December  2017, 12(4): 683-705. doi: 10.3934/nhm.2017028

Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces

1. 

Department of Mathematics, University of Maryland, College Park, MD 20742, USA

2. 

Current address: Foundation for Research and Technology Hellas, Institute of Applied and Computational Mathematics, N. Plastira 100, Vassilika Vouton, Heraklion 70013, Greece

Received  July 2016 Revised  February 2017 Published  October 2017

In this paper we study the hydrodynamic (small mass approximation) limit of a Fokker-Planck equation. This equation arises in the kinetic description of the evolution of a particle system immersed in a viscous Stokes flow. We discuss two different methods of hydrodynamic convergence. The first method works with initial data in a weighted L2 space and uses weak convergence and the extraction of convergent subsequences. The second uses entropic initial data and gives an L1 convergence to the solution of the limit problem via the study of the relative entropy.

Citation: Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
References:
[1]

C. BardosF. GolseB. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1. Google Scholar

[2] B. BirdR. ArmstrongC. Curtiss and O. Hassager, Dynamics of Polymeric Liquids: Kinetic Theory, vol 2, 2nd edition, John Wiley & Sons, 1994. Google Scholar
[3]

B. CichockiB. U. FelderhofK. HinsenE. Wajnryb and J. Blawzdziewicz, Friction and mobility of many spheres in Stokes flow, J. Chem. Phys., 100 (1994), 3780-3790. doi: 10.1063/1.466366. Google Scholar

[4]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Stud. Sci. Math. Hung., 2 (1967), 299-318. Google Scholar

[5]

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

[6]

P. Degond and H. Liu, Kinetic models for polymers with inertial effects, Netw. Heterog. Media, 4 (2009), 625-647. doi: 10.3934/nhm.2009.4.625. Google Scholar

[7] M. Doi, Introduction to Polymer Physics, Oxford University Press, 1996. Google Scholar
[8] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, New York, 1986. Google Scholar
[9]

J. DolbeaultP. A. MarkowichD. Oelz and C. Schmeiser, Non linear diffusions as limit of kinetic equations with relaxation collision kernels, Arch. Ration. Mech. Anal., 186 (2007), 133-158. doi: 10.1007/s00205-007-0049-5. Google Scholar

[10]

A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys., 322 (1905), 549-560. doi: 10.1002/andp.19053220806. Google Scholar

[11]

M. Freidlin, Some remarks on the Smoluchowski-Kramers approximation, J. Stat. Phys., 117 (2004), 617-634. doi: 10.1007/s10955-004-2273-9. Google Scholar

[12]

N. Ghani and N. Masmoudi, Diffusion limit of The Vlassov-Poisson-Fokker-Planck system, Commun. Math. Sci., 8 (2010), 463-479. doi: 10.4310/CMS.2010.v8.n2.a9. Google Scholar

[13]

F. Golse, C. D. Levermore and L. Saint-Raymond, La Méthode de L'entropie Relative Pour les Limites Hydrodynamiques de Modéles Cinétiques Séminaire Equations aux Derivées Partielles, Exp. No. XIX, Ecole Polytechnique, 2000. Google Scholar

[14]

F. Golse and F. Poupaud, Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, Asympot. Anal., 6 (1992), 135-160. Google Scholar

[15]

T. Goudon, Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci., 15 (2005), 737-752. doi: 10.1142/S021820250500056X. Google Scholar

[16]

T. GoudonP.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equation. Part Ⅰ: Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508. Google Scholar

[17]

T. GoudonP.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equation. Part Ⅱ: Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509. Google Scholar

[18]

P. -E. Jabin, private communication.Google Scholar

[19]

P.-E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432. doi: 10.1007/s00220-004-1126-3. Google Scholar

[20]

P. -E. Jabin and B. Perthame, Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid, in Modeling in Applied Sciences, a Kinetic Theory Approach (eds. N. Bellomo and M. Pulvirenti), Birkhäuser, (2000), 111-147. Google Scholar

[21] G. Jannick and J. des Cloizeaux, Polymers in Solution: Their Modelling and Structure, Oxford University Press, 1990. Google Scholar
[22]

