Chapter 5 Mathematical Issues Concerning The Navier-Stokes Equations and Some Of Its Generalizations

Handbook of Differential Equations: Evolutionary Equations

Volumn 2, Issue , 2005, Pages 371-373

  Malek J ^a   Rajagopal, K R ^b  

^a CHARLES UNIVERSITY   (Czech Republic)

^b Texas AandM University   (United States)

Author keywords

Incompressible fluid; Mathematical analysis; Navier Stokes fluid; Non Newtonian fluid; Rheology

Indexed keywords

EID: 66049099719     PISSN: 18745717     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5717(06)80008-3     Document Type: Chapter

References (153)

1. Amann H. Stability of the rest state of a viscous incompressible fluid. Arch. Ration. Mech. Anal. 126 (1994) 231-242
2. Anand M., and Rajagopal K.R. A mathematical model to describe the change in the constitutive character of blood due to platelet activation. C. R. Méc. 330 (2002) 557-562
3. to appear. Anand M., and Rajagopal K.R. A note on the flow of inhomogeneous fluids with shear dependent viscosities. Arch. Mech. (2005)
4. Andrade C. Viscosity of liquids. Nature 125 (1930) 309-310
5. Antontsev S.N., Kazhikhov A.V., and Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Stud. Math. Appl. Vol. 22 (1990), North-Holland, Berlin-Heidelberg Transl. from the Russian
6. Babin A.V. Attractors of Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics (2003), North-Holland, Amsterdam 169-222
7. n. Chinese Ann. Math. 16 (1995)
8. Ser. A. A summary. Chinese Ann. Math. 16 (1995) 797
9. Beirao da Veiga H. On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Comm. Pure Appl. Math. 58 4 (2005) 552-577
10. Bellout H., Bloom F., and Ne
11. (electronic). Berselli L.C., and Galdi G.P. Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations. Proc. Amer. Math. Soc. 130 (2002) 3585-3595
12. Bingham E.C. Fluidity and plasticity. In: Alexander J. (Ed). Colloid Chemistry (1928), Chemical Catalog Co., Amsterdam 720-726
13. Boccardo L., and Murat F. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19 (1992) 581-597
14. Bridgman P.W. The Physics of High Pressure (1931), MacMillan, New York
15. M. Bulí
16. prepared by the committee for the study of viscosity of the Academy of Sciences at Amsterdam. Burgers J.M. Mechanical consideration - model systems - phenomenological theories of relaxation and of viscosity. First Report on Viscosity and Plasticity. 2nd Edition (1939), Nordemann, New York 5-72
17. L. Caffarelli, A Clay mathematical institute millennium prize problem, existence and smoothness of the Navier-Stokes equations, Lecture (video). Available at
18. http://www.claymath.org/millennium/Navier-Stokes_Equations
19. Caffarelli L., Kohn R., and Nirenberg L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771-831
20. with a preface by Yves Meyer. Cannone M. Ondelettes, Paraproduits et Navier-Stokes (1995), Diderot
21. 1999. Chae D., and Choe H.-J. Regularity of solutions to the Navier-Stokes equation, Electron. J. Differential Equations 05 (1999) 1-7
22. Choe H.J., and Lewis J.L. On the singular set in the Navier-Stokes equations. J. Funct. Anal. 175 (2000) 348-369
23. Coleman B.D., and Noll W. The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13 (1963) 167-178
24. Constantin P., and FoiaŞ C. Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations. Comm. Pure Appl. Math. 38 (1985) 1-27
25. Constantin P., and FoiaŞ C. Navier-Stokes Equations, Chicago Lectures in Math (1988), University of Chicago Press, Paris
26. DiBenedetto E. Degenerate Parabolic Equations (1993), Springer-Verlag, Chicago, IL
27. to appear. Diening L., and Rů
28. Doering C.R., and Gibbon J.D. Applied Analysis of the Navier-Stokes Equations. Cambridge Texts Appl. Math. (1995), Cambridge University Press, New York
29. Paris. Du Buat P.L.G. Principes d'hydraulique, L'imprimerie de minsieur (1779)
