On Stability and Instability of $C^{1,\alpha }$ Singular Solutions to the 3D Euler and 2D Boussinesq Equations

Abstract

Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging (Majda and Bertozzi in Vorticity and incompressible flow, Cambridge University Press, Cambridge, vol 27, 2002; Gibbon in Physica D 237(14):1894–1904, 2008; Kiselev, in: Proceedings of the international congress of mathematicians, vol 3, 2018; Drivas and Elgindi in EMS Surv Math Sci 10(1):1–100, 2023; Constantin in Bull Am Math Soc 44(4):603–621, 2007). In Elgindi (Ann Math 194(3):647–727, 2021) (see also Elgindi et al. in Camb J Math 9(4), 2021), Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,\alpha }$ initial velocity develops a finite time singularity. Inspired by Elgindi’s work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with $C^{1,\alpha }$ initial velocity and boundary develop a stable asymptotically self-similar (or approximately self-similar) finite time singularity (Chen and Hou in Commun Math Phys 383(3):1559–1667, 2021) in the same setting as the Hou-Luo blowup scenario (Luo and Hou in Proc Natl Acad Sci 111(36):12968–12973, 2014; Luo and Hou in SIAM Multiscale Model Simul 12(4):1722–1776, 2014). On the other hand, the authors of Vasseur and Vishik (Commun Math Phys 378(1):557–568, 2020) and Lafleche et al. (Journal de Mathématiques Pures et Appliquées 155:140–154, 2021) recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in Vasseur and Vishik (2020) and Lafleche et al. (2021) require some strong regularity assumption on the initial data, which is not satisfied by the $C^{1,\alpha }$ velocity field. In this paper, we generalize the analysis of Elgindi (Ann Math 194(3):647–727, 2021), Chen and Hou (Commun Math Phys 383(3):1559–1667, 2021), Vasseur and Vishik (2020) and Lafleche et al. (2021) to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with $C^{1,\alpha }$ velocity are unstable under the notion of stability introduced in Vasseur and Vishik (2020) and Lafleche et al. (2021). These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions.

Appendices

Appendix A. Review of the Construction of Unstable Solutions

We provide a brief review of the construction of the unstable solution in [46, 64] via a WKB expansion and explain the connections among the WKB expansion, the bicharacteristics-amplitude ODE system (3.6)–(3.8), and the growth of the unstable solution.

1.1 Construction of the approximate solution

Suppose that ${\textbf{u}}(t, x)$ is a singular solution of (1.1). Denote by $\gamma _t(x)$ the flow map

$$\begin{aligned} \frac{d}{dt} \gamma _t(x) = {\textbf{u}}( t, \gamma _t(x)), \quad \gamma _0(x) = x. \end{aligned}$$

(A.1)

The main idea in [64] is to construct an approximate solution to (1.2) using a WKB expansion

$$\begin{aligned} v(t, x) \approx b(t, x) \exp ( \frac{ i S(t, x)}{\varepsilon } ) \end{aligned}$$

(A.2)

for sufficiently small $\varepsilon$ and the characteristics of the flow, where $b(t, x) \in {\mathbb {R}}^3$ and S is a scalar. Plugging the above ansatz into (1.2), we obtain

$$\begin{aligned} R_{\varepsilon } = ( \partial _t + {\textbf{u}}\cdot \nabla + \nabla {\textbf{u}}) v = \frac{i}{\varepsilon }( \partial _t + {\textbf{u}}\cdot \nabla ) S \cdot b e^{ iS / \varepsilon } + ( \partial _t + {\textbf{u}}\cdot \nabla + \nabla {\textbf{u}}) b \cdot e^{i S / \varepsilon }, \end{aligned}$$

where $(\nabla {\textbf{u}}) f = f \cdot \nabla {\textbf{u}}= f_j \partial _j u_i e_i$. To eliminate the $O(\varepsilon ^{-1})$ term, one requires

$$\begin{aligned} (\partial _t + {\textbf{u}}\cdot \nabla ) S =0. \end{aligned}$$

(A.3)