D. Jeffrey and Y. Onishi, Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow, J. Fluid Mech., 139 (1984), 261-290. doi: 10.1017/S0022112084000355. Google Scholar

[23] G. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann, Boston, 1991. Google Scholar
[24]

J. G. Kirkwood, John Gamble Kirkwood Collected Works: Macromolecules, vol 3, Documents on modern physics, Gordon and Breach, 1967.Google Scholar

[25]

S. Kullback, A lower bound for discrimination information in terms of variation, IEEE Trans. Inform. Theory, 13 (1967), 126-127. Google Scholar

[26]

C. Le Bris and P.-L. Lions, Renormalized solutions of some transport equations with partially $W^{1, 1}$ velocities and applications, Ann. Mat. Pura Appl., 183 (2004), 97-130. doi: 10.1007/s10231-003-0082-4. Google Scholar

[27]

C. Le Bris and P.-L. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations, 33 (2008), 1272-1317. doi: 10.1080/03605300801970952. Google Scholar

[28] M. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. Google Scholar
[29]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: analysis of boundary layers, Asympot. Anal., 4 (1991), 293-317. Google Scholar

[30]

F. Poupaud and C. Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Methods Appl. Sci., 14 (1991), 301-318. doi: 10.1002/mma.1670140503. Google Scholar

[31]

M. Reichert, Hydrodynamic Interactions in Colloidal and Biological Systems, Ph. D thesis, University Konstanz, 2006.Google Scholar

[32]

H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications, in Springer Series in Synergetics, 18 2nd edition, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. Google Scholar

[33]

J. Rotne and S. Prager, Variational treatment of hydrodynamic iteractions in polymers, J. Chem. Phys., 50 (1969), 4831-4837. Google Scholar

[34]

S. Varadhan, Entropy methods in hydrodynamic scaling, Proceedings of the International Congress of Mathematicians, Birkhäuser, Basel, 1 (1995), 196-208 Google Scholar

[35]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. Phys., 326 (1906), 756-780. doi: 10.1002/andp.19063261405. Google Scholar

[36]

H. Yamakawa, Transport properties of polymer chains in dilute solutions: Hydrodynamic interactions, J. Chem. Phys., 53 (1970), 436-443. doi: 10.1063/1.1673799. Google Scholar

[37]

H. T. Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys., 22 (1991), 63-80. doi: 10.1007/BF00400379. Google Scholar

show all references

References:
[1]

C. BardosF. GolseB. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1. Google Scholar

[2] B. BirdR. ArmstrongC. Curtiss and O. Hassager, Dynamics of Polymeric Liquids: Kinetic Theory, vol 2, 2nd edition, John Wiley & Sons, 1994. Google Scholar
[3]

B. CichockiB. U. FelderhofK. HinsenE. Wajnryb and J. Blawzdziewicz, Friction and mobility of many spheres in Stokes flow, J. Chem. Phys., 100 (1994), 3780-3790. doi: 10.1063/1.466366. Google Scholar

[4]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Stud. Sci. Math. Hung., 2 (1967), 299-318. Google Scholar

[5]

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

[6]

P. Degond and H. Liu, Kinetic models for polymers with inertial effects, Netw. Heterog. Media, 4 (2009), 625-647. doi: 10.3934/nhm.2009.4.625. Google Scholar

[7] M. Doi, Introduction to Polymer Physics, Oxford University Press, 1996. Google Scholar
[8] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, New York, 1986. Google Scholar
[9]

J. DolbeaultP. A. MarkowichD. Oelz and C. Schmeiser, Non linear diffusions as limit of kinetic equations with relaxation collision kernels, Arch. Ration. Mech. Anal., 186 (2007), 133-158. doi: 10.1007/s00205-007-0049-5. Google Scholar

[10]

A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys., 322 (1905), 549-560. doi: 10.1002/andp.19053220806. Google Scholar

[11]

M. Freidlin, Some remarks on the Smoluchowski-Kramers approximation, J. Stat. Phys., 117 (2004), 617-634. doi: 10.1007/s10955-004-2273-9. Google Scholar

