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Published in:
Analysis of Viscous Fluid Flows: An Approach by Evolution Equations Read chapter Read first chapter

2020 | OriginalPaper | Chapter

2. Partial Regularity for the 3D Navier–Stokes Equations

Author : James C. Robinson

Published in: Mathematical Analysis of the Navier-Stokes Equations

Publisher: Springer International Publishing

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Abstract

These notes give a relatively quick introduction to some of the main results for the three-dimensional Navier–Stokes equations, concentrating in particular on ‘partial regularity’ results that limit the size of the set of (potential) singularities, both in time and in space-time.

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previous chapter Analysis of Viscous Fluid Flows: An Approach by Evolution Equations
next chapter Boundedness, Maximal Regularity and Free Boundary Problems for the Navier Stokes Equations
Footnotes
1
This is a particular case of the very useful anti-symmetry property
$$\displaystyle \begin{aligned} \langle(u\cdot\nabla)v,w\rangle=-\langle(u\cdot\nabla)w,v\rangle, \end{aligned} $$
(2.4)
which we will use from time to time in what follows.
 
2
Leray’s 1934 paper treats the equations on the whole space and does not use the Galerkin approach (see Ożański and Pooley [25], for a modern treatment of the methods in his paper). A ‘Galerkin-like’ argument for the equations on the whole space can be found in the book by Chemin et al. [4]. The first proof of existence of solutions on bounded domains was due to Hopf [17].
 
3
A more careful argument proceeds by contradiction: if (2.19) does not hold then ∥∇u(t)∥4 <  RHS of (2.19). In this case it follows from (2.17) that a strong solution v with v(t) = u(t) exists on the time interval [t, t j] and ∥∇v(t j)∥ < ∥∇u(t j)∥. But by weak-strong uniqueness we must have u = v on [t, t j], which yields a contradiction.
 
4
Here we use a colon for the matrix product, i.e. \(A:B=\sum _{i,j=1}^3A_{ij}B_{ij}\).
 
5
The vorticity equation follows on taking the curl of the Navier–Stokes equations and using the two vector identities
$$\displaystyle \begin{aligned}\frac{1}{2}{\nabla}|u|{}^2=(u\cdot{\nabla})u+u\times\omega\quad \mbox{and}\quad \nabla\times(a\times b)=a(\nabla\cdot b)-b(\nabla\cdot a)+(b\cdot{\nabla})a-(a\cdot{\nabla})b \end{aligned}$$
along with the fact that both u and ω are divergence free.
 
6
This is the basis of the proof given in the paper by Takahashi [39], although rather than following exactly the argument here he works with the equation for ϕω, where ϕ is a cutoff function.
 
7
Here our argument has what is potentially a fatal flaw: in the inequality (2.42) we have assumed that ω ∈ L r, which is in fact what we want to prove (the inequality is trivially true if ω∉L r since then both sides are infinite). However, this can be circumvented by considering estimates for the equation
$$\displaystyle \begin{aligned}\partial_tW^\varepsilon-\Delta W^\varepsilon=\mathrm{div}(W^\varepsilon u_\varepsilon-u_\varepsilon W^\varepsilon), \end{aligned}$$
where u ε is a mollified version of the original function u, and then taking limits as ε → 0.
 
8
Wojciech Ożański recently remarked to me that the pressure term usually included in hypothesis (A n) (as in CKN or RRS, for example) is not in fact necessary: the pressure estimates required in the course of the proof rely only on the estimates for u in our (2.50) and (2.51) and on the initial smallness assumption on p in (2.44). A very elegant version of the full inductive argument, including the pressure, is presented in his monograph [24]).
 
