On the Euler equations of incompressible fluids

Constantin, Peter

doi:10.1090/S0273-0979-07-01184-6

On the Euler equations of incompressible fluids

Author: Peter Constantin
Journal: Bull. Amer. Math. Soc. 44 (2007), 603-621
MSC (2000): Primary 76B47; Secondary 35Q30
DOI: https://doi.org/10.1090/S0273-0979-07-01184-6
Published electronically: July 5, 2007
MathSciNet review: 2338368
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Abstract: Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

1. H. Abidi and R. Danchin, Optimal bounds for the inviscid limit of Navier-Stokes equations, Asymptot. Anal. 38 (2004), no. 1, 35–46. MR 2060619
2. A. Alexakis, C. Doering, Energy and enstrophy dissipation in steady state 2D turbulence. Physics Lett. A 359 (2006), 652-657.
3. David M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal. 35 (2003), no. 1, 211–244. MR 2001473, https://doi.org/10.1137/S0036141002403869
4. David M. Ambrose and Nader Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math. 58 (2005), no. 10, 1287–1315. MR 2162781, https://doi.org/10.1002/cpa.20085
5. Alexandre Arenas and Alexandre J. Chorin, On the existence and scaling of structure functions in turbulence according to the data, Proc. Natl. Acad. Sci. USA 103 (2006), no. 12, 4352–4355. MR 2213976, https://doi.org/10.1073/pnas.0600482103
6. Vladimir I. Arnold and Boris A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. MR 1612569
7. V. Barcilon, P. Constantin, and E. S. Titi, Existence of solutions to the Stommel-Charney model of the Gulf Stream, SIAM J. Math. Anal. 19 (1988), no. 6, 1355–1364. MR 965256, https://doi.org/10.1137/0519099
8. C. Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux, J. Math. Anal. Appl. 40 (1972), 769–790 (French). MR 333488, https://doi.org/10.1016/0022-247X(72)90019-4
9. C. Bardos, Y. Guo, and W. Strauss, Stable and unstable ideal plane flows, Chinese Ann. Math. Ser. B 23 (2002), no. 2, 149–164. Dedicated to the memory of Jacques-Louis Lions. MR 1924132, https://doi.org/10.1142/S0252959902000158
10. C. Bardos, E. Titi, Euler equations of incompressible ideal fluids. Preprint 2007.
11. Grigory Isaakovich Barenblatt, Scaling, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2003. With a foreword by Alexandre Chorin. MR 2034052
12. G. I. Barenblatt and Alexandre J. Chorin, Scaling laws and vanishing-viscosity limits for wall-bounded shear flows and for local structure in developed turbulence, Comm. Pure Appl. Math. 50 (1997), no. 4, 381–398. MR 1438152, https://doi.org/10.1002/(SICI)1097-0312(199704)50:4<381::AID-CPA5>3.0.CO;2-6
13. G. I. Barenblatt and Alexandre J. Chorin, Scaling laws and vanishing viscosity limits in turbulence theory, Recent advances in partial differential equations, Venice 1996, Proc. Sympos. Appl. Math., vol. 54, Amer. Math. Soc., Providence, RI, 1998, pp. 1–25. MR 1492690, https://doi.org/10.1090/psapm/054/1492690
14. G. I. Barenblatt, A. J. Chorin, and V. M. Prostokishin, Comment on the paper: “On the scaling of three-dimensional homogeneous and isotropic turbulence” [Phys. D. 80 (1995), no. 4, 385–398; MR1312600 (95i:76046)] by R. Benzi, S. Ciliberto, C. Baudet and G. Ruiz-Chavarría, Phys. D 127 (1999), no. 1-2, 105–110. MR 1677445, https://doi.org/10.1016/S0167-2789(98)00289-9
15. J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR 763762
16. Roberto Benzi, Sergio Ciliberto, Cristophe Baudet, and Gerardo Ruiz-Chavarría, On the scaling of three-dimensional homogeneous and isotropic turbulence, Phys. D 80 (1995), no. 4, 385–398. MR 1312600, https://doi.org/10.1016/0167-2789(94)00190-2
17. Henri Berestycki, François Hamel, and Nikolai Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), no. 2, 451–480. MR 2140256, https://doi.org/10.1007/s00220-004-1201-9
