Uniform Stabilization of Navier–Stokes Equations in Critical $L^q$-Based Sobolev and Besov Spaces by Finite Dimensional Interior Localized Feedback Controls

Abstract

We consider 2- or 3-dimensional incompressible Navier–Stokes equations defined on a bounded domain $\varOmega$, with no-slip boundary conditions and subject to an external force, assumed to cause instability. We then seek to uniformly stabilize such N–S system, in the vicinity of an unstable equilibrium solution, in critical $L^q$-based Sobolev and Besov spaces, by finite dimensional feedback controls. These spaces are ‘close’ to $L^3(\varOmega )$ for $d=3$. This functional setting is significant. In fact, in the case of the uncontrolled N–S dynamics, extensive research efforts have recently lead to the space $L^3(\mathbb {R}^3)$ as being a critical space for the issue of well-posedness in the full space. Thus, our present work manages to solve the stated uniform stabilization problem for the controlled N–S dynamics in a correspondingly related function space setting. In this paper, the feedback controls are localized on an arbitrarily small open interior subdomain $\omega$ of $\varOmega$. In addition to providing a solution of the uniform stabilization problem in such critical function space setting, this paper manages also to much improve and simplify, at both the conceptual and computational level, the solution given in the more restrictive Hilbert space setting in the literature. Moreover, such treatment sets the foundation for the authors’ final goal in a subsequent paper. Based critically on said low functional level where compatibility conditions are not recognized, the subsequent paper solves in the affirmative a presently open problem: whether uniform stabilization by localized tangential boundary feedback controls, which—in addition—are finite dimensional, is also possible in dim $\varOmega = 3$.

Appendices

On Helmholtz Decomposition

We return to the Helmholtz decomposition in (1.4), (1.5) and provide additional information.

For $M \subset L^q(\varOmega ), \quad 1< q < \infty$, we denote the annihilator of M by

$$\begin{aligned} M^{\perp } = \bigg \{ f \in L^{q'}(\varOmega ) : \int _{\varOmega } fg \ d \varOmega = 0, \text { for all } g \in M \bigg \} \end{aligned}$$

(A.1)

where $q'$ is the dual exponent of $\displaystyle q: \ {}^{1}\!/_{q} + {}^{1}\!/_{q'} = 1$.

Proposition A.1

[26, Prop. 2.2.2, p. 6], [21, Ex. 16, p. 115], [17] Let $\varOmega \subset \mathbb {R}^d$ be an open set and let $1< q < \infty$.

• (a) The Helmholtz decomposition exists for $L^q(\varOmega )$ if and only if it exists for $L^{q'}(\varOmega )$, and we have: (adjoint of $P_q$) = $P^*_q = P_{q'}$ (in particular $P_2$ is orthogonal), where $P_q$ is viewed as a bounded operator $\displaystyle L^q(\varOmega ) \longrightarrow L^q(\varOmega )$, and $\displaystyle P^*_q = P_{q'}$ as a bounded operator $\displaystyle L^{q'}(\varOmega ) \longrightarrow L^{q'}(\varOmega ), \ {}^{1}\!/_{q} + {}^{1}\!/_{q'} = 1$.

• (b) Then, with reference to (1.5)

$$\begin{aligned} \Big [ L^{q}_{\sigma }(\varOmega )\Big ]^{\perp } = G^{q'}(\varOmega ) \text { and } \Big [ G^q(\varOmega ) \Big ]^{\perp } = L^{q'}_{\sigma }(\varOmega ). \end{aligned}$$

(A.2a)

Remark A.1

Throughout the paper we shall use freely that

$$\begin{aligned} \big ( L^{q}_{\sigma }(\varOmega )\big )' = L^{q'}_{\sigma }(\varOmega ), \quad \frac{1}{q} + \frac{1}{q'} = 1. \end{aligned}$$

(A.2b)

Thus can be established as follows. From (1.5) write $\displaystyle L^{q}_{\sigma }(\varOmega )$ as a factor space $\displaystyle L^{q}_{\sigma }(\varOmega )= L^q(\varOmega ) / G^q(\varOmega ) \equiv X/M$ so that [55, p. 135].

$$\begin{aligned} \big ( L^{q}_{\sigma }(\varOmega )\big )' = \big ( L^q(\varOmega ) / G^q(\varOmega ) \big )' = \big ( X/M \big )' = M^{\perp } = \Big [ G^q(\varOmega ) \Big ]^{\perp } = L^{q'}_{\sigma }(\varOmega ). \end{aligned}$$

