复杂流体动力学中FENE模型的适定性研究
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摘要
本文主要考虑复杂流体动力学中的一些数学问题.复杂流体属于非牛顿流体的范畴,是介于流体和固体之间的,具有复杂本构关系的物质.高分子流体是一类重要的复杂流体,引起了物理学家、化学家、流变学家的广泛关注.本文对高分子流体中著名的FENE (Finite Extensible Nonlinear Elasticity)模型的适定性进行了研究.
全文共分四章.
第一章,首先介绍了复杂流体的数学背景和FENE模型.FENE模型是一个多尺度模型,它是由描述宏观流体流动的Navier-Stokes方程和微观分子运动的Fokker-Planck方程耦合而成.随后介绍了FENE模型国内外研究现状和本文的主要工作.
第二章,因为FENE模型在边界附近有奇性,为了研究解在边界附近的状态,并准确的刻画边值问题的适定性,作者对一类退化抛物型方程初边值问题做了深入的研究.利用正则化的方法,证明了在不需事先给定边界条件的情况下,方程存在唯一的光滑解.
第三章,单独考察了Fokker-Planck方程.首先证明了Fokker-Planck方程弱解的存在性,然后利用第二章结论,证明了当非量纲参数b>2时,Fokker-Planck方程在某个加权的Sobolev空间中存在唯一的光滑解.得到的正则性结果是本质的并且当b不是偶数时是最优的.
第四章,利用压缩映像原理,证明了当参数b>2时,FENE模型方程组局部光滑解的存在性.
The present Ph.D. thesis is concerned with some mathematical problems on hydro-dynamic system for complex fluids. Complex fluids are non-Newtonian fluids, they can be viewed as the intermediate states between the fluids and the solids. An important flu-id of the complex fluid is the polymer fluid, which has aroused extensive attention of physicists, chemists and rheologists. The author considers the finite extensible nonlinear elasticity (FENE) systems for polymeric fluids.
This thesis has four parts.
In Chapter1, we introduce the background of complex fluids and the FENE model. This system couples the incompressible Navier-Stokes equation for the fluid velocity with the Fokker-Planck equation describing the evolution of the polymer density. Moreover, this chapter also includes some recent results and our main works.
In Chapter2, since the FENE model is of singularity at the boundary, in order to discuss the behavior of solutions near the boundary to the FENE model and the exact formulation of the well-posedness of boundary value problems, an initial boundary value problem for some modeling degenerate parabolic equation is introduced and its existence is proved by the method of regularization, and the higher order regularity of solutions is studied in Sobolev spaces without any boundary condition.
In Chapter3, we focus our attention on the microscopic description in the FENE model. We prove the existence of global smooth solutions in some weighted Sobolev spaces to the microscopic FENE model if the non-dimensional parameter b>2. The regularity obtained here is intrinsic and the best when b is not even.
In Chapter4, we prove the local existence of smooth solutions for the FENE dumb-bell model of polymeric flows in some weighted spaces by the fixed-point theorem if the non-dimensional parameter b>2.
全文共分四章.
第一章,首先介绍了复杂流体的数学背景和FENE模型.FENE模型是一个多尺度模型,它是由描述宏观流体流动的Navier-Stokes方程和微观分子运动的Fokker-Planck方程耦合而成.随后介绍了FENE模型国内外研究现状和本文的主要工作.
第二章,因为FENE模型在边界附近有奇性,为了研究解在边界附近的状态,并准确的刻画边值问题的适定性,作者对一类退化抛物型方程初边值问题做了深入的研究.利用正则化的方法,证明了在不需事先给定边界条件的情况下,方程存在唯一的光滑解.
第三章,单独考察了Fokker-Planck方程.首先证明了Fokker-Planck方程弱解的存在性,然后利用第二章结论,证明了当非量纲参数b>2时,Fokker-Planck方程在某个加权的Sobolev空间中存在唯一的光滑解.得到的正则性结果是本质的并且当b不是偶数时是最优的.
第四章,利用压缩映像原理,证明了当参数b>2时,FENE模型方程组局部光滑解的存在性.
