Mathematical analysis of pulsatile flow, vortex breakdown and instantaneous blow-up for the axisymmetric Euler equations

TL;DR

This paper analyzes the stability and blow-up phenomena in axisymmetric Euler flows, showing rapid inflow can destabilize laminar profiles and non-zero axis vorticity prevents steady states, indicating potential singularities.

Contribution

It provides a mathematical framework demonstrating conditions leading to instability and blow-up in axisymmetric Euler equations, using Frenet-Serret formulas and moving frames.

Findings

Rapid inflow destabilizes laminar flow profiles.

Non-zero vorticity on the axis prevents steady solutions.

Conditions for instantaneous blow-up are identified.

Abstract

The dynamics along the particle trajectories for the 3D axisymmetric Euler equations are considered. It is shown that if the inflow is rapidly increasing (pushy) in time, the corresponding laminar profile of the incompressible Euler flow is not (in some sense) stable provided that the swirling component is not zero. It is also shown that if the vorticity on the axis is not zero (with some extra assumptions), then there is no steady flow. We can rephrase these instability to an instantaneous blow-up. In the proof, Frenet-Serret formulas and orthonormal moving frame are essentially used.

Equations

$${\partial }_{t}u+(u\cdot \nabla )u-\nu \Delta u=-\nabla p,\nabla \cdot u=0\text{in}{\Omega }_{\mathcal{R}},u=0\text{on}\partial {\Omega }_{\mathcal{R}}$$

$$u_s(r)=(u_1,u_2,u_3)=(0,0,\frac{p_s}{4\nu \ell }({\mathcal{R}}^2-r^2)),$$

$$\frac{p_1(t)-p_2(t)}{\ell }=p_o{e}^{iNt}$$

$$\alpha =\mathcal{R}\sqrt{\frac{N}{\nu }}.$$

$${\int }_{{\Omega }_{\mathcal{R}}}{u}_3(x_1,x_2,x_3,t{)}^{2}d{x}_{1}d{x}_2=a{\left({\int }_{{\Omega }_{\mathcal{R}}}{u}_3(x_1,x_2,x_3,t)d{x}_{1}d{x}_2\right)}^2$$

$$D_{t}u:={\partial }_{t}u+(u\cdot \nabla )u=-\nabla p,\nabla \cdot u=0\text{in}{\Omega }_{\mathcal{R}},$$

$$u{|}_{t=0}=u_0,u\cdot n=0\text{on}\partial {\Omega }_{\mathcal{R}},$$

$$u(x,t){|}_{x_3=0}=(0,0,U_{in}(r,t))\text{with}{U}_{in}(r,t)>0,$$

$$C^{-1}a\le b\le Ca,$$

$$a\le Cb.$$

$$|U_o(r,t)|\approx 1,\underset{1\le j+k\le 2}{\sup }\left(|{\partial }_r^j{\partial }_t^k{U}_o(r,t)|+|{\partial }_r^j{\partial }_t^k{U}_s(r,t)|\right)\lesssim 1,$$

$$\frac{d}{dt}{\Phi }_{\ast }(x,t)=u({\Phi }_{\ast }(x,t),t),$$

$${\Phi }_{\ast }(x,0)=x\in {\Omega }_{\mathcal{R}}.$$

$$\partial_{z}\tilde{\Phi}(z)=\frac{u(x,t)}{u_{3}(x,t)}\bigg{|}_{x=\tilde{\Phi}(z)}\quad$$

$$\frac{d}{dt}{Z}_{\ast }(t)=v_z(R_{\ast }(t),Z_{\ast }(t),t),$$

$$Z_{\ast }(0)=z_0$$

$$\frac{d}{dt}{R}_{\ast }(t)=v_r(R_{\ast }(t),Z_{\ast }(t),t),$$

$$R_{\ast }(0)=r_0$$

$${\partial }_t{v}_r+v_r{\partial }_r{v}_r+v_z{\partial }_z{v}_r-\frac{v_{\theta }^2}{r}+{\partial }_{r}p$$

$${\partial }_t{v}_{\theta }+v_r{\partial }_r{v}_{\theta }+v_z{\partial }_z{v}_{\theta }+\frac{v_r{v}_{\theta }}{r}$$

$${\partial }_t{v}_z+v_r{\partial }_r{v}_z+v_z{\partial }_z{v}_z+{\partial }_{z}p$$

