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 & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
References:
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R. A. Adams, "Sobolev Spaces," volume 65 of "Pure and Applied Mathematics,", Academic Press, (1975).   Google Scholar

[2]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin., "Optimal Control,", Contemporary Soviet Mathematics. Consultants Bureau, (1987).   Google Scholar

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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.  doi: 10.3792/pjaa.67.171.  Google Scholar

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

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

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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.   Google Scholar

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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, (1997), 35.   Google Scholar

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

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M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, In, (2004), 161.   Google Scholar

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J.-Y. Chemin, Le système de Navier-Stokes incompressible soixante dix ans apr\`es Jean Leray,, In, (2004), 99.   Google Scholar

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

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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 (): 35.  doi: 10.1051/cocv:1996102.  Google Scholar

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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.   Google Scholar

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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.  doi: 10.1016/j.jde.2008.10.019.  Google Scholar

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

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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, (1985).   Google Scholar

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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.   Google Scholar

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

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[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, (2001).   Google Scholar

[27]

A. V. Fursikov, M. Gunzburger, L. S. Hou and S. Manservisi, Optimal control problems for the Navier-Stokes equations,, In, (1999), 143.   Google Scholar

[28]

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

[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, (1996).   Google Scholar

[30]

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

[31]

M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system,, In, 1540 (1991), 55.   Google Scholar

[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, (2008).   Google Scholar

[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.   Google Scholar

[34]

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

[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), (2003).   Google Scholar

[36]

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

[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.  doi: 10.1137/0719018.  Google Scholar

[38]

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

[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.  doi: 10.1155/S107379280321117X.  Google Scholar

[40]

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

[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.  doi: 10.1137/0518007.  Google Scholar

[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, (2009), 2.   Google Scholar

[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, (2002).   Google Scholar

[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.   Google Scholar

[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, (1996).   Google Scholar

[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.  doi: 10.1081/PDE-100107819.  Google Scholar

[47]

N. Masmoudi, Uniqueness results for some PDEs,, In, (2003).   Google Scholar

[48]

R. L. Panton, "Incompressible Flow,", A Wiley-Interscience Publication, (1984).   Google Scholar

[49]

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

[50]

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

[51]

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

[52]

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

[53]

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

[54]

H. Schlichting, "Boundary Layer Theory,", McGraw-Hill, (1955).   Google Scholar

[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, (1989).   Google Scholar

[56]

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

[57]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," volume 2 of "Studies in Mathematics and its Applications,", North-Holland Publishing Co., (1984).   Google Scholar

[58]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," volume 68 of "Applied Mathematical Sciences,", Springer-Verlag, (1997).   Google Scholar

[59]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. IV,", Springer-Verlag, (1988).   Google Scholar

show all references

References:
[1]

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

[2]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin., "Optimal Control,", Contemporary Soviet Mathematics. Consultants Bureau, (1987).   Google Scholar

[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.  doi: 10.3792/pjaa.67.171.  Google Scholar

[4]

S. Anita, V. Arnautu and V. Capasso, "An Introduction to Optimal Control Problems in Life Sciences and Economics,", Modeling and Simulation in Science, (2011).   Google Scholar

[5]

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

[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, (2003).   Google Scholar

[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.   Google Scholar

[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, (1997), 35.   Google Scholar

[9]

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

[10]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations,, In, (2004), 161.   Google Scholar

[11]

J.-Y. Chemin, Le système de Navier-Stokes incompressible soixante dix ans apr\`es Jean Leray,, In, (2004), 99.   Google Scholar

[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.  doi: 10.1081/PDE-120021191.  Google Scholar

[13]

A. J. Chorin and J. E. Marsden, "A Mathematical Introduction to Fluid Mechanics," volume 4 of "Texts in Applied Mathematics,", Springer-Verlag, (1993).   Google Scholar

[14]

P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988).   Google Scholar

[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 (): 35.  doi: 10.1051/cocv:1996102.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jde.2008.10.019.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.09.007.  Google Scholar

[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, (1985).   Google Scholar

[21]

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

[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.   Google Scholar

[23]

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

[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.   Google Scholar

[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.  doi: 10.1137/04061965X.  Google Scholar

[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, (2001).   Google Scholar

[27]

A. V. Fursikov, M. Gunzburger, L. S. Hou and S. Manservisi, Optimal control problems for the Navier-Stokes equations,, In, (1999), 143.   Google Scholar

[28]

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

[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, (1996).   Google Scholar

[30]

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

[31]

M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system,, In, 1540 (1991), 55.   Google Scholar

[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, (2008).   Google Scholar

[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.   Google Scholar

[34]

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

[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), (2003).   Google Scholar

[36]

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

[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.  doi: 10.1137/0719018.  Google Scholar

[38]

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

[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.  doi: 10.1155/S107379280321117X.  Google Scholar

[40]

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

[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.  doi: 10.1137/0518007.  Google Scholar

[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, (2009), 2.   Google Scholar

[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, (2002).   Google Scholar

[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.   Google Scholar

[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, (1996).   Google Scholar

[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.  doi: 10.1081/PDE-100107819.  Google Scholar

[47]

N. Masmoudi, Uniqueness results for some PDEs,, In, (2003).   Google Scholar

[48]

R. L. Panton, "Incompressible Flow,", A Wiley-Interscience Publication, (1984).   Google Scholar

[49]

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

[50]

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

[51]

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

[52]

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

[53]

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

[54]

H. Schlichting, "Boundary Layer Theory,", McGraw-Hill, (1955).   Google Scholar

[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, (1989).   Google Scholar

[56]

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

[57]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," volume 2 of "Studies in Mathematics and its Applications,", North-Holland Publishing Co., (1984).   Google Scholar

[58]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," volume 68 of "Applied Mathematical Sciences,", Springer-Verlag, (1997).   Google Scholar

[59]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. IV,", Springer-Verlag, (1988).   Google Scholar

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