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On the controllability of the Navier-Stokes equation in spite of boundary layers (Mathematical Analysis of Viscous Incompressible Fluid)

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(1)162. 数理解析研究所講究録 第2058巻 2017年 162-180. On the. of the Navier‐Stokes. controllability. in. equation. spite of. boundary layers Jean‐Michel CORON Sorbonne Universités, UPMC Univ Paris 06 Lab.. Jacques‐Louis Lions,. UMR CNRS 7598. coron@ann.jussieu.fr Frédéric MARBACH. Sorbonne. Universités, UPMC. Univ Paris 06. Lab. Jacques‐Louis Lions, UMR CNRS 7598. marbach@ann.jussieu.fr Franck SUEUR Institut de. de. Mathématiques. Bordeaux, UMR CNRS 5251, Université. de Bordeaux. Franck. Sueur@math. \mathrm{u} ‐bordeaux. fr we expose a particular case of a recent result obtained in [6] by the incompressible Navier‐Stokes equations in a smooth bounded and. Abstract: In this note authors. regarding. the. simply connected bounded domain, either in 2\mathrm{D} or in 3\mathrm{D} with a Navier slip‐with‐friction boundary condition except on a part of the boundary. This under‐determination encodes ,. \mathrm{T}. that. one. initial this. has control. data, for. given. any. the. remaining part of the boundary. We prove that for any positive time, there exists a weak Leray solution which vanishes at over. time.. 2010 Mathematics. Subject. Classification:. Primary. 93\mathrm{B}05 ;. Secondary 35\mathrm{Q}30.. Keywords: Navier‐Stokes equations, Controllability, Navier slip with friction boundary condition, boundary layers, return method, well‐prepared dissipation method. Geometric. 1. We consider. d=3. .. a. smooth bounded and. Inside this. equations.. setting. domain,. We will. an. name u. simply connected1. incompressible viscous. its. velocity. domain $\Omega$ in. \mathbb{R}^{d} with d=2 ,. or. fluid evolves under the Navier‐Stokes. field and p the associated pressure. The equations. read:. \partial_{t}u+(u\cdot\nabla)u- $\Delta$ u+\nabla p=0 '. Indeed. our. i \mathrm{n}. analysis also. eboundaxy .tersecting general c \mathrm{a} $\epsilon$. covers. the. case. of. a. and. multiply connected. all its connected components, but. we. \mathrm{d}\mathrm{i}\mathrm{v}u=0. domain for. will stick here to this. some. in $\Omega$. (1.1). .. controls located. simple. case. and. we. part of the refer to [6] for the on. a.

(2) 163. Let. emphasize. us. for the sake of. that the fluid. For where. the. an. impermeable wall,. n. denotes the outward. case a. it is natural to. prescribe. pointing normal. to the. a. perfect fluid,. driven. by. the Euler. the. \bullet. to the. no‐slip. previous condition). to. domain, which. u\cdot n. means. =. 0. denotes the. boundary. Indeed in than the Navier‐Stokes, Cauchy problem. to the. uniqueness. here of the Navier‐Stokes. case. following propositions. back to Stokes in. the. of the vector field. tangential part. slip‐with‐friction. condition. N(u)=0. N(u):=[D(u)n+ $\alpha$ u]_{\mathrm{t}\mathrm{a}\mathrm{J}1} (or. the rate of strain tensor. simplicity2.. are. equations,. the most used. with. shear. the. ,. 1851),. where. where. D(u). stress). :=. and. Let. The Let. us. is. a. boundary.. real constant coefficient for. observe. no‐slip. (2.2). (see [26]).. This coefficient. that, formally, when. $\alpha$\rightarrow. condition.. Cauchy problem. recall the. divergence. (\displaystyle \frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i}) _{1\leq i,j\leq d} $\alpha$. us. +\infty , the Navier condition reduces to the usual. 3. (in. u.. ’. near. in. (2.1). This condition dates baek to Navier \mathrm{i}\mathrm{n} 1833. describes the friction. \partial $\Omega$,. on. that the fluid. [u]_{\tan} :=u-(u\cdot n)n. \bullet. one. :. [u]_{\tan}=0 (dating. condition. the condition. equations rather. appropriate functional settings. For the. extra condition has to be added. The two. complement. equal. cavitation at the. no. condition is sufficient to have existence and. various an. of. set. are. conditions. cannot escape the domain and that there is. such. viscosity coefficient. clarity.. Boundary. 2. and the. density. following result,. free vector fields which. Theorem 3.1 Let. u_{0}\in L_{ $\sigma$}^{2}( $\Omega$). .. where are. L_{ $\sigma$}^{2}( $\Omega$). tangent. denotes the closure in. L^{2}( $\Omega$). of smooth. to \partial $\Omega$.. Then there exists. a. weak solution. global. u. associated with. the initial data u_{0}.. This result dates back to the u. \in. pioneering. work. C_{w}^{0}([0+\infty);L_{ $\sigma$}^{2}( $\Omega$))\cap L^{2}((0, +\infty);H^{1}( $\Omega$)). partial regularity property: for. almost every t in. \overline{2\mathrm{O}\mathrm{u}\mathrm{r}} the where isa smooth anaJysis ako covers. case. .. $\alpha$. [21] by Leray. where it is. Moreover, Leray proved. (0, +\infty) u(t, \cdot) ,. matrix‐valued function.. is. C^{\infty}( $\Omega$). .. proved that the. following.

