Weak and Strong Solutions of the 3D Navier–Stokes Equations and Their Relation to a Chessboard of Convergent Inverse Length Scales

Abstract

Using the scale invariance of the Navier–Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the 3D Navier–Stokes equations, on a periodic domain $\mathcal {V} =[0,\,L]^{3}$ an infinite ‘chessboard’ of estimates for these inverse length scales is displayed in terms of labels $(n,\,m)$ corresponding to n derivatives of the velocity field in $L^{2m}(\mathcal {V})$. The $(1,\,1)$ position corresponds to the inverse Kolmogorov length $Re^{3/4}$. These estimates ultimately converge to a finite limit, $Re^3$, as $n,\,m\rightarrow \infty$, although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by $(n,\,m)$. In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by $(n,\,m)$, the only difference being a factor of 2 in the exponent. This appears to be a generalization of the Prodi–Serrin conditions for $n\ge 1$.

Appendices

Appendix : Proof of Theorem 1

The following proof is based around the independent result of Foias et al. (1981) on the bounded hierarchy of time averages

$$\begin{aligned} \left<H_{n,1}^{\frac{1}{2n-1}}\right>_{T} \le c_{n}L^{-1}\nu ^{\frac{2}{2n-1}}Re^{3} + O\left( T^{-1}\right) . \end{aligned}$$

(28)

$H_{n,1}$ is defined by (8) with $m=1$. Given the nature of the $H_{n,m}$ defined in (8) we are dealing with $W^{n,2m}$-spaces with initial data ${\varvec{u}}_{0} \in \dot{W}^{n,2m}$. The aim is to show that ${\varvec{u}}\in L^{\alpha _{n,m}}\left( [0,\,T],;~\dot{W}^{n,2m}(\mathbb {R}^{3})\right)$ for $n\ge 1$ with $1 \le m \le \infty$ and for $n=0$ with $3 < m \le \infty$.

1.1 The Case $n\ge 1$ and $1\le m \le \infty$

In terms of time averages we wish to estimate $\left<\Vert \nabla ^{n}{\varvec{u}}\Vert _{L^{2m}}^{\alpha _{n,m}}\right>_{T}$ when $n\ge 1$. We first use the Gagliardo–Nirenberg inequality (with $\mathcal {A}\equiv \nabla {\varvec{u}}$) to interpolate between $\Vert \nabla ^{n-1}\mathcal {A}\Vert _{L^{2m}}$ and $\Vert \nabla ^{N}\mathcal {A}\Vert _{L^{2}}$ (Adams 1975)

$$\begin{aligned} \Vert \nabla ^{n-1}\mathcal {A}\Vert _{L^{2m}} \le C\, \Vert \nabla ^{N}\mathcal {A}\Vert _{L^{2}}^{a}\Vert \mathcal {A}\Vert _{L^{2}}^{1-a}, \end{aligned}$$

(29)

where the standard dimensional formula for a is

$$\begin{aligned} a = \frac{m(2n+1)-3}{2mN}. \end{aligned}$$

(30)

We require $(n-1)/N \le a < 1$ so N must be chosen such that $N > {\scriptstyle \frac{1}{2}}(2n+1)-3/2m$. The other end of the inequality is automatically satisfied for $m \ge 1$. Now we introduce the exponent $\alpha _{n,m}$ and time average :

$$\begin{aligned} \left<\Vert \nabla ^{n}{\varvec{u}}\Vert _{L^{2m}}^{\alpha _{n,m}}\right>_{T}\le & {} c_{N,n,m} \left<\left( H_{N+1,1}^{a/2}H_{1,1}^{(1-a)/2}\right) ^{\alpha _{n,m}}\right>_{T}\nonumber \\= & {} c_{N,n,m}\left<\left\{ \left( H_{N+1,1}^{\frac{1}{2N+1}}\right) ^{(2N+1)a/2} H_{1,1}^{(1-a)/2}\right\} ^{\alpha _{n,m}}\right>_{T}\nonumber \\\le & {} c_{N,n,m}\left<H_{N+1,1}^{\frac{1}{2N+1}}\right>_{T}^{\frac{(2N+1)a\alpha _{n,m}}{2}} \left<H_{1,1}^{\frac{(1-a)\alpha _{n,m}}{2-(2N+1)a\alpha _{n,m}}}\right>_{T}^{\frac{2-(2N+1)a\alpha _{n,m}}{2}} \end{aligned}$$

