-
1
-
-
58149341340
-
The 3D Navier-Stokes problem
-
10.1146/annurev.fluid.010908.165218
-
Doering C.R. The 3D Navier-Stokes problem. Annu. Rev. Fluid Mech. 2009, 41:1. 10.1146/annurev.fluid.010908.165218.
-
(2009)
Annu. Rev. Fluid Mech.
, vol.41
, pp. 1
-
-
Doering, C.R.1
-
2
-
-
77957198447
-
Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations
-
10.1098/rspa.2009.0642
-
Gibbon J.D. Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations. Proc. R. Soc. London, Ser. A 2010, 466:2587. 10.1098/rspa.2009.0642.
-
(2010)
Proc. R. Soc. London, Ser. A
, vol.466
, pp. 2587
-
-
Gibbon, J.D.1
-
5
-
-
0003610387
-
-
(Cambridge University Press, Cambridge, England, ).
-
Foias C. Manley O. Rosa R. Temam R. Navier-Stokes Equations and Turbulence 2001, and (Cambridge University Press, Cambridge, England, ).
-
(2001)
Navier-Stokes Equations and Turbulence
-
-
Foias, C.1
Manley, O.2
Rosa, R.3
Temam, R.4
-
6
-
-
84861990595
-
Some rigorous results about 3D Navier-Stokes
-
Text of two lectures delivered at NATO-ASI meeting "Turbulence in spatially ordered systems," Les Houches.
-
Gallavotti G. Some rigorous results about 3D Navier-Stokes. 1992, Text of two lectures delivered at NATO-ASI meeting "Turbulence in spatially ordered systems," Les Houches.
-
(1992)
-
-
Gallavotti, G.1
-
7
-
-
84861962141
-
-
(Springer, Berlin, ), Chap. 3, Sec. 3.5
-
Gallavotti G. Foundations of Fluid Dynamics 2002, 209-214. (Springer, Berlin, ), Chap. 3, Sec. 3.5.
-
(2002)
Foundations of Fluid Dynamics
, pp. 209-214
-
-
Gallavotti, G.1
-
8
-
-
77955339508
-
On the size of the Navier-Stokes singular set
-
10.3934/dcds.2010.28.1165
-
Arnold M. Craig W. On the size of the Navier-Stokes singular set. Discrete Contin. Dyn. Syst. 2010, 28:3. 10.3934/dcds.2010.28.1165.
-
(2010)
Discrete Contin. Dyn. Syst.
, vol.28
, pp. 3
-
-
Arnold, M.1
Craig, W.2
-
9
-
-
84855200936
-
Bounds on Kolmogorov spectra for the Navier-Stokes equations
-
10.1016/j.physd.2011.10.013
-
Biryuk A. Craig W. Bounds on Kolmogorov spectra for the Navier-Stokes equations. Phys. D 2011, 241:426. 10.1016/j.physd.2011.10.013.
-
(2011)
Phys. D
, vol.241
, pp. 426
-
-
Biryuk, A.1
Craig, W.2
-
10
-
-
61649125593
-
On the partial regularity of a 3D model of the Navier-Stokes equations
-
10.1007/s00220-008-0689-9
-
Hou T.Y. Lei Z. On the partial regularity of a 3D model of the Navier-Stokes equations. Commun. Math. Phys. Differ. Equ. 2009, 287:2. 10.1007/s00220-008-0689-9.
-
(2009)
Commun. Math. Phys. Differ. Equ.
, vol.287
, pp. 2
-
-
Hou, T.Y.1
Lei, Z.2
-
11
-
-
58149342173
-
-
(American Mathematical Society, Providence, RI, )
-
Fefferman C.L. The Millennium Prize Problems 2006, 57-70. (American Mathematical Society, Providence, RI, )
-
(2006)
The Millennium Prize Problems
, pp. 57-70
-
-
Fefferman, C.L.1
-
12
-
-
85066965397
-
Fully developed turbulence and statistical mechanics
-
10.1051/jphyslet:019780039010500
-
Rose H.A. Sulem P.L. Fully developed turbulence and statistical mechanics. J. Phys. (Paris) 1978, 39:5. 10.1051/jphyslet:019780039010500.
