December  2012, 5(6): 1021-1090. doi: 10.3934/dcdss.2012.5.1021

Motivation, analysis and control of the variable density Navier-Stokes equations

1. 

Departamento EDAN, University of Sevilla, 41012 Sevilla, Spain

Received  December 2011 Revised  March 2012 Published  August 2012

The main objective of these Notes is to provide an introduction to variable density NS: their motivation, some of the main mathematical problems connected with them, the main techniques used to solve these problems, the main results and open questions. First, we will describe the physical origin of the equations. Then, we will be concerned with existence, uniqueness, regularity and control of initial-boundary value problems in cylindrical domains $ Ω $ $\times (0,T)$; as usual, $ Ω $ is the spatial domain, an open set in $\mathbb{R}$2 or $\mathbb{R}$3 ``filled'' by the fluid particles and (0,T) is the time observation interval. Some open problems (not all them of the same difficulty) are also recalled.
Citation: Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
References:
[1]

R. A. Adams, "Sobolev Spaces," volume 65 of "Pure and Applied Mathematics," Academic Press, 1975.

[2]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin., "Optimal Control," Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. Translated from the Russian by V. M. Volosov.

[3]

C. Amrouche and V. Girault, On the existence and regularity of the solutions of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175. doi: 10.3792/pjaa.67.171.

[4]

S. Anita, V. Arnautu and V. Capasso, "An Introduction to Optimal Control Problems in Life Sciences and Economics," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2011.

[5]

F. Araruna and E. Fernández-Cara, On the controllability of nonhomogeneous viscous fluids, To appear, 2012.

[6]

V. Arnautu and P. Neittaanmaki, "Optimal Control From Theory to Computer Programs," volume 111 of "Solid Mechanics and its Applications," Kluwer Academic Publishers Group, Dordrecht, 2003.

[7]

J. L. Boldrini and M. A. Rojas-Medar, Global solutions to the equations for the motion of stratified incompressible fluids, Mat. Contemp., 3 (1992), 1-8.

[8]

J. L. Boldrini and M. A. Rojas-Medar, Global strong solutions of the equations for the motion of nonhomogeneous incompressible fluids, In C. Conca and G. N. Gatica, editors, "Numerical Methods in Mechanics," number 371 in Pitman Res. Notes Math. Ser., pages 35-45, Harlow, 1997, Longman.

[9]

R. C. Cabrales and E. Fernández-Cara, Numerical control of some solidification processes, To appear, 2012.

[10]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, In "Handbook of Mathematical Fluid Dynamics, Vol. III," pages 161-244, North-Holland, Amsterdam, 2004.

[11]

J.-Y. Chemin, Le système de Navier-Stokes incompressible soixante dix ans apr\`es Jean Leray, In "Actes des Journées Mathématiques à la Mémoire de Jean Leray," volume 9 of "Sémin. Congr.," pages 99-123, Soc. Math. France, Paris, 2004.

[12]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201. doi: 10.1081/PDE-120021191.

[13]

A. J. Chorin and J. E. Marsden, "A Mathematical Introduction to Fluid Mechanics," volume 4 of "Texts in Applied Mathematics," Springer-Verlag, New York, third edition, 1993.

[14]

P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.

[15]

J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Contrôle Optim. Calc. Var., 1 (1995/96), 35-75 (electronic). doi: 10.1051/cocv:1996102.

[16]

J.-M. Coron and A. V. Fursikov, Global exact controllability of the $2$D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys., 4 (1996), 429-448.

[17]

J.-M. Coron and S. Guerrero, Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl. (9), 92 (2009), 528-545.

[18]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921. doi: 10.1016/j.jde.2008.10.019.

[19]

R. Danchin, On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework, J. Differential Equations, 248 (2010), 2130-2170. doi: 10.1016/j.jde.2009.09.007.

[20]

R. Dautray and J.-L. Lions, "Analyse mathématique et calcul numérique pour les sciences et les techniques. Tomes 1, 2, 3," Collection du Commissariat à l'Énergie Atomique: Série Scientifique, [Collection of the Atomic Energy Commission: Science Series]. Masson, Paris, 1985.

