$$\begin{aligned}{}[\mathbf {A} y,y]_\infty =&-(1-2M\gamma _s)(\nabla p,\nabla u)+2M(M-\gamma _w)(-p,u)+M(-p,p)\,{+}\\&+(-\varDelta u+2\gamma _w p-2\gamma _s \varDelta p, M u+p)\!{=}\! M\Vert \nabla u\Vert ^2{+}4M\gamma _s(\nabla p,\nabla u)\,{+} \\&+2\gamma _s\Vert \nabla p\Vert ^2+2M(2\gamma _w-M)(u,p)+ (2\gamma _w-M)\Vert p\Vert ^2. \end{aligned}$$

It follows that

$$\begin{aligned}{}[\mathbf {A}y,y]_\infty -M[y,y]_\infty&= 2M^2\gamma _s\Vert \nabla u\Vert ^2+ 4M\gamma _s(\nabla p, \nabla u)+2\gamma _s\Vert \nabla p\Vert ^2\,{-}\\&\quad -2M^2(M-\gamma _w)\Vert u\Vert ^2+2M(2\gamma _w-2M)(u, p)\,{+}\\&\quad +(2\gamma _w-2M)\Vert p\Vert ^2\,{=}\\&= 2\gamma _s\Vert M\nabla u+\nabla p\Vert ^2- 2\gamma _s\lambda _M\Vert M u+p\Vert ^2. \end{aligned}$$

The last expression is nonnegative by (14.14).

4.4 End of the Proof of Theorem 14.3

Denote $\fancyscript{H}^{\eta } =\langle \eta _1e_1, \ldots , \eta _{k_1}e_{k_1}\rangle$, $\fancyscript{H}^{\xi } = \langle \xi _1e_1, \ldots , \xi _{k_1}e_{k_1} \rangle$, $\fancyscript{H}^{I} = \fancyscript{H}^{\eta }\oplus \fancyscript{H}_{k_1+1}\oplus \ldots \fancyscript{H}_{k_2}$, $\fancyscript{H}^{II}=\fancyscript{H}^{\xi } \oplus \fancyscript{H}_\infty$. The spaces $\fancyscript{H}^I$ and $\fancyscript{H}^{II}$ are orthogonal to each other with respect to the new inner product.

Since $\mathbf {A}( \xi _k e_k)=\mu _k( \xi _k e_k)$, $\mathbf {A}(\eta _k e_k)=\nu _k(\eta _k e_k)$ for $k=1,\dots ,k_1$, it follows that

$$\begin{aligned}&[\mathbf {A} y,y]\le \max _{1\le k\le k_1}\mu _k\cdot [y,y] =\mu _{k_1}[y,y] \qquad \forall y\in \fancyscript{H}^{\xi },\end{aligned}$$

(14.16)

$$\begin{aligned}&[\mathbf {A} y,y]\ge \min _{1\le k\le k_1}\nu _k\cdot [y,y] =\nu _{k_1}[y,y] \qquad \, \forall y\in \fancyscript{H}^{\eta }. \end{aligned}$$

(14.17)

It follows from condition (14.16), Lemma 14.3, and the inequality $m >\mu _{k_1}$ that

$$\begin{aligned}{}[\mathbf {A}y,y]\le m[y,y] \qquad \, \forall y\in \fancyscript{H}^{I}. \end{aligned}$$

(14.18)

Also, condition (14.17), Lemma 14.7, and the inequality $M < \nu _{k_1}$ imply that

$$\begin{aligned}{}[\mathbf {A}y,y]\ge M [y,y] \qquad \, \forall y\in \fancyscript{H}^{II}\cap \fancyscript{D}(\mathbf {A}). \end{aligned}$$

(14.19)

Since the vector $F(y)$ has zero $u$-component, it follows that

(14.20)

By estimates (14.12), (14.13), (14.15) of the vector $y = y_1-y_2 = y_1+\ldots +y_{k_2}+y_\infty$, $y_k \in \fancyscript{H}_k$, $y_\infty \in \fancyscript{H}_\infty$, we obtain

(14.21)

It follows from inequalities (14.20) and (14.21) that

Thus the global Lipschitz constant $L$ for the function $F(y)$ is equal to

$$ L =\frac{l}{\min \{\varkappa _I, \varkappa _{II}, \varkappa _{III}, \varkappa _{IV}\}}. $$

Let us define the orthogonal projection to the $(2k_2-k_1)$-dimensional space $\fancyscript{H}^I=P(\fancyscript{H})$ and denote it by $P$ and define the orthogonal projection $Q=\mathrm{Id}-P$ to $\fancyscript{H}^{II}\oplus \fancyscript{H}_\infty =Q(\fancyscript{H})$. Then the inequalities (14.18) and (14.19) acquire the form (14.3), and the spectral gap condition (14.4) is equivalent to condition (14.11).

Thus, all conditions of Theorem 14.1 are satisfied, and thus the space $\fancyscript{H}$ contains an integral manifold which dimension is equal to that of the subspace $\fancyscript{H}^I$, i. e., to $2k_2-k_1$. This completes the proof of the theorem.