(A.2c)

In the last step, we have invoked (A.2a), which is also established in [21, Lemma 2.1, p. 116]. Similarly

$$\begin{aligned} \big ( G^q(\varOmega ) \big )' = \big ( L^q(\varOmega ) / L^{q}_{\sigma }(\varOmega )\big )' = \Big [ L^{q}_{\sigma }(\varOmega )\Big ]^{\perp } = G^{q'}(\varOmega ). \end{aligned}$$

(A.2d)

Proof of Theorem 1.6: Maximal Regularity of the Oseen Operator ${\mathcal {A}}_q$ on $L^{q}_{\sigma }(\varOmega )$, $1< p,q< \infty , T < \infty$

Part I: (1.46). By (1.41) with $\psi _0 = 0$

$$\begin{aligned} \psi (t) = \int _{0}^{t} e^{{\mathcal {A}}_q(t-\tau )} F_{\sigma }(\tau ) d \tau . \end{aligned}$$

(B.1)

where by the statement preceding Theorem 1.4

$$\begin{aligned} \left||e^{{\mathcal {A}}_q(t-\tau )}\right||_{{\mathcal {L}}(L^{q}_{\sigma }(\varOmega ))} \le M e^{b (t - \tau )}, \quad 0 \le \tau \le t \end{aligned}$$

(B.2)

for $M \ge 1, \ b$ possibly depending on q.

Step 1 We have the following estimate

$$\begin{aligned} \int _{0}^{T} \left||\psi (t)\right||_{L^{q}_{\sigma }(\varOmega )}^p dt \le C_T \int _{0}^{T} \left||F_{\sigma }(t)\right||_{L^{q}_{\sigma }(\varOmega )}^p dt \end{aligned}$$

(B.3)

where the constant $C_T$ may depend also on p, q, b. This follows at once from the Young’s inequality for convolutions [43, p. 26]:

$$\begin{aligned} \left||\psi (t)\right||_{L^{q}_{\sigma }(\varOmega )} \le M \int _{0}^{t} e^{b (t - \tau )} \left||F_{\sigma }(\tau )\right||_{L^{q}_{\sigma }(\varOmega )} d \tau \in L^p(0,T), \ T < \infty , \end{aligned}$$

and the convolution of the $L^p(0,T)$-function $\displaystyle \left||F_{\sigma }\right||_{L^{q}_{\sigma }(\varOmega )}$ and the $\displaystyle L^1(0,T)$-function $e^{bt}$ is in $L^p(0,T)$. More elementary, one can use Hölder inequality with $\displaystyle {}^{1}\!/_{p} + {}^{1}\!/_{{\tilde{p}}} = 1$ and obtain an explicit constant.

Step 2 Claim: Here we shall next complement (B.3) with the estimate

$$\begin{aligned} \int _{0}^{T} \left||A_q \psi (t)\right||_{L^{q}_{\sigma }(\varOmega )}^p dt \le C \int _{0}^{T} \left||\psi (t)\right||_{L^{q}_{\sigma }(\varOmega )}^p dt + C \int _{0}^{T} \left||F_{\sigma }(t)\right||_{L^{q}_{\sigma }(\varOmega )}^p dt \end{aligned}$$

(B.4)

to be shown below. Using (B.3) in (B.4) then yields

$$\begin{aligned} \int _{0}^{T} \left||A_q \psi (t)\right||_{L^{q}_{\sigma }(\varOmega )}^p dt \le C_T \int _{0}^{T} \left||F_{\sigma }(t)\right||_{L^{q}_{\sigma }(\varOmega )}^p dt. \end{aligned}$$

(B.5)

With respect to (1.41) with $\psi _0 = 0$, then (B.5) says

$$\begin{aligned} F_{\sigma }\in L^p(0,T; L^{q}_{\sigma }(\varOmega )) \longrightarrow \psi \in L^p(0,T;{\mathcal {D}}(A_q) = {\mathcal {D}}({\mathcal {A}}_q)) \end{aligned}$$

(B.6)

while (1.40) then yields via (B.6)

$$\begin{aligned} F_{\sigma }\in L^p(0,T; L^{q}_{\sigma }(\varOmega )) \longrightarrow \psi _t \in L^p(0,T;L^{q}_{\sigma }(\varOmega )) \end{aligned}$$

(B.7)

continuously. Then, (B.6), (B.7) shows part (i) of Theorem 1.6.

