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Global existence of weak solution to the free boundary problem for compressible Navier-Stokes
Kinetic derivation of fractional Stokes and Stokes-Fourier systems
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 |
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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