Control Problems for the Navier–Stokes System with Nonlocal Spatial Terms

Abstract

We consider the local null controllability of a modified Navier–Stokes system where we include nonlocal spatial terms. We generalize a previous work where the nonlocal spatial term is given by the linearization of a Ladyzhenskaya model for a viscous incompressible fluid. Here, the nonlocal spatial term is more general and we consider a control with one vanishing component. The proof of the result is based on a Carleman estimate where the main difficulty consists in handling the nonlocal spatial terms. One key point corresponds to a particular decomposition of the solution of the adjoint system that allows us to overcome regularity issues. With a similar approach, we also show the existence of insensitizing controls for the same system.

A Proof of Proposition 2.1

We give here a sketch of the proof of Proposition 2.1 that is quite standard. The proof of Proposition 2.2 can be obtained with a similar method.

Proof of Proposition 2.1

The proof of Proposition 2.1 can be obtained by using the standard Galerkin method and by following for instance the proof of [25, Proposition 1.2, pp. 267–268] for the Stokes system. The main idea is to obtain uniform a priori estimates for the approximate solutions. Here, to simplify, we only show a priori estimates on the solutions of (2.2), but one can recover similar estimates for the Galerkin approximation.

First, by multiplying the first equation of (2.3) by $\varphi$ and integrating in $(t,T)\times \Omega$, we obtain that for $t\in (0,T)$

$$\begin{aligned}{} & {} \int _{\Omega } \left| \varphi (t,x)\right| ^2 \ \textrm{d}x + 2\int _t^T\int _{\Omega } \left| \nabla \varphi \right| ^2 \ \textrm{d}x \, \textrm{d}s \leqslant \int _{\Omega } \left| \varphi _T\right| ^2 \ \textrm{d}x + \int _0^T\int _{\Omega } \left| g\right| ^2 \ \textrm{d}x \, \textrm{d}s \\{} & {} \quad + \left( 1+\sum _{i=1}^n \left\| a^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))} \left\| b^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))}\right) \int _t^T\int _{\Omega } \left| \varphi \right| ^2 \ \textrm{d}x \, \textrm{d}s \end{aligned}$$

and with the Grönwall lemma, this yields that for $t\in (0,T)$

$$\begin{aligned} \int _{\Omega } \left| \varphi (t,x)\right| ^2 \ \textrm{d}x + 2\int _t^T\int _{\Omega } \left| \nabla \varphi \right| ^2 \ \textrm{d}x \, \textrm{d}s \lesssim \int _{\Omega } \left| \varphi _T\right| ^2 \ \textrm{d}x + \int _0^T\int _{\Omega } \left| g\right| ^2 \ \textrm{d}x \, \textrm{d}s, \end{aligned}$$

where the constant of the above inequality depends on T and on the norms of $a^{(i)}$ and $b^{(i)}$ in $H^2(0,T;L^2(\Omega ))$. With the Poincaré inequality, we thus deduce that,

$$\begin{aligned} \left\| \varphi \right\| _{L^\infty (0,T;L^2(\Omega ))} + \left\| \varphi \right\| _{L^2(0,T;H^1(\Omega ))} \lesssim \left\| \varphi _T \right\| _{L^2(\Omega )} + \left\| g \right\| _{L^2(0,T;L^2(\Omega ))}. \end{aligned}$$

(A.1)

Second, by multiplying the first equation of (2.3) by $-\partial _t \varphi$ and integrating in $(t,T)\times \Omega$, we obtain that for $t\in (0,T)$

$$\begin{aligned}{} & {} \int _t^T\int _{\Omega } \left| \partial _t \varphi \right| ^2 \ \textrm{d}x \, \textrm{d}s+\frac{1}{2}\int _{\Omega } \left| \nabla \varphi (t,x)\right| ^2 \ \textrm{d}x \leqslant \frac{1}{2}\int _{\Omega } \left| \nabla \varphi _T\right| ^2 \ \textrm{d}x+\int _0^T\int _{\Omega } \left| g\right| ^2 \ \textrm{d}x \, \textrm{d}s\\{} & {} \quad +\int _t^T \sum _{i=1}^n \left\| a^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))}^2 \left\| b^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))}^2 \left\| \varphi (s,\cdot ) \right\| _{L^2(\Omega )}^2 \, \textrm{d}s \end{aligned}$$

and combining the above relation with (A.1) implies

$$\begin{aligned} \left\| \varphi \right\| _{H^1(0,T;L^2(\Omega ))} +\left\| \varphi \right\| _{L^\infty (0,T;H^1(\Omega ))} \lesssim \left\| \varphi _T \right\| _{H^1(\Omega )} + \left\| g \right\| _{L^2(0,T;L^2(\Omega ))}. \end{aligned}$$