(3) 164. Even. though Leray’s almost. paper tackled the. right. away to the. The control. problem. adapted. of the. case. no‐slip condition, this. of the Navier. case. slip‐with‐friction. result. condition. can. be. (see [17,. Section 3. 4. We. that. now assume. ary \partial $\Omega$. In. .. particular. we are. have in mind. now. \partial $\Omega$\backslash $\Sigma$. On the part. \bullet. condition u=0 source once. .. terms. and for. is the. or. non‐empty. open. that the fluid is. part $\Sigma$ of the full bound‐ same. volume. incompressible).. Then the. setting. boundary. prescribed,. either the. no‐slip. conditions. are. N(u). the Navier condition u\cdot n=0 and. ability. or. on a. fluid enter into the domain and the. following (see Figure 1).. some. ,. some. (recall. of fluid go out of the domain we. able to act. may let. we. to. modify. the. slip. =0. (that. is without any. which is assumed to be. given. condition which is relevant for. some. coefficient. $\alpha$. all).. On the part $\Sigma$. we are. ,. free to choose. a. boundary. purpose.. 1 : The control. More some. precisely. given. we. state at. problem. have in mind to drive the system from. some. Lions in the late 80 ’s. given. (cf.. time. The. for instance. following goal,. [22]). tackles the. an. first case. arbitrary. initial data to. suggested by Jacques‐Louis where the. target. is the rest. state.. Open. Problem. (OP).. For any T>0 and u_{0} in. the Navier‐Stokes system with. u(0, \cdot)=u_{0}. Above the Navier‐Stokes system to which Navier‐Stokes. \partial $\Omega$\backslash $\Sigma$. ,. equations (1.1). but without any. such that. (OP). in $\Omega$ and of the. boundary. L_{ $\sigma$}^{2}( $\Omega$). condition. ,. does there exist. u(T, \cdot)=0. no‐slip. prescribed. on. or. solution to. ?. refers is constituted of the condition. a. incompressible. the Navier condition. on. the controlled part $\Sigma$ of the.

(4) 165. boundary. Such. system is therefore under‐determined. a. expected (even. not. uncontrolled. in the 2\mathrm{D}. case. be recovered. for which uniqueness of. setting corresponding. above the control is implicit:. a. the trace. by taking. so. to the. case. $\Sigma$ of. a. Leray. a. solution is. solutions is known in the. where $\Sigma$=\emptyset ). Indeed in the formulation. relevant condition to on. that uniqueness of. prescribe. as. control. a. on. $\Sigma$. can. convenient solution to the under‐determined. system. Observe that there is. no. restriction. L_{ $\sigma$}^{2}( $\Omega$). the initial data u_{0} in. .. In the. amounts to the small‐time. precisely. the small‐time. .rest state and has to be. the sizes neither of the time T>0. of control. exact null. controllability. a. since the. nor. of. positive result to this. terminology theory global controllability of the Navier‐Stokes,. question. global. regarding. target. in. or more. (OP). is the. reached exactly.. Our result. 5. original question, the boundary condition on the uncontrolled part \partial $\Omega$\backslash $\Sigma$ of boundary is the no‐slip boundary condition. Our goal here is to present the following. In Lions’ the. result are. establishing. prescribed. on. positive. a. answer. to. (OP). in the. case. where. some. Navier conditions. \partial $\Omega$\backslash $\Sigma$.. Theorem 5.1 Let T>0 and. u_{0}\in L_{ $\sigma$}^{2}( $\Omega$). There exists. .. a. weak solution. u. to. \left{bginary}{l \partil_{}u+(\cdotnabl)u-$\Deltau+\nblap=0&in$\Omega$,\ mathr{d}\mathr{i}\mathr{v}u=0&in$\Omega$,\ ucdotn=0&o\partil$\Omega$\bckslah$\Sigma,\ N(u)=0&on\partil$\Omega$\bckslah$\Sigma \end{ary}\ight.. satisfying u(0, \cdot)=u_{0}. and. u(T, \cdot)=0.. Theorem 5.1 does not require any condition tion. (2.2). of N. .. Indeed,. observe that there is. above. Still the next lines about the Let. that. us. one. intercept. on. no. the coefficient. appearing. asymptotic parameter. of it will be full of. [6]. $\alpha$. are more. in the defini‐. in the statement. $\epsilon$.. general,. in. particular they. prove. at time T any smooth uncontrolled solution to the Navier‐Stokes. system with Navier condition. 6. proof. also mention that the results in may. (5.1). on. the full. boundary \partial $\Omega$.. Earlier results When. since the. Jacques‐Louis answer was. Lions formulated it in the late 80' \mathrm{s} ,. not. even. known in the. case. (OP). was. pretty impressive. of the heat equation. For this equation.

(5) 166. the first. key breakthroughs. respectively associated. obtained. with. been then extended to the Stokes. in the. case. that the can. of small initial data. quadratic. by [20]. [19]. equations and later. [18].. Imanuvilov in. by. convective term may be. be obtained from the. and. thanks to Carleman. inequalities parabolic and elliptic second order operators. The latter has. were. seen as a. on. to the Navier‐Stokes. The smallness. perturbation. equations. assumption implies. term. so. that the result. controllability of the Stokes equations by a fixed point strategy. improved in [7] by Fernández‐Cara, Guerrero, Imanuvilov and. This result has since been Puel.. All these works deal with the. case. local null The time. let. no‐slip boundary condition. For Navier slip‐. us. of large initial data was first tackled in. case. global result. in. a. 2\mathrm{D}. sufficient to conclude using. a. is. (for large smooth. initial. [5]. a source. the return method and to the. good,. particular. where the first author proves. but the estimates up to the. known local result. In. first author and Fursikov prove in. setting, the control. [3],. which prove in. a. small‐. setting with Navier boundary conditions: the smallness obtained. within the inside of the domain is. this. of the. mention [14] and [16] boundary conditions, controllability when the initial data is small.. with‐friction. a. small‐time. global. term located in. a. global controllability. data). Likewise,. in. [8],. fact, when there exact null. is. no. are. not. boundary,. the. controllability. small subset of the of the. boundary. domain). result. (in. thanks to. incompressible Euler equations. Fursikov and Imanuvilov prove small‐time. supported on the whole boundary (i.e. controllability obtains null exact global controllability for Navier‐Stokes in a $\Gamma$=\partial $\Omega$) [1], Chapouly 2\mathrm{D} rectangular domain under Lions’ boundary condition (corresponding to the case where global. exact null. when the control is. In. .. $\alpha$=0 in the Navier. Still the. condition). approaches. on. the uncontrolled part of. boundary.. used in the aforementioned papers failed to deal with the viscous. boundary layers appearing near the uncontrolled part of the boundary. This is precisely the goal of this paper to promote the well‐prepared dissipation method in order to obtain some. controllability results despite. introduced in 1\mathrm{D}. [24] by. be crucial in. viscosity. a. our. a. This method. controllability. The extension of this method to the Navier‐Stokes. proof of Theorem. multi‐scale. 5.1. In. particular. expansion describing. limit of the Navier‐Stokes. in Section 12.. boundary layers.. the second author in order to deduce. Burgers equation.. thanks to. the presence of. the. was. result for the. equations. here the method will be. first. will. implemented. in the. boundary layer occurring vanishing equations. The application of the method is presented.