$$\frac{{\partial }_r(r{v}_r)}{r}+{\partial }_z{v}_z$$

$$\tilde{\Phi }({\tilde{r}}_0,z,t)=\tilde{\Phi }(z):=(\tilde{R}(z)\cos \tilde{\Theta }(z),\tilde{R}(z)\sin \tilde{\Theta }(z),z)$$

$${\partial }_z\tilde{\Phi }\cdot e_z=1,{\partial }_z\tilde{\Phi }\cdot e_r={\partial }_z\tilde{R}=\frac{v_r}{v_z}\text{and}{\partial }_z\tilde{\Phi }\cdot e_{\theta }=\tilde{R}{\partial }_z\tilde{\Theta }=\frac{v_{\theta }}{v_z}.$$

$$\Phi (z)=(R(z)\cos \Theta (z),R(z)\sin \Theta (z),z)$$

$$L^0({\tilde{r}}_0,z,t)$$

$$,L^x({\tilde{r}}_0,z,t)$$

$$L^t({\tilde{r}}_0,z,t)$$

$$L^0\equiv 2,L^x\equiv 0\text{and}{L}^t\equiv 0$$

$$\tilde{\Phi }({\tilde{r}}_0(t),z(t),t)={\Phi }_{\ast }(x,t)$$

$$|g|\approx 1,g^{\prime }(t_j)\to \infty ,g^{\prime \prime }(t_j)\to \infty \text{and}f(t_j)\to \infty (t_j\to T\text{as}j\to \infty ),$$

$$L^t({\tilde{r}}_0(t_j),z(t_j),t_j)\ge f(t_j)\text{for}x\in D_{\gamma }\text{and}j=1,2,\cdots .$$

$$|v_z(0,z,0)|$$

$$|{\partial }_r{v}_{\theta }(0,z,0)|$$

$$\sum _{\stackrel{0\le j+k\le 3}{(j,k)\ne (1,0)}}|{\partial }_r^j{\partial }_z^k{v}_{\theta }(0,z,0)|$$

$$\sum _{0\le j+k\le 2}|{\partial }_r^j{\partial }_z^k{v}_r(0,z,0)|$$

$$\sum _{1\le j+k\le 2}|{\partial }_r^j{\partial }_z^k{v}_z(0,z,0)|$$

Full text

1. Introduction

We study the dynamics along the particle trajectories for the 3D axisymmetric Euler equations. Such Lagrangian dynamics of the 3D axisymmetric Euler flow (inviscid flow) have already been studied in mathematics (see [6, 7, 8]). For example, in [7], Chae considered a blow-up problem for the axisymmetric 3D incompressible Euler equations with swirl. More precisely, he showed that under some assumption of local minima for the pressure on the axis of symmetry with respect to the radial variations along some particle trajectory, the solution blows up in finite time.

Although the blowup problem of the 3D incompressible Euler equations (also the Navier-Stokes equations) is still an outstanding open problem, in this paper, we focus on a different problem in physics, especially, “pulsatile flow” and “vortex breakdown”. In the pulsatile flow study field, Womersley number is the key. The Wormersley number comes from oscillating (in time) solutions to the incompressible Navier-Stokes equations in a tube. Let us explain more precisely. We define a pipe $\Omega_{\mathcal{R}}$ as $\Omega_{\mathcal{R}}:=\{x\in\mathbb{R}^{3}:\sqrt{x_{1}^{2}+x_{2}^{2}}<\mathcal{R},\ 0<x_{3}<\ell\}$ with its side-boundary $\partial\Omega_{\mathcal{R}}=\{x\in\mathbb{R}^{3}:\sqrt{x_{1}^{2}+x_{2}^{2}}=\mathcal{R},\ 0<x_{3}<\ell\}$, top and bottom boundaries: $\partial\Omega_{\mathcal{R}}^{top}:=\{x\in\mathbb{R}^{3}:0\leq\sqrt{x_{1}^{2}+x_{2}^{2}}<\mathcal{R},\ x_{3}=\ell\}$ and $\partial\Omega_{\mathcal{R}}^{bottom}:=\{x\in\mathbb{R}^{3}:0\leq\sqrt{x_{1}^{2}+x_{2}^{2}}<\mathcal{R},\ x_{3}=0\}$. The incompressible Navier-Stokes equations are described as follows:

[TABLE]

with $u=u(x,t)=(u_{1}(x_{1},x_{2},x_{3},t),u_{2}(x_{1},x_{2},x_{3},t),u_{3}(x_{1},x_{2},x_{3},t))$ and $p=p(x,t)$.