The present Ph.D. thesis is concerned with some mathematical problems on hydro-dynamic system for complex fluids. Complex fluids are non-Newtonian fluids, they can be viewed as the intermediate states between the fluids and the solids. An important flu-id of the complex fluid is the polymer fluid, which has aroused extensive attention of physicists, chemists and rheologists. The author considers the finite extensible nonlinear elasticity (FENE) systems for polymeric fluids.
This thesis has four parts.
In Chapter1, we introduce the background of complex fluids and the FENE model. This system couples the incompressible Navier-Stokes equation for the fluid velocity with the Fokker-Planck equation describing the evolution of the polymer density. Moreover, this chapter also includes some recent results and our main works.
In Chapter2, since the FENE model is of singularity at the boundary, in order to discuss the behavior of solutions near the boundary to the FENE model and the exact formulation of the well-posedness of boundary value problems, an initial boundary value problem for some modeling degenerate parabolic equation is introduced and its existence is proved by the method of regularization, and the higher order regularity of solutions is studied in Sobolev spaces without any boundary condition.
In Chapter3, we focus our attention on the microscopic description in the FENE model. We prove the existence of global smooth solutions in some weighted Sobolev spaces to the microscopic FENE model if the non-dimensional parameter b>2. The regularity obtained here is intrinsic and the best when b is not even.
In Chapter4, we prove the local existence of smooth solutions for the FENE dumb-bell model of polymeric flows in some weighted spaces by the fixed-point theorem if the non-dimensional parameter b>2.
引文
[1]伍卓群,尹景学,王春朋.椭圆与抛物型方程引论[M].北京：科学出版社,2003.
[2]R. A. Adams. Sobolev Spaces. Academic Press,1975.
[3]A. Arnold, J. A. Carrillo and C. Manzini. Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci.8 (2010), no.3,763-782.
[4]J. W. Barrett, C. Schwab and E. Suli. Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci.15 (2005), no.6,939-983.
[5]J. W. Barrett and E. Suli. Existence of global weak solutions to kinetic models of dilute polymers. Multiscale Model. Simul 6 (2007),506-546.
[6]J. W. Barrett and E. Suli. Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off. Math. Models Methods Appl. Sci.18 (2008),935-971.
[7]J. W. Barrett and E. Siili. Existence of equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers. Math. Models Methods Appl. Sci.21 (2011), no.6,1211-1289.
[8]R. B. Bird, C. Curtiss, R. C. Armstrong and O. Hassager. Dynamics of Polymeric Liquids, Volume 2:Kinetic Theory. Wiley Interscience, New York,1987.
[9]R. B. Bird and H. C. Ottinger. Transport properties of polymeric liquids. Annual Rev. Phys. Chem.54 (1992),371-406.
[10]C. Chauviere and A. Lozinski. Simulation of complex viscoelastic flows using Fokker-Planck equation:3D FENE model. J. Non-Newtonian Fluid Mech. 111(2004),201-214.
[11]C. Chauviere and A. Lozinski. Simulation of dilute polymer solutions using a Fokker-Planck equation. J. Comput. Fluids.33 (2004),687-696.
[12]J. Y. Chemin and N. Masmoudi. About lifespan of regular solutions of equations related to viscoelastic fluids. SIAMJ. Math. Anal.33 (2001), no.1,84-112.
[13]Y. Chen and P. Zhang. The global existence of small solutions to the incompressible vis-coelastic fluid system in 2 and 3 space dimensions. Comm. Partial Differential Equations. 31 (2006), no.11-12,1793-1810.
[14]P. Constantin. Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci.3 (2005), no.4,531-544.
[15]P. Constantin, C. Fefferman, E. Titi and A. Zarnescu. Regularity for coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes system. Comm. Math. Phys.270 (2007),789-811.
[16]P. Constantin and C. Foias. Navier-Stokes Equations. The University of Chicago Press,1988.
[17]P. Constantin and N. Masmoudi. Global well-posedness for a Smoluchowski equation cou-pled with Navier-Stokes Equations in 2D. Comm. Math. Phys.278 (2008),179-191.
[18]M. Doi and S. F. Edwards. The Theory of Polymer Dynamics. Oxford University Press, Oxford,1986.
[19]Q. Du, C. Liu and P. Yu. FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul.4 (2005), no.3,709-731.
[20]W. E, T. J. Li and P. W. Zhang. Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys.248 (2004), no.2,409-427.