-
(1978)
J. Phys. (Paris)
, vol.39
, pp. 5
-
-
Rose, H.A.1
Sulem, P.L.2
-
13
-
-
34250772284
-
A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations
-
10.1063/1.2372512
-
Chernyshenko S.I. Constantin P. Robinson J.C. Titi E.S. A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations. J. Math. Phys. 2007, 48:6. 10.1063/1.2372512.
-
(2007)
J. Math. Phys.
, vol.48
, pp. 6
-
-
Chernyshenko, S.I.1
Constantin, P.2
Robinson, J.C.3
Titi, E.S.4
-
14
-
-
0003261515
-
Incipient singularities in the Navier-Stokes equations
-
10.1103/PhysRevLett.55.1749
-
Pumir A. Siggia E.D. Incipient singularities in the Navier-Stokes equations. Phys. Rev. Lett. 1985, 55:1749. 10.1103/PhysRevLett.55.1749.
-
(1985)
Phys. Rev. Lett.
, vol.55
, pp. 1749
-
-
Pumir, A.1
Siggia, E.D.2
-
15
-
-
0023101871
-
Vortex dynamics and the existence of solutions to the Navier-Stokes equations
-
10.1063/1.866226
-
Pumir A. Siggia E.D. Vortex dynamics and the existence of solutions to the Navier-Stokes equations. Phys. Fluids 1987, 30:5. 10.1063/1.866226.
-
(1987)
Phys. Fluids
, vol.30
, pp. 5
-
-
Pumir, A.1
Siggia, E.D.2
-
16
-
-
11944253796
-
Finite-time singularities in the axisymmetric three-dimension Euler equations
-
10.1103/PhysRevLett.68.1511
-
Pumir A. Siggia E.D. Finite-time singularities in the axisymmetric three-dimension Euler equations. Phys. Rev. Lett. 1992, 68:10. 10.1103/PhysRevLett.68.1511.
-
(1992)
Phys. Rev. Lett.
, vol.68
, pp. 10
-
-
Pumir, A.1
Siggia, E.D.2
-
17
-
-
0001149206
-
Singularities of the equations of fluid motion
-
10.1103/PhysRevA.38.6287
-
Sreenivasan K.R. Meneveau C. Singularities of the equations of fluid motion. Phys. Rev. A. 1988, 38:6287. 10.1103/PhysRevA.38.6287.
-
(1988)
Phys. Rev. A.
, vol.38
, pp. 6287
-
-
Sreenivasan, K.R.1
Meneveau, C.2
-
18
-
-
3042986399
-
Navier-Stokes initial value problem for boundary-free incompressible fluid flow
-
10.1063/1.1692879
-
Rosen G. Navier-Stokes initial value problem for boundary-free incompressible fluid flow. Phys. Fluids 1970, 12:2891. 10.1063/1.1692879.
-
(1970)
Phys. Fluids
, vol.12
, pp. 2891
-
-
Rosen, G.1
-
19
-
-
1342279309
-
Remarks on the possible global regularity of solutions of the three-dimensional Navier-Stokes equations
-
Heywood J.G. Remarks on the possible global regularity of solutions of the three-dimensional Navier-Stokes equations. Pitman Res. Notes Math. Ser. 1994, 308:1.
-
(1994)
Pitman Res. Notes Math. Ser.
, vol.308
, pp. 1
-
-
Heywood, J.G.1
-
20
-
-
39049108490
-
On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid
-
Kiselev A.A. Ladyzhenskaya O.A. Kiselev A.A. Ladyzhenskaya O.A. On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid. Am. Math. Soc. Transl. 1957, 24:79.
-
(1957)
Am. Math. Soc. Transl.
, vol.24
, pp. 79
-
-
Kiselev, A.A.1
Ladyzhenskaya, O.A.2
Kiselev, A.A.3
Ladyzhenskaya, O.A.4
-
21
-
-
84861969024
-
Euler equation and burger equation - Relation with turbulence
-
in, Lecture Notes in Mathematics (Springer Verlag, Berlin)
-
Bardos C. Euler equation and burger equation - Relation with turbulence. Nonlinear Partial Differential Equations and Applications 1978, 648:1. in Lecture Notes in Mathematics Vol. (Springer Verlag, Berlin, )
-
(1978)
Nonlinear Partial Differential Equations and Applications
, vol.648
, pp. 1
-
-
Bardos, C.1
-
22
-
-
84905124317
-
Equations de Navier-Stokes et modéle de la turbulence
-
10.1051/jphyscol:1978518
-
Bardos C. Equations de Navier-Stokes et modéle de la turbulence. J. Phys. (Paris), Colloq. 1978, 39:C5-53. 10.1051/jphyscol:1978518.