(6) 167. A few words of caution. 7. Next sections. few. a. the. clarity.. whereas. sense. [6]. We refer to. that it will be relevant. for. one. an. on. highlight. purpose for. complete proof. not. on some. on. going. to. really. use a. control all the time in. time intervals to choose. \partial $\Omega$\backslash $\Sigma$. so. as. boundary. condition. that the system \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} coincides with. $\Sigma$=\emptyset.. for which. Reduction to. a. we are. Navier condition than. same. the uncontrolled. 8. technical difficulties will be omitted. some. also mention here that. us. $\Sigma$ the. on. devoted to the scheme of proof of Theorem 5.1. We will try to. key ingredients. the sake of Let. are. from. approximate controllability problem. a. smooth initial data In this section. starting from in any. we are. an. going to. prove that it is sufficient to have the existence of. smooth initial data and. arbitrary. time in order to conclude the. positive. Leray’s partial regularity. result hinted above. (0, T/2). is. of. a. enough. u(t^{*}, \cdot). such that. solution to. starting. zero. from. L_{ $\sigma$}^{2}( $\Omega$). in. null. controllability. In order to. simplify. a. problem,. (3T/4, T). L_{ $\sigma$}^{2}( $\Omega$). assumed to be in. 2).. the notations let. in order to. below Theorem. controllability. 3.1),. in. L_{ $\sigma$}^{2}( $\Omega$). according. ,. to. there exists t^{*} in. we. .. us. on. only. to obtain. it from the. be. approximate. the intermediate time interval. pretend. can. Then the concatenation of this three. that this interval is. will denote u_{*} the initial. distinguish. results mentioned above. Our task is therefore. smooth initial data. sections. On the other hand new. zero. ,. time interval. from. (cf.. state close to. .. such that the local. the. proof. a. of Theorem 5.1. Indeed. solution. C^{\infty}( $\Omega$) Let us assume that we are able to prove the existence u(t^{*}, \cdot) at time t^{*} and reaching, say at time 3T/4 a state close. applied3 remaining steps yields Theorem 5.1 (see Figure on. reaching. a. original. data,. (0, T). (t^{*}, 3T/4). .. in the next. which is smooth, for this. initial data u_{0} which. was. only. .. A fast and furious control. 9. In order to take. matively. zero. and duration. '. profit. thanks to are. of the a. nonlinearity. at. our. advantage. we. control which is fast and furious in the. scaled with respect to. a. aim at sense. small positive parameter. $\epsilon$. reaching approxi‐ that its. amplitude. which is introduced. Results available in the local controllability literature require to start with an initial data which is more regular than L^{2} but Leray’s partial regularization of the uncontrolled Navier‐Stokes equations can be used again in order to glue these steps. Here we have to pay attention to the preservation of the smallness assumption in this regularization argument (cf. [6]. for. more)..

(7) 168. E2: Reduction. by. force and will be. (5.1). ultimately. a. global approximate controllability problem. taken small. Indeed. enough.. look for. we. a. solution. u. to. of the form. having on. to. in mind to look for. time interval of order. fast transitions. underlying. on. a. family. O(1). .. of functions u^{ $\epsilon$}. This. time interval of order. with, typically,. for the. means. O(\displaystyle \frac{1}{ $\Xi$}). (9.1). ,. original. with furious. variations of order. searched solution. amplitudes. of order. idea is to start with the ambitious idea to try to control the system. shorter time interval. Regarding. principle. u(t,x)=\displaystyle\frac{1}{$\epsilon$}u^{$\epsilon$}(\frac{t} $\epsilon$},x). [0, $\epsilon$ T]. with. forcing. the system to evolve in. the pressure p associated with the. original. solution. u,. which associates the pressure with the square of the. a. u. O(1). having. O(\displaystyle\frac{1}{$\epsilon$}). .. The. during the. high Reynolds regime.. having in mind Bernoulli’s velocity,. we. look for. an. ansatz of the form. This translates the. i). The. new. iii). original system. unknowns u^{ $\epsilon$} and p^{ $\epsilon$}. coefficient. ii). p(t,x)=\displaystyle\frac{1}{$\epsilon$^{2} p^{$\epsilon$}(\frac{t} $\epsilon$^{2} ,x) a new. system with four main changes:. satisfy the Navier‐Stokes equations with. a. small. viscosity. $\epsilon$.. The initial data for u^{ $\xi$ j} is The time interval is of the system. matter.. in. .. In. now. now. (0,\displaystyle\frac{T}{$\epsilon$}). small so. equal. that. we. particular, although. to $\epsilon$ u_{*}.. have to. investigate. the initial data is. the. large. small,. time behaviour. nonlinearities will.