[21]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[22]

C. Fabre, Résultats d'unicité pour les équations de Stokes et applications au contrôle, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1191-1196.

[23]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446 (electronic).

[24]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501-1542.

[25]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier-Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173 (electronic). doi: 10.1137/04061965X.

[26]

C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," volume 83 of "Encyclopedia of Mathematics and its Applications," Cambridge University Press, Cambridge, 2001.

[27]

A. V. Fursikov, M. Gunzburger, L. S. Hou and S. Manservisi, Optimal control problems for the Navier-Stokes equations, In "Lectures on Applied Mathematics (Munich, 1999)," pages 143-155, Springer, Berlin, 2000.

[28]

A. V. Fursikov and O. Yu. Imanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54 (1999), 93-146.

[29]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," volume 34 of "Lecture Notes Series," Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.

[30]

P. Germain, Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier-Stokes system, J. Anal. Math., 105 (2008), 169-196. doi: 10.1007/s11854-008-0034-4.

[31]

M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system, In "Functional Analysis and Related Topics, 1991 (Kyoto)," volume 1540 of "Lecture Notes in Math.," pages 55-67, Springer, Berlin, 1993.

[32]

R. Glowinski, J.-L. Lions and J. He, "Exact and Approximate Controllability for Distributed Parameter Systems," volume 117 of "Encyclopedia of Mathematics and its Applications," Cambridge University Press, Cambridge, 2008. A numerical approach.

[33]

M. González-Burgos, S. Guerrero and J.-P. Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., 8 (2009), 311-333.

[34]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61. doi: 10.1016/j.anihpc.2005.01.002.

[35]

M. D. Gunzburger, "Perspectives in Flow Control and Optimization," volume 5 of "Advances in Design and Control," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.

[36]

J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048.

[37]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. doi: 10.1137/0719018.

[38]

O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72 (electronic).

[39]

O. Yu. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., 16 (2003), 883-913. doi: 10.1155/S107379280321117X.

[40]

T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova, 32 (1962), 243-260.

[41]

J. U. Kim, Weak solutions of an initial-boundary value problems for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96. doi: 10.1137/0518007.

[42]

K. Kunisch, G. Leugering, J. Sprekels and F. Tröltzsch, editors, "Optimal Control of Coupled Systems of Partial Differential Equations," volume 158 of "International Series of Numerical Mathematics," Birkhäuser Verlag, Basel, 2009. Papers from the International Conference held in Oberwolfach, March 2-8, 2008.

[43]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," volume 431 of "Chapman & Hall/CRC Research Notes in Mathematics," Chapman & Hall/CRC, Boca Raton, FL, 2002.

[44]

J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkin's approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 605-621.

[45]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol I: Incompressible Models," volume 3 of "Oxford Lecture Series in Mathematics and its Applications," The Clarendon Press, Oxford University Press, New York, 1996.

[46]

P.-L. Lions and N. Masmoudi, Uniqueness of mild solutions of the Navier-Stokes system in $L^N$, Comm. Partial Differential Equations, 26 (2001), 2211-2226. doi: 10.1081/PDE-100107819.

[47]

N. Masmoudi, Uniqueness results for some PDEs, In "Journées ‘Équations aux Dérivées Partielles’," pages Exp. No. X, 13. Univ. Nantes, Nantes, 2003.

[48]

R. L. Panton, "Incompressible Flow," A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984.

[49]

F. Rempfer, On boundary conditions for incompressible navier-stokes problems, Applied Mechanics Reviews, 59 (2006), 107-126. doi: 10.1115/1.2177683.

[50]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52 (1973), 189-211.

[51]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[52]

R. Salvi, The equations of viscous incompressible nonhomogeneous fluid: on the existence and regularity, J. Austral. Math. Soc. Ser. B, 33 (1991), 94-110. doi: 10.1017/S0334270000008651.

[53]

J. C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[54]

H. Schlichting, "Boundary Layer Theory," McGraw-Hill, New York, 1955, Translated by J. Kestin.

[55]

J. Simon, "Existencia de solución del problema de Navier-Stokes con densidad variable," (spanish) [existence of solution for the variable density Navier-Stokes problem], Lectures at the University of Sevilla, Spain, 1989.