-
(1978)
J. Phys. (Paris), Colloq.
, vol.39
-
-
Bardos, C.1
-
23
-
-
77953597029
-
Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows
-
10.1063/1.3327284
-
Tran C.V. Dritschel D.G. Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows. Phys. Fluids 2010, 22:037102. 10.1063/1.3327284.
-
(2010)
Phys. Fluids
, vol.22
, pp. 037102
-
-
Tran, C.V.1
Dritschel, D.G.2
-
24
-
-
80052922755
-
On maximum enstrophy growth in a hydrodynamics system
-
10.1016/j.physd.2011.07.003
-
Ayala D. Protas B. On maximum enstrophy growth in a hydrodynamics system. Physica D 2011, 240:1553. 10.1016/j.physd.2011.07.003.
-
(2011)
Physica D
, vol.240
, pp. 1553
-
-
Ayala, D.1
Protas, B.2
-
25
-
-
58149337141
-
A miscellany of basic issues on incompressible fluid equations
-
10.1088/0951-7715/21/12/T02
-
Ohkitani K. A miscellany of basic issues on incompressible fluid equations. Nonlinearity 2008, 21:1. 10.1088/0951-7715/21/12/T02.
-
(2008)
Nonlinearity
, vol.21
, pp. 1
-
-
Ohkitani, K.1
-
26
-
-
0345099790
-
The Cole-Hopf and Miura transformations revisited
-
S. Albeverio, P. Blanchard, L. Ferreira, T. Hida, Y. Kondratiev, R. Vilela Mendes, in edited by , and (World Scientific, Singapore, )
-
Gesztesy F. Holden H. The Cole-Hopf and Miura transformations revisited. Mathematical Physics and Stochastic Analysis: Essays in Honour of Ludwig Streit 2000, 198. S. Albeverio, P. Blanchard, L. Ferreira, T. Hida, Y. Kondratiev, R. Vilela Mendes, in edited by and (World Scientific, Singapore, ), p.
-
(2000)
Mathematical Physics and Stochastic Analysis: Essays in Honour of Ludwig Streit
, pp. 198
-
-
Gesztesy, F.1
Holden, H.2
-
27
-
-
3042686501
-
Statistics of a passive scalar in homogeneous turbulence
-
10.1088/1367-2630/6/1/040, and
-
Watanabe T. Gotoh T. Statistics of a passive scalar in homogeneous turbulence. New J. Phys. 2004, 6:40. 10.1088/1367-2630/6/1/040.
-
(2004)
New J. Phys.
, vol.6
, pp. 40
-
-
Watanabe, T.1
Gotoh, T.2
-
28
-
-
84861966953
-
-
Consider an equation which is linear in θ;∂/θt + (u Δ)θ = vΔθ + f (x,t) with a "forcing" f (x, t) = -px , where u(s,t) and p(x,t) are taken from the Naver-Stokes solutions. By a uniqueness argument, if θ = u1 at t = 0, then θ = u1 for all t > 0 as long as u remains smooth. If we drop "forcing" we obtain.
-
Consider an equation which is linear in θ;∂/θt + (u Δ)θ = vΔθ + f (x,t) with a "forcing" f (x, t) = -px , where u(s,t) and p(x,t) are taken from the Naver-Stokes solutions. By a uniqueness argument, if θ = u1 at t = 0, then θ = u1 for all t > 0 as long as u remains smooth. If we drop "forcing" we obtain.
-
-
-
-
29
-
-
0016847289
-
On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars
-
10.1017/S0022112075003047
-
Mandlebrot B.B. On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars. J. Fluid Mech. 1975, 72:401. 10.1017/S0022112075003047.
-
(1975)
J. Fluid Mech.
, vol.72
, pp. 401
-
-
Mandlebrot, B.B.1
-
30
-
-
59549098973
-
Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations
-
10.1512/iumj.2008.57.3716
-
Lu L. Doering C.R. Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J. 2008, 57:6. 10.1512/iumj.2008.57.3716.