18. D. Bernard, Influence of friction on the direct cascade of 2D forced turbulence. Europhys. Lett. 50 (2000), 333-339.
19. Hugo Beirão da Veiga and Luigi C. Berselli, On the regularizing effect of the vorticity direction in incompressible viscous flows, Differential Integral Equations 15 (2002), no. 3, 345–356. MR 1870646
20. L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. MR 673830, https://doi.org/10.1002/cpa.3160350604
21. L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. ArXiv: Math.AP/0608447 (2006).
22. Russel E. Caflisch, Singularity formation for complex solutions of the 3D incompressible Euler equations, Phys. D 67 (1993), no. 1-3, 1–18. MR 1234435, https://doi.org/10.1016/0167-2789(93)90195-7
23. Marco Sammartino and Russel E. Caflisch, Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, Comm. Math. Phys. 192 (1998), no. 2, 433–461. MR 1617542, https://doi.org/10.1007/s002200050304
24. Marco Sammartino and Russel E. Caflisch, Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, Comm. Math. Phys. 192 (1998), no. 2, 433–461. MR 1617542, https://doi.org/10.1007/s002200050304
25. D. Chae, Nonexistence of self-similar singularities for the 3D incompressible Euler equations. Commun. Math. Phys. (2007) (to appear).
26. Dongho Chae, On the conserved quantities for the weak solutions of the Euler equations and the quasi-geostrophic equations, Comm. Math. Phys. 266 (2006), no. 1, 197–210. MR 2231970, https://doi.org/10.1007/s00220-006-0018-0
27. Dongho Chae, On the regularity conditions for the dissipative quasi-geostrophic equations, SIAM J. Math. Anal. 37 (2006), no. 5, 1649–1656. MR 2215601, https://doi.org/10.1137/040616954
28. Jean-Yves Chemin, Perfect incompressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. MR 1688875
29. A. Cheskidov, P. Constantin, S. Friedlander, R. Shvydkoy, Energy conservation and Onsager's conjecture for the Euler equations. ArXiv; Math.AP/0704.0759 (2007).
30. A. Cheskidov, S. Friedlander, N. Pavlovic, An inviscid dyadic model of turbulence: the fixed point and Onsager's conjecture. Journal of Mathematical Physics, to appear.
31. A. Cheskidov, S. Friedlander, N. Pavlovic, An inviscid dyadic model of turbulence: the global attractor. Preprint (2007).
32. S. Childress, G. R. Ierley, E. A. Spiegel, and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech. 203 (1989), 1–22. MR 1002875, https://doi.org/10.1017/S0022112089001357
33. Alexandre J. Chorin and Jerrold E. Marsden, A mathematical introduction to fluid mechanics, 3rd ed., Texts in Applied Mathematics, vol. 4, Springer-Verlag, New York, 1993. MR 1218879
34. Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, https://doi.org/10.1017/S0022112073002016
35. Guido Schneider and C. Eugene Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math. 53 (2000), no. 12, 1475–1535. MR 1780702, https://doi.org/10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V
36. Peter Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Comm. Math. Phys. 104 (1986), no. 2, 311–326. MR 836008
37. Peter Constantin, Geometric statistics in turbulence, SIAM Rev. 36 (1994), no. 1, 73–98. MR 1267050, https://doi.org/10.1137/1036004