(31)

where a Hölder inequality has been used at the last step. We know the first term on the last line of the right-hand side is bounded using (28). The last term is bounded only if the exponent of $H_{1,1}$ inside the average is unity

$$\begin{aligned} \frac{(1-a)\alpha _{n,m}}{2-(2N+1)a\alpha _{n,m}} = 1\qquad \Rightarrow \qquad \alpha _{n,m} = \frac{2}{2Na+1}. \end{aligned}$$

(32)

From (30) we note that the combination 2Na is a function of m only, and gives the correct formula for $\alpha _{n,m}$, uniform in N. Checking that the coefficients in L and $\nu$ are correct is an exercise in algebra. $\square$

1.2 The Case $n = 0$ and $3 < m \le \infty$

Firstly, we prove a generalized inequality of the type first used by Tartar (1978) in bounding $\left<\Vert {\varvec{u}}\Vert _{\infty }\right>_{T}$. We use the Gagliardo–Nirenberg inequality

$$\begin{aligned} \Vert {\varvec{u}}\Vert _{L^{p}} \le c\,\Vert \nabla {\varvec{u}}\Vert ^{a}_{L^{2}}\Vert {\varvec{u}}\Vert _{L^{2m}}^{1-a},\qquad \qquad a = \frac{3(2m-p)}{p(m-3)}, \end{aligned}$$

(33)

where $m > 3$ and $6< p < 2m$. This ensures that $a < 1$. Next we use another inequality for $N > 3/2$

$$\begin{aligned} \Vert {\varvec{u}}\Vert _{L^{2m}} \le c\,\Vert \nabla ^{N}{\varvec{u}}\Vert ^{A}_{L^{2}}\Vert {\varvec{u}}\Vert _{L^{p}}^{1-A},\qquad \qquad A = \frac{3(2m-p)}{m[p(2N-3)+6]}. \end{aligned}$$

(34)

Taken together these give : for $m > 3$ and $N > \frac{3(m-1)}{2m}$

$$\begin{aligned} \Vert {\varvec{u}}\Vert _{L^{2m}} \le c\,\Vert \nabla ^{N}{\varvec{u}}\Vert ^{B}_{L^{2}}\Vert \nabla {\varvec{u}}\Vert _{L^{2}}^{1-B},\qquad \qquad B = \frac{m-3}{2m(N-1)}. \end{aligned}$$

(35)

When $m\rightarrow \infty$ we recover the $L^{\infty }$-inequality, where $B = \frac{1}{2(N-1)}$.

Next we proceed to prove (16) for $n=0$. For an exponent $\alpha _{0,m}$ to be determined we use (35) and write

$$\begin{aligned} \left<\Vert {\varvec{u}}\Vert _{L^{2m}}^{\alpha _{0,m}}\right>_{T}\le & {} c_{N,m}\left<H_{N,1}^{B\alpha _{0,m}/2} H_{1,1}^{(1-B)\alpha _{0,m}/2}\right>_{T}\nonumber \\\le & {} c_{N,m}\left<\left( H_{N,1}^{\frac{1}{2N-1}}\right) ^{B(2N-1)\alpha _{0,m}/2} H_{1,1}^{(1-B)\alpha _{0,m}/2}\right>_{T}\nonumber \\\le & {} c_{N,m}\left<H_{N,1}^{\frac{1}{2N-1}}\right>_{T}^{B(2N-1)\alpha _{0,m}/2}\nonumber \\&\times \left<H_{1,1}^{\frac{(1-B)\alpha _{0,m}}{2 - B(2N-1)\alpha _{0,m}}}\right>_{T}^{1 - B[2N-1]\alpha _{0,m}/2}, \end{aligned}$$

(36)

To be able to bound the right-hand side of (36) from above we use the result of Foias et al. (1981) expressed in (28). In addition, to be able to use the upper bound on $\left<H_{1,1}\right>_{T}$ we set

$$\begin{aligned} \frac{(1-B)\alpha _{0,m}}{2 - B(2N-1)\alpha _{0,m}} = 1, \end{aligned}$$