To give the Womersley number, we need to focus on the axisymmetric Navier-Stokes flow without swirl (see [34]). If $p_{1}$ and $p_{2}$ are the pressure at the ends of the pipe $\Omega_{\mathcal{R}}$, namely, $\partial\Omega_{\mathcal{R}}^{top}$ and $\partial\Omega_{\mathcal{R}}^{bottom}$, the pressure gradient can be expressed as $(p_{1}-p_{2})/\ell$ (for the study of pressure boundary conditions on $\partial\Omega_{\mathcal{R}}^{top}$ and $\partial\Omega_{\mathcal{R}}^{bottom}$, see [21] for example). If the pressure gradient is time-independent, $(p_{1}-p_{2})/\ell=:p_{s}$, then we can find a stationary Navier-Stokes flow (Poiseuille flow):

[TABLE]

where $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$. Note that $u_{s}$ is also a solution to the linearized Navier-Stokes equations. Next we consider the oscillating pressure gradient case,

[TABLE]

which is periodic in the time. Then its corresponding solution $u_{o}=u_{o}(r,t)$ can be written explicitly by using a Bessel function (see [34, (8)] and [32, (1)]) with $u_{1}=u_{2}=0$. Thus $u_{o}$ is also a solution to the linearized Navier-Stokes equations. Note that $u_{o}+u_{s}$ is a time-periodic solution to the Navier-Stokes equations. In this study field, the following Womersley number $\alpha$ is the key:

[TABLE]

In [32], they also defined the oscillatory Reynolds number and the mean Reynolds number by using $u_{o}$ and $u_{s}$ respectively, and they investigated how the transition of pulsatile flow $u_{o}+u_{s}$ from the laminar to the turbulent (critical Reynolds number) is affected by the Womersley number and the oscillatory Reynolds number. According to their experiment, measurement at different Womersley numbers yield similar transition behavior, and variation of the oscillatory Reynolds number also appear to have little effect. Thus they conclude that the transition seems to be determined only by the mean Reynolds number. However it seems they did not investigate the effect of the swirl component (azimuthal component), and our aim here is to show that the non-zero swirl component induces an instability of the laminar profile which is, at a glance, nothing to do with wall turbulence.

On the other hand, in the study of vortex breakdown, determining the possible flow topologies of the steady axisymmetric Navier-Stokes flow in a cylindrical container (such as $\Omega_{\mathcal{R}}$) with rotating end-covers (on $\partial\Omega^{top}_{\mathcal{R}}$ and $\partial\Omega^{bottom}_{\mathcal{R}}$) has been the main subject (see [5, 16, 19, 30] for example, see also [20]). The flow structures and the stability of the flow turns out to be sensitive to changes in the rotation ratio of the two covers. Using a combination of bifurcation theory for two-dimensional dynamical systems and numerical computations, Brons-Voigt and Sorensen [5] systematically determined the possible flow topologies of the steady vortex breakdown in the axisymmetric flow. Their basic idea is to analyze the streamlines of the ordinary differential equations (c.f. the definition of axis-length streamline (2.3) and axis-length trajectory: Definition 2.8 in this paper). For the detail, see Figure 1 and Section 3 in [5]. Our aim here is to show that non-zero swirl component with laminar profile on the axis (with some extra assumptions) creates unsteady flow.

Remark 1.1*.*

These mathematical analysis must be applicable to a study of reduced cardiovascular 1D model [17, Section 10]. If the blood flow is in large and medium sized vessels, the flow is governed by the usual incompressible Navier-Stokes equations. To obtain the reduced model from the Navier-Stokes equations, we need to assume the flow is always unilateral laminar flow, especially, the axis direction of the flow $u_{3}$ is assumed to satisfy

[TABLE]

for some positive constant $a>0$ (see [17, (10.18)]). However, in this setting, it is unclear whether or not such condition (1.4) is always valid. For example, if the flow is not unilateral, containing the reverse flow (possibly, turbulence), then $a$ may become infinity.

Since we would not like to take the boundary layer into account (instead, we focus on behavior of the interior flow), it is still valid to consider a simpler model: the inviscid flow in $\Omega_{\mathcal{R}}$. The incompressible Euler equations (inviscid flow) are expressed as follows:

[TABLE]

where $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$ and $n$ is a unit normal vector on $\partial\Omega_{\mathcal{R}}$. Note that the boundary condition here is not important anymore.