(A.2)

Then, we can see (2.3) as a stationary Stokes system and apply the elliptic regularity of such a system (see, for instance, [25, Proposition 2.2, p.33]):

$$\begin{aligned}{} & {} \left\| \varphi \right\| _{L^2(0,T;H^2(\Omega ))} +\left\| \pi \right\| _{L^2(0,T;H^1(\Omega )/ \mathbb {R})} \lesssim \left\| \partial _t \varphi \right\| _{L^2(0,T;L^2(\Omega ))} \nonumber \\{} & {} \quad + \sum _{i=1}^n \left\| a^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))} \left\| b^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))} \left\| \varphi \right\| _{L^2(0,T;L^2(\Omega ))}. \end{aligned}$$

(A.3)

Combining (A.1), (A.2) and (A.3), we deduce (2.4).

Then to obtain (2.5), we differentiate (2.3) in time to obtain

$$\begin{aligned} \left\{ \begin{array}{rl} -\partial _{t}\left( \partial _t \varphi \right) -\Delta \left( \partial _t \varphi \right) +\sum _{i=1}^n \left( \int _{\Omega } a^{(i)}\cdot \partial _t \varphi \ dx\right) b^{(i)}+\nabla \left( \partial _t \pi \right) =&{} g^{(1)}~\text{ in } \ (0,T)\times \Omega ,\\ {\text {div}}\left( \partial _t \varphi \right) =&{}0~\text{ in } \ (0,T)\times \Omega ,\\ \partial _t \varphi =&{} 0 ~\text{ on }\ (0,T)\times \partial \Omega ,\\ \left( \partial _t \varphi \right) (T,\cdot )=&{} 0 ~\text{ in } \ \Omega , \end{array} \right. \end{aligned}$$

(A.4)

with

$$\begin{aligned} g^{(1)}:=\partial _t g -\sum _{i=1}^n \left( \int _{\Omega } \partial _t a^{(i)}\cdot \varphi \ \textrm{d}x\right) b^{(i)}-\sum _{i=1}^n \left( \int _{\Omega } a^{(i)}\cdot \varphi \ \textrm{d}x\right) \partial _t b^{(i)}. \end{aligned}$$

From the first part of the proof, we have

$$\begin{aligned} \left\| \partial _t \varphi \right\| _{H^1(0,T;L^2(\Omega ))} +\left\| \partial _t \varphi \right\| _{L^2(0,T;H^2(\Omega ))} \lesssim \left\| g^{(1)} \right\| _{L^2(0,T;L^2(\Omega ))}. \end{aligned}$$

(A.5)

We can check that

$$\begin{aligned}{} & {} \left\| g^{(1)} \right\| _{L^2(0,T;L^2(\Omega ))}\\{} & {} \lesssim \left\| g \right\| _{H^1(0,T;L^2(\Omega ))} + \sum _{i=1}^n \left\| a^{(i)}\right\| _{H^1(0,T;L^2(\Omega ))}\left\| b^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))} \left\| \varphi \right\| _{L^\infty (0,T;L^2(\Omega ))}\\{} & {} \quad + \sum _{i=1}^n \left\| b^{(i)}\right\| _{H^1(0,T;L^2(\Omega ))}\left\| a^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))} \left\| \varphi \right\| _{L^\infty (0,T;L^2(\Omega ))}. \end{aligned}$$

Combining the above estimate with (A.5) and (2.4) implies

$$\begin{aligned} \left\| \partial _t \varphi \right\| _{H^1(0,T;L^2(\Omega ))} +\left\| \partial _t \varphi \right\| _{L^2(0,T;H^2(\Omega ))} \lesssim \left\| g \right\| _{H^1(0,T;L^2(\Omega ))}. \end{aligned}$$

(A.6)

Then, using the elliptic regularity of the Stokes system (see, for instance, [25, Proposition 2.2, p.33]), we deduce

$$\begin{aligned}{} & {} \left\| \varphi \right\| _{L^2(0,T;H^4(\Omega ))} +\left\| \pi \right\| _{L^2(0,T;H^3(\Omega )/ \mathbb {R})} \lesssim \left\| \partial _t \varphi \right\| _{L^2(0,T;H^2(\Omega ))} \\{} & {} + \sum _{i=1}^n \left\| a^{(i)}\right\| _{L^\infty (0,T;L^2(\Omega ))} \left\| b^{(i)}\right\| _{L^2(0,T;H^2(\Omega ))} \left\| \varphi \right\| _{L^\infty (0,T;L^2(\Omega ))}. \end{aligned}$$

Gathering (A.6), (2.4) and the above estimate yields (2.5).