[21]A. Friendman. Partial Differential Equation. Holt, Rinehart and Winston, New York,1969.
[22]C. H. Gu. On certain differential equations of mixed type in n-dimensionsl space. (Russian) Sci. Sinica.14 (1965),1574-1581.
[23]C. H. Gu. Boundary value problems for linear equations of mixed type in n-independent variables. Kexue Tongbao.23 (1978), no.6,335-339.
[24]Y. Guo. The Vlasov-Maxwell-Boltzman system near Maxwellians. Invent. Math.253 (2003), 593-630.
[25]Y. He. Well-posedness and regularity of boundary value problems for a class of second-order degenerate semilinear elliptic equations. Chinese Ann. Math. Ser. A.25 (2004), no.2,225-242.
[26]L. B. He and P. Zhang. L2 decay of solutions to a micro-macro model for polymeric fluids near equilibrium. SUM J. Math. Anal.40 (2008/09), no.5,1905-1922.
[27]L. B. He and Z. F. Zhang. Regularity of the solutions for Micro-Macro models near equilib-rium. Jour. Math. Anal. Appl.348 (2008),419-432.
[28]J. G. Heywood. The Navier-Stokes equations:on the existence regularity and decay of so-lutions, Indiana Univ. Math. J.29 (1980), no.5,639-681.
[29]J. X. Hong. On boundary value problems for mixed equations with characteristic degenerate surfaces. Chinese Ann. Math.2 (1981), no.4,407-424.
[30]J. X. Hong and G. Yang. On the regularity of solutions to FENE models, submitted..
[31]B. Jourdain and T. Lelievre. Mathematical analysis of a stochastic differential equation aris-ing in the micro-macro modelling of polymeric fluids. In Probabilistic Methods in Fluids. World Sci. Publ. River Edge, NJ,2003,205-223.
[32]B. Jourdain, T. Lelievre and C. Le Bris. Existence of solution for a micro-macro model of polymeric fluid:the FENE model. J. Fund. Anal.209 (2004), no.1,162-193.
[33]B. Jourdain, T. Lelievre, C. Le Bris and F. Otto. Long-time asymptotics of amultiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal.181 (2006),97-148.
[34]M. Kroger. Simple models for complex nonequilibrium fluids. Physics Reports.390 (2004), 453-551.
[35]D. J. Knezevic and E. Siili. Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift. ESAIM:M2AN.42 (2009), no.3,445-485.
[36]O. A. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. Second Edition. New York, Gordon and Breach,1969.
[37]Z. Lei and Y. Zhou. Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit. SIAMJ. Math. Anal.37 (2005), no.3,797-814.
[38]Z. Lei, C. Liu and Y. Zhou. Global solutions for incompressible viscoelastic fluids. Arch. RationalMech. Anal.188 (2008),371-398.
[39]Z. Lei and Y. Wang. Global solutions for micro-macro models of polymeric fluids. J. Dif-ferential Equations.250 (2011),3813-3830.
[40]T. J. Li and P. W. Zhang. Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci.5 (2007),1-51.
[41]F. H. Lin, C. Liu and P. Zhang. On hydrodynamics of viscoelastic fluids. Comm. PureAppl. Math.58 (2005), no.11,1437-1471.
[42]F. H. Lin, C. Liu and P. Zhang. On a micro-macro model for polymeric fluids near equilib-rium. Comm. Pure Appl. Math.60 (2007), no.6,838-866.
[43]F. H. Lin and P. Zhang. On the initial boundary value problem of the incompressible vis-coelastic fluid system. Comm. Pure Appl. Math.61 (2008), no.4,539-558.
[44]F. H. Lin and P. Zhang. The FENE dumbbell model near equilibrium. Acta Math. Sin. (Engl. Ser.) 24 (2008), no.4,529-538.
[45]F. H. Lin, P. Zhang and Z. F. Zhang. On the global existence of smooth solution to the 2-D FENE dumbbell model. Comm. Math. Phys. 111(2008), no.2,531-553.
[46]P. L. Lions and N. Masmoudi. Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B.21 (2000), no.2,131-146.
[47]P. L. Lions and N. Masmoudi. Global existence of weak solutions to some micro-macro models. C. R Math. Acad. Sci. Paris.345 (2007), no.1,15-20.