Then we can rewrite $R_{\varepsilon }$ as follows

$$\begin{aligned} R_{\varepsilon } = (\partial _t + {\textbf{u}}\cdot \nabla + \nabla {\textbf{u}}) b \cdot e^{ i S/ \varepsilon } \triangleq F(t, x) \cdot e^{ i S/ \varepsilon } , \quad F \triangleq (\partial _t + {\textbf{u}}\cdot \nabla + \nabla {\textbf{u}}) b . \end{aligned}$$

(A.4)

An important observation in [64] is that for high frequency oscillation, i.e. small $\varepsilon$, the pressure term in (1.2) is almost local. We would like to construct (v, Q) such that

$$\begin{aligned} R_{\varepsilon } = F(t, x) e^{i S/ \varepsilon } = \nabla Q + E_{\varepsilon }, \end{aligned}$$

where $E_{\varepsilon }$ is a small error term. This is possible since Q is one order more regular than a highly oscillatory function $F(t, x) e^{i S / \varepsilon }$. By integration and exploiting the cancellation, Q can be of order $O(\epsilon )$. In fact, taking $\nabla \times$ on both sides, we obtain

$$\begin{aligned} \nabla \times R_{\varepsilon } = ( \nabla \times F) e^{i S / \varepsilon } + \frac{i}{\varepsilon } (\nabla S \times F) e^{i S / \varepsilon } = \nabla \times ( \nabla Q + E_{\varepsilon } ) = \nabla \times E_{\varepsilon }. \end{aligned}$$

To eliminate the $O(\varepsilon ^{-1})$ term, we require $\nabla S \times F = 0$, which implies $F = c(t, x) \nabla S$ for some scalar c(t, x). In this case, one can construct the pressure Q as follows

$$\begin{aligned} Q = - i \varepsilon c(t, x) e^{i S / e}. \end{aligned}$$

As a result, the error is given by

$$\begin{aligned} E_{\varepsilon } = R_{\varepsilon } - \nabla Q = c \nabla S e^{i S / \varepsilon } + i \varepsilon \cdot \nabla c \cdot e^{i S / \varepsilon } + i \varepsilon c \frac{ i \nabla S}{\varepsilon } e^{iS / \varepsilon } = i \varepsilon \cdot \nabla c \cdot e^{i S / \varepsilon }. \end{aligned}$$

(A.5)

Suppose that c is smooth, then the $L^p$ norm of the error $E_{\varepsilon }$ is small as $\varepsilon \rightarrow 0$.

From $F = c(t, x) \nabla S$ and (A.4), we yield

$$\begin{aligned} ( \partial _t + {\textbf{u}}\cdot \nabla + \nabla {\textbf{u}}) b = F(t, x) = c(t, x) (\nabla S)( t, x). \end{aligned}$$

Using the Lagrangian coordinates and the flow map $\gamma _t$ (A.1), we get

$$\begin{aligned} \partial _t b(t, \gamma _t(x)) = - ( \nabla {\textbf{u}}) b(t, \gamma _t(x)) + c(t, x) (\nabla S)(t, \gamma _t(x)). \end{aligned}$$

Denote

$$\begin{aligned} \xi _t( x) \triangleq (\nabla S)( t, \gamma _t(x)) , \quad b_t(x) \triangleq b(t, \gamma _t(x)) \;. \end{aligned}$$

(A.6)

The above equation reduces to

$$\begin{aligned} \frac{d}{dt} b_t = - (\nabla {\textbf{u}}) b_t + c(t, x) \xi _t . \end{aligned}$$

(A.7)

Next, we determine the equations for $b, \xi$. In order for v(t, x) to be incompressible, from the ansatz (A.2) and

$$\begin{aligned} \nabla \cdot v(t, x) = ( \nabla \cdot b ) e^{i S / \varepsilon } + \frac{i}{\varepsilon } b \cdot \nabla S e^{ i S /\varepsilon } \;, \end{aligned}$$

we require $b(t, x) \cdot (\nabla S)(t, x) = 0$ to eliminate the $O(\varepsilon ^{-1})$ term. In the Lagrangian coordinates, this condition is equivalent to enforcing

$$\begin{aligned} b(t, \gamma _t(x)) \cdot (\nabla S)( t, \gamma _t(x)) = b_t(x) \cdot \xi _t(x) = 0. \end{aligned}$$

(A.8)

Taking the gradient in the transport equation (A.3), we get

$$\begin{aligned} ( \partial _t + {\textbf{u}}\cdot \nabla ) \nabla S = - (\nabla {\textbf{u}})^T \nabla S. \end{aligned}$$