(8) 169. The system for. iv). Last more. (u^{ $\epsilon$},p^{ $\epsilon$}). therefore reads:. \left{bginary}l \pt_{u^$esilon}+({\p$cdotnabl)u^{$\epsio}-Dlta$u^{\epsion}+abl^{$\epsion}=0&mathr{i\ n}(0,T/$\epsilo)tm$\Oega, mthr{d}\aimthr{v}u^$\epsilon=0&mathr{i}\ n(0,T/$\epsilo)tm$\Oega, u^{$\psilon}cdt=0&\mahr{o}tn(0,T/$\epsilo)tmarl$\Oegbackslh$\Sigm, N(u^{$\epsilon})=0&mathr{o\ n}(0,T/$\epsilo)tmarl$\Oegbackslh$\Sigm, u^{$\epsilon}|_t=0$\epsilonu_{*}&mathri\ {n}$Omega. \nd{ry}ight.. change. but not. least,. precision. Indeed. underdetermined system. (9.2). in order to deduce from. \Vert u(T, \cdot)\Vert_{L^{2}( $\Omega$)} =o(1) arbitrarily. Inviscid. When. $\epsilon$. is. the final time. plan. a. \displaystyle\Vertu^{$\epsilon$}(\frac{T} $\epsilon$},\cdot)\Vert_{L^{2}($\Omega$)}=o($\epsilon$). (9.1) .. it is. global following counterpart. with. that there exists. This will. small. $\epsilon$. provide. a. the. (9.3). solution. enough. u. (5.1). to. such that. allows to reach. a. approximate controllability. state. result. section.. expected that the analysis of the system (9.2). exact. controllability. of the system. (9.2). As the initial data is of order at least for times of order. O( $\epsilon$) O(1). where the. viscosity. in L^{\infty} it is natural to for. ,. into. on. term has been. a. solution. dropped. out:. (10.1). u^{E}. to. (10.1). which. of the form:. u^{E}= $\epsilon$ u^{1}+o( $\epsilon$) Plugging expansions (10.2). may be built. of Euler equations. We therefore consider the. \left{bginary}{l \parti_{}u^E+({}\cdotnabl)u^{E}+\nablp^{E}=0\mathr{i}\mathr{n}$\Omega,\ mthr{d}\mathr{i}\mathr{v}u^E=0\mathr{i}\mathr{n}$\Omega,\ u^{E}cdotn=0\mahr{o}\mathr{n}\patil$Omega\bckslh$\Sigma, u^{E}|_t=0$\epsilonu_{*}\mathr{i}\mathr{n}$\Omega. \nd{ray}\ight.. is,. T/e). solution u^{ $\epsilon$} to the. such that. particular, choosing. previous. targeted (at. now. to prove that there exists. flushing. small,. the small‐time. In. close to 0 in L^{2}. mentioned in the. 10. .. the rest state is. we now. (9.2). (10.1). and. and. p^{E}= $\epsilon$ p^{1}+o( $\epsilon$). grouping. terms of order. \left{bginary}l \ptia_{}u^1+\nblap{}=0&\mathr{i} mn$\Oega, mthr{d}\ami thr{v}u^1=0&\mathr{i} mn$\Oega, u^{1}\cdotn=0&mahr{o}\tmnparil$\Omegbackslh$\Sigma, u^{1}|_t=0 *&\mathr{i} mn$\Oega. nd{ry}\ight.. (10.2). .. O_{L}\infty( $\epsilon$) yields:. (10.3).

(9) 170. By elementary combinations of the equations not admit any. harmonic. so. ution. function,. controllability In order to. case. which will prevent from. this. overcome. by the first author auxiliary controlled will rather look for. difficulty. [2].. in. in. general. System (10.3). using. we are. it for. going. to. This method takes. (u^{0},p^{0}). where the extra‐term. gradient of. suffers from. a. our. purposes.. use. the return method first introduced an we. the form:. p^{E}=p^{0}+\in p^{1}+o( $\epsilon$). and. a. lack of. of the nonlinearity thanks to Indeed, instead expansions (10.2),. asymptotic expansions of. u^{E}=u^{0}+ $\epsilon$ u^{1}+o(\mathrm{s}). does. profit. solution to the Euler system. some. (10.3). observe that the system. 0 unless the initial data u_{*} is the. reaching exactly. which is not the. we. is introduced in order to. help. (10.4). ,. to control. (u^{1},p^{1}). .. Of. course. (u^{0},p^{0}) O(1) which appear when plugging the expansions (10.4) into the first three equations of (10.1). Moreover the last equation yields the initial data u^{0}|_{t=0}=0 in $\Omega$ The interest is that the equations obtained by gathering the terms of order O( $\epsilon$) are now: has to be solution to the Euler system in order to cancel out the terms of order. .. \left{bginary}{l \partil_{}u^1+({0}\cdotnabl)u^{1}+( \cdotnabl)u^{0}+\nablp^{1}=0&\mathr{i}\mathr{n}$\Omega,\ mathr{d}\mathr{i}\mathr{v}\mathr{u}\mathr{l}=0&\mathr{i}\mathr{n}$\Omega,\ u^{1}cdotn=0&\mathr{o}\mathr{n}\patil$Omega\bckslah.$\Sigma,\ u^{1}|_t=0u{*}&\mathr{i}\mathr{n}$\Omega. \nd{ary}\ight.. This is the linearisation of the Euler. (10.3)).. We may. u_{*} to 0 ,. see. rely. by u^{0}. to set. is if any fluid. some. time t_{x}\in. equal. which is. to 0. .. on. the. u^{1}. physical. (0, T). non. x. in $\Omega$. ,. a. we. a. course. obtain that if the. moves. we. u^{1} from. want to. use. (10.5). the system. incompressibility fluid particles are. time interval of order. for which it reaches $\Sigma$ with. Observe that this requires. around 0 like in. to drive. precisely. local features due to the. vorticity of u^{1} at. (10.5).. More. out of the domain. Of. domain within. particle initially. of. equation. transport aspect,. Still, reasoning. flushed outside of the. (that. equations around u^{0} (rather than by u^{0} (see Figure 3) in order. the transport. in order to flush. in addition to the. condition.. on. the second term in the first. the transport. has,. now. (10.5). O(1). ,. say. [0, T]. with the flow associated with u^{0} up a. positive velocity),. then u^{1}. can. time interval far smaller than the allotted. be. one. [0, T/ $\epsilon$].. On the other hand this to construct such. the Euler. a. field,. system enjoy. scalar function. $\alpha$(x). a. auxiliary a. field u^{0} also has to vanish at the final time. In order. crucial observation is that the. lot of freedom. regarding. potential flows. as. their behaviour in time.. solutions to. Indeed if. a. satisfies. \left{\begin{ar y}{l $\Delta$_{x} \alph$=0&\mathrm{i}\ athrm{n}$\Omega$,\ partil_{n}$\alph$=0&\mathrm{o}\mathrm{n}\partil$\Omega$\bckslah$\Sigma$, \end{ar y}\right.. (10.6).