30. Dunn J.E., and Serrin J. On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88 (1985) 95-133
31. Eckart C. The thermodynamics of irreversible processes. IV The theory of elasticity and anelasticity. Phys. Rev. 73 (1948) 373-382
32. Eden A., FoiaŞ C., Nicolaenko B., and Temam R. Exponential Attractors for Dissipative Evolution Equations. Res. in Appl. Math. Vol. 37 (1994), Masson, Cambridge
33. Escauriaza L., Serëgin G., and
34. Escauriaza L., Serëgin G., and
35. 3 ∞-solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat. Nauk. 58 (2003) 3-44
36. C.L. Fefferman, Existence and smoothness of the Navier-Stokes equation, Official Problem Description of the Third Clay Mathematical Institute Millennium Problem. Available at
37. http://www.claymath.org/millennium/Navier-Stokes_Equations
38. Feireisl E. Dynamics of Viscous Compressible Fluids (2004), Oxford University Press, Paris
39. to appear. Feireisl E. Mathematical theory of compressible, viscous and heat conducting fluids. Comput. Math. Appl. (2005)
40. Fernández-Cara E., Guillén F., and Ortega R.R. Some theoretical results for viscoplastic and dilatant fluids with variable density. Nonlinear Anal. 28 (1997) 1079-1100
41. FoiaŞ C., Guillopé C., and Temam R. New a priori estimates for Navier-Stokes equations in dimension 3. Comm. Partial Differential Equations 6 (1981) 329-359
42. FoiaŞ C., and Olson E. Finite fractal dimension and Hölder-Lipschitz parametrization. Indiana Univ. Math. J. 45 (1996) 603-616
43. FoiaŞ C., and Temam R. A generic property of the set of stationary solutions of Navier-Stokes equations. Turbulence and Navier-Stokes Equations, Proc. Conf., Univ. Paris-Sud, Orsay, 1975, Lecture Notes in Math. (1976), Springer-Verlag, Oxford 24-28
44. FoiaŞ C., and Temam R. Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations. J. Math. Pures Appl. 58 9 (1979) 339-368
45. Franta M., Málek J., and Rajagopal K.R. On steady flows of fluids with pressure- and shear-dependent viscosities. Proc. Roy. Soc. London Ser. A 461 (2005) 651-670
46. Frehse J., and Málek J. Problems due to the no-slip boundary in incompressible fluid dynamics, Geometric Analysis and Nonlinear Partial Differential Equations (2003), Springer-Verlag, Berlin 559-571
47. Frehse J., Malek J., and Steinhauer M. On existence results for fluids with shear dependent viscosity-unsteady flows. Partial Differential Equations, Praha 1998, Chapman & Hall/CRC Res. Notes Math (2000), Chapman & Hall/CRC, Berlin 121-129
48. Frehse J., Málek J., and Steinhauer M. On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34 (2003) 1064-1083
49. Galdi G.P. An introduction to the Navier-Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics. Adv. Math. Fluid Mech., Birkhauser, Basel (2000) 1-70
50. North-Holland, Amsterdam. Ghidaglia J.-M., and Temam R. Lower bound on the dimension of the attractor for the Navier-Stokes equations in space dimension 3. Mechanics, Analysis and Geometry: 200 Years after Lagrange, North-Holland Delta Ser. (1991) 33-60
51. Goldstein S. Modern Developments in Fluid Dynamics (1938), Oxford University Press, Boca Raton, FL
52. Green A.E., and Naghdi P.M. On thermodynamics and nature of the second law. Proc. Roy. Soc. London Ser. A 357 (1977) 253-270
53. Guillén-González F. Density-dependent incompressible fluids with non-Newtonian viscosity. Czechoslovak Math. J. 54 (2004) 637-656
54. Hale J.K. Asymptotic Behavior of Dissipative Systems. Math. Surveys Monogr. Vol. 25 (1988), Amer. Math. Soc., Oxford
55. Heywood J.G. Open problems in the theory of the Navier-Stokes equations for viscous incompressible flow. The Navier-Stokes Equations, Oberwolfach 1988, Lecture Notes in Math. (1990), Springer-Verlag, Providence, RI 1-22
56. Hopf E. Uber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951) 213-231
