Reading List on incompressible Navier-Stokes

The following list of references and topics  is given by Zhen Lei:

[1] 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.
[2] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771–831.
[3] C.-C. Chen, R. M. Strain, T.-P. Tsai and H.-T. Yau, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. Int. Math. Res. Not., 9 (2008),

[4] C.-C. Chen, R. M. Strain, T.-P. Tsai and H.-T. Yau, Lower bounds on the blow- up rate of the axisymmetric Navier-Stokes equations II. Comm. Partial Differential Equations, 34 (2009), 203–232.

[5] P. Constantin and C. Foias, Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.
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[6] L. Escauriaza, G. A. Seregin and V. Sverák, L3,∞ -solutions of Navier-Stokes equations and backward uniqueness. (Russian) Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3–44; translation in Russian Math. Surveys 58 (2003), no. 2, 211–250
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[7] L. Escauriaza, G. A. Seregin and V. Sver ́k, Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169 (2003), no. 2, 147–157.
[8] C. Fefferman, Available online at http://www.claymath.org/millennium/Navier-Stokes equations.
[9] E. Hopf, uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen. (German) Math. Nachr. 4, (1951). 213–231.
[10] Thomas Y. Hou, Zhen Lei and Congming Li, Global regularity of the 3D axisymmetric Navier-Stokes equations with anisotropic data. Comm. Partial Differential Equations 33 (2008), no. 7-9, 1622–1637.

[11] Thomas Y. Hou and Congming Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl. Comm. Pure Appl. Math. 61 (2008), no. 5, 661–697.
[12] T. Kato, Strong Lp -solutions of the Navier-Stokes equation in Rm , with applications to weak solutions. Math. Z. 187 (1984), no. 4, 471–480.

[13] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (2001), no. 1, 22–35.
[14] Ladyzhenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 155–177 (Russian).

[15] O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1 (1999), no. 4, 356–387.
[16] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta. Math. 63 (1934), 193–248.
[17] F.-H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51 (1998), no. 3, 241–257.

[18] A. J. Majda and A. L. Bertozzi, Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.
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[19] J. Necas, M. Ruzic̆ka and V. Sverák, On Leray’s self-similar solutions of the Navier-Stokes equations, Acta Math. 176 (1996), 283–294.
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[20] G. Koch, N. Nadirashvili, G. A. Seregin, and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications. Acta Math. 203 (2009), no. 1, 83–105.
[21] M. Struwe, On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), no. 4, 437–458.
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[22] G. A. Seregin and V. Sverak, On type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations. Comm. Partial Differential Equations 34 (2009), no. 1-3, 171–201.
[23] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal. 9 1962 187–195.
[24] R. Temam, Navier-Stokes Equations. Second Edition, AMS Chelsea Publishing, Providence, RI, 2001.
[25] G. Tian and Zhouping Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom. 7 (1999), no. 2, 221–257.
[26] T. P. Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rational Mech. Anal., 143 (1998), 29–51.
[27] M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous
fluids filling the whole space, J. Appl. Math. Mech. 32 (1968), 52–61.

Topics covered:

– Weak solutions on bounded domain, periodic domain and the whole space, suitable weak solutions. References: Leray [16], Hopf [9], Caffarelli-Kohn-Nirenberg [2], Books of Majda-Bertozzi [18] and Temam [24].
– Local strong and classical solutions, Serrin’s criterion, BKM’s criterion. References: Serrin [23], Struwe [21], Kato [12], Beale-Kato-Majda [1], Koch-Tararu [13], Majda-Bertozzi [18].

– Partial regularity theory. References: Caffarelli-Kohn-Nirenberg [2], Lin [17], Ladyzhenskaya-Seregin [15], Tian-Xin [25].
– Non-existence of self-similar solutions. References: Necas-Ruzicka-Sverak [19], Tsai [26].

– Axis-symmetric Navier-Stokes equations without swirl, blowup rate of axis-symmetric Navier-Stokes equations with swirl. References: Ukhovskii-Iudovich [27], Ladyzhenskaya [14], Hou-Lei-Li [10], Chen-Strain-Tsai-Yau [3, 4], Koch-Nadirashvili-Seregin-Sverák [20], Zhang [28], Seregin-Sverak [22].
– L∞ (L3 )-solutions, backward uniqueness and unique continuation. References: Escauriaza-Seregin-Sverák [6], Escauriaza-Seregin-Sverak [7].

– Explicit examples. References: Majda-Bertozzi [18], Hou-Li [11].

More references:

A. Vasseur, A new proof of partial regularity of solutions to Navier Stokes equations, NoDEA, 2007

The above paper of Vasseur paper looks quite interesting to me. Though it doesn’t contain any final new result, it gives a nice new proof of Caffarelli-Kohn-Nirenberg partial regularity theorem (which says that the singular set has “parabolic Hausdorff dimension” less than 1). The two key steps in Vasseur’s proof are:

Theorem 1. For every p > 1 there exists a universal constant C such that any suitable weak solution of  INS (incompressible Navier Stokes equation, with zero boundary condition) in [-1,1] \times B(1) (where B(1) is the ball of radius 1 in the 3-dim space) verifying

\sup_{t \in [-1,1]} (\int_{B(1)} |u|^2 dx) + \int_{-1}^1 |\nabla u|^2 dx dt + [\int_{-1}^1 (\int_{B(1)} |P| dx)^p]^{2/p} < C

(where P is the pressure function) is bounded by 1 on [-1/2, 1] \times B(1/2)

Theorem 2. There exists a universal constant C_2  such that the following property holds for any suitable weak solution of INS in [0,+\infty [ \times \Omega: Let (t_0,x_0) be a point lying in the interior of  [0,+\infty [ \times \Omega  such that

\limsup_{\epsilon \to 0+} (1/\epsilon) \int_{t_0 - \epsilon^2}^{t_0 + \epsilon^2} \int_{x_0 + B(\epsilon)} |\nabla u|^2 dx dt < C_2

Then the solution is bounded in a neighborhood of (t_0,x_0).

It’s wellknown (Caffarelli-Kohn-Nirenberg) that Theorem 2 + some  simple covering arguments lead to the partial regularity result (parabolic Hausdorff dimension of the singular set is less than 1).

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