Proof of (B.4):

In this step, with $\psi _0 = 0$, we shall employ the alternative formula, via (1.42) ($\nu = 1$, wlog)

$$\begin{aligned} \psi (t) = \int _{0}^{t} e^{-A_q(t-\tau )} (-A_{o,q}) \psi (\tau ) d \tau + \int _{0}^{t} e^{-A_q(t-\tau )} F_{\sigma }(\tau ) d \tau , \end{aligned}$$

(B.8)

where by maximal regularity of the Stokes operator $-A_q$ on the space $L^{q}_{\sigma }(\varOmega )$, as asserted in Theorem 1.5.ii, Eq (1.35), we have in particular

$$\begin{aligned} F_{\sigma }\in L^p(0,T; L^{q}_{\sigma }(\varOmega )) \longrightarrow \int _{0}^{t} e^{-A_q(t - \tau )} F_{\sigma }(\tau ) d \tau \in L^p(0,T;{\mathcal {D}}(A_q)) \quad \text {continuously.} \end{aligned}$$

(B.9)

Regarding the first integral term in (B.8) we shall employ the (complex) interpolation formula (1.22), and recall from (1.9) that ${\mathcal {D}}(A_{o,q}) = {\mathcal {D}}(A^{{}^{1}\!/_{2}}_q)$:

$$\begin{aligned} {\mathcal {D}}(A_{o,q}) = {\mathcal {D}}(A^{{}^{1}\!/_{2}}_q) = \left[ {\mathcal {D}}(A_q), L^{q}_{\sigma }(\varOmega )\right] _{{}^{1}\!/_{2}} \end{aligned}$$

(B.10)

so that the interpolation inequality [62, Theorem p 53, Eq. (3)] with $\theta = {}^{1}\!/_{2}$ yields from (B.10)

$$\begin{aligned} \begin{aligned} \left||a\right||_{{\mathcal {D}}(A_{o,q})} = \left||a\right||_{{\mathcal {D}}\big (A_q^{{}^{1}\!/_{2}} \big )}&\le C \left||a\right||_{{\mathcal {D}}(A_q)}^{{}^{1}\!/_{2}} \left||a\right||_{L^{q}_{\sigma }(\varOmega )}^{{}^{1}\!/_{2}}\\&\le \varepsilon \left||a\right||_{{\mathcal {D}}(A_q)} + C_{\varepsilon } \left||a\right||_{L^{q}_{\sigma }(\varOmega )}. \end{aligned} \end{aligned}$$

(B.11)

$\Big [$Since ${\mathcal {D}}(A_q^{{}^{1}\!/_{2}}) = W^{1,q}_0(\varOmega ) \cap L^{q}_{\sigma }(\varOmega )$ by (1.22), then for $a \in {\mathcal {D}}(A_q) = W^{2,q}(\varOmega ) \cap W^{1,q}_0(\varOmega ) \cap L^{q}_{\sigma }(\varOmega )$, see (1.17), we may as well invoke the interpolation inequality for W-spaces. [1, Theorem 4.13, p. 74]:

$$\begin{aligned} \left||a\right||_{W^{1,q}_0(\varOmega )} \le \varepsilon \left||a\right||_{W^{2,q}(\varOmega )} + C_{\varepsilon } \left||a\right||_{L^{q}_{\sigma }(\varOmega )} \end{aligned}$$

We return to (B.8) and obtain

$$\begin{aligned} A_q \psi (t) = A_q \int _{0}^{t} e^{-A_q(t-\tau )} (-A_{o,q}) \psi (\tau ) d \tau + A_q \int _{0}^{t} e^{-A_q(t-\tau )} F_{\sigma }(\tau ) d \tau . \end{aligned}$$

(B.12)

Hence via the maximal regularity of the uniformly stable Stokes semigroup $e^{-A_q t}$, Eq. (1.35), (B.11) yields

$$\begin{aligned} \left||A_q \psi \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))}&\le C \Big \{ \left||A_{o,q} \psi \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} + \left||F_{\sigma }\right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \Big \} \end{aligned}$$

(B.13)

$$\begin{aligned} \text {by (B.11)} \quad&\le \varepsilon ' \left||A_q \psi \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} + C_{\varepsilon '} \left||\psi \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \nonumber \\&\quad + C \left||F_{\sigma }\right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \end{aligned}$$

(B.14)

$\varepsilon ' = \varepsilon C > 0$ arbitrarily small. Hence (B.14) yields

$$\begin{aligned} \left||A_q \psi \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \le \frac{C_{\varepsilon '}}{1-\varepsilon '} \left||\psi \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} + \frac{C}{1-\varepsilon '} \left||F_{\sigma }\right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \end{aligned}$$

(B.15)

and estimate (B.4) of Step 2 is established. Part I of Theorem 1.6 is proved.