57. Hron J., Málek J., Ne
58. Huilgol R.R. Continuum Mechanics of Viscoelastic Liquids (1975), Hindusthan Publishing Co., Berlin
59. Kaniel S. A sufficient condition for smoothness of solutions of Navier-Stokes equations. Israel J. Math. 6 (1968) 354-358
60. Kaniel S. On the initial value problem for an incompressible fluid with nonlinear viscosity. J. Math. Mech. 19 (1970) 681-707
61. Kaplický P. Some remarks to regularity of flow of generalized Newtonian fluid. In: Dumortier F., Broer H., Mawhin J., Vanderbauwhede A., and Verduyn Lunel S. (Eds). Proc. Conf. Equadiff 2003 (2005), World Scientific, Delhi 377-380
62. to appear. Kaplický P. Regularity of flows of a non-Newtonian fluid in two dimensions subject to Dirichlet boundary conditions. Z. Anal. Anwendungen (2005)
63. Kaplicy P., Málek J., and Stará J. Global-in-time Holder continuity of the velocity gradients for fluids with shear-dependent viscosities. Nonlinear Differential Equations Appl. 9 (2002) 175-195
64. Kiselev A.A., and Ladyzhenskaya O.A. On existence and uniqueness of the solution of the nonstationary problem for a viscous incompressible fluid. Izv. Akad. Nauk SSSR 21 (1957) 655-680
65. Kozono H. On well-posedness of the Navier-Stokes equations, Mathematical Fluid Mechanics. Adv. Math. Fluid Mech., Birkhäuser, Basel (2001) 207-236
66. Kozono H., and Taniuchi Y. Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000) 173-194
67. Murali Krishnan J., and Rajagopal K.R. A thermomechanical framework for the constitutive modeling of asphalt concrete: Theory and applications. ASCE J. Materials in Civil Engineering 16 (2004) 155-166
68. Ladyzhenskaya O.A. On some new equations describing dynamics of incompressible fluids and on global solvability of boundary value problems to these equations. Trudy Mat. Inst. Steklov 102 (1967) 85-104
69. Ladyzhenskaya O.A. Example of nonuniqueness in the Hopf class of weak solutions for the Navier-Stokes equations. Izv. Akad. Nauk SSSR 3 (1969) 229-236
70. Ladyzhenskaya O.A. The Mathematical Theory of Viscous Incompressible Flow. In: Silverman R.A. (Ed). 2nd English Edition, revised and enlarged. J. Chu, Math. Appl. Vol. 2 (1969), Gordon and Breach, Singapore
71. Ladyzhenskaya O.A. A dynamical system that is generated by the Navier-Stokes equations. Dokl. Akad. Nauk SSSR 205 (1972) 318-320
72. Ladyzhenskaya O.A. Limit states for modified Navier-Stokes equations in three-dimensional space. Zap. Nauchn. Sem. LOMI 84 (1979) 131-146
73. Ladyzhenskaya O.A. Finite-dimensionality of bounded invariant sets for Navier-Stokes systems and other dissipative systems. Zap. Nauchn. Sem. LOMI 115 (1982) 137-155
74. Ladyzhenskaya O.A. Attractors for Semigroups and Evolution Equations, Lezioni Lincee (1991), Cambridge University Press, New York
75. Ladyzhenskaya O.A. Attractors for the modifications of the three-dimensional Navier-Stokes equations. Phil. Trans. Roy. Soc. London Ser. A 346 (1994) 173-190
76. Ladyzhenskaya O.A., and Seregin G.A. On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1 (1999) 356-387
77. Lemarié-Rieusset P.G. Recent Developments in the Navier-Stokes Problem. Chapman & Hall/CRC Res. Notes Math. (2002), Chapman & Hall/CRC, Cambridge
78. Leray J. Sur le mouvement d'un liquide visqueux emplissant l, space. Acta Math. 63 (1934) 193-248
79. Lin F.H. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51 (1998) 241-257
80. Lions J.L. Quelques méthodes de résolution des probémes aux limites non lineáires (1969), Dunod, Boca Raton, FL
81. Lions P.L. Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models. Oxford Lecture Ser. Math. AppL (1996), Clarendon Press/Oxford University Press, Paris
82. Málek J., and Ne
83. Málek J., Ne
84. Málek J., Ne
85. Málek J., Ne
86. Málek J., Ne
87. Málek J., Ne-2. Adv. Differential Equations 6 (2001) 257-302