Notations “$\approx$” and “$\lesssim$” are convenient. The notation “$a\approx b$” means there is a positive constant $C>0$ such that

[TABLE]

and “$a\lesssim b$” means that there is a positive constant $C>0$ such that

[TABLE]

In the pulsatile flow case, we consider the following inflow setting:

• •

$U_{in}=U_{s}(r)+U_{o}(r)g(t)$ with rapidly increasing $g$ (in time) and

[TABLE]

Throughout this paper we always assume existence of smooth solutions in $\Omega_{\mathcal{R}}\times[0,\infty)$ (we can regard nonuniqueness, nonexistence and blowup as some kind of “ strong instability”).

Remark 1.2*.*

According to the boundary layer theory, outside the boundary layer the fluid motion is accurately described by the Euler flow. Thus the above simplification seems (more or less) valid. For the recent progress on the mathematical analysis of the boundary layer, see [25].

2. Geometry setting and the main results

To describe the main theorems, we need to give a geometry setting. First we define the particle trajectory.

Definition 2.1**.**

(Particle trajectory $\Phi_{*}$.) For given time-dependent smooth vector field $u=u(x,t)$, the associated Lagrangian flow $\Phi_{*}(t)$ is a solution of the following initial value problem

[TABLE]

Throughout this paper we always assume the vector field $u$ is unilateral, that is, $u_{3}>0$. Also define the axis-length streamline $\tilde{\Phi}(z)$.

Definition 2.2**.**

(Axis-length streamline $\tilde{\Phi}$.) for fixed $t>0$, let $\tilde{\Phi}$ be such that

[TABLE]

with the initial point $\tilde{\Phi}(0)=(x_{1},x_{2},0).$

Later, we use the following axis-length trajectory $\Phi$.

Definition 2.3**.**

(Axis-length trajectory $\Phi$.) Let $Z_{*}(t):=\Phi_{*}(t)\cdot e_{z}$ (with $e_{z}=(0,0,1)$) and since the flow is unilateral, we can define its inverse $Z^{-1}_{{}_{*}t}(z)=t$. In this case we see $\partial_{z}Z^{-1}_{*t}=1/\partial_{t}Z_{*}=1/u_{3}$. Let $\Phi$ be such that $\Phi(z)=\Phi_{*}(x,Z^{-1}_{*t}(z))$.

We restrict our vector field to the axi-symmetric one. Let $e_{r}:=x_{h}/|x_{h}|$, $e_{\theta}:=x_{h}^{\perp}/|x_{h}|$ and $e_{z}=(0,0,1)$ with $x_{h}=(x_{1},x_{2},0)$, $x_{h}^{\perp}=(-x_{2},x_{1},0)$. The vector valued function $u$ can be rewritten as $u=v_{r}e_{r}+v_{\theta}e_{\theta}+v_{z}e_{z}$, where $v_{r}=v_{r}(r,z,t)$, $v_{\theta}=v_{\theta}(r,z,t)$, $v_{z}=v_{z}(r,z,t)$, $v_{r,0}=v_{r}(r,z,0)$, $v_{\theta,0}=v_{\theta}(r,z,0)$ and $v_{z,0}=v_{z}(r,z,0)$ with $r=|x_{h}|$ and $z=x_{3}$.

We define a Lagrangian flow on the meridian plane ($r$-$z$ plane).

Definition 2.4**.**

(Lagrangian flow on the meridian plane.) Let $Z_{*}$ and $R_{*}$ be such that

[TABLE]

and

[TABLE]

with $Z_{*}(t)=Z_{*}(r_{0},z_{0},t)$ and $R_{*}(t)=R_{*}(r_{0},z_{0},t)$.

Note $Z_{*}$ is already defined in Definition 2.8.

Remark 2.5*.*

We can rephrase $Z_{*}$ and $R_{*}$ by using the stream function (see (2.2) in [5] for example).

Remark 2.6*.*

The axisymmetric Euler equations can be expressed as follows:

[TABLE]

In this paper we use (2.7) which is independent of the pressure term.

Remark 2.7*.*

(Axisymmetric axis-length streamline.) For fixed $t>0$, $\tilde{\Phi}(z)$ can be explicitly expressed as

[TABLE]

with $\tilde{R}(z)=\tilde{R}(\tilde{r}_{0},z,t)$, $\tilde{R}(\tilde{r}_{0},0,t)=\tilde{r}_{0}$, $\tilde{\Theta}(z)=\tilde{\Theta}(z,t)$. We easily see

[TABLE]

Since $\partial_{\tilde{r}_{0}}\tilde{R}>0$ by the smoothness, we have its inverse $\tilde{r}_{0}=\tilde{R}^{-1}(r,z,t)$.