9
We define \(\langle f,g\rangle _{H^2}=\sum _k (1+|k|{ }^2)^2\hat f_k\hat g_k\).
 
Literature
1.
T. Buckmaster, V. Vicol, Nonuniquenness of weak solutions to the Navier–Stokes equation. arXiv:1709.10033 (2017)
2.
T. Buckmaster, C. De Lellis, L. Székelyhidi Jr., V. Vicol, Onsager’s conjecture for admissible weak solutions. arXiv:1701.08678 (2017)
3.
L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure. Appl. Math. 35, 771–931 (1982)MathSciNetCrossRef
4.
J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Mathematical Geophysics (Oxford University Press, Oxford, 2006)MATH
5.
P. Constantin, C. Foias, Navier–Stokes Equations (University of Chicago Press, Chicago, 1988)MATH
6.
P. Constantin, E. Weinan, E.S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165, 207–209 (1994)MathSciNetCrossRef
7.
M. Dashti, J.C. Robinson, A simple proof of uniqueness of the particle trajectories for solutions of the Navier–Stokes equations. Nonlinearity 22, 735–746 (2009)MathSciNetCrossRef
8.
L. Escauriaza, G. Seregin, V. Šverák, L 3,∞-solutions of Navier–Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211–250 (2003)MathSciNetCrossRef
9.
L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence, 2010)MATH
10.
E.B. Fabes, B.F. Jones, N.M. Rivière, The initial value problem for the Navier–Stokes equations with data in L p. Arch. Ration. Mech. Anal. 45, 222–240 (1972)MathSciNetCrossRef
11.
12.
C.L. Fefferman, Existence and smoothness of the Navier–Stokes equation, in The Millennium Prize Problems (Clay Mathematics Institute, Cambridge, 2000), pp. 57–67
13.
C. Foias, C. Guillopé, R. Temam, New a priori estimates for Navier–Stokes equations in dimension 3. Comm. Partial Diff. Equ. 6, 329–359 (1981)MathSciNetCrossRef
14.
15.
G.P. Galdi, An introduction to the Navier–Stokes initial-boundary value problem, in ed. by G.P. Galdi, J.G. Heywood, R. Rannacher. Fundamental Directions in Mathematical Fluid Dynamics (Birkhäuser, Basel, 2000), pp. 1–70
16.
P. Hartman, Ordinary Differential Equations (Wiley, Baltimore, 1973)MATH
17.
18.
19.
H. Jia, V. Šverák, Are the incompressible 3d Navier–Stokes equations locally ill-posed in the natural energy space? J. Func. Anal. 268, 3734–3766 (2015)MathSciNetCrossRef
20.
I. Kukavica, Partial regularity results for solutions of the Navier–Stokes system, in ed. by J.C. Robinson, J.L. Rodrigo, Partial Differential Equations and Fluid Mechanics (Cambridge University Press, Cambridge, 2009), pp. 121–145
21.
I. Kukavica, The fractal dimension of the singular set for solutions of the Navier–Stokes system. Nonlinearity 22, 2889–2900 (2009)MathSciNetCrossRef
22.
23.
24.
W. Ożański, The Partial Regularity Theory of Caffarelli, Kohn, and Nirenberg and Its Sharpness. Lecture Notes in Mathematical Fluid Mechanics (Birkhäuser/Springer, 2019)
25.
W. Ożański, B. Pooley, Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquid visqueux emplissant l’espace”. in ed. by C.L. Fefferman, J.L. Rodrigo, J.C. Robinson. Partial Differential Equations in Fluid Mechanics. LMS Lecture Notes (Cambridge University Press, Cambridge, 2018)
26.
27.
J.C. Robinson, Infinite-Dimensional Dynamical Systems (Cambridge University Press, Cambridge, 2001)CrossRef
28.
J.C. Robinson, Dimensions, Embeddings, and Attractors (Cambridge University Press, Cambridge, 2011)MATH
29.
J.C. Robinson, W. Sadowski, Decay of weak solutions and the singular set of the three-dimensional Navier–Stokes equations. Nonlinearity 20, 1185–1191 (2007)MathSciNetCrossRef
30.
J.C. Robinson, W. Sadowski, Almost everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier–Stokes equations. Nonlinearity 22, 2093–2099 (2009)MathSciNetCrossRef
31.
J.C. Robinson, J.L. Rodrigo, W. Sadowski, The Three-Dimensional Navier–Stokes Equations (Cambridge University Press, Cambridge, 2016)CrossRef
32.
V. Scheffer, Turbulence and Hausdorff dimension, in Turbulence and Navier–Stokes Equation, Orsay 1975. Springer Lecture Notes in Mathematics, vol. 565 (Springer, Berlin, 1976), pp. 174–183
33.
V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations. Pacific J. Math. 66, 535–552 (1976)MathSciNetCrossRef
34.
35.
36.
V. Scheffer, Nearly one-dimensional singularities of solutions to the Navier–Stokes inequality. Comm. Math. Phys. 110, 525–551 (1987)MathSciNetCrossRef
37.
J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962)MathSciNetCrossRef
38.
M. Struwe, On partial regularity result for the Navier–Stokes equations. Comm. Pure Appl. Math. 41, 437–458 (1988)MathSciNetCrossRef
39.
S. Takahashi, On interior regularity criteria for weak solutions of the Navier–Stokes equations. Manuscripta Math. 69, 237–254 (1990)MathSciNetCrossRef
Metadata
Title
Partial Regularity for the 3D Navier–Stokes Equations
Author
James C. Robinson
Copyright Year
2020
Book
Mathematical Analysis of the Navier-Stokes Equations

Print ISBN: 978-3-030-36225-6

Electronic ISBN: 978-3-030-36226-3

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-36226-3_2

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