[48]J. L. Lion and E. Magenes. Non-Homogeneous Boundary Value problem and Applications, Vol.2. Springer-Verlag, Berlin Herdelberg, New York,1973.
[49]C. Liu and H. L. Liu. Boundary conditions for the microscopic FENE models. SIAMJ. Appl. Math.68 (2008), no.5,1304-1315.
[50]H. L. Liu and J. Shin. Global well-posedness for the microscopic FENE model with a sharp boundary condition. J. Differential Equations.252 (2012), no.1,641-662.
[51]H. L. Liu and J. Shin. The Cauchy-Dirichlet problem for the FENE dumbbell model of polymeric flows. Preprint,2011.
[52]H. L. Liu and H. Yu. An entropy satisfying finite volume method for the Fokker-Planck equation of FENE dumbbell model for polymers. Preprint,2010.
[53]A. A. Majda and L. A. Bertozzi. Vorticity and Incompressible Flow. Cambridge University Press, Cambridge,2002.
[54]N. Masmoudi. Well-posedness for the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math.61 (2008), no.12,1685-1714.
[55]N. Masmoudi, P. Zhang and Z. F. Zhang. Global well-posedness for 2D polymeric fluid models and growth estimate. Phys. D.237 (2008), no.10-12,1663-1675.
[56]N. Masmoudi. Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Preprint,2011.
[57]L. Nirenberg. On elliptic partial differential equation. Ann. Sc. Norm. Super. Pisa.13 (1959), 115-162.
[58]O. A. Olelnik and E. V. Radkevic. Second Order Equations with Nonnegative Characteristic Form. AMS, Providence, Rhode Island and Plenum Press, New York,1973.
[59]R. G. Owens and T. N. Phillips. Computational Rheology. Imperial College Press, London, 2002.
[60]H. C. Ottinger. Stochastic Processes in Polymeric Liquids. Springer-Verlag, Berlin and New York,1996.
[61]M. Renardy. An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAMJ. Math. Anal.22 (1991), no.2,313-327.
[62]M. E. Schonbek. Existence and decay of polymeric flows. SIAMJ. Math. Anal.41 (2009), no.2,564-587.
[63]J. Shen and H. J. Yu. On the approximation of the Fokker-Planck equation of FENE dumbbell model, I:A new weighted formulation and an optimal Spectral-Galerkin algorithm in 2-D. Preprint,2010.
[64]E. M. Stein. Harmonic Analysis. Princetion University Press, New Jersey,1993.
[65]R. Teman. Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland,1984.
[66]G. Yang. Local existence of smooth solutions to the FENE dumbbell model. Chinese Ann. Math. Ser. B., to appear.
[67]P. T. Yue, J. J. Feng, C. Liu and J. Shen. A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech.,2004,515:293-317.
[68]H. Zhang and P. W. Zhang. Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal.181 (2006), no.2,373-400.
[69]W. J. Zhao. The global existence of small solutions to the Oldroyd-B model. Chinese Ann. Math. Ser. B.32 (2011), no.2,215-222.
[2]R. A. Adams. Sobolev Spaces. Academic Press,1975.
[3]A. Arnold, J. A. Carrillo and C. Manzini. Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci.8 (2010), no.3,763-782.
[4]J. W. Barrett, C. Schwab and E. Suli. Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci.15 (2005), no.6,939-983.
[5]J. W. Barrett and E. Suli. Existence of global weak solutions to kinetic models of dilute polymers. Multiscale Model. Simul 6 (2007),506-546.
[6]J. W. Barrett and E. Suli. Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off. Math. Models Methods Appl. Sci.18 (2008),935-971.
[7]J. W. Barrett and E. Siili. Existence of equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers. Math. Models Methods Appl. Sci.21 (2011), no.6,1211-1289.
[8]R. B. Bird, C. Curtiss, R. C. Armstrong and O. Hassager. Dynamics of Polymeric Liquids, Volume 2:Kinetic Theory. Wiley Interscience, New York,1987.
[9]R. B. Bird and H. C. Ottinger. Transport properties of polymeric liquids. Annual Rev. Phys. Chem.54 (1992),371-406.
[10]C. Chauviere and A. Lozinski. Simulation of complex viscoelastic flows using Fokker-Planck equation:3D FENE model. J. Non-Newtonian Fluid Mech. 111(2004),201-214.