38. P. Constantin, The Littlewood-Paley spectrum in 2D turbulence. Theor. Comp. Fluid Dyn. 9 (1997), 183-189.
39. Peter Constantin, The Euler equations and nonlocal conservative Riccati equations, Internat. Math. Res. Notices 9 (2000), 455–465. MR 1756944, https://doi.org/10.1155/S1073792800000258
40. Peter Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory, J. Amer. Math. Soc. 14 (2001), no. 2, 263–278. MR 1815212, https://doi.org/10.1090/S0894-0347-00-00364-7
41. Peter Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations, Comm. Math. Phys. 216 (2001), no. 3, 663–686. MR 1815721, https://doi.org/10.1007/s002200000349
42. Peter Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., vol. 1871, Springer, Berlin, 2006, pp. 1–43. MR 2196360, https://doi.org/10.1007/11545989_1
43. Peter Constantin, Nonlinear Fokker-Planck Navier-Stokes systems, Commun. Math. Sci. 3 (2005), no. 4, 531–544. MR 2188682
44. Peter Constantin, Diego Cordoba, and Jiahong Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 50 (2001), no. Special Issue, 97–107. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). MR 1855665, https://doi.org/10.1512/iumj.2008.57.3629
45. Peter Constantin, Weinan E, and Edriss S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Comm. Math. Phys. 165 (1994), no. 1, 207–209. MR 1298949
46. Peter Constantin and Charles Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J. 42 (1993), no. 3, 775–789. MR 1254117, https://doi.org/10.1512/iumj.1993.42.42034
47. Peter Constantin, Charles Fefferman, and Andrew J. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Partial Differential Equations 21 (1996), no. 3-4, 559–571. MR 1387460, https://doi.org/10.1080/03605309608821197
48. P. Constantin, C. Fefferman, E. S. Titi, and A. Zarnescu, Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems, Comm. Math. Phys. 270 (2007), no. 3, 789–811. MR 2276466, https://doi.org/10.1007/s00220-006-0183-1
49. Peter Constantin and Gautam Iyer, Stochastic Lagrangian transport and generalized relative entropies, Commun. Math. Sci. 4 (2006), no. 4, 767–777. MR 2264819
50. P. Constantin, A. Kiselev, L. Ryzhik, A. Zlatos, Diffusion and mixing in fluid flow. Annals of Math., to appear (2007).
51. P. Constantin, B. Levant, E. Titi, Regularity of inviscid shell models of turbulence. Physical Review E 75 1 (2007), 016305.
52. P. Constantin, N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D. Commun. Math. Phys., to appear (07-08).
53. Peter Constantin, Andrew J. Majda, and Esteban Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), no. 6, 1495–1533. MR 1304437
54. P. Constantin, F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in ${\mathbb{R}}^2$ . Commun. Math. Phys., to appear (2007).
55. P. Constantin, L. Ryzhik, A. Novikov, Relaxation in reactive flows. GAFA (2007) (to appear).
56. Peter Constantin and Jiahong Wu, Inviscid limit for vortex patches, Nonlinearity 8 (1995), no. 5, 735–742. MR 1355040
57. Peter Constantin and Jiahong Wu, The inviscid limit for non-smooth vorticity, Indiana Univ. Math. J. 45 (1996), no. 1, 67–81. MR 1406684, https://doi.org/10.1512/iumj.1996.45.1960
58. Peter Constantin and Jiahong Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 (1999), no. 5, 937–948. MR 1709781, https://doi.org/10.1137/S0036141098337333
59. P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. ArXiv: Math.AP/0701592 (2007).
60. P. Constantin, J. Wu, Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. ArXiv: Math.AP/0701594 (2007).
61. Diego Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math. (2) 148 (1998), no. 3, 1135–1152. MR 1670077, https://doi.org/10.2307/121037
62. Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511–528. MR 2084005, https://doi.org/10.1007/s00220-004-1055-1
63. D. Cordoba, F. Gancedo, Contour dynamics of incompressible 3D fluids in a porous medium with different densities. Commun. Math. Phys., to appear (2007).