Using the Lagrangian coordinates and (A.6), we derive

$$\begin{aligned} \frac{d}{d t} \xi _t = \frac{d}{dt} (\nabla S)(t, \gamma _t(x)) = - (\nabla {\textbf{u}})^T (\nabla S )( t ,\gamma _t(x)) = - (\nabla {\textbf{u}})^T \xi _t. \end{aligned}$$

(A.9)

The incompressible condition (A.8) implies $\frac{d}{dt} (b_t \cdot \xi _t ) = 0$. Thus, from (A.7) and (A.9), we get

$$\begin{aligned} \langle c(t, x) \xi _t, \xi _t \rangle - \langle (\nabla u) b_t, \xi _t \rangle - \langle (\nabla {\textbf{u}})^T \xi _t, b_t \rangle = 0, \end{aligned}$$

where $\langle p, q \rangle = p_i q_i$. It follows that

$$\begin{aligned} c(t, x) = 2 \frac{ \xi _t^T ( \nabla {\textbf{u}}) b_t}{ | \xi _t|^2}. \end{aligned}$$

Thus, from (A.1),(A.7),(A.9), $\gamma _t, \xi _t, b_t$ satisfy the bicharacteristics-amplitude ODE system (3.6)–(3.8) of (1.1) [46, 64]

The above derivation reveals the main idea behind the construction of an approximate solution to (1.2) in [64] and the relationship between the WKB expansion (A.2) and the bicharacteristics-amplitude ODEs (3.6)–(3.8). The last step is to localize the solution v(t, x) to some trajectory and add a correction to v(t, x) (A.2) so that it is incompressible. We refer to [64] for the details.

1.2 Growth of the solution

The solution v(t, x) satisfies (1.2) up to an error similar to (A.5). Since $E_{\varepsilon }$ contains the highly oscillatory phase $e^{i S / \varepsilon }$, the error may not be small in $C^{k, \alpha }$ or $H^s$ norm. In [64], based on the WKB construction (A.2) and using the smallness of the error in the $L^p$ norm, the authors constructed an approximate solution to (1.2) with error controlled by $\varepsilon$. To prove the instability, they further showed the growth of v(t, x). From (A.2), the growth of $|| v||_p$ is due to $|| b_t||_p$. The authors showed that if the velocity ${\textbf{u}}(t, x)$ is smooth, the system (3.6)–(3.8) satisfies the following conservations along the characteristic $\gamma _t(x)$

$$\begin{aligned} \begin{aligned} \omega (t, \gamma _t(x) ) \cdot \xi _t&= \omega _0(x) \cdot \xi _0, \quad b_t \cdot \xi _t = \tilde{b}_t \cdot \xi _t, \quad ( b_t \times \tilde{b}_t) \cdot \xi _t = (b_0 \times \tilde{b}_0) \cdot \xi _0, \end{aligned} \end{aligned}$$

where $\omega = \nabla \times {\textbf{u}}$ is the vorticity of the blowup solution ${\textbf{u}}$, $\xi _t, b_t, \tilde{b}_t$ are the solution to (3.6)–(3.8) with initial data $x_0, \xi _0, b_0, \tilde{b}_0$, $b_0 \cdot \xi _0 = \tilde{b}_0 \cdot \xi _0 = 0$ and $b_0, \tilde{b}_0, \xi _0$ being linearly independent.

From the first and the third identity, formally, $b_t \times \tilde{b}_t$ plays a role similar to $\omega (t, \gamma _t(x))$. Indeed, using the above conservations, the authors further proved

$$\begin{aligned} || \omega (t, \cdot )||_{\infty } \le || \omega _0||_{L^{\infty }} \Big ( \sup _{ |b_0| = |\xi _0| = 1, x_0 \in \Omega , b_0 \cdot \xi _0 = 0} | b_t(x_0, \xi _0, b_0)| \Big )^2. \end{aligned}$$

(A.10)

According to the BKM blowup criterion, $|| \omega (t)||_{\infty }$ must blowup, which leads to the growth of $b_t$ and $|| v(t)||_{L^p}$ and implies linear instability.