-
(2008)
Indiana Univ. Math. J.
, vol.57
, pp. 6
-
-
Lu, L.1
Doering, C.R.2
-
31
-
-
38949135270
-
Local and nonlocal strain rate fields and vorticity alignment in turbulent flows
-
10.1103/PhysRevE.77.026303
-
Hamlington P.E. Schumacher J. Dahm W.J. A. Local and nonlocal strain rate fields and vorticity alignment in turbulent flows. Phys. Rev. E. 2008, 77:11. 10.1103/PhysRevE.77.026303.
-
(2008)
Phys. Rev. E.
, vol.77
, pp. 11
-
-
Hamlington, P.E.1
Schumacher, J.2
Dahm, W.J.A.3
-
32
-
-
57149103133
-
Direct assessment of vorticity alignment with local and nonlocal strain rates in turbulent flows
-
10.1063/1.3021055
-
Hamlington P.E. Schumacher J. Dahm W.J. A. Direct assessment of vorticity alignment with local and nonlocal strain rates in turbulent flows. Phys. Fluids 2008, 20:111703. 10.1063/1.3021055.
-
(2008)
Phys. Fluids
, vol.20
, pp. 111703
-
-
Hamlington, P.E.1
Schumacher, J.2
Dahm, W.J.A.3
-
33
-
-
73249140158
-
Extreme vorticity growth in Navier-Stokes turbulence
-
10.1016/j.physleta.2009.11.078
-
Schumacher J. Eckhardt B. Doering C.R. Extreme vorticity growth in Navier-Stokes turbulence. Phys. Lett. A 2010, 374:861. 10.1016/j.physleta.2009.11.078.
-
(2010)
Phys. Lett. A
, vol.374
, pp. 861
-
-
Schumacher, J.1
Eckhardt, B.2
Doering, C.R.3
-
34
-
-
77949423904
-
Vorticity dynamics in turbulence growth
-
10.1007/s00162-009-0100-2
-
Orlandi P. Sergio S.P. Vorticity dynamics in turbulence growth. Theor. Comput. Fluid Dyn. 2010, 24:247. 10.1007/s00162-009-0100-2.
-
(2010)
Theor. Comput. Fluid Dyn.
, vol.24
, pp. 247
-
-
Orlandi, P.1
Sergio, S.P.2
-
35
-
-
0000863080
-
Oscillations and concentrations in weak solutions of the incompressible fluid equations
-
10.1007/BF01214424
-
DiPerna R.J. Majda A.J. Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 1987, 108:667. 10.1007/BF01214424.
-
(1987)
Commun. Math. Phys.
, vol.108
, pp. 667
-
-
DiPerna, R.J.1
Majda, A.J.2
-
36
-
-
33846553005
-
Statistical properties of four-dimensional turbulence
-
10.1103/PhysRevE.75.016310
-
Gotoh T. Watanabe Y. Shiga Y. Nakano T. Suzuki E. Statistical properties of four-dimensional turbulence. Phys. Rev. E 2007, 75:016310. 10.1103/PhysRevE.75.016310.
-
(2007)
Phys. Rev. E
, vol.75
, pp. 016310
-
-
Gotoh, T.1
Watanabe, Y.2
Shiga, Y.3
Nakano, T.4
Suzuki, E.5
-
37
-
-
24144464002
-
Energy transfer and intermittency in four-dimensional turbulence
-
10.1063/1.2001692
-
Suzuki E. Nakano T. Takahashi N. Gotoh T. Energy transfer and intermittency in four-dimensional turbulence. Phys. Fluids 2005, 17:081702. 10.1063/1.2001692.
-
(2005)
Phys. Fluids
, vol.17
, pp. 081702
-
-
Suzuki, E.1
Nakano, T.2
Takahashi, N.3
Gotoh, T.4
-
38
-
-
34347240410
-
Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time
-
10.1007/s00220-007-0259-6
-
Dong H. Du D. Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time. Commun. Math. Phys. 2007, 273:785. 10.1007/s00220-007-0259-6.
-
(2007)
Commun. Math. Phys.
