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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Notes
- 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 \(\langle f,g\rangle _{H^2}=\sum _k (1+|k|{ }^2)^2\hat f_k\hat g_k\).
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Robinson, J.C. (2020). Partial Regularity for the 3D Navier–Stokes Equations. In: Galdi, G., Shibata, Y. (eds) Mathematical Analysis of the Navier-Stokes Equations. Lecture Notes in Mathematics(), vol 2254. Springer, Cham. https://doi.org/10.1007/978-3-030-36226-3_2
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