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March  2016, 9(1): 105-129. doi: 10.3934/krm.2016.9.105

Kinetic derivation of fractional Stokes and Stokes-Fourier systems

Sabine Hittmeir 1, and  Sara Merino-Aceituno 2,

1. 

RICAM Linz, Austrian Academy of Sciences, Altenberger Str. 69, 4040 Linz, Austria
2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  September 2014 Revised  July 2015 Published  October 2015

In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmann-type equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavy-tailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision frequency for small velocities. This work aims to an extension in the sense that a linear BGK-type equation conserving not only mass, but also momentum and energy, for both mentioned regimes of equilibrium distributions is considered. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria.
Keywords: hydrodynamic limit, anomalous diffusive time scale, Stokes-Fourier system., incompressible Stokes equation, fractional diffusion, linear BGK model, Kinetic transport equation.
Mathematics Subject Classification: Primary: 35Q20, 35Q30, 35Q35, 35R1.
Citation: Sabine Hittmeir, Sara Merino-Aceituno. Kinetic derivation of fractional Stokes and Stokes-Fourier systems. Kinetic and Related Models, 2016, 9 (1) : 105-129. doi: 10.3934/krm.2016.9.105
References:
[1]

C. Bardos, P. Penel, U. Frisch and P. L. Sulem, Modified dissipativity for a non-linear evolution equation arising in turbulence, Arch. Ration. Mech. Anal., 71 (1979), 237-256. doi: 10.1007/BF00280598.

[2]

C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257. doi: 10.1142/S0218202591000137.

[3]

N. Ben Abdallah, P. Degond, F. Deluzet, V. Latocha, R. Talaalout and M. H. Vignal, Diffusion limits of kinetic models, Hyperbolic problems: theory, numerics, applications, Springer, (2003), 3-17.

[4]

N. Ben Abdallah, A. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models Methods Appl. Sci., 21 (2011), 2249-2262. doi: 10.1142/S0218202511005738.

[5]

N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: a Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900. doi: 10.3934/krm.2011.4.873.

[6]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Boundary layer analysis in homogeneization of diffusion equations with Dirichlet conditions in the half space. Proc. int. Symp. on stochastic differential equations, Kyoto, (1978), 21-40.

[7]

A. V. Bobylev and I. M. Gamba, Boltzmann equations for mixtures of Maxwell gases: Exact solutions and power like tails, J. Stat. Phys., 124 (2006), 497-516. doi: 10.1007/s10955-006-9044-8.

[8]

N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the Vlasov-BGK equation, J. Comput. Phys., 203 (2005), 572-601. doi: 10.1016/j.jcp.2004.09.006.

[9]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198.

[10]

B. Düring and G. Toscani, International and domestic trading and wealth distribution. Commun. Math. Sci., 6 (2008), 1043-1058. doi: 10.4310/CMS.2008.v6.n4.a12.

[11]

F. Golse, Hydrodynamic Limits, European Congress of Mathematics, Eur. Math. Soc., Zürich, (2005), 699-717.

[12]

F. Golse and C. D. Levermore, Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs. Comm. Pure Appl. Math., 55 (2002), 336-393. doi: 10.1002/cpa.3011.

[13]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. Appl. Probab., 19, (2009), 2270-2300. doi: 10.1214/09-AAP610.

[14]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.

[15]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360. doi: 10.1512/iumj.2010.59.4128.

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[17]

L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-92847-8.

[18]

H. Struchtrup, The BGK-model with velocity-dependent collision frequency, Contin. Mech. Thermodyn., 9 (1997), 23-31. doi: 10.1007/s001610050053.

[19]

D. Summers and R. M. Thorne, The modified plasma dispersion function, Phys. Fluids B, 3 (1991), 1835-1847. doi: 10.1063/1.859653.

[20]

J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Commun. Math. Phys., 263 (2006), 803-831. doi: 10.1007/s00220-005-1483-6.

show all references

References:
[1]

C. Bardos, P. Penel, U. Frisch and P. L. Sulem, Modified dissipativity for a non-linear evolution equation arising in turbulence, Arch. Ration. Mech. Anal., 71 (1979), 237-256. doi: 10.1007/BF00280598.

[2]

C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257. doi: 10.1142/S0218202591000137.

[3]

N. Ben Abdallah, P. Degond, F. Deluzet, V. Latocha, R. Talaalout and M. H. Vignal, Diffusion limits of kinetic models, Hyperbolic problems: theory, numerics, applications, Springer, (2003), 3-17.

[4]

N. Ben Abdallah, A. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models Methods Appl. Sci., 21 (2011), 2249-2262. doi: 10.1142/S0218202511005738.

[5]

N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: a Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900. doi: 10.3934/krm.2011.4.873.

[6]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Boundary layer analysis in homogeneization of diffusion equations with Dirichlet conditions in the half space. Proc. int. Symp. on stochastic differential equations, Kyoto, (1978), 21-40.

[7]

A. V. Bobylev and I. M. Gamba, Boltzmann equations for mixtures of Maxwell gases: Exact solutions and power like tails, J. Stat. Phys., 124 (2006), 497-516. doi: 10.1007/s10955-006-9044-8.

[8]

N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the Vlasov-BGK equation, J. Comput. Phys., 203 (2005), 572-601. doi: 10.1016/j.jcp.2004.09.006.

[9]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198.

[10]

B. Düring and G. Toscani, International and domestic trading and wealth distribution. Commun. Math. Sci., 6 (2008), 1043-1058. doi: 10.4310/CMS.2008.v6.n4.a12.

[11]

F. Golse, Hydrodynamic Limits, European Congress of Mathematics, Eur. Math. Soc., Zürich, (2005), 699-717.

[12]

F. Golse and C. D. Levermore, Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs. Comm. Pure Appl. Math., 55 (2002), 336-393. doi: 10.1002/cpa.3011.

[13]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. Appl. Probab., 19, (2009), 2270-2300. doi: 10.1214/09-AAP610.

[14]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.

[15]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360. doi: 10.1512/iumj.2010.59.4128.

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[17]

L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-92847-8.

[18]

H. Struchtrup, The BGK-model with velocity-dependent collision frequency, Contin. Mech. Thermodyn., 9 (1997), 23-31. doi: 10.1007/s001610050053.

[19]

D. Summers and R. M. Thorne, The modified plasma dispersion function, Phys. Fluids B, 3 (1991), 1835-1847. doi: 10.1063/1.859653.

[20]

J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Commun. Math. Phys., 263 (2006), 803-831. doi: 10.1007/s00220-005-1483-6.

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