Remark 2.8*.*

(Axisymmetric axis-length trajectory.) Also $\Phi$ can be explicitly expressed as

[TABLE]

with $R(z)=R(r_{0},\theta_{0},z_{0},z)$, $\Theta(z)=\Theta(r_{0},\theta_{0},z_{0},z)$, $R(r_{0},\theta_{0},z_{0},z_{0})=r_{0}$ and $\Theta(r_{0},\theta_{0},z_{0},z_{0})=\theta_{0}$. Note that $\tilde{R}(t)|_{t=Z^{-1}_{*t}(z)}=R(z)$.

In order to show that the non-zero swirl component induces the instability, we need to measure appropriately the rate of disturbing laminar profile of the flow. We now give the key definition.

Definition 2.9**.**

(Rate of disturbing laminar profile.) We define “rate of disturbing laminar profile” $L^{0}$, $L^{x}$ and $L^{t}$ as follows: for $(\tilde{r}_{0},z)\in[0,\mathcal{R})\times(0,\ell)$,

[TABLE]

Note that $L^{0}$ and $L^{x}$ do not include any time derivative, while, $L^{t}$ includes it. We can see that if $L^{t}$ is not zero, then the flow cannot be any steady flow.

Remark 2.10*.*

Minumum value of $L^{0}$ is $2$, since $|\partial_{r}\tilde{R}^{-1}|=1/|\partial_{\bar{r}_{0}}\tilde{R}|$.

Remark 2.11*.*

The typical Euler flow $u(x,t)=(0,0,g(t))$, namely, a bunch of stationary straight tubes $\tilde{R}(\tilde{r}_{0},z,t)\equiv\tilde{r}_{0}$ is the typical laminar flow. In this case

[TABLE]

for any $g$.

Remark 2.12*.*

Streamlines of outside bubbles which are attaching on the axis (see $B_{1}$, $B_{2}$, $B_{3}$, $C_{1}$, $C_{2}$, $D$, $E$, $F$ in Figure 1 in [5]) may create large $L^{x}$ and/or $L^{0}$. Moreover, at a hyperbolic saddle (or stagnation point), they may be infinity.

Now we give the main theorems.

Theorem 2.13**.**

(Pulsatile flow case.) Let $\tilde{r}_{0}(t)$ and $z(t)$ be another expression of particle trajectory such that

[TABLE]

and let $D_{\gamma}$ be a non-zero swirl region such that $D_{\gamma}:=\{x:|u_{0}(x)\cdot e_{\theta}|>\gamma\}$. Assume $D_{\gamma}\not=\emptyset$ for the corresponding initial data, and assume there is a unique smooth solution to the Euler equations (1.5) in $t\in[0,T)$. Then there is a smooth function $g$, $f$ and discrete-time $\{t_{j}\}_{j}$ such that

[TABLE]

and the following case must occur:

[TABLE]

Theorem 2.14**.**

(Vortex breakdown case.) Assume there is a unique smooth solution to the Euler equations (1.5). For any $\epsilon_{1}>0$ and $\epsilon_{2}>0$ ($\epsilon_{1}\ll\epsilon_{2}$), there is $\delta>0$ such that if

[TABLE]

for some $z$, then there is no stationary Euler flow near the initial time, that is,

[TABLE]

Note that $\delta$ becomes smaller if $\epsilon_{1}$ becomes smaller, to the contrary, $\delta$ becomes larger if $\epsilon_{2}$ becomes smaller.

Remark 2.15*.*

Roughly saying, $\partial_{r}v_{\theta,0}$ on the axis should be corresponding to rotating top and bottom boundaries: $\Omega_{1}$ in [5, (2.1)].

3. Explicit formulas of $v_{r}$, $v_{z}$, $v_{\theta}$ and $D_{t}|u|$.

Before we prove the main theorems, in this section, we give explicit formulas of $v_{r}$, $v_{z}$, $v_{\theta}$ and $\partial_{t}|u(\Phi(x,t),t)|$ by using $\tilde{R}$ and $U_{in}$. First we construct $v_{z}$ and $v_{r}$. To do so, we define the cross section of the stream-tube (annulus). Let $B_{0}(\tilde{r}_{0})=\{x\in\mathbb{R}^{3}:|x_{h}|<\bar{r}_{0},\ x_{3}=0\}$ and let

[TABLE]

We see that its measure is

[TABLE]

Definition 3.1**.**

(Inflow propagation.) Let $\rho$ be such that

[TABLE]

We see that

[TABLE]

Remark 3.2*.*

Since $\tilde{R}(0,z,t)\equiv 0$, we see that

[TABLE]

on the axis.