[12]

N. Ghani and N. Masmoudi, Diffusion limit of The Vlassov-Poisson-Fokker-Planck system, Commun. Math. Sci., 8 (2010), 463-479. doi: 10.4310/CMS.2010.v8.n2.a9. Google Scholar

[13]

F. Golse, C. D. Levermore and L. Saint-Raymond, La Méthode de L'entropie Relative Pour les Limites Hydrodynamiques de Modéles Cinétiques Séminaire Equations aux Derivées Partielles, Exp. No. XIX, Ecole Polytechnique, 2000. Google Scholar

[14]

F. Golse and F. Poupaud, Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, Asympot. Anal., 6 (1992), 135-160. Google Scholar

[15]

T. Goudon, Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci., 15 (2005), 737-752. doi: 10.1142/S021820250500056X. Google Scholar

[16]

T. GoudonP.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equation. Part Ⅰ: Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508. Google Scholar

[17]

T. GoudonP.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equation. Part Ⅱ: Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509. Google Scholar

[18]

P. -E. Jabin, private communication.Google Scholar

[19]

P.-E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432. doi: 10.1007/s00220-004-1126-3. Google Scholar

[20]

P. -E. Jabin and B. Perthame, Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid, in Modeling in Applied Sciences, a Kinetic Theory Approach (eds. N. Bellomo and M. Pulvirenti), Birkhäuser, (2000), 111-147. Google Scholar

[21] G. Jannick and J. des Cloizeaux, Polymers in Solution: Their Modelling and Structure, Oxford University Press, 1990. Google Scholar
[22]

D. Jeffrey and Y. Onishi, Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow, J. Fluid Mech., 139 (1984), 261-290. doi: 10.1017/S0022112084000355. Google Scholar

[23] G. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann, Boston, 1991. Google Scholar
[24]

J. G. Kirkwood, John Gamble Kirkwood Collected Works: Macromolecules, vol 3, Documents on modern physics, Gordon and Breach, 1967.Google Scholar

[25]

S. Kullback, A lower bound for discrimination information in terms of variation, IEEE Trans. Inform. Theory, 13 (1967), 126-127. Google Scholar

[26]

C. Le Bris and P.-L. Lions, Renormalized solutions of some transport equations with partially $W^{1, 1}$ velocities and applications, Ann. Mat. Pura Appl., 183 (2004), 97-130. doi: 10.1007/s10231-003-0082-4. Google Scholar

[27]

C. Le Bris and P.-L. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations, 33 (2008), 1272-1317. doi: 10.1080/03605300801970952. Google Scholar

[28] M. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. Google Scholar
[29]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: analysis of boundary layers, Asympot. Anal., 4 (1991), 293-317. Google Scholar

[30]

F. Poupaud and C. Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Methods Appl. Sci., 14 (1991), 301-318. doi: 10.1002/mma.1670140503. Google Scholar

[31]

M. Reichert, Hydrodynamic Interactions in Colloidal and Biological Systems, Ph. D thesis, University Konstanz, 2006.Google Scholar

[32]

H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications, in Springer Series in Synergetics, 18 2nd edition, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. Google Scholar

[33]

J. Rotne and S. Prager, Variational treatment of hydrodynamic iteractions in polymers, J. Chem. Phys., 50 (1969), 4831-4837. Google Scholar

[34]

S. Varadhan, Entropy methods in hydrodynamic scaling, Proceedings of the International Congress of Mathematicians, Birkhäuser, Basel, 1 (1995), 196-208 Google Scholar

[35]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. Phys., 326 (1906), 756-780. doi: 10.1002/andp.19063261405. Google Scholar

[36]

H. Yamakawa, Transport properties of polymer chains in dilute solutions: Hydrodynamic interactions, J. Chem. Phys., 53 (1970), 436-443. doi: 10.1063/1.1673799. Google Scholar

[37]

H. T. Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys., 22 (1991), 63-80. doi: 10.1007/BF00400379. Google Scholar

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