64. Diego Córdoba, Charles Fefferman, and Rafael de la Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal. 36 (2004), no. 1, 204–213. MR 2083858, https://doi.org/10.1137/S0036141003424095
65. D. Coutaud, S. Shkoller, Well-posedness of the free surface incompressible Euler equations with or without surface tension. JAMS (2007).
66. Walter Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations 10 (1985), no. 8, 787–1003. MR 795808, https://doi.org/10.1080/03605308508820396
67. C. De Lellis, L. Szekelyhidi, The Euler equations as differential inclusions. Preprint (2007).
68. Jean-Marc Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991), no. 3, 553–586 (French). MR 1102579, https://doi.org/10.1090/S0894-0347-1991-1102579-6
69. Jian Deng, Thomas Y. Hou, and Xinwei Yu, Geometric properties and nonblowup of 3D incompressible Euler flow, Comm. Partial Differential Equations 30 (2005), no. 1-3, 225–243. MR 2131052, https://doi.org/10.1081/PDE-200044488
70. R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, https://doi.org/10.1007/BF01393835
71. Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643
72. Ronald J. DiPerna and Andrew Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow, J. Amer. Math. Soc. 1 (1988), no. 1, 59–95. MR 924702, https://doi.org/10.1090/S0894-0347-1988-0924702-6
73. Jean Duchon and Raoul Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity 13 (2000), no. 1, 249–255. MR 1734632, https://doi.org/10.1088/0951-7715/13/1/312
74. Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 2, 207–218. MR 1778702, https://doi.org/10.1007/s101140000034
75. Weinan E and Bjorn Engquist, Blowup of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1287–1293. MR 1476316, https://doi.org/10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4
76. Weinan E, Tiejun Li, and Pingwen Zhang, Well-posedness for the dumbbell model of polymeric fluids, Comm. Math. Phys. 248 (2004), no. 2, 409–427. MR 2073140, https://doi.org/10.1007/s00220-004-1102-y
77. L. Euler, Principes généraux du mouvement des fluides. Mémoires de L'Académie Royale des Sciences et des Belles-Lettres de Berlin 11 (4 September 1755, printed 1757), 217-273; reprinted in Opera Omnia ser. 2 12, 219-250.
78. Gregory L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer, Phys. D 78 (1994), no. 3-4, 222–240. MR 1302409, https://doi.org/10.1016/0167-2789(94)90117-1
79. Gregory L. Eyink, Dissipation in turbulent solutions of 2D Euler equations, Nonlinearity 14 (2001), no. 4, 787–802. MR 1837638, https://doi.org/10.1088/0951-7715/14/4/307
80. Gregory L. Eyink and Katepalli R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Rev. Modern Phys. 78 (2006), no. 1, 87–135. MR 2214822, https://doi.org/10.1103/RevModPhys.78.87
81. C. Foiaş, Statistical study of Navier-Stokes equations. I, II, Rend. Sem. Mat. Univ. Padova 48 (1972), 219–348 (1973); ibid. 49 (1973), 9–123. MR 352733
82. C. Foiaş, Statistical study of Navier-Stokes equations. I, II, Rend. Sem. Mat. Univ. Padova 48 (1972), 219–348 (1973); ibid. 49 (1973), 9–123. MR 352733
83. Susan Friedlander and Alexander Lipton-Lifschitz, Localized instabilities in fluids, Handbook of mathematical fluid dynamics, Vol. II, North-Holland, Amsterdam, 2003, pp. 289–354. MR 1984155, https://doi.org/10.1016/S1874-5792(03)80010-1
84. Uriel Frisch, Turbulence, Cambridge University Press, Cambridge, 1995. The legacy of A. N. Kolmogorov. MR 1428905
85. J. D. Gibbon, D. R. Moore, and J. T. Stuart, Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations, Nonlinearity 16 (2003), no. 5, 1823–1831. MR 1999581, https://doi.org/10.1088/0951-7715/16/5/315
86. J. D. Gibbon, A. S. Fokas, and C. R. Doering, Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations, Phys. D 132 (1999), no. 4, 497–510. MR 1704825, https://doi.org/10.1016/S0167-2789(99)00067-6
87. Thomas Y. Hou and Ruo Li, Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci. 16 (2006), no. 6, 639–664. MR 2271429, https://doi.org/10.1007/s00332-006-0800-3
88. G. Iyer, A stochastic Lagrangian formulation of the incompressible Navier-Stokes and related transport equations. PhD Thesis, The University of Chicago (2006).