, vol.273
, pp. 785
-
-
Dong, H.1
Du, D.2
-
40
-
-
0002465925
-
The Navier-Stokes equations in space dimension four
-
10.1007/BF01609467
-
Scheffer V. The Navier-Stokes equations in space dimension four. Commun. Math. Phys. 1978, 61:41. 10.1007/BF01609467.
-
(1978)
Commun. Math. Phys.
, vol.61
, pp. 41
-
-
Scheffer, V.1
-
41
-
-
34250261427
-
Stationary solutions to the Navier-Stokes equations in dimension four
-
10.1007/BF01182469
-
Gerhardt C. Stationary solutions to the Navier-Stokes equations in dimension four. Math. Z. 1979, 165:193. 10.1007/BF01182469.
-
(1979)
Math. Z.
, vol.165
, pp. 193
-
-
Gerhardt, C.1
-
42
-
-
67349176199
-
Existence and regularity of very weak solutions of the stationary Navier-Stokes equations
-
10.1007/s00205-008-0168-7
-
Kim H. Existence and regularity of very weak solutions of the stationary Navier-Stokes equations. Arch. Ration. Mech. Anal. 2009, 193:117. 10.1007/s00205-008-0168-7.
-
(2009)
Arch. Ration. Mech. Anal.
, vol.193
, pp. 117
-
-
Kim, H.1
-
43
-
-
78650493482
-
Global regularity of the 4D restricted Euler equations
-
10.1016/j.physd.2009.07.009
-
Liu H. Tadmor E. Wei D. Global regularity of the 4D restricted Euler equations. Physica D 2010, 239:1225. 10.1016/j.physd.2009.07.009.
-
(2010)
Physica D
, vol.239
, pp. 1225
-
-
Liu, H.1
Tadmor, E.2
Wei, D.3
-
44
-
-
84861972973
-
-
Note
-
The reason for this is that at d = 4 the Gagliardo-Nirenberg inequality reduces to the Sobolev lemma. Hence, no application of the Hölder inequality is possible, because P would get an exponent 1.
-
-
-
-
45
-
-
0016602026
-
Bounds on energy flux for finite energy turbulence
-
10.1017/S0022112075003059
-
Sulem P.L. Frisch U. Bounds on energy flux for finite energy turbulence. J. Fluid Mech. 1975, 72:417. 10.1017/S0022112075003059.
-
(1975)
J. Fluid Mech.
, vol.72
, pp. 417
-
-
Sulem, P.L.1
Frisch, U.2
-
46
-
-
0037093148
-
Bounds on moments of the energy spectrum for weak solutions of the three-dimensional Navier-Stokes equations
-
10.1016/S0167-2789(02)00427-X
-
Doering C.R. Gibbon J.D. Bounds on moments of the energy spectrum for weak solutions of the three-dimensional Navier-Stokes equations. Physica D 2002, 165:163. 10.1016/S0167-2789(02)00427-X.
-
(2002)
Physica D
, vol.165
, pp. 163
-
-
Doering, C.R.1
Gibbon, J.D.2
-
47
-
-
21844522201
-
Onsager's conjecture on the energy conservation for solutions of Euler's equation
-
10.1007/BF02099744
-
Constantin P. Weinan E. Titi E.S. Onsager's conjecture on the energy conservation for solutions of Euler's equation. Commun. Math. Phys. 1994, 165:207. 10.1007/BF02099744.
-
(1994)
Commun. Math. Phys.
, vol.165
, pp. 207
-
-
Constantin, P.1
Weinan, E.2
Titi, E.S.3
-
48
-
-
0002205604
-
Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer
-
10.1016/0167-2789(94)90117-1
-
Eyink G.L. Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer. Physica D 1994, 78:222. 10.1016/0167-2789(94)90117-1.
-
(1994)
Physica D
, vol.78
, pp. 222
-
-
Eyink, G.L.1
-
49
-
-
0032654650
-
Impulse formulation of the Euler equations: general properties and numerical methods
-
10.1017/S0022112099005170
-
Russo G. Smereka P. Impulse formulation of the Euler equations: general properties and numerical methods. J. Fluid Mech. 1991, 391:189. 10.1017/S0022112099005170.
-
(1991)
J. Fluid Mech.
, vol.391
, pp. 189
-
-
Russo, G.1
Smereka, P.2
|