(10) 171. \mathbb{H}3 : Inviscid flushing. $\eta$(t). then for any function. ,. $\eta$(t)\nabla_{x} $\alpha$(x). the vector field. satisfies the first three equations of. (10.1) for an appropriate pressure. In particular it is possible to choose a nonzero function $\eta$(t) satisfying $\eta$(0)= $\eta$(T) =0 so that this process leads to a field u^{0} starting from zero ,. initial data and which vanishes at time T. satisfying. the underdetermined Neumann. appropriate gluing strategy,. vector fields. .. Moreover the set of the scalar functions. problem (10.6). is rich. enough. flushing the whole domain. on. to. provide, by. an. the time interval. [0, T]. Lemma 10.1 There exists. a. smooth solution. (u^{0}, p^{0}). to. \left{bginary}{l \ptia_{}u^0+( \cdotnabl)u^{0}+\nablp^{0}=&in$\Omega, \thrm{d}a i\mthr{v}u^0=&in$\Omega, u^{0}\cdotn=&\partil$Omega\bckslh$\Sigma, u^{0}|_t= &i{7}$\Omega, u^{0}|_t=T&i{71}$\Omega, nd{ray}\ight.. such that the smooth solution This lemma is. respectively. proved. for 2\mathrm{D}. sects all connected. the. corresponding. In the. by. zero. The a. simply. the. one. to. system. hand. ,. when. on. in $\Omega$.. the first author in the papers. connected domains and for. components of \partial $\Omega$ and cases. by. (10.5) satisfies u^{1}|_{t=T}=0 general. the other hand. [2]. and. [4]. 2\mathrm{D} domains when $\Sigma$ inter‐. by Glass. in. [9]. and. [10]. for. in 3\mathrm{D}.. we. need. it,. we. will. implicitly. extend the. previous fields u^{0} and u^{1}. after T.. Boundary layer. 11. of. sequel,. on. (u^{1},p^{1}). (10.7). difficulty. comes. perfect fluid,. not. from the fact that the Euler. subject. to. friction,. is. only. equation,. which models the behavior. associated with the. boundary condition.

(11) 172. u\cdot n. 0 for. =. friction. boundary. condition. Navier‐Stokes equation where. expansion. wall and does not. impermeable. an. \partial $\Omega$\backslash $\Sigma$ An \partial $\Omega$\backslash $\Sigma$ even for. on. near. In the uncontrolled. setting for which $\Sigma$. Navier‐Stokes equation under the Navier. cosity limit. performed by. was. asymptotic expansion involving ness. \mathcal{O}(\sqrt{ $\epsilon$}). for. extended in the Let. equations. boundary \partial $\Omega$. temporarily again. The. .. $\epsilon$. .. to the controlled. the time interval. slip‐with‐friction. boundary layer. vanishing viscosity. sequel. here. us use on. a. [0, T]. Let. slip‐with‐. solution of the. a. to. use an. equation.. \emptyset the description of the behavior of the. =. condition in the. [17]. Iftimie and the third author in a. of. description. solution to the Euler. a. the Navier. general. in. viscosity, has therefore. small. a. ,. corrector is added to. a. satisfy. accurate. .. term. first. us. v. of. vanishing. thanks to. amplitude \mathcal{O}(\sqrt{ $\epsilon$}). briefly. vis‐. multiscale. a. and of thick‐. recall this result which will be. case.. the notation u^{0} for. with the. a. smooth solution to the Euler. impermeability condition u^{0}\cdot n=0 on the full. boundary layer corrector will involve an extra variable describing the velocity in the normal direction near the boundary and will be. fast variations of the fluid. given. as a. solution to. on. \partial $\Omega$, $\varphi$>0. in. a. small. in $\Omega$ and. (2.1),. u_{\mathrm{b}^{0} u^{0}\cdot n\prime=0 \partial $\Omega$,. .. a. of \partial $\Omega$. value. is extended. is not. singular. boundary layer. the. near. be. | $\varphi$(x)|=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x, $\Omega$). that. -\nabla $\varphi$. close. The notation. [\cdot]_{\tan},. computed .. $\chi$. =. 1. on. \partial $\Omega$. .. because of the. Taylor. for. Even. by a. a. x. in. v. expressed. z= $\varphi$(x)/\sqrt{ $\epsilon$}. ,. \overline{$\Omega$}. in. where. satisfies the equation:. \overline{$\Omega$} and. We refer to. only.. u_{\mathrm{b}^{0}. condition. is smooth in. smooth vector field. fast scalar variable. z. in. \mathbb{R}_{+}. ,. with the. following boundary condition. \partial_{z}v(t;x, 0)=g^{0}(t, x) us. $\varphi$ vanishes. impermeability. \partial_{t}v+ [(u^{0}\cdot\nabla)v+(v\cdot\nabla)u^{0}]_{\tan}+u_{\mathrm{b}}^{0}z\partial_{z}v-\partial_{zz}v=0 for. definitions:. x\in $\Omega$,. though. expansion proves that. corrector will be described. as. following. g^{0}(t, x):=2 $\chi$(x)N(u^{0})(t, x). terms both of the slow space variable x\in $\Omega$ and. v(t, x, z). \mathbb{R}^{d}\rightar ow \mathbb{R} such that $\varphi$=0. :. We also introduce the. boundary. smooth,. n can. condition with. boundary. within the full domain $\Omega$. satisfying a. a. we assume. .. accordingly. and. with. smooth function $\varphi$. \overline{$\Omega$} Moreover,. smoothly. smooth cut‐off function. Indeed since u^{0} is. a. problem. Hence, the normal. .. -\displaystyle \frac{u^{0}(t,x)\cdot n(x)}{ $\varphi$(x)}. u_{\mathrm{b} ^{0}(t, x. The. boundary. outside of. and extended. boundary. where $\chi$ is. $\varphi$<0. neighborhood. introduced in. on. initial. to this extra variable. We introduce. respect. to the. an. [17,. Section. 2]. for.. a. mention here that these. at z=0 :. (11.2). .. detailed heuristic of the equations. equations. are. obtained. u^{0}(t, x)+\displaystyle \sqrt{ $\epsilon$}v(t, x, \frac{ $\varphi$(x)}{\sqrt{ $\epsilon$} ). (11.1). ,. (11.1). by plugging. instead of. u^{ $\epsilon$}(t, x). and. (11.2).. Let.