88. Málek J., and Pra
89. Málek J., and Pra
90. to appear. Málek J., Pra
91. to appear. Málek J., and Rajagopal K.R. On the modeling of inhomogeneous incompressible fluid-like bodies. Mech. Mater. (2005)
92. Málek J., Rajagopal K.R., and Rů
93. Maxwell J.C. Theory of Heat (1970), Greenwood Press, New York
94. Miller J.R., O'Leary M., and Schonbek M. Nonexistence of singular pseudo-self-similar solutions of the Navier-Stokes system. Math. Ann. 319 (2001) 809-815
95. Mosolov P.P., and Mjasnikov V.P. A proof of Korn's inequality. Dokl. Akad. Nauk SSSR 201 (1971) 36-39
96. Navier C.L.M.H. Mémoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Paris 6 (1823) 389-416
97. p(ω) et sur la coercivite des formes formellement positives. Séminaire Equations aux Dérivées Partielles, Les Presses de l'Université de Montréal (1966), Montréal, Westport, CT 102-128
98. Ne
99. Ne
100. Neustupa J., Novotný A., and Penel P. An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity. Topics in Mathematical Fluid Mechanics, Quad. Mat. (2002), Seconda Univ. Napoli 163-183
101. Neustupa J., and Penel P. Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations. Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhauser, Basel (2001) 237-265
102. Neustupa J., and Penel P. The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier-Stokes equations. C. R. Math. Acad. Sci. Paris 336 (2003) 805-810
103. London. Newton I. Philosophiα Naturalis Principia Mathematica. J. Societatis Regiae ac Typis J. Streater (1687)
104. Noll W. A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2 (1957) 197-226
105. Noll W. A new mathematical theory of simple materials. Arch. Ration. Mech. Anal. 48 (1972) 1-50
106. Oseen C.W. Hydrodynamik (1927), Akademische Verlagsgesellschaft M.B.H., Caserta
107. Parés C. Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids. Appl. Anal. 43 (1992) 245-296
108. Poisson S.D. Mémoire sur les Équations générates de lequilibre et du moivement des corps solides élastiques et des fluides. Journal de l'École Polytechnique 13 (1831) 1-174
109. Prasad S.C., Rao I.J., and Rajagopal K.R. A continuum model for the creep of single crystal nickel-base superalloys. Acta Materilia 53 (2005) 669-679
110. Prodi G. Un teorema di unicitá per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48 4 (1959) 173-182
111. Rajagopal K.R. Mechanics of non-Newtonian fluids, Recent Developments in Theoretical Fluid Mechanics. In: Galdi G.P., and Ne
112. Rajagopal K.R. Multiple natural configurations in continuum mechanics. Technical Report 6, Institute for Computational and Applied Mechanics (1995), University of Pittsburgh, Harlow-Essex
113. Rajagopal K.R., and Srinivasa A.R. On the inelastic behavior of solids - Part 1: Twinning, Internat. J. Plasticity 11 (1995) 653-678
114. Rajagopal K.R., and Srinivasa A.R. Inelastic behavior of materials - Part II: Energetics associated with discontinuous deformation twinning. Internat. J. Plasticity 13 (1997) 1-35
115. Rajagopal K.R., and Srinivasa A.R. Mechanics of the inelastic behavior of materials: Part I - Theoretical underpinnings. Internat. J. Plasticity 14 (1998) 945-967
116. Rajagopal K.R., and Srinivasa A.R. Mechanics of the inelastic behavior of materials: Part II - Inelastic response. Internat. J. Plasticity 14 (1998) 969-995
117. Rajagopal K.R., and Srinivasa A.R. On the thermodynamics of shape memory wires. Z. Angew. Math. Phys. 50 (1999) 459-496
118. Rajagopal K.R., and Srinivasa A.R. A thermodynamic framework for rate type fluid models. J. Non-Newt. Fluid Mech. 88 (2000) 207-227
119. Rajagopal K.R., and Srinivasa A.R. Modeling anisotropic fluids within the framework of bodies with multiple natural configurations. J. Non-Newt. Fluid Mech. 99 (2001) 1-16
