February  2019, 12(1): 133-158. doi: 10.3934/krm.2019006

Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations

1. 

Ecole Normale Supérieure de Lyon, UMPA, ENS-Lyon, 46, allée d'Italie, 69364-Lyon Cedex 07, France

2. 

Istituto per le Applicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, via dei Taurini 19, I-00185 Rome, Italy

* Corresponding author: Roberto Natalini

Received  July 2017 Published  July 2018

Fund Project: The first author was supported by a Ph. D. grant of University of Rome Tor Vergata.

We present a rigorous convergence result for smooth solutions to a singular semilinear hyperbolic approximation, called vector-BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof deeply relies on the dissipative properties of the system and on the use of an energy which is provided by a symmetrizer, whose entries are weighted in a suitable way with respect to the singular perturbation parameter. This strategy allows us to perform uniform energy estimates and to prove the convergence by compactness.

Citation: Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006
References:
[1]

D. Aregba-Driollet and R. Natalini, Discrete Kinetic Schemes for Multidimensional Conservation Laws, SIAM J. Num. Anal., 37 (2000), 1973-2004.  doi: 10.1137/S0036142998343075.  Google Scholar

[2]

D. Aregba-DriolletR. Natalini and S. Q. Tang, Diffusive kinetic explicit schemes for nonlinear degenerate parabolic systems, Math. Comp., 73 (2004), 63-94.   Google Scholar

[3]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of hyperbolic equations. Ⅰ. Formal derivations, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations - Ⅱ Convergence proofs for the Boltzmann-equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[5]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007  Google Scholar

[6]

A. Bertozzi and A. Majda, Vorticity and Incompressible Flow, Cambridge University Press, 2002.  Google Scholar

[7]

S. BianchiniB. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622.  doi: 10.1002/cpa.20195.  Google Scholar

[8]

F. Bouchut, Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws, J. of Stat. Phys., 95 (1999), 113-170.  doi: 10.1023/A:1004525427365.  Google Scholar

[9]

F. BouchutF. Guarguaglini and R. Natalini, Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations, Indiana Univ. Math. J., 49 (2000), 723-749.  doi: 10.1512/iumj.2000.49.1811.  Google Scholar

[10]

F. BouchutY. JobicR. NataliniR. Occelli and V. Pavan, Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations, SMAI Journal of Computational Mathematics, 4 (2018), 1-56.  doi: 10.5802/smai-jcm.28.  Google Scholar

[11]

Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), 1013-1037.  doi: 10.1137/0721063.  Google Scholar

[12]

Y. BrenierR. Natalini and M. Puel, On a relaxation approximation of the incompressible Navier-Stokes equations, Proc. Amer. Math. Soc., 132 (2004), 1021-1028.  doi: 10.1090/S0002-9939-03-07230-7.  Google Scholar

[13]

M. Carfora and R. Natalini, A discrete kinetic approximation for the incompressible Navier-Stokes equations, ESAIM: Math. Modelling Numer. Anal., 42 (2008), 93-112.  doi: 10.1051/m2an:2007055.  Google Scholar

[14]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[15]

A. DeMasiR. Esposito and J. Lebowitz, Incompressible Navier-Stokes and Euler Limits of the Boltzmann equation, Comm. Pure Appl. Math., 42 (1989), 1189-1214.  doi: 10.1002/cpa.3160420810.  Google Scholar

[16]

F. Golse, C. D. Levermore and L. Saint-Raymond, La méthode de l'entropie relative pour les limites hydrodynamiques de modèles cinétiques, Séminaire Équations Aux Dérivées Partielles (2000), 23pp.  Google Scholar

[17]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5.  Google Scholar

[18]

I. Hachicha, Approximations Hyperboliques des Équations de Navier-Stokes, Ph. D. Thesis, Université d'Évry-Val d'Essone, 2013. Google Scholar

[19]

I. Hachicha, Global existence for a damped wave equation and convergence towards a solution of the Navier-Stokes problem, Nonlinear Anal., 96 (2014), 68-86.  doi: 10.1016/j.na.2013.10.020.  Google Scholar

[20]

B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Nonlinear Anal., 96 (2014), 68-86.  doi: 10.1007/s00205-003-0257-6.  Google Scholar

[21]

S. Jin and Z. Xin, The relaxation schemes for system of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[22]

M. Junk and W.-A- Yong, Rigorous Navier-Stokes Limit of the Lattice Boltzmann Equation, Asymptotic Anal., 35 (2003), 165-185.   Google Scholar

[23]

C. Lattanzio and R. Natalini, Convergence of diffusive BGK approximations for nonlinear strongly parabolic systems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 341-358.  doi: 10.1017/S0308210500001669.  Google Scholar

[24]

P. L. Lions and G. Toscani, Diffusive limits for finite velocity Boltzmann kinetic models, Revista Mat. Iberoamer., 13 (1997), 473-513.  doi: 10.4171/RMI/228.  Google Scholar

[25]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[26]

R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Diff. Eq., 148 (1998), 292-317.  doi: 10.1006/jdeq.1998.3460.  Google Scholar

[27]

M. Paicu and G. Raugel, A hyperbolic perturbation of the Navier-Stokes equations, (Une perturbation hyperbolique des equations de Navier-Stokes.), ESAIM, Proc., 21 (2007), 65-87.  doi: 10.1051/proc:072106.  Google Scholar

[28]

B. Perthame, Kinetic Formulation of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 21, Oxford University Press, 2002.  Google Scholar

[29]

L. Saint-Raymond, From the BGK model to the Navier-Stokes equations, Annales Scientifiques de l'École Normale Supérieure, 36 (2003), 271-317.  doi: 10.1016/S0012-9593(03)00010-7.  Google Scholar