(12) 173. equations of (9.2) and keeping the. into the first and fourth. into account that u^{0} satisfies the Euler. expanded. as. the latter. can. and. well,. into the. sum. Cauchy problem. Moreover for any. (11.1). associated with. x\in\overline{ $\Omega$},. boundary layer. p^{ $\epsilon$} has. z\geq 0 and t\geq 0. ,. we. and. as a. to be. term but. boundary layer. term acts. the normal. projection. (11.1) only tangential. (11.2) is well‐posed in Sobolev. the second term in. why. a. (taking. order. higher. resulting equation by distinguishing. Thus this pressure. convective terms and this is. The. of the Euler pressure and of. be eliminated from the. tangential parts.. equations).. terms of. Indeed the pressure. on. spaces.. have. v(t, x, z)\cdot n(x)=0. (11.3). It is easy to check that the solution inherits this condition from the initial and. orthogonality property is the reason why equation (11.1) quadratic term (v\cdot n)\partial_{z}v should have been taken into account if. data. the. This. Thanks to the cut‐off function $\chi$ near. \partial $\Omega$ while ensuring that ,. v. ,. the. is. satisfying $\chi$=1. on. \partial $\Omega$,. v. boundary. is linear.. Indeed,. it did not vanish. in. compactly supported. is. compensates the Navier slip‐with‐friction boundary. x. trace. of u^{0}. Then it is. proved. be described. in. the. by. [17]. that the. Leray. following expansion. solutions u^{ $\epsilon$} to the Navier‐Stokes. in. L^{\infty}((0, T);L^{2}( $\Omega$)). u^{ $\epsilon$}(t, x)=u^{0}(t, x)+\displaystyle \sqrt{ $\epsilon$}v(t, x, \frac{ $\varphi$(x)}{\sqrt{ $\epsilon$} ) +O( $\epsilon$) Let. us. that this. highlight. solution to the Euler. fails to describe the of order Now. O($\xi$\displaystyle\frac{1}{j ). ,. even. going back. expansion holds. equations. on. vanishing viscosity in the. case. [0, T]. to the controlled. setting for which. describe the behavior of the Navier‐Stokes equation of the since. boundary. we. aim at. in the. finding. vanishing viscosity a. u^{0} is. smooth. analysis equation for large times. stays smooth for all. \neq \emptyset. $\Sigma$. near. we. expect. times.. to be able to. the uncontrolled part. limit thanks to. Navier‐Stokes solution. a. On the other hand this. .. limit of the Navier‐Stokes. where the Euler solution. can. .. up to any time T>0 for which. the time interval. equation. :. a. similar. satisfying (9.3). we. \partial $\Omega$\backslash $\Sigma$. expansion. Indeed. consider the. following. refined expansion:. u^{\mathrm{g} (t, x)=u^{0}(t, x)+\displaystyle \sqrt{ $\epsilon$}v(t, x, \frac{ $\varphi$(x)}{\sqrt{ $\epsilon$} ) + $\epsilon$ u^{1}(t, x)+ $\epsilon$ r^{ $\epsilon$}(t, x) where u^{0} and u^{1}. \displayte\frac{T}$\epsilon$}. .. are as. Lemma 10.1 and the vector field r^{ $\epsilon$} is wished to be. If so, and since the fields u^{0} and u^{1}. theexpansion (11.4). (11.4). ,. after T is given. therefore must understand the. large. are zero. by. after the time T , the. the second term in the. time behavior of this. o(1). at time. leading part. of the. hand side and. right boundary layer.. For. we. t\geq T the ,.