Appendix B. Embedding Inequalities and Estimates of Nonlinear Terms

We have the following estimates for different norms. The first and last inequality generalize Proposition 7.6 in [11]. The second inequality is exactly Proposition 7.7 in [11]. The third inequality in (B.1) generalizes Lemma 7.11 in [11]. Since the proof essentially uses the estimates in [11], we omit the proof here and refer it to Appendix B in the arXiv version of this paper [12].

Proposition B.1

Let ${{\mathcal {C}}}^k$ and ${{\mathcal {W}}}^{k,\infty }$ be the norms defined in (4.15) and (4.17). For $k \ge 1$,

$$\begin{aligned} \begin{aligned} || f g||_{{{\mathcal {C}}}^k}&\lesssim || f||_{{{\mathcal {C}}}^k} || g||_{{{\mathcal {C}}}^k}, \quad || f g ||_{{{\mathcal {W}}}^{k,\infty }} \lesssim || f ||_{{{\mathcal {W}}}^{k,\infty }} || g||_{{{\mathcal {W}}}^{k, \infty }} , \\ || f||_{{{\mathcal {C}}}^{ k}}&\lesssim \alpha ^{-1/2} || f||_{ {{\mathcal {H}}}^{k+2}}, \quad || f||_{{{\mathcal {C}}}^k} \lesssim || \frac{1+R}{R} f ||_{{{\mathcal {W}}}^{k, \infty }}. \end{aligned} \end{aligned}$$

(B.1)

We have the following elliptic estimates for the stream function (4.10), (4.6).

Proposition B.2

Assume that $\alpha \le \frac{1}{4}$ and $\Omega \in {{\mathcal {H}}}^k, k \ge 3$. Let $\Psi$ be the solution to (4.6) with boundary condition (4.7). Then we have

$$\begin{aligned} \begin{aligned}&\alpha ^2 || R^2 \partial _{RR} \Psi ||_{{{\mathcal {H}}}^k} + \alpha || R \partial _{R \beta } \Psi ||_{{{\mathcal {H}}}^k} +|| \partial _{\beta \beta } (\Psi - \frac{1}{\alpha \pi } \sin (2\beta ) L_{12}(\Omega )) ||_{{{\mathcal {H}}}^k} \lesssim _k || \Omega ||_{{{\mathcal {H}}}^k}. \end{aligned} \end{aligned}$$

The above estimate with $k =3$ has been established in [11]. The general case $k \ge 3$ can be proved similarly. See also [26].

We have the following estimates for the velocity ${{\bar{u}}}$ of the approximate steady state.

Proposition B.3

For $\alpha \le \frac{1}{4}$ and $k \ge 5$, we have

$$\begin{aligned} \begin{aligned}&|| \frac{1+R}{ R} \partial _{\beta \beta } ( {\bar{\Psi }} - \frac{\sin (2\beta )}{\pi \alpha } L_{12}({\bar{\Omega }})) ||_{{{\mathcal {W}}}^{k+2,\infty }} \lesssim \alpha , \qquad || L_{12}({\bar{\Omega }})||_{{{\mathcal {W}}}^{k+2,\infty }} \lesssim \alpha , \\&\alpha || \frac{1+R}{R} D_R^2 {\bar{\Psi }} ||_{{{\mathcal {W}}}^{k,\infty }} + \alpha || \frac{1+R}{R} \partial _{\beta } D_R {\bar{\Psi }} ||_{{{\mathcal {W}}}^{k,\infty }} \\&\quad \quad + || \frac{1+R}{ R} \partial _{\beta \beta } ( {\bar{\Psi }} - \frac{\sin (2\beta )}{\pi \alpha } L_{12}({\bar{\Omega }})) ||_{{{\mathcal {W}}}^{k,\infty }} \lesssim \alpha \nonumber . \end{aligned} \end{aligned}$$

The case of $k = 5$ has been proved in Proposition 7.8 [11]. The general case $k \ge 5$ follows from a similar argument. See also [26].