(37)

which determines $\alpha _{0,m}$. Given that $B = \frac{m-3}{2m(N-1)}$ from (35), we recover

$$\begin{aligned} \alpha _{0,m} = \frac{2m}{2m-3},\qquad m > 3, \end{aligned}$$

(38)

uniform in N, which is the result as advertised. $\square$

Appendix : Proof of Theorem 2

For the first parts of the theorem (i) and (ii), inequality (22) has already been proved in Theorem 1. For parts (iii) and (iv), involving (23), consider n such that for $n=0$ with m lying in the range $3/2 \le m\le \infty$, and $n\ge 1$ with m lying the range $1 \le m \le \infty$ ; we wish to prove that strong solutions exist if ${\varvec{u}}\in L^{2\alpha _{n,m}}\left( [0,\,T]\,; ~\dot{W}^{n,2m}(\mathbb {R}^{3})\right)$. Consider (12) written down again as

$$\begin{aligned} M_{n,m,T}({\varvec{u}}) := \int _{0}^{T} \left\| \nabla ^n {\varvec{u}} \right\| _{L^{2m}}^{2 \alpha _{n,m}}\,\mathrm{d}t. \end{aligned}$$

(39)

Does the assumption $M_{n,m,T}({\varvec{u}}) < \infty$ imply there is a smooth solution on $[0,\,T]$?

The case$n=0$, $3/2 < m \le \infty$ : Does the above assumption imply that ${\varvec{u}}\in L^{q}\left( [0,\,T];\,L^{p}\right)$? For completeness we repeat the argument given in item 1, equation (25). For $n=0$, let $p=2m$ and $q= 2\alpha _{0,m} = 2p/(p-3)$ then $2/q + 3/p = (p-3)/p + 3/p = 1$, which is the Prodi–Serrin criterion (Prodi 1959; Serrin 1962) – see line 3 of Table 2 and also (24). The special case $n=0$, $m=3/2$ is dealt with below.

The case$n\ge 1$, $1 \le m \le \infty$ : As already pointed out, a simple time integration of equation (10) verifies that $M_{n,m,T}({\varvec{u}})$ is scaling invariant for any $n\ge 0$ ; that is, if ${\varvec{u}}({\varvec{x}},t)$ is a solution of 3D Navier–Stokes on $\mathbb {R}^3 \times [0,\,T]$, then under ${\varvec{u}}_{\lambda }({\varvec{x}},t) =\lambda ^{-1} {\varvec{u}}\left( \frac{{\varvec{x}}}{\lambda },\frac{t}{\lambda ^2}\right)$  we have $M_{n,m,T}({\varvec{u}}) = M_{n,m,\lambda ^2 T}({\varvec{u}}_{\lambda })$. Thus, requiring the boundedness of the quantity $M_{n,m,T}$ from (39) appears to be the generalization of the Prodi–Serrin criterion for $n\ge 1$. The following is an adaptation of the standard proof of this criterion to the case $n\ge 1$.

For initial data ${\varvec{u}}_{0} \in \dot{W}^{n,2m}$, the norm which is integrated in time in (39), it is not difficult to prove a local existence in time result, with the time of existence depending only on the norm of the initial data. Then the proof is by contradiction using the standard maximal in time $T_{*} = T_{*}\left( \left\| {\varvec{u}}_{0} \right\| _{\dot{W}^{n,2m}}\right) >0$ argument. Assume that a weak solution ${\varvec{u}}$ of 3D Navier–Stokes obeys $M_{n,m,T}({\varvec{u}}) < \infty$, but that it blows up at time T. In particular, this means that for any increasing sequence of times $t_{n} \nearrow T$ we must have

$$\begin{aligned} \lim _{n\rightarrow \infty } \left\| {\varvec{u}}(\cdot ,\,t_n) \right\| _{\dot{W}^{n,2m}} = \infty . \end{aligned}$$

(40)

Otherwise, by the local existence theorem, one can extend the solution past time the putative blowup time T. Using this sequence of times we renormalize the solution ${\varvec{u}}$ according to the sequence of times $t_n$ as

$$\begin{aligned} {\varvec{u}}_{n}({\varvec{x}},t) = \frac{1}{\Lambda _n} {\varvec{u}}\left( \frac{{\varvec{x}}}{\Lambda _n},\,t_{n} + \frac{t}{\Lambda _{n}^{2}}\right) \end{aligned}$$

(41)

where $\Lambda _n>0$ is defined³ such that

$$\begin{aligned} \left\| {\varvec{u}}_{n}({\varvec{x}},\,0) \right\| _{\dot{W}^{n,2m}} = 1. \end{aligned}$$