[11]C. Chauviere and A. Lozinski. Simulation of dilute polymer solutions using a Fokker-Planck equation. J. Comput. Fluids.33 (2004),687-696.
[12]J. Y. Chemin and N. Masmoudi. About lifespan of regular solutions of equations related to viscoelastic fluids. SIAMJ. Math. Anal.33 (2001), no.1,84-112.
[13]Y. Chen and P. Zhang. The global existence of small solutions to the incompressible vis-coelastic fluid system in 2 and 3 space dimensions. Comm. Partial Differential Equations. 31 (2006), no.11-12,1793-1810.
[14]P. Constantin. Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci.3 (2005), no.4,531-544.
[15]P. Constantin, C. Fefferman, E. Titi and A. Zarnescu. Regularity for coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes system. Comm. Math. Phys.270 (2007),789-811.
[16]P. Constantin and C. Foias. Navier-Stokes Equations. The University of Chicago Press,1988.
[17]P. Constantin and N. Masmoudi. Global well-posedness for a Smoluchowski equation cou-pled with Navier-Stokes Equations in 2D. Comm. Math. Phys.278 (2008),179-191.
[18]M. Doi and S. F. Edwards. The Theory of Polymer Dynamics. Oxford University Press, Oxford,1986.
[19]Q. Du, C. Liu and P. Yu. FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul.4 (2005), no.3,709-731.
[20]W. E, T. J. Li and P. W. Zhang. Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys.248 (2004), no.2,409-427.
[21]A. Friendman. Partial Differential Equation. Holt, Rinehart and Winston, New York,1969.
[22]C. H. Gu. On certain differential equations of mixed type in n-dimensionsl space. (Russian) Sci. Sinica.14 (1965),1574-1581.
[23]C. H. Gu. Boundary value problems for linear equations of mixed type in n-independent variables. Kexue Tongbao.23 (1978), no.6,335-339.
[24]Y. Guo. The Vlasov-Maxwell-Boltzman system near Maxwellians. Invent. Math.253 (2003), 593-630.
[25]Y. He. Well-posedness and regularity of boundary value problems for a class of second-order degenerate semilinear elliptic equations. Chinese Ann. Math. Ser. A.25 (2004), no.2,225-242.
[26]L. B. He and P. Zhang. L2 decay of solutions to a micro-macro model for polymeric fluids near equilibrium. SUM J. Math. Anal.40 (2008/09), no.5,1905-1922.
[27]L. B. He and Z. F. Zhang. Regularity of the solutions for Micro-Macro models near equilib-rium. Jour. Math. Anal. Appl.348 (2008),419-432.
[28]J. G. Heywood. The Navier-Stokes equations:on the existence regularity and decay of so-lutions, Indiana Univ. Math. J.29 (1980), no.5,639-681.
[29]J. X. Hong. On boundary value problems for mixed equations with characteristic degenerate surfaces. Chinese Ann. Math.2 (1981), no.4,407-424.
[30]J. X. Hong and G. Yang. On the regularity of solutions to FENE models, submitted..
[31]B. Jourdain and T. Lelievre. Mathematical analysis of a stochastic differential equation aris-ing in the micro-macro modelling of polymeric fluids. In Probabilistic Methods in Fluids. World Sci. Publ. River Edge, NJ,2003,205-223.
[32]B. Jourdain, T. Lelievre and C. Le Bris. Existence of solution for a micro-macro model of polymeric fluid:the FENE model. J. Fund. Anal.209 (2004), no.1,162-193.
[33]B. Jourdain, T. Lelievre, C. Le Bris and F. Otto. Long-time asymptotics of amultiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal.181 (2006),97-148.
[34]M. Kroger. Simple models for complex nonequilibrium fluids. Physics Reports.390 (2004), 453-551.
[35]D. J. Knezevic and E. Siili. Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift. ESAIM:M2AN.42 (2009), no.3,445-485.
[36]O. A. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. Second Edition. New York, Gordon and Breach,1969.
[37]Z. Lei and Y. Zhou. Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit. SIAMJ. Math. Anal.37 (2005), no.3,797-814.
[38]Z. Lei, C. Liu and Y. Zhou. Global solutions for incompressible viscoelastic fluids. Arch. RationalMech. Anal.188 (2008),371-398.