(13) 174. equations (11. 1) and (11.2) reduce. where the slow variables. \overline{v}(x, z) :=v(T, x, z) decay (that. \left\{ begin{ar y}{l \parti l_{t}v-\parti l_{z}v=0,&\mathrm{f}\mathrm{o}\mathrm{r}z\in\mathb {R}_{+},\ \parti l_{Z}v(t,x 0)= ,& \end{ar y}\right. \in. x. \mathcal{O}. the role of parameters. play. dissipates towards the null equilibrium. is without any. assumption. on. v). fact that the average of. v. is. preserved. fact that the energy contained. by. low. frequency. overcome. the previous. (11.6). ,. (9.3). Physically,. modes. will here in the. adjust. vanishing viscosity. difficulty. we are. going. to. use. boundary layers. .. the. well‐prepared. in order to obtain. in the presence of. a. dis‐. a new. boundary layer.. We. associated with the Navier conditions. limit of the Navier‐Stokes. control strategy in order to enhance the natural the time T. and to the. method. Burgers equation. the method to the. this is due to the. by equation (11.5) decays slowly.. sipation method first introduced in [24] by the second author result of the 1\mathrm{D}. the nat‐. t=T/ $\epsilon$ only yields. under its evolution. Well‐prepared dissipation. controllability. the “initial” data. Unfortunately. at the final time. which is not sufficient in view of the wished estimate. In order to. through. state.. \displayst le\Vert\sqrt{$\epsilon$}v(\frac{T} $\epsilon$},\cdot,\frac{$\varphi$(\cdot)}{\sqrt{$\epsilon$})\Vert_{L^2}($\Omega$)}=\mathcal{O}($\epsilon$). 12. (11.5). .. This heat system ural. to. equations. The idea is. dissipation. Our strategy will be to guarantee that \overline{v} satisfies. of the. to. design. boundary layer. a. after. finite number of vanishing. a. moment conditions for k\in \mathbb{N} of the form:. \displaystyle \foral x\in $\Omega$, \int_{\mathb {R}_{+} z^{k}\overline{v}(x, z)\mathrm{d}z=0 This will allow to enhance the. dissipation. and to. improve the. \displayst le\Vert\sqrt{$\epsilon$}v(\frac{T} $\epsilon$},\cdot,\frac{$\varphi$(\cdot)}{\sqrt{$\epsilon$})\Vert_{L^2}($\Omega$)}=o($\epsilon$) Actually physical across. $\Sigma$. we. aim at. constructing. (and. can. we. intend to find. a. analogous problems. be chosen smooth, bounded and. compactly supported. estimate. in the added. solution that. in. still denote. an. so. far. larger. by. of the. equations:. \partial_{t}u^{ $\epsilon$}+(u^{ $\epsilon$}\cdot\nabla)u^{ $\epsilon$}- $\epsilon \Delta$ u^{ $\epsilon$}+\nabla p^{ $\epsilon$}=$\zeta$^{ $\epsilon$}. restriction to the. domain \mathcal{O} extended. simply connected). (u^{ $\epsilon$},p^{ $\epsilon$}). into. (12.2). portion of the domain. This we. (11.6). .. the different fields mentioned. domain $\Omega$ of solutions to O. (12.1). .. means. with in. source. terms. particular that. following Navier‐Stokes.

(14) 175. for. x. in \mathcal{O} where the. source. term. $\zeta$^{ $\epsilon$}(t, x). is. vector field. a. supported for. in. x. \mathcal{O}\backslash\overline{$\Omega$}. ,. of the. form. $\zeta$^{$\epsilon$}=$\zeta$^{0}(t,x)+\displaystyle\sqrt{$\epsilon$} \zeta$_{v}(t,x,\frac{$\varphi$(x)}{\sqrt{$\xi$\mathrm{i} )+$\epsilon\zeta$^{1}(t,x) $\zeta$^{0}. where. ,. $\zeta$^{1} are smooth vector fields used in order to insure Lemma 10.1 whereas $\zeta$_{v}(t, x, z) is devoted to the control of the moments of the boundary layer. now aim at obtaining a profile v solution to the following equation:. and. the vector field Indeed. we. \partial_{t}v+ [(u^{0}\cdot\nabla)v+(v\cdot\nabla)u^{0}]_{\tan}+u_{\mathrm{b}}^{0}z\partial_{z}v-\partial_{zz}v=$\zeta$_{v} for. x. in Ư and. z. of. an. (see. the. Since the initial. .. (see. addend due to the. control are. \mathbb{R}_{+}. in. moments at time T. right. right hand. according interval. [0, T]. ,. value satisfied. (12.1)), (11.2) and of. hand side of. side of. convected inside the domain to Duhamel’s. boundary. the left hand side of. formula,. (12.3)),. by. can. an. which generates. the field u^{0} ,. see. by. decomposed. some. given by. (12.1). is. v. linear, the. as. its. sum. addend due to the outside moments. the second term in. the second addend is. which allows to insure the condition. be. (12.3). ,. an. for all. x. outside,. and. (11.1). Indeed,. integral. over. the time. in $\Omega$.. \mathrm{H}4 : The kaitenzushi strategy. Let. us use. here the. following metaphor:. see. the extended domain. sushi restaurant, the added part of the extended domain as. the. plates (see Figure4).. the chef into the serve. dining. room,. the wished moments. boundary). all. to the time T. along .. the. In order to send. (compensating. boundary. some. what. by. comes. linearity (otherwise (v\cdot n)\partial_{z}v mentioned above).. boundary layer. of the. v. equation. Let. us. because the. plates. need. seems. some. a. conveyor‐Uelt. kitchen, as a. without. sending. conveyor belt to. from the uncontrolled part of the. service, which corresponds here. dramatically. observe that it. the. the field u^{0}. before the end of the. would. as. the kitchen and the moments. plates from. the transport. In this process it is crucial to maintain the. the. the. we use. as. orthogonality. condition. (11.3). fall down because of the term. impossible. time to be. to control. conveyed. completely. from the kitchen.