Appendix C. Estimate of the Approximate Steady State

Recall from (4.4) that ${{\bar{\Omega }}}, {{\bar{\eta }}}, {{\bar{\xi }}}$ denote the approximate steady state ${{\bar{\omega }}}, {{\bar{\theta }}}_x, \bar{\theta }_y$ under the coordinates $(R, \beta )$, and the formula of ${\bar{\Omega }}, {\bar{\eta }}$ in (4.8).

$$\begin{aligned} {\bar{\Omega }} = \frac{\alpha }{c} \frac{3 R \Gamma (\beta ) }{(1+R)^2}, \quad {\bar{\eta }} = \frac{\alpha }{c} \frac{6 R \Gamma (\beta ) }{(1+R)^3} . \end{aligned}$$

(C.1)

We generalize Lemma A.6 in [11] from $k \le 3$ to any k below.

Lemma C.1

The following results apply to any $k \ge 0, 0 \le i + j \le k, j \ne 0$. (a) For $f = {\bar{\Omega }}, {\bar{\eta }}, {\bar{\Omega }} - D_R {\bar{\Omega }}, {\bar{\eta }} - D_R {\bar{\eta }}$, we have

$$\begin{aligned} | D_R^k f | \lesssim _k f , \quad |D_R^i D^j_{\beta } f | \lesssim _k \alpha \sin (\beta ) f. \end{aligned}$$

(C.2)

Recall the ${{\mathcal {W}}}^{k, \infty }$ norm (4.17). We generalize Lemma A.7 in [11] from $k = 7$ to any $k \ge 7$.

Lemma C.2

For any $k \ge 7$, it holds true that $\Gamma (\beta ),{\bar{\Omega }}, {\bar{\eta }} \in {{\mathcal {W}}}^{k, \infty }$ with

$$\begin{aligned} \begin{aligned} ||\frac{(1+R)^2}{R} {\bar{\Omega }}||_{{{\mathcal {W}}}^{k,\infty }} + || \frac{(1+R)^2}{R}{\bar{\eta }} ||_{{{\mathcal {W}}}^{k,\infty }} \lesssim _k \alpha , \quad || D_{\beta } {\bar{\Omega }}||_{{{\mathcal {W}}}^{k,\infty }} + || D_{\beta }{\bar{\eta }} ||_{{{\mathcal {W}}}^{k,\infty }} \lesssim _k \alpha ^2. \end{aligned} \end{aligned}$$

Recall the ${{\mathcal {K}}}^k$ norm (4.15). We generalize Lemma A.8 in [11] from $k= 5$ to any $k \ge 5$ below.

Lemma C.3

Assume that $0\le \alpha \le \frac{1}{1000}$. For $R \ge 0, \beta \in [0, \pi /2], k \ge 1$ and $i + j \le k$, we have

$$\begin{aligned}&| D^i_R D^j_{\beta } {\bar{\xi }} | \lesssim _k - {\bar{\xi }}, \quad | D^i_R D^j_{\beta } (3{\bar{\xi }} - R\partial _R {\bar{\xi }}) | \lesssim _k -{\bar{\xi }}, \\&|| {\bar{\xi }} ||_{{{\mathcal {C}}}^k} \lesssim || \frac{1+R}{R} ( 1 + ( R \sin ( 2 \beta )^{\alpha } )^{-\frac{1}{40} } ) {\bar{\xi }} ||_{L^{\infty }} \lesssim \alpha ^2, \end{aligned}$$

The proofs of Lemmas C.1–C.3 follows from the argument in [11], and thus are omitted.

For the $L_{12}$ operator (4.10), we generalize Lemma A.4 in [11] from ${{\mathcal {H}}}^3$ to its ${{\mathcal {H}}}^k$ version. The proof follows from a similar argument.

Lemma C.4

Let $\chi (\cdot ): [0, \infty ) \rightarrow [0, 1]$ be a smooth cutoff function, such that $\chi (R) = 1$ for $R \le 1$ and $\chi (R) = 0$ for $R \ge 2$. For $0 \le k \le n, 0\le l \le n-1, n \ge 3$, we have

$$\begin{aligned} \begin{aligned}&|| L_{12}(\Omega ) - L_{12}(\Omega )(0) \chi ||_{{{\mathcal {H}}}^n} + || D_R( L_{12}(\Omega ) - L_{12}(\Omega )(0) \chi ) ||_{{{\mathcal {H}}}^n} \lesssim _n || \Omega ||_{{{\mathcal {H}}}^n}, \\&|| D^k_R L_{12}(\Omega ) ||_{\infty } + || D^k_R ( L_{12}(\Omega ) -\chi L_{12}(\Omega )(0)) ||_{\infty } \lesssim _n || \Omega ||_{{{\mathcal {H}}}^n}. \end{aligned} \end{aligned}$$