Since

[TABLE]

by divergence-free and Gauss’ divergence theorem, we can figure out $v_{z}$ by using the inflow propagation $\rho$,

[TABLE]

Thus we have the following proposition.

Proposition 3.3**.**

We have the following formulas of $v_{z}$ and $v_{r}$:

[TABLE]

and

[TABLE]

Remark 3.4*.*

Recall that $e_{\theta}=(-\sin\Theta(z),\cos\Theta(z),0)$ and $e_{r}=(\cos\Theta(z),\sin\Theta(z),0)$. We also have the following explicit formulas of $\Theta^{\prime}$ and $R^{\prime}$ ($\Theta$ and $R$ already appeared in the axis-length trajectory. See Remark 2.8):

[TABLE]

Moreover, along the axis,

[TABLE]

Remark 3.5*.*

For the vortex breakdown case, we have the following estimates on $\Theta^{\prime}$, $\Theta^{\prime\prime}$, $\Theta^{\prime\prime\prime}$, $R^{\prime}$, $R^{\prime\prime}$ and $R^{\prime\prime\prime}$:

[TABLE]

Let $r=R(r_{0},z_{0},t)$. Moreover we have that

[TABLE]

where $C(\delta)$ is a positive constant depending only on $\delta$ (if $\delta\to 0$, then $C(\delta)\to 0$).

Next we construct $v_{\theta}$. By (2.7) we see that

[TABLE]

Applying the Gronwall equality, we see

[TABLE]

and then

[TABLE]

and

[TABLE]

with $r_{0}=R^{-1}_{*}(r,z,t)$ and $z_{0}=Z^{-1}_{*}(r,z,t)$ (distinguish with $Z^{-1}_{*t}$). In order to estimate spatial derivatives on $v_{\theta}$, first we consider a non-incompressible 2D-flow composed by $R_{*}$ and $Z_{*}$. Let us denote $\phi_{2D}=\phi_{2D}(t)=(R_{*}(t),Z_{*}(t))$, $\phi_{2D}^{-1}=(R^{-1}_{*},Z^{-1}_{*})$ and $D\phi_{2D}$ be its Lagrangian deformation:

[TABLE]

We see $\det(D\phi_{2D})=\partial_{r_{0}}R_{*}\partial_{z_{0}}Z_{*}-\partial_{z_{0}}R_{*}\partial_{r_{0}}Z_{*}$ and thus we have

[TABLE]

A direct calculation with (2.9), (2.4) and (2.5) yields

[TABLE]

Thus

[TABLE]

Since $|v_{r}/r|\approx|\partial_{r}v_{r}|\lesssim 1$ near the axis, we have

[TABLE]

Since we have already controlled $\det D\phi_{2D}$, it suffices to estimate $\partial_{r_{0}}R_{*}$, $\partial_{r_{0}}Z_{*}$, $\partial_{z_{0}}R_{*}$ and $\partial_{z_{0}}Z_{*}$ respectively. From Proposition 3.3, We see the following:

[TABLE]

[TABLE]

Then we can construct a Gronwall’s inequality of $|\partial_{z_{0}}Z|+|\partial_{z_{0}}R|$, that is

[TABLE]

where $C$ is depending on $L^{0}$, $L^{x}$ and $L^{t}$. Again, we just take integration in time, we have

[TABLE]

and this is the explicit formula of $\partial_{z_{0}}Z_{*}$. In a small time interval, we have $\partial_{z_{0}}Z_{*}\approx 1$ and by the same calculation, $\partial_{z_{0}}R_{*}\approx 0$, $\partial_{r_{0}}Z_{*}\approx 0$ and $\partial_{r_{0}}R_{*}\approx 1$. By the above estimates, we can estimate derivatives on $v_{\theta}$.