Finally, to obtain (2.6), we differentiate (A.4) in time to obtain

$$\begin{aligned}\left\{ \begin{array}{rl} -\partial _{t}\left( \partial _t^2 \varphi \right) -\Delta \left( \partial _t^2 \varphi \right) +\sum _{i=1}^n \left( \int _{\Omega } a^{(i)}\cdot \partial _t^2 \varphi \ dx\right) b^{(i)}+\nabla \left( \partial _t^2 \pi \right) =&{} g^{(2)}~\text{ in } \ (0,T)\times \Omega ,\\ {\text {div}}\left( \partial _t^2 \varphi \right) =&{} 0~\text{ in } \ (0,T)\times \Omega ,\\ \partial _t^2 \varphi =&{} 0 ~\text{ on }\ (0,T)\times \partial \Omega ,\\ \left( \partial _t^2 \varphi \right) (T,\cdot )=&{} 0 ~\text{ in } \ \Omega , \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} g^{(2)}:= & {} \partial _t^2 g -\sum _{i=1}^n \left[ \left( \int _{\Omega } \partial _t^2 a^{(i)}\cdot \varphi \ \textrm{d}x\right) b^{(i)} +\left( \int _{\Omega } a^{(i)}\cdot \varphi \ \textrm{d}x\right) \partial _t^2 b^{(i)} \right. \\{} & {} \left. +2 \left( \int _{\Omega } \partial _t a^{(i)}\cdot \partial _t \varphi \ \textrm{d}x\right) b^{(i)} +2 \left( \int _{\Omega } \partial _t a^{(i)}\cdot \varphi \ \textrm{d}x\right) \partial _t b^{(i)} \right. \\{} & {} \left. +2 \left( \int _{\Omega } a^{(i)}\cdot \partial _t \varphi \ \textrm{d}x\right) \partial _t b^{(i)} \right] . \end{aligned}$$

We can check that

$$\begin{aligned} \left\| g^{(2)} \right\| _{L^2(0,T;L^2(\Omega ))}\lesssim & {} \left\| g \right\| _{H^2(0,T;L^2(\Omega ))} \\{} & {} + \sum _{i=1}^n \left\| a^{(i)}\right\| _{H^2(0,T;L^2(\Omega ))}\left\| b^{(i)}\right\| _{H^2(0,T;L^2(\Omega ))} \left\| \varphi \right\| _{L^\infty (0,T;L^2(\Omega ))}. \end{aligned}$$

From the first part of the proof and the above estimate, we have

$$\begin{aligned} \left\| \partial _t^2 \varphi \right\| _{H^1(0,T;L^2(\Omega ))} +\left\| \partial _t^2 \varphi \right\| _{L^2(0,T;H^2(\Omega ))} \lesssim \left\| g \right\| _{H^2(0,T;L^2(\Omega ))}. \end{aligned}$$

(A.7)

Using the elliptic regularity of the Stokes system (see, for instance, [25, Proposition 2.2, p.33]) on (A.4), we deduce from the above estimate that

$$\begin{aligned} \left\| \partial _t \varphi \right\| _{L^2(0,T;H^4(\Omega ))} \lesssim \left\| g \right\| _{H^2(0,T;L^2(\Omega ))}+\left\| g \right\| _{H^1(0,T;H^2(\Omega ))}. \end{aligned}$$

Then, using the elliptic regularity of the Stokes system (see, for instance, [25, Proposition 2.2, p.33]) on (2.3) and the above estimate, we obtain

$$\begin{aligned} \left\| \varphi \right\| _{L^2(0,T;H^6(\Omega ))} \lesssim \left\| g \right\| _{H^2(0,T;L^2(\Omega ))}+\left\| g \right\| _{H^1(0,T;H^2(\Omega ))}+\left\| g \right\| _{L^2(0,T;H^4(\Omega ))}. \end{aligned}$$

Combining the above relation and (A.7) gives (2.6).