Local controllability to trajectories for non-homogeneous incompressible Navier–Stokes equations
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 529-574.

The goal of this article is to show a local exact controllability to smooth (C2) trajectories for the density dependent incompressible Navier–Stokes equations. Our controllability result requires some geometric condition on the flow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the flow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations.

DOI : 10.1016/j.anihpc.2014.11.006
Mots-clés : Non-homogeneous Navier–Stokes equations, Local exact controllability to trajectories, Carleman estimates 
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     title = {Local controllability to trajectories for non-homogeneous incompressible {Navier{\textendash}Stokes} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Badra, Mehdi; Ervedoza, Sylvain; Guerrero, Sergio. Local controllability to trajectories for non-homogeneous incompressible Navier–Stokes equations. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 529-574. doi : 10.1016/j.anihpc.2014.11.006. https://www.numdam.org/articles/10.1016/j.anihpc.2014.11.006/

[1] Albano, P.; Tataru, D. Carleman estimates and boundary observability for a coupled parabolic–hyperbolic system, Electron. J. Differ. Equ., Volume 22 (2000) 15 pp. (electronic) | MR | Zbl

[2] Boyer, F. Trace theorems and spatial continuity properties for the solutions of the transport equation, Differ. Integral Equ., Volume 18 (2005) no. 8, pp. 891–934 | MR | Zbl

[3] Boyer, F.; Fabrie, P. Outflow boundary conditions for the incompressible non-homogeneous Navier–Stokes equations, Discrete Contin. Dyn. Syst., Ser. B, Volume 7 (2007) no. 2, pp. 219–250 (electronic) | MR | Zbl

[4] Boyer, F.; Fabrie, P. Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013 | DOI | MR | Zbl

[5] Chaves-Silva, F.W.; Rosier, L.; Zuazua, E. Null controllability of a system of viscoelasticity with a moving control | arXiv | DOI | Zbl

[6] Coron, J.-M. On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var., Volume 1 (1995/96), pp. 35–75 (electronic) | Numdam | MR | Zbl

[7] Coron, J.-M. On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. (9), Volume 75 (1996) no. 2, pp. 155–188 | MR | Zbl

[8] Coron, J.-M.; Fursikov, A.V. Global exact controllability of the 2D Navier–Stokes equations on a manifold without boundary, Russ. J. Math. Phys., Volume 4 (1996) no. 4, pp. 429–448 | MR | Zbl

[9] Desjardins, B. Linear transport equations with initial values in Sobolev spaces and application to the Navier–Stokes equations, Differ. Integral Equ., Volume 10 (1997) no. 3, pp. 577–586 | MR | Zbl

[10] DiPerna, R.J.; Lions, P.-L. Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989) no. 3, pp. 511–547 | DOI | MR | Zbl

[11] Ervedoza, S.; Glass, O.; Guerrero, S.; Puel, J.-P. Local exact controllability for the one-dimensional compressible Navier–Stokes equation, Arch. Ration. Mech. Anal., Volume 206 (2012) no. 1, pp. 189–238 | DOI | MR | Zbl

[12] Fernández-Cara, E. Motivation, analysis and control of the variable density Navier–Stokes equations, Discrete Contin. Dyn. Syst., Ser. S, Volume 5 (2012) no. 6, pp. 1021–1090 | MR | Zbl

[13] Fernández-Cara, E.; Guerrero, S. Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., Volume 45 (2006) no. 4, pp. 1399–1446 (electronic) | DOI | MR | Zbl

[14] Fernández-Cara, E.; Guerrero, S.; Imanuvilov, O.Y.; Puel, J.-P. Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl. (9), Volume 83 (2004) no. 12, pp. 1501–1542 | DOI | MR | Zbl

[15] Fernández-Cara, E.; Guerrero, S.; Imanuvilov, O.Y.; Puel, J.-P. Some controllability results for the N-dimensional Navier–Stokes and Boussinesq systems with N−1 scalar controls, SIAM J. Control Optim., Volume 45 (2006) no. 1, pp. 146–173 (electronic) | DOI | MR | Zbl

[16] Fursikov, A.V.; Èmanuilov, O.Yu. Exact controllability of the Navier–Stokes and Boussinesq equations, Usp. Mat. Nauk, Volume 54 (1999) no. 3(327), pp. 93–146 | MR | Zbl

[17] Fursikov, A.V.; Imanuvilov, O.Y. Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996 | MR | Zbl

[18] Fursikov, A.V.; Imanuvilov, O.Yu. Local exact boundary controllability of the Boussinesq equation, SIAM J. Control Optim., Volume 36 (1998) no. 2, pp. 391–421 | DOI | MR | Zbl

[19] Girault, V.; Raviart, P.-A. Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer, Berlin, 1986 | MR | Zbl

[20] Glass, O. Exact boundary controllability of 3-D Euler equation, ESAIM Control Optim. Calc. Var., Volume 5 (2000), pp. 1–44 (electronic) | DOI | Numdam | MR | Zbl

[21] González-Burgos, M.; Guerrero, S.; Puel, J.-P. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., Volume 8 (2009) no. 1, pp. 311–333 | MR | Zbl

[22] Imanuvilov, O.Y. Remarks on exact controllability for the Navier–Stokes equations, ESAIM Control Optim. Calc. Var., Volume 6 (2001), pp. 39–72 (electronic) | DOI | Numdam | MR | Zbl

[23] Imanuvilov, O.Y.; Puel, J.-P. Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., Volume 16 (2003), pp. 883–913 | MR | Zbl

[24] Imanuvilov, O.Y.; Puel, J.-P.; Yamamoto, M. Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math., Ser. B, Volume 30 (2009) no. 4, pp. 333–378 | DOI | MR | Zbl

[25] Imanuvilov, O.Y.; Yamamoto, M. Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., Volume 39 (2003) no. 2, pp. 227–274 | DOI | MR | Zbl

[26] Martin, P.; Rosier, L.; Rouchon, P. Null controllability of the structurally damped wave equation with moving control, SIAM J. Control Optim., Volume 51 (2013) no. 1, pp. 660–684 | DOI | MR | Zbl

[27] Tucsnak, M.; Weiss, G. Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts, vol. XI, Springer, 2009 | MR | Zbl

[28] Zuazua, E. Log-Lipschitz regularity and uniqueness of the flow for a field in (Wlocn/p+1,p(Rn))n , C. R. Math. Acad. Sci. Paris, Volume 335 (2002) no. 1, pp. 17–22 | DOI | MR | Zbl

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