Now we figure out the explicit formula of $\partial_{t}|u(\Phi_{*},t)|$. Recall that the particle trajectory $\Phi_{*}(x,t)$ satisfies

[TABLE]

Then, by $u=v_{r}e_{r}+v_{\theta}e_{\theta}+v_{z}e_{z}$ with $e_{\theta}=(-\sin\Theta_{*}(t),\cos\Theta_{*}(t),0)$ and $e_{r}=(\cos\Theta_{*}(t),\sin\Theta_{*}(t),0)$, we see that

[TABLE]

along the trajectory. In fact, since

[TABLE]

and

[TABLE]

we see $\partial_{t}\Theta_{*}=v_{\theta}/R_{*}$. We multiply $u=v_{r}e_{r}+v_{\theta}e_{\theta}+v_{z}e_{z}$ to

[TABLE]

then we have (3.3). Thus we have the following explicit formula:

[TABLE]

Combining the Lagrangian deformation on $R_{*}$ and $Z_{*}$, we also have the explicit formulas of $\partial_{z}\partial_{t}|u(\Phi(x,t),t)|$ and $\partial_{r}\partial_{t}|u(\Phi(x,t),t)|$.

4. Estimates on curvature and torsion along particle trajectory.

Let us define the arc-length trajectory $\phi(s):=\Phi(z(s))$ with smooth function $z(s)$ such that $z^{\prime}(s)=|(\partial_{z}\Phi)(z(s))|^{-1}$. We also define the unit tangent vector $\tau$ as

[TABLE]

the unit curvature vector $n$ as $\kappa n=\partial_{s}\tau$ with a curvature function $\kappa(s)>0$, the unit torsion vector $b$ as : $b(s):=\pm\tau(s)\times n(s)$ ($\times$ is an exterior product) with a torsion function to be positive $T(s)>0$ (once we restrict $T$ to be positive, then the direction of $b$ can be uniquely determined). From $\kappa n$, we can figure out the curvature constant $\kappa:=|\partial_{s}^{2}\phi|$ and corresponding unit normal vector: $n=\partial_{s}^{2}\phi/|\partial_{s}^{2}\phi|$. Thus, theoretically, we can explicitly figure out $\kappa$ and $\partial_{s}\kappa$ by using $R$ and $\Theta$. First, $\tau$ and $\kappa n$ are expressed as

[TABLE]

Then direct calculations yield

[TABLE]

Therefore

[TABLE]

From the above explicit formulas of $\kappa$, we can figure out the explicit formula of $\partial_{s}\kappa$ (omit its detail) which will be important in the proof of the main theorems.

Remark 4.1*.*

• •

(The vortex breakdown case.) If $\Theta^{\prime}$ is larger than the other terms, we have

[TABLE]

which is a controllable term.

• •

(Instantaneous blowup case in Appendix.) If $\Theta^{\prime\prime}$ is larger than $\Theta^{\prime}$, and $\Theta^{\prime\prime\prime}$ is larger than $\Theta^{\prime\prime}$, then we have

[TABLE]

which will be the dominant term.

5. Rewrite Euler equations by using curvature and torsion

In this section we rewrite the Euler equations by using curvature and torsion. The basic idea comes from Chan-Czubak-Y [9, Section 2.5], more originally, see Ma-Wang [24, (3.7)]. They considered 2D separation phenomena using elementary differential geometry. The key idea here is “local pressure estimate” on a normal coordinate in $\bar{\theta}$, $\bar{r}$ and $\bar{z}$ valuables. Two derivatives to the scalar function $p$ on the normal coordinate is commutative, namely, $\partial_{\bar{r}}\partial_{\bar{\theta}}p(\bar{\theta},\bar{r},\bar{z})-\partial_{\bar{\theta}}\partial_{\bar{r}}p(\bar{\theta},\bar{r},\bar{z})=0$. This fundamental observation is the key to extract the local property of the pressure.

Remark 5.1*.*

It should be noticed that Enciso and Peralta-Salas [15] considered the existence of Beltrami fields $u$ with a nonconstant proportionality factor $f$:

[TABLE]

It is well known that a Beltrami field is also a solution of the steady Euler equation in $\mathbb{R}^{3}$. They showed that for a generic function $f$, the only vector field $u$ satisfying (5.1) is the trivial one $u\equiv 0$. See (2.12), (3.4) and (3.6) in [15] for the specific condition on $f$. Note that $g_{ij}$ (induced metric of the level set of $f$) is the fundamental component of the condition. It would be also interesting to consider whether we can apply their method to our unsteady flow problem, and compare with our method.

For any point $x\in\mathbb{R}^{3}$ near the arc-length trajectory $\phi$ is uniquely expressed as $x=\phi(\bar{\theta})+\bar{r}n(\bar{\theta})+\bar{z}b(\bar{\theta})$ with $(\bar{\theta},\bar{r},\bar{z})\in\mathbb{R}^{3}$ (the meaning of the parameters $s$ and $\bar{\theta}$ are the same along the arc-length trajectory). By the Frenet-Serret formulas, we have that

[TABLE]

This means that

[TABLE]

Remark 5.2*.*

For any smooth scalar function $f$, we have

[TABLE]

$\nabla f$ itself is essentially independent of any coordinates, thus we can regard a partial derivative as the corresponding vector.