(15) 176. and. therefore. are. they. are. data. on. a. strongly regularized. supposed. to be. compensation is only possible for. types of contributions and. (the equation (11.2). we. a. precisely. on a. of. use. the. functional space. some. in. parabolic. and is therefore far less. projection make. instantly from. comes. boundary. is. z). when. nonhomogeneous Thus. regularized.. containing the. finite dimensions. two. projections by. finite number of moments.. a. Estimates of the remainder. 13. Going the. z. compensating what. the uncontrolled part of the. adjusting. in. velocity expansion (11.4). back to the. time behaviour of the remainder r^{ $\epsilon$}. large. solution to. .. led. are. we. The field ré. now can. to the issue of. be. naturally. estimating. defined. as. the. Navier‐Stokes type equation of the form:. a. \partial_{t}r^{ $\epsilon$}+(u^{ $\epsilon$}\cdot\nabla)r^{ $\epsilon$}-\mathrm{e} $\Delta$ r^{ $\epsilon$}+\nabla$\pi$^{ $\epsilon$}+A^{ $\epsilon$}r^{ $\epsilon$}=f^{ $\epsilon$}. (13.1). ,. where $\pi$^{ $\epsilon$} denotes the pressure associated with the vector field r^{ $\epsilon$} the notation A^{ $\epsilon$} stands ,. for. an. amplification operator. \mathrm{o}\mathrm{m}_{\backslash} itted in the equations of. and. f^{ $\epsilon$} for. u^{1} and. u^{0},. v. term both due to the terms which. a source. for. of. being. bears the initial data u_{*} , this remainder starts with. higher. a zero. order in. initial data. $\epsilon$. .. were. Since the field u^{1}. (taking. into account that u^{0} and. the trace at. zero equality (11.4) taking initial data) but is generated by the source term f^{ $\epsilon$} and possibly amplified by mean of the term A^{ $\xi$ j}r^{ $\epsilon$} Of course the equation (13.1) is completed with the divergence free condition. the initial time of the. and. v. start with. .. and. some. of r^{ $\epsilon$}. initial and. are on. the. one. the third term of. boundary. conditions. Here the. hand that the. (13.1). is tamed. key points. quadratic nonlinearity. by. a. factor. $\epsilon$ , see. f^{ $\epsilon$}. are. hinted in Section 12. Let. [6]. for. refer. conclude here that the result of. once more. an. to. in term of r^{ $\epsilon$} which is hidden in. more on. the technicalities and. only. energy estimate is that. (13.2). .. Conclusion. Taking plugged. into account that the fields u^{0} and u^{1} vanish after T , estimates. into. expansion (11.4) yield (9.3). thanks to the state. time estimate. velocity expansion (11.4), and on tamed by the enhanced dissipation. \displaystyle\Vert ^{$\epsilon$}(\frac{T}{$\epsilon$},\cdot)\Vert_{L^{2}($\Omega$)}=o(1) 14. large. the. the other hand that the effects of both A^{ $\epsilon$} and us. in the. during. and therefore conclude the. The main steps of the. strategy is pictured. proof. can. in. proof. and. (13.2). of Theorem 5.1. in Sections 8 and 9. The evolution of the. preliminary reductions performed. the control. (12.2). Figure. be summarized. as. 5.. in Table 1..

(16) 177. \otimes 5 : Four main steps of the evolution. \ovalbox{\t smal REJ CT}. 1 : Main features of the control. steps. Perspectives. 15 Let. us. mention. 1. Provided. a. a. few questions. smooth initial. system reaching 2. Is it. possible. zero. inspired by. data,. is there. at time T ? The 2\mathrm{D}. to deduce from the. this work: a. strong solution. case. previous analysis. results? This would extend the results obtained in Euler. equations. and in. related to the previous. [13]. for the. one as. to the 3\mathrm{D} Navier‐Stokes. follows from Theorem 5.1. some. Lagrangian controllability. [11], [12]. for the. Stokes. incompressible. stationary equation.This actually Lagrangian setting requires enough regularity for the issue is. flow to be controlled. 3. Last but not. least,. is it. possible. no‐slip boundary condition, very. challenging. boundary layers. open. at least for. problem. that have. a. to tackle some. because the. (OP). in the. favorable. more. difficult. case. geometric settings? condition. of the. This is. a. rise to. no‐slip boundary gives larger amplitude than Navier slip‐with‐friction boundary.

(17) 178. layers. on. We refer to the nice recent survey. boundary layers. in the. no‐slip. [23] by. Maekawa and Mazzucato for. more. case.. Acknowledgements The two first authors. were. partly supported by ERC Advanced Grant. 266907. (CP‐. Programme (FP7). DENL) Agence Nationale de la Recherche, Project DYFICOLTI, grant ANR‐13‐BS01‐0003‐01, Project IFSMACS, grant ANR‐15‐CE40‐0010 for their financial support and the third author thanks the hospitality of RIMS during the workshop on “Mathematical Analysis of the 7th Research Framework. of Viscous. Incompressible. The third author thanks the. Fluid”. References. [1]. Marianne. Chapouly. On. with Navier. the. global. null. controllability of. conditions.. slip boundary. J.. a. Navier‐Stokes system. Differential Equations, 247(7):2094-. 2123, 2009.. [2]. Jean‐Michel Coron. Contrôlabilité exacte frontière de. ides. parfaits incompressibles. 317(3):271-276, [3]. Jean‐Michel Coron.. Optim. Calc. Var.,. On the controllability of the 2‐D incompressible Navier‐. [5]. 1:35−75. Jean‐Michel Coron. J. Math. Pures. [6]. slip boundary conditions.. ESAIM Contrôle. (electronic), 1995/96.. On the controllability of 2‐D incompressible. Appl. (9), 75(2):155-188. ,. on a. perfect. fluids.. 1996.. Jean‐Michel Coron and Andrei Fursikov. Global exact. Navier‐Stokes equations. 4(4):429-448,. d’Euler des flu‐. 1993.. Stokes equations with the Navier. [4]. l’équation. bidimensionnels. C. R. Acad. Sci. Paris Sér. I Math.,. manifold without. controllability of. boundary.. Russian J. Math.. the 2D. Phys.,. 1996.. Jean‐Michel Coron, Frédéric Marbach, Franck Sueur, Small time null. controllability. boundary. of the Navier‐Stokes equation with Navier. conditions.. Preprint. 2016.. http: // arxiv. \mathrm{o}\mathrm{r}\mathrm{g}/\mathrm{a}\mathrm{b}\mathrm{s}/1612.08087.. [7] Enrique Fernández‐Cara, Sergio Guerrero, Oleg Imanuvilov, Puel.. Local exact. controllability. Appl. (9), 83(12):1501-1542. ,. 2004.. global exact slip‐with‐friction. of the Navier‐Stokes system.. and Jean‐Pierre. J. Math. Pures.

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