120. to appear. Rajagopal K.R., and Srinivasa A.R. On a geometric approach to the treatment of constraints in the mechanics of continua. Proc. Roy. Soc. London Ser. A (2005)
121. Rajagopal K.R., and Tao L. Modeling of microwave drying process of aqueous dielectrics. Z. Angew. Math. Phys. 9 (2004) 305-312
122. Rao I.J., and Rajagopal K.R. A study of strain-induced crystallization of polymers. International J. Solids Structures 38 (2001) 1149-1166
123. Rao I.J., and Rajagopal K.R. A thermodynamic framework for the study of crystallization in polymers. Z. Angew. Math. Phys. 53 (2002) 365-406
124. Renardy M. Some remarks on the Navier-Stokes equations with a pressure-dependent viscosity. Comm. Partial Differential Equations 11 (1986) 779-793
125. Saint-Venant A.J.C.B. Note ájoindre au mémoire sur la dynamique desfluides. C. R. Acad. Sci. 17 (1843) 1240-1243
126. Scheffer V. Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66 (1976) 535-552
127. Scheffer V. Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55 (1977) 97-112
128. Scheffer V. The Navier-Stokes equations on a bounded domain. Comm. Math. Phys. 73 (1980) 1-42
129. Scheffer V. Nearly one-dimensional singularities of solutions to the Navier-Stokes inequality. Comm. Math. Phys. 110 (1987) 525-551
130. Schowalter W.R. Mechanics of Non-Newtonian Fluids (1978), Pergamon Press
131. Sell G.R. Global attractors for the three-dimensional Navier-Stokes equations. J. Dynam. Differential Equations 8 (1996) 1-33
132. Serëgin G.A., and
133. Seregin G.A., and
134. 231. Seregin G., and
135. Serrin J. The initial value problem for the Navier-Stokes equations, Nonlinear Problems. Proc. Sympos., Madison, Wisconsin (1963), University of Wisconsin Press, New York 69-98
136. Shinbrot M., and Kaniel S. The initial value problem for the Navier-Stokes equations. Arch. Rational Mech. Anal. 21 (1966) 270-285
137. In: Simpson J., and Weiner E. (Eds). The Oxford English Dictionary. 2nd Edition (1989), Oxford University Press, Madison, WI
138. Smagorinsky J.S. General circulation model of the atmosphere. Mon. Weather Rev. 91 (1963) 99-164
139. Sohr H. The Navier-Stokes Equations. Birkhauser Adv. Texts Basler Lehrb., Birkhäuser, Basel (2001)
140. Spencer A.J.M. Theory of invariants. In: Eringen I.A.C. (Ed). Continuum Physics (1971), Academic Press, New York 292-352
141. Steinhauer M. On uniqueness-and regularity criteria for the Navier-Stokes equations. Geometric Analysis and Nonlinear Partial Differential Equations (2003), Springer-Verlag, New York 543-557
142. Stokes G.G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc. 8 (1845) 287-305
143. Szeri A.Z. Fluid Film Lubrication: Theory and Design (1998), Cambridge University Press, Berlin
144. Temam R. Attractors for Navier-Stokes equations. Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar. Res. Notes Math. Vol. 122 (1985), Pitman, Paris 1983-1984 272-292
145. Temam R. Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Regional Conf. Ser. in Appl. Math. 2nd Edition (1995), SIAM, Boston, MA
146. Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci. 2nd Edition (1997), Springer-Verlag, Philadelphia, PA
147. Reprint of the 1984 edition. Temam R. Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, RI (2001)
148. Trouton F.T. On the coefficient of viscous traction and its relation to that of viscosity. Proc. Roy. Soc. London Ser. A 77 (1906) 426-440
149. Truesdell C. A First Course in Rational Mechanics (1991), Academic Press, New York
150. 3 (1965), Springer-Verlag, Boston
151. Tsai T.-R. On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Ration. Mech. Anal. 143 (1998) 29-51
152. Vieweg, Braunschweig. von Wahl W. The Equations of Navier-Stokes and Abstract Parabolic Equations. Aspects Math. Vol. E8 (1985)
153. Wiegner M. The Navier-Stokes equations - A neverending challenge?. Jahresber. Deutsch. Math.-Verein. 101 (1999) 1-25