# 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:

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).