[39]Z. Lei and Y. Wang. Global solutions for micro-macro models of polymeric fluids. J. Dif-ferential Equations.250 (2011),3813-3830.
[40]T. J. Li and P. W. Zhang. Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci.5 (2007),1-51.
[41]F. H. Lin, C. Liu and P. Zhang. On hydrodynamics of viscoelastic fluids. Comm. PureAppl. Math.58 (2005), no.11,1437-1471.
[42]F. H. Lin, C. Liu and P. Zhang. On a micro-macro model for polymeric fluids near equilib-rium. Comm. Pure Appl. Math.60 (2007), no.6,838-866.
[43]F. H. Lin and P. Zhang. On the initial boundary value problem of the incompressible vis-coelastic fluid system. Comm. Pure Appl. Math.61 (2008), no.4,539-558.
[44]F. H. Lin and P. Zhang. The FENE dumbbell model near equilibrium. Acta Math. Sin. (Engl. Ser.) 24 (2008), no.4,529-538.
[45]F. H. Lin, P. Zhang and Z. F. Zhang. On the global existence of smooth solution to the 2-D FENE dumbbell model. Comm. Math. Phys. 111(2008), no.2,531-553.
[46]P. L. Lions and N. Masmoudi. Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B.21 (2000), no.2,131-146.
[47]P. L. Lions and N. Masmoudi. Global existence of weak solutions to some micro-macro models. C. R Math. Acad. Sci. Paris.345 (2007), no.1,15-20.
[48]J. L. Lion and E. Magenes. Non-Homogeneous Boundary Value problem and Applications, Vol.2. Springer-Verlag, Berlin Herdelberg, New York,1973.
[49]C. Liu and H. L. Liu. Boundary conditions for the microscopic FENE models. SIAMJ. Appl. Math.68 (2008), no.5,1304-1315.
[50]H. L. Liu and J. Shin. Global well-posedness for the microscopic FENE model with a sharp boundary condition. J. Differential Equations.252 (2012), no.1,641-662.
[51]H. L. Liu and J. Shin. The Cauchy-Dirichlet problem for the FENE dumbbell model of polymeric flows. Preprint,2011.
[52]H. L. Liu and H. Yu. An entropy satisfying finite volume method for the Fokker-Planck equation of FENE dumbbell model for polymers. Preprint,2010.
[53]A. A. Majda and L. A. Bertozzi. Vorticity and Incompressible Flow. Cambridge University Press, Cambridge,2002.
[54]N. Masmoudi. Well-posedness for the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math.61 (2008), no.12,1685-1714.
[55]N. Masmoudi, P. Zhang and Z. F. Zhang. Global well-posedness for 2D polymeric fluid models and growth estimate. Phys. D.237 (2008), no.10-12,1663-1675.
[56]N. Masmoudi. Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Preprint,2011.
[57]L. Nirenberg. On elliptic partial differential equation. Ann. Sc. Norm. Super. Pisa.13 (1959), 115-162.
[58]O. A. Olelnik and E. V. Radkevic. Second Order Equations with Nonnegative Characteristic Form. AMS, Providence, Rhode Island and Plenum Press, New York,1973.
[59]R. G. Owens and T. N. Phillips. Computational Rheology. Imperial College Press, London, 2002.
[60]H. C. Ottinger. Stochastic Processes in Polymeric Liquids. Springer-Verlag, Berlin and New York,1996.
[61]M. Renardy. An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAMJ. Math. Anal.22 (1991), no.2,313-327.
[62]M. E. Schonbek. Existence and decay of polymeric flows. SIAMJ. Math. Anal.41 (2009), no.2,564-587.
[63]J. Shen and H. J. Yu. On the approximation of the Fokker-Planck equation of FENE dumbbell model, I:A new weighted formulation and an optimal Spectral-Galerkin algorithm in 2-D. Preprint,2010.
[64]E. M. Stein. Harmonic Analysis. Princetion University Press, New Jersey,1993.
[65]R. Teman. Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland,1984.
[66]G. Yang. Local existence of smooth solutions to the FENE dumbbell model. Chinese Ann. Math. Ser. B., to appear.
[67]P. T. Yue, J. J. Feng, C. Liu and J. Shen. A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech.,2004,515:293-317.
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