[56]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061.

[57]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," volume 2 of "Studies in Mathematics and its Applications," North-Holland Publishing Co., Amsterdam, third edition, 1984.

[58]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," volume 68 of "Applied Mathematical Sciences," Springer-Verlag, New York, second edition, 1997.

[59]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. IV," Springer-Verlag, New York, 1988, Applications to mathematical physics, Translated from the German and with a preface by Juergen Quandt.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," volume 65 of "Pure and Applied Mathematics," Academic Press, 1975.

[2]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin., "Optimal Control," Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. Translated from the Russian by V. M. Volosov.

[3]

C. Amrouche and V. Girault, On the existence and regularity of the solutions of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175. doi: 10.3792/pjaa.67.171.

[4]

S. Anita, V. Arnautu and V. Capasso, "An Introduction to Optimal Control Problems in Life Sciences and Economics," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2011.

[5]

F. Araruna and E. Fernández-Cara, On the controllability of nonhomogeneous viscous fluids, To appear, 2012.

[6]

V. Arnautu and P. Neittaanmaki, "Optimal Control From Theory to Computer Programs," volume 111 of "Solid Mechanics and its Applications," Kluwer Academic Publishers Group, Dordrecht, 2003.

[7]

J. L. Boldrini and M. A. Rojas-Medar, Global solutions to the equations for the motion of stratified incompressible fluids, Mat. Contemp., 3 (1992), 1-8.

[8]

J. L. Boldrini and M. A. Rojas-Medar, Global strong solutions of the equations for the motion of nonhomogeneous incompressible fluids, In C. Conca and G. N. Gatica, editors, "Numerical Methods in Mechanics," number 371 in Pitman Res. Notes Math. Ser., pages 35-45, Harlow, 1997, Longman.

[9]

R. C. Cabrales and E. Fernández-Cara, Numerical control of some solidification processes, To appear, 2012.

[10]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, In "Handbook of Mathematical Fluid Dynamics, Vol. III," pages 161-244, North-Holland, Amsterdam, 2004.

[11]

J.-Y. Chemin, Le système de Navier-Stokes incompressible soixante dix ans apr\`es Jean Leray, In "Actes des Journées Mathématiques à la Mémoire de Jean Leray," volume 9 of "Sémin. Congr.," pages 99-123, Soc. Math. France, Paris, 2004.

[12]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201. doi: 10.1081/PDE-120021191.

[13]

A. J. Chorin and J. E. Marsden, "A Mathematical Introduction to Fluid Mechanics," volume 4 of "Texts in Applied Mathematics," Springer-Verlag, New York, third edition, 1993.

[14]

P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.

[15]

J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Contrôle Optim. Calc. Var., 1 (1995/96), 35-75 (electronic). doi: 10.1051/cocv:1996102.

[16]

J.-M. Coron and A. V. Fursikov, Global exact controllability of the $2$D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys., 4 (1996), 429-448.

[17]

J.-M. Coron and S. Guerrero, Local null controllability of the two-dimensional Navier-Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl. (9), 92 (2009), 528-545.

[18]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921. doi: 10.1016/j.jde.2008.10.019.

[19]

R. Danchin, On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework, J. Differential Equations, 248 (2010), 2130-2170. doi: 10.1016/j.jde.2009.09.007.

[20]

R. Dautray and J.-L. Lions, "Analyse mathématique et calcul numérique pour les sciences et les techniques. Tomes 1, 2, 3," Collection du Commissariat à l'Énergie Atomique: Série Scientifique, [Collection of the Atomic Energy Commission: Science Series]. Masson, Paris, 1985.

[21]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[22]

C. Fabre, Résultats d'unicité pour les équations de Stokes et applications au contrôle, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1191-1196.

[23]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446 (electronic).

[24]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501-1542.

[25]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier-Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173 (electronic). doi: 10.1137/04061965X.

[26]

C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," volume 83 of "Encyclopedia of Mathematics and its Applications," Cambridge University Press, Cambridge, 2001.