Part II: (1.49). For simplicity of notation, we shall write the proof on $\displaystyle {\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )$ i.e. for $\displaystyle 1< q, p < {}^{2q}\!/_{2q-1}$. The proof on $\displaystyle \big ( L^{q}_{\sigma }(\varOmega ), {\mathcal {D}}(A_q) \big )_{1-\frac{1}{p},p}$ in the other case ${}^{2q}\!/_{2q-1} < p$ is exactly the same.

Step 1 Let $\eta _0 \in {\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )$ and consider the s.c. analytic Oseen semigroup $e^{{\mathcal {A}}_q t}$ on the space ${\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )$, as asserted by Theorem 1.4.ii (take $\nu = 1$ wlog):

$$\begin{aligned} \eta (t) = e^{{\mathcal {A}}_q t} \eta _0, \quad \text {or } \eta _t = {\mathcal {A}}_q \eta = -A_q \eta - A_{o,q} \eta . \end{aligned}$$

(B.16)

Then we can rewrite $\eta$ as

$$\begin{aligned} \eta (t)&= e^{-A_q t} \eta _0 + \int _{0}^{t} e^{-A_q (t - \tau )}(-A_{o,q})\eta (\tau ) \quad d \tau \end{aligned}$$

(B.17)

$$\begin{aligned} A_q \eta (t)&= A_q e^{-A_q t} \eta _0 + A_q \int _{0}^{t} e^{-A_q (t - \tau )}(-A_{o,q})\eta (\tau ) \quad d \tau . \end{aligned}$$

(B.18)

We estimate, recalling the maximal regularity (1.35), (1.36) as well as the uniform decay (1.25) of the Stokes operator.

$$\begin{aligned} \left||A_q \eta \right||_{L^p(0,T;L^q(\varOmega ))}&\le C \left||\eta _0\right||_{{\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )} + C \left||A_{o,q} \eta \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \end{aligned}$$

(B.19)

$$\begin{aligned}&\le C \left||\eta _0\right||_{{\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )} + \varepsilon {\widetilde{C}} \left||A_q \eta \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))}\nonumber \\&\quad + C_{\varepsilon }\left||\eta \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \end{aligned}$$

(B.20)

after invoking, in the last step, the interpolation inequality (B.11). Thus (B.20) yields via (1.18)

$$\begin{aligned} \left||A_q \eta \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))}&= \left||{\mathcal {A}}_q \eta \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))} \nonumber \\&\le \frac{C}{1 - \varepsilon {\widetilde{C}}} \left||\eta _0\right||_{{\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )} + \frac{C_{\varepsilon }}{1 - \varepsilon {\widetilde{C}}} \left||\eta \right||_{L^p(0,T;L^{q}_{\sigma }(\varOmega ))}. \end{aligned}$$

(B.21)

Step 2 With $\displaystyle \eta _0 \in {\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )$, since $\displaystyle e^{{\mathcal {A}}_q t}$ generates a s.c (analytic) semigroup on $\displaystyle {\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )$, Theorem 1.4.ii, we have

$$\begin{aligned} \eta (t) = e^{{\mathcal {A}}_qt} \eta _0 \in C \Big (0,T; {\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )\Big ) \subset L^p \Big (0,T; {\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )\Big ) \subset L^p \big (0,T; L^{q}_{\sigma }(\varOmega )\big ) \end{aligned}$$

(B.22)

continuously, where in the last step, we have recalled that $\displaystyle {\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )$ is the interpolation between $\displaystyle L^q(\varOmega )$ and $\displaystyle W^{2,q}(\varOmega )$, see (1.16b). (B.22) says explicitly

$$\begin{aligned} \left||\eta \right||_{L^p \big (0,T; L^{q}_{\sigma }(\varOmega )) \big )} \le C \left||\eta _0\right||_{{\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )} \end{aligned}$$

(B.23)

Step 3 Substituting (B.23) in (B.21) yields

$$\begin{aligned} \left||A_q \eta \right||_{L^p \big (0,T; L^{q}_{\sigma }(\varOmega )\big )} \le C \left||\eta _0\right||_{{\widetilde{B}}^{2-{}^{2}\!/_{p}}_{q,p}(\varOmega )} \end{aligned}$$

(B.24)

and (1.49) is established, from which (1.50) follows at once. Thus Theorem 1.6 is proved. $\square$