89. Benjamin Jourdain, Tony Lelièvre, and Claude Le Bris, Existence of solution for a micro-macro model of polymeric fluid: the FENE model, J. Funct. Anal. 209 (2004), no. 1, 162–193. MR 2039220, https://doi.org/10.1016/S0022-1236(03)00183-6
90. V. Kamotski and G. Lebeau, On 2D Rayleigh-Taylor instabilities, Asymptot. Anal. 42 (2005), no. 1-2, 1–27. MR 2133872
91. Tosio Kato, Nonstationary flows of viscous and ideal fluids in 𝑅³, J. Functional Analysis 9 (1972), 296–305. MR 0481652, https://doi.org/10.1016/0022-1236(72)90003-1
92. A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), no. 3, 445–453. MR 2276260, https://doi.org/10.1007/s00222-006-0020-3
93. A. Kolmogoroff, The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers, C. R. (Doklady) Acad. Sci. URSS (N.S.) 30 (1941), 301–305. MR 0004146
94. Hideo Kozono and Yasushi Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000), no. 1, 191–200. MR 1794270, https://doi.org/10.1007/s002200000267
95. R. H. Kraichnan, Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (1967), 1417-1423.
96. Sergei Kuksin and Armen Shirikyan, Some limiting properties of randomly forced two-dimensional Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 4, 875–891. MR 2006207, https://doi.org/10.1017/S0308210500002729
97. Gilles Lebeau, Régularité du problème de Kelvin-Helmholtz pour l’équation d’Euler 2d, ESAIM Control Optim. Calc. Var. 8 (2002), 801–825 (French, with English and French summaries). A tribute to J. L. Lions. MR 1932974, https://doi.org/10.1051/cocv:2002052
98. C. Le Bris, P-L. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Rapport de receherche du CEREMICS 349, April 2007.
99. Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, https://doi.org/10.1007/BF02547354
100. Fang-Hua Lin, Chun Liu, and Ping Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math. 58 (2005), no. 11, 1437–1471. MR 2165379, https://doi.org/10.1002/cpa.20074
101. F.-H. Lin, P. Zhang, Z. Zhang.
On the global existence of smooth solution to the 2-d FENE dumbbell model.
Preprint, 2007.
102. Hans Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2) 162 (2005), no. 1, 109–194. MR 2178961, https://doi.org/10.4007/annals.2005.162.109
103. P.-L. Lions, N. Masmoudi.
Global existence of weak solutions to micro-macro models.
C. R. Math. Acad. Sci. Paris, 2007.
104. Milton C. Lopes Filho, Anna L. Mazzucato, and Helena J. Nussenzveig Lopes, Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence, Arch. Ration. Mech. Anal. 179 (2006), no. 3, 353–387. MR 2208320, https://doi.org/10.1007/s00205-005-0390-5
105. Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
106. A. Majda, X. Wang, Nonlinear dynamics and statistical theories for basic geophysical flows. CUP, Cambridge (2006).
107. Carlo Marchioro and Mario Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Applied Mathematical Sciences, vol. 96, Springer-Verlag, New York, 1994. MR 1245492
108. Nader Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm. Math. Phys. 270 (2007), no. 3, 777–788. MR 2276465, https://doi.org/10.1007/s00220-006-0171-5
109. J. Mattingly, T. Suidan, E. Vanden-Eijnden, Simple systems with anomalous dissipation and energy cascade. Commun. Math. Phys., to appear 2007-08.