[30]

S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Numerical Mathematics and Scientific Computation, Oxford Science Publications, the Clarendon Press, Oxford University Press, New York, 2001.  Google Scholar

[31]

D. A. Wolf-Gladrow, Lattice-gas Cellular Automata and Lattice Boltzmann models. An introduction, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/b72010.  Google Scholar

show all references

References:
[1]

D. Aregba-Driollet and R. Natalini, Discrete Kinetic Schemes for Multidimensional Conservation Laws, SIAM J. Num. Anal., 37 (2000), 1973-2004.  doi: 10.1137/S0036142998343075.  Google Scholar

[2]

D. Aregba-DriolletR. Natalini and S. Q. Tang, Diffusive kinetic explicit schemes for nonlinear degenerate parabolic systems, Math. Comp., 73 (2004), 63-94.   Google Scholar

[3]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of hyperbolic equations. Ⅰ. Formal derivations, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations - Ⅱ Convergence proofs for the Boltzmann-equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[5]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007  Google Scholar

[6]

A. Bertozzi and A. Majda, Vorticity and Incompressible Flow, Cambridge University Press, 2002.  Google Scholar

[7]

S. BianchiniB. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622.  doi: 10.1002/cpa.20195.  Google Scholar

[8]

F. Bouchut, Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws, J. of Stat. Phys., 95 (1999), 113-170.  doi: 10.1023/A:1004525427365.  Google Scholar

[9]

F. BouchutF. Guarguaglini and R. Natalini, Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations, Indiana Univ. Math. J., 49 (2000), 723-749.  doi: 10.1512/iumj.2000.49.1811.  Google Scholar

[10]

F. BouchutY. JobicR. NataliniR. Occelli and V. Pavan, Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations, SMAI Journal of Computational Mathematics, 4 (2018), 1-56.  doi: 10.5802/smai-jcm.28.  Google Scholar

[11]

Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), 1013-1037.  doi: 10.1137/0721063.  Google Scholar

[12]

Y. BrenierR. Natalini and M. Puel, On a relaxation approximation of the incompressible Navier-Stokes equations, Proc. Amer. Math. Soc., 132 (2004), 1021-1028.  doi: 10.1090/S0002-9939-03-07230-7.  Google Scholar

[13]

M. Carfora and R. Natalini, A discrete kinetic approximation for the incompressible Navier-Stokes equations, ESAIM: Math. Modelling Numer. Anal., 42 (2008), 93-112.  doi: 10.1051/m2an:2007055.  Google Scholar

[14]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[15]

A. DeMasiR. Esposito and J. Lebowitz, Incompressible Navier-Stokes and Euler Limits of the Boltzmann equation, Comm. Pure Appl. Math., 42 (1989), 1189-1214.  doi: 10.1002/cpa.3160420810.  Google Scholar

[16]

F. Golse, C. D. Levermore and L. Saint-Raymond, La méthode de l'entropie relative pour les limites hydrodynamiques de modèles cinétiques, Séminaire Équations Aux Dérivées Partielles (2000), 23pp.  Google Scholar

[17]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5.  Google Scholar

[18]

I. Hachicha, Approximations Hyperboliques des Équations de Navier-Stokes, Ph. D. Thesis, Université d'Évry-Val d'Essone, 2013. Google Scholar

[19]

I. Hachicha, Global existence for a damped wave equation and convergence towards a solution of the Navier-Stokes problem, Nonlinear Anal., 96 (2014), 68-86.  doi: 10.1016/j.na.2013.10.020.  Google Scholar

[20]

B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Nonlinear Anal., 96 (2014), 68-86.  doi: 10.1007/s00205-003-0257-6.  Google Scholar

[21]

S. Jin and Z. Xin, The relaxation schemes for system of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276.  doi: 10.1002/cpa.3160480303.  Google Scholar

[22]

M. Junk and W.-A- Yong, Rigorous Navier-Stokes Limit of the Lattice Boltzmann Equation, Asymptotic Anal., 35 (2003), 165-185.   Google Scholar

[23]

C. Lattanzio and R. Natalini, Convergence of diffusive BGK approximations for nonlinear strongly parabolic systems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 341-358.  doi: 10.1017/S0308210500001669.  Google Scholar

[24]

P. L. Lions and G. Toscani, Diffusive limits for finite velocity Boltzmann kinetic models, Revista Mat. Iberoamer., 13 (1997), 473-513.  doi: 10.4171/RMI/228.  Google Scholar

[25]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[26]

R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Diff. Eq., 148 (1998), 292-317.  doi: 10.1006/jdeq.1998.3460.  Google Scholar

[27]

M. Paicu and G. Raugel, A hyperbolic perturbation of the Navier-Stokes equations, (Une perturbation hyperbolique des equations de Navier-Stokes.), ESAIM, Proc., 21 (2007), 65-87.  doi: 10.1051/proc:072106.  Google Scholar

[28]

B. Perthame, Kinetic Formulation of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, 21, Oxford University Press, 2002.  Google Scholar

[29]

L. Saint-Raymond, From the BGK model to the Navier-Stokes equations, Annales Scientifiques de l'École Normale Supérieure, 36 (2003), 271-317.  doi: 10.1016/S0012-9593(03)00010-7.  Google Scholar

[30]

S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Numerical Mathematics and Scientific Computation, Oxford Science Publications, the Clarendon Press, Oxford University Press, New York, 2001.  Google Scholar

[31]

D. A. Wolf-Gladrow, Lattice-gas Cellular Automata and Lattice Boltzmann models. An introduction, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/b72010.  Google Scholar

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