(C.3)

Appendix D. Derivation of $u_r^r( 0,0)$

We derive the formula (5.28) for $u_r^r(0, 0)$ using the formula

$$\begin{aligned} {\textbf{u}}(x) = \nabla \times (-\Delta )^{-1} \omega = \frac{1}{4\pi } \int _{{\mathbb {R}}^3} \frac{\omega (y) \times (x- y)}{ |x-y|^3} dy. \end{aligned}$$

Recall the coordinates and change of variables (5.27)

$$\begin{aligned} \beta = \arctan ( z / r), \quad \rho = (r^2 + z^2)^{1/2}, \quad R = \rho ^{\alpha }, \quad \Omega (R, \beta ) = \omega ^{\vartheta }(\rho , \beta ), \end{aligned}$$

where $(r, \vartheta , z)$ is the cylindrical coordinates in ${\mathbb {R}}^3$ (3.1). Note that $u_r^r(0,0) = -\frac{1}{2} u_z^z(0,0)$ (3.10), we compute $u_z^z(0, 0)$. Since there is no swirl $u^{\vartheta } \equiv 0$, we get

$$\begin{aligned}{} & {} \omega = \omega ^{\vartheta } e_{\vartheta } = ( - \omega ^{\vartheta } \sin \vartheta , \omega ^{\vartheta } \cos \vartheta , 0), \\{} & {} \quad ( \omega \times (x - y) )_3 = - \omega ^{\vartheta } \sin (\vartheta ) (x_2 - y_2 ) - \omega ^{\vartheta } \cos (\vartheta ) (x_1 - y_1). \end{aligned}$$

Since the above formula is independent of $z= x_3$ and $\omega ^{\vartheta }(y)$ is odd in $y_3$, we yield

$$\begin{aligned} \partial _3 u^3 = \frac{1}{4\pi } \int _{{\mathbb {R}}^3} (\omega \times (x - y))_3 \partial _{x_3} \frac{1}{|x-y|^3} dy = \frac{1}{4\pi } \int _{{\mathbb {R}}^3} (\omega \times (x - y))_3 \frac{-3( x_3 - y_3)}{ |x-y|^5} dy. \end{aligned}$$

Evaluating at $x = 0$ and using

$$\begin{aligned} ( \omega \times ( - y) )_3 = \omega ^{\vartheta }( y) \sin (\vartheta ) y_2 + \omega ^{\vartheta } \cos (\vartheta ) y_1, = \omega ^{\vartheta }(y) r \end{aligned}$$

and $r = \rho \cos \beta , z = \rho \sin \beta , \beta \in [-\pi /2, \pi /2]$, we obtain

$$\begin{aligned} \begin{aligned} \partial _3 u^3(0, 0)&= \frac{3 }{4\pi } \int _{{\mathbb {R}}^3} \frac{\omega ^{\vartheta }(y) r y_3}{ |y|^5} d y = \frac{3}{4\pi } \int _0^{\infty }\int _0^{2\pi }\int _{{\mathbb {R}}} \frac{\omega ^{\vartheta }(y) r z }{ |y|^5} r d r d \vartheta d z \\&=\frac{3}{2} \int _{{\mathbb {R}}_+\times {\mathbb {R}}} \frac{\omega ^{\vartheta }(y) r^2 z }{ \rho ^5} d r d z \\&= \frac{3}{2} \int _0^{\infty } \int _{-\pi /2}^{\pi /2} \frac{\omega ^{\vartheta }(\rho , \beta ) \cos ^2(\beta ) \sin (\beta ) }{ \rho } d \rho d \beta \\&= 3 \int _0^{\infty } \int _{0}^{\pi /2} \frac{\omega ^{\vartheta }(\rho , \beta ) \cos ^2( \beta ) \sin (\beta ) }{ \rho } d \rho d \beta . \end{aligned} \end{aligned}$$

Using $u_r^r(0,0) = -\frac{1}{2}u_z^z(0,0)$ (3.10) and $\frac{ d \rho }{\rho } = \frac{1}{\alpha } \frac{dR}{R}$, we prove (5.28).