110. Philippe Michel, Stéphane Mischler, and Benoît Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris 338 (2004), no. 9, 697–702 (English, with English and French summaries). MR 2065377, https://doi.org/10.1016/j.crma.2004.03.006
111. H. K. Moffatt and A. Tsinober, Helicity in laminar and turbulent flow, Annual review of fluid mechanics, Vol. 24, Annual Reviews, Palo Alto, CA, 1992, pp. 281–312. MR 1145012
112. L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), no. Suppl, Supplemento, no. 2 (Convegno Internazionale di Meccanica Statistica), 279–287. MR 36116
113. Koji Ohkitani and John D. Gibbon, Numerical study of singularity formation in a class of Euler and Navier-Stokes flows, Phys. Fluids 12 (2000), no. 12, 3181–3194. MR 1796376, https://doi.org/10.1063/1.1321256
114. Koji Ohkitani and Michio Yamada, Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. Fluids 9 (1997), no. 4, 876–882. MR 1437554, https://doi.org/10.1063/1.869184
115. F. Otto, A.E. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules. SFB preprint Nr. 141 (2004).
116. S. Resnick, Dynamical problems in nonlinear advective partial differential equations. Ph.D. Thesis, University of Chicago, 1995.
117. Raoul Robert, Statistical hydrodynamics (Onsager revisited), Handbook of mathematical fluid dynamics, Vol. II, North-Holland, Amsterdam, 2003, pp. 1–54. MR 1983588, https://doi.org/10.1016/S1874-5792(03)80003-4
118. V. Rom-Kedar, A. Leonard, and S. Wiggins, An analytical study of transport, mixing and chaos in an unsteady vortical flow, J. Fluid Mech. 214 (1990), 347–394. MR 1054106, https://doi.org/10.1017/S0022112090000167
119. Vladimir Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal. 3 (1993), no. 4, 343–401. MR 1231007, https://doi.org/10.1007/BF02921318
120. Herrmann Schlichting and Klaus Gersten, Boundary-layer theory, Eighth revised and enlarged edition, Springer-Verlag, Berlin, 2000. With contributions by Egon Krause and Herbert Oertel, Jr.; Translated from the ninth German edition by Katherine Mayes. MR 1765242
121. A. Shnirelman, On the nonuniqueness of weak solution of the Euler equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1261–1286. MR 1476315, https://doi.org/10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.3.CO;2-4
122. J. T. Stuart, Nonlinear Euler partial differential equations: singularities in their solution, Applied mathematics, fluid mechanics, astrophysics (Cambridge, MA, 1987) World Sci. Publishing, Singapore, 1988, pp. 81–95. MR 973917
123. H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in 𝑅₃, Trans. Amer. Math. Soc. 157 (1971), 373–397. MR 277929, https://doi.org/10.1090/S0002-9947-1971-0277929-7
124. Misha Vishik, Spectrum of small oscillations of an ideal fluid and Lyapunov exponents, J. Math. Pures Appl. (9) 75 (1996), no. 6, 531–557. MR 1423046
125. M. Vishik, A. Fursikov, Mathematical problems of statistical hydrodynamics. Kluwer AP, Dordrecht (1988).
126. Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math. 130 (1997), no. 1, 39–72. MR 1471885, https://doi.org/10.1007/s002220050177
127. Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc. 12 (1999), no. 2, 445–495. MR 1641609, https://doi.org/10.1090/S0894-0347-99-00290-8
128. Sijue Wu, Mathematical analysis of vortex sheets, Comm. Pure Appl. Math. 59 (2006), no. 8, 1065–1206. MR 2230845, https://doi.org/10.1002/cpa.20110
129. V. I. Yudovich, The linearization method in hydrodynamical stability theory, Translations of Mathematical Monographs, vol. 74, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by J. R. Schulenberger. MR 1003607
130. V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 1032–1066 (Russian). MR 158189
131. V. E. Zakharov (ed.), What is integrability?, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. MR 1098334
132. V. Zakharov, talk at the University of Chicago, May 2007.