[27]

A. V. Fursikov, M. Gunzburger, L. S. Hou and S. Manservisi, Optimal control problems for the Navier-Stokes equations, In "Lectures on Applied Mathematics (Munich, 1999)," pages 143-155, Springer, Berlin, 2000.

[28]

A. V. Fursikov and O. Yu. Imanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54 (1999), 93-146.

[29]

A. V. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations," volume 34 of "Lecture Notes Series," Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.

[30]

P. Germain, Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier-Stokes system, J. Anal. Math., 105 (2008), 169-196. doi: 10.1007/s11854-008-0034-4.

[31]

M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system, In "Functional Analysis and Related Topics, 1991 (Kyoto)," volume 1540 of "Lecture Notes in Math.," pages 55-67, Springer, Berlin, 1993.

[32]

R. Glowinski, J.-L. Lions and J. He, "Exact and Approximate Controllability for Distributed Parameter Systems," volume 117 of "Encyclopedia of Mathematics and its Applications," Cambridge University Press, Cambridge, 2008. A numerical approach.

[33]

M. González-Burgos, S. Guerrero and J.-P. Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., 8 (2009), 311-333.

[34]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61. doi: 10.1016/j.anihpc.2005.01.002.

[35]

M. D. Gunzburger, "Perspectives in Flow Control and Optimization," volume 5 of "Advances in Design and Control," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.

[36]

J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048.

[37]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. doi: 10.1137/0719018.

[38]

O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72 (electronic).

[39]

O. Yu. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., 16 (2003), 883-913. doi: 10.1155/S107379280321117X.

[40]

T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova, 32 (1962), 243-260.

[41]

J. U. Kim, Weak solutions of an initial-boundary value problems for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96. doi: 10.1137/0518007.

[42]

K. Kunisch, G. Leugering, J. Sprekels and F. Tröltzsch, editors, "Optimal Control of Coupled Systems of Partial Differential Equations," volume 158 of "International Series of Numerical Mathematics," Birkhäuser Verlag, Basel, 2009. Papers from the International Conference held in Oberwolfach, March 2-8, 2008.

[43]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," volume 431 of "Chapman & Hall/CRC Research Notes in Mathematics," Chapman & Hall/CRC, Boca Raton, FL, 2002.

[44]

J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkin's approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 605-621.

[45]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol I: Incompressible Models," volume 3 of "Oxford Lecture Series in Mathematics and its Applications," The Clarendon Press, Oxford University Press, New York, 1996.

[46]

P.-L. Lions and N. Masmoudi, Uniqueness of mild solutions of the Navier-Stokes system in $L^N$, Comm. Partial Differential Equations, 26 (2001), 2211-2226. doi: 10.1081/PDE-100107819.

[47]

N. Masmoudi, Uniqueness results for some PDEs, In "Journées ‘Équations aux Dérivées Partielles’," pages Exp. No. X, 13. Univ. Nantes, Nantes, 2003.

[48]

R. L. Panton, "Incompressible Flow," A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984.

[49]

F. Rempfer, On boundary conditions for incompressible navier-stokes problems, Applied Mechanics Reviews, 59 (2006), 107-126. doi: 10.1115/1.2177683.

[50]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52 (1973), 189-211.

[51]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[52]

R. Salvi, The equations of viscous incompressible nonhomogeneous fluid: on the existence and regularity, J. Austral. Math. Soc. Ser. B, 33 (1991), 94-110. doi: 10.1017/S0334270000008651.

[53]

J. C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[54]

H. Schlichting, "Boundary Layer Theory," McGraw-Hill, New York, 1955, Translated by J. Kestin.

[55]

J. Simon, "Existencia de solución del problema de Navier-Stokes con densidad variable," (spanish) [existence of solution for the variable density Navier-Stokes problem], Lectures at the University of Sevilla, Spain, 1989.

[56]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061.

[57]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," volume 2 of "Studies in Mathematics and its Applications," North-Holland Publishing Co., Amsterdam, third edition, 1984.

[58]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," volume 68 of "Applied Mathematical Sciences," Springer-Verlag, New York, second edition, 1997.

[59]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. IV," Springer-Verlag, New York, 1988, Applications to mathematical physics, Translated from the German and with a preface by Juergen Quandt.

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