(42)

In particular, we have that $\Lambda _{n} \rightarrow \infty$ as $n\rightarrow \infty$, because (10) and (42) together show that

$$\begin{aligned} \Lambda _{n} = \left\| {\varvec{u}}(\cdot ,\,t_{n}) \right\| _{\dot{W}^{n,2m}}^{\alpha _{n,m}}. \end{aligned}$$

(43)

Note that the functions ${\varvec{u}}_{n}$ also solve the 3D Navier–Stokes equations, but since the solution ${\varvec{u}}$ lives only up to time T, we know that the solutions ${\varvec{u}}_{n}$ live up to time $(T-t_{n}) \Lambda _{n}^{2}$. Thus, ${\varvec{u}}_{n}$ solves the 3D Navier–Stokes equations on $\mathbb {R}^{3} \times [0,\,(T-t_{n}) \Lambda _{n}^{2})$. However, by (42) and the local existence result, we know that ${\varvec{u}}_{n}$ does not blow up before the local existence time $T_{*}(1)>0$. Therefore, we must have

$$\begin{aligned} (T-t_{n}) \Lambda _{n}^{2} \ge T_{*}(1). \end{aligned}$$

(44)

By (43), the above condition implies that

$$\begin{aligned} \left\| {\varvec{u}}(\cdot ,\,t_{n}) \right\| _{\dot{W}^{n,2m}}^{2\alpha _{n,m}} \ge \frac{T_*(1)}{T-t_n}. \end{aligned}$$

(45)

However, the sequence $t_{n}\nearrow T$ was arbitrary, and thus (45) implies a minimal blowup rate

$$\begin{aligned} \left\| {\varvec{u}}(\cdot ,\,t) \right\| _{\dot{W}^{n,2m}}^{2\alpha _{n,m}} \ge \frac{T_*(1)}{T-t}. \end{aligned}$$

(46)

The contradiction is now immediate : we have made the assumption $M_{n,m,T}({\varvec{u}}) < \infty$, and thus the left side of (46) is integrable on $[0,\,T)$. On the other hand, the right side of (46) is not integrable on $[0,\,T)$, which is the desired contradiction.

The special case$n=0$, $m=3/2$ : The $\Vert {\varvec{u}}(\cdot ,\,t)\Vert _{L^3}$ result, excluded by the Prodi–Serrin conditions, but proved by Escauriaza et al. (2003), (see row 4 of Table 2) can be shown to be controlled by (23) when $n=1$ and $m=3/2$. To prove this we ignore the Laplacian term and write

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Vert {\varvec{u}}\Vert _{L^{2m}} \le \Vert \nabla p\Vert _{L^{2m}}. \end{aligned}$$

(47)

We use the Sobolev embedding

$$\begin{aligned} \Vert A\Vert _{L^{q}} \le c\Vert \nabla A\Vert _{L^{p}}\qquad \qquad \frac{1}{q} = \frac{1}{p} - \frac{1}{3} \end{aligned}$$

(48)

in three dimensions. This fits nicely for $p=3/2$ and $q=3$. In other words $\Vert A\Vert _{L^{3}} \le c\Vert \nabla A\Vert _{L^{3/2}}$. Thus, with $m=3/2$, we can write

$$\begin{aligned} \Vert \nabla p\Vert _{L^{3}} \le \Vert \nabla ^{2} p\Vert _{L^{3/2}} \le c\, \Vert \nabla ^{2}\Delta ^{-1}u_{i,j}u_{j,i}\Vert _{L^{3/2}} \le c\, \Vert \nabla {\varvec{u}}\Vert _{L^{3}}^{2}, \end{aligned}$$

(49)

having used a Riesz transform. Thus

$$\begin{aligned} \Vert {\varvec{u}}(\cdot ,\,t)\Vert _{L^{3}} \le c\,\int _{0}^{t}\Vert \nabla {\varvec{u}}\Vert _{L^{3}}^{2}\,\mathrm{d}\tau . \end{aligned}$$

(50)

Finally $\alpha _{1,3/2} = \frac{3}{6-3} = 1$, and so $\Vert {\varvec{u}}(\cdot ,\,t)\Vert _{L^{3}}$ is controlled by (23) when $n=1,~m=3/2$. $\square$