Then we have the following inverse matrix:

[TABLE]

Therefore we have the following orthonormal moving frame: $\partial_{\bar{r}}=n$, $\partial_{\bar{z}}=b$ and

[TABLE]

Lemma 5.3**.**

We see $-\nabla p\cdot\tau=D_{t}|u|:=\partial_{t}|u(\Phi_{*}(x,t),t)|$ along the trajectory.

Proof.

Let us define a unit tangent vector $\tilde{\tau}$ (in time $t$) as follows:

[TABLE]

Note that there is a re-parametrize factor $s(t)$ such that

[TABLE]

Since $u\cdot\partial_{s}\tau=0$, we see that

[TABLE]

By the above calculation we have

[TABLE]

∎

We now rewrite the Euler equations by using curvature and torsion.

Lemma 5.4**.**

Along the arc-length trajectory, we have

[TABLE]

and

[TABLE]

Proof.

Let us re-define $\phi(s)=\Phi_{*}(x,t(s))$ with smooth function $t(s)$ satisfying $\partial_{s}t=|u|^{-1}$. We see that

[TABLE]

By the unit normal vector with the curvature constant, we see

[TABLE]

Recall the Euler equation: $\partial_{t}^{2}\Phi_{*}=-\nabla p$. Then we have

[TABLE]

Note that $\partial_{s}^{2}t$ is unknown, so we now figure out it by Lemma 5.3 and the above third equality:

[TABLE]

Along the arc-length trajectory, we have (recall $\partial_{\bar{\theta}}=\partial_{s}$)

[TABLE]

Since $\nabla p\cdot b=\partial_{\bar{z}}p\equiv 0$ along the trajectory, then

[TABLE]

By Lemma 5.3 along the arc-length trajectory $\phi$, we have

[TABLE]

and

[TABLE]

∎

6. Proof of the main theorem (the pulsatile flow case).

To prove the main theorem, it is enough to show the following lemma:

Lemma 6.1**.**

Let $t_{j}>0$ ($j=1,2,\cdots$) be fixed. For any $x\in\Phi(D_{\gamma},t_{j})$, there is $\beta>0$ such that $\beta\lesssim u(x,t_{j})\cdot e_{\theta}\lesssim\beta^{-1}$, $x\cdot e_{r}>2\beta$ and $|L^{0}(\tilde{r}_{0}(t_{j}),z(t_{j}),t_{j})|+|L^{x}(\tilde{r}_{0}(t_{j}),z(t_{j}),t_{j})|\leq 1/(2\beta)$. For any $\epsilon>0$, then there is $\delta>0$ such that for any small time interval $I$ with initial time $t_{j}$, at least either of the following four cases must happen:

• •

$L^{x}(\tilde{r}_{0}(t),z(t),t),L^{0}(\tilde{r}_{0}(t),z(t),t)>1/\beta$,

• •

$L^{t}(\tilde{r}_{0}(t),z(t),t)\gtrsim 1/\epsilon$,

• •

$|\Phi_{*}(x,t)\cdot e_{r}|<\beta$,

• •

$\tilde{r}_{0}(t)<\beta$,

for some $t\in I$, with any inflow $g(t)$ satisfying

[TABLE]

where $\tilde{r}_{0}(t)$ and $z(t)$ are determined by $\tilde{\Phi}(\tilde{r}_{0}(t),z(t),t)=\Phi_{*}(x,t)$ (in this case $\Phi_{*}(x,t_{j})=x$). Since $\Phi(D_{\gamma},t)$ is always compact and the solution is always smooth, $\delta$ can be independent of the choice of $x\in\Phi(D_{\gamma},t)$.

Since the time interval $I$ is arbitrary, we see that $L^{0}$ or $L^{x}$ or $\Phi_{*}\cdot e_{r}$ or $\tilde{r}_{0}$ is not continuous at $t_{j}$, or $L^{t}\gtrsim 1/\epsilon$ for some $t\in I$. The discontinuity contradicts the smoothness property, thus

[TABLE]

only occurs.

Proof.

In what follows, we prove the above lemma. For any small time interval $I$, assume that the axisymmetric smooth Euler flow satisfies the following conditions:

• •

$L^{x}(\tilde{r}_{0}(t),z(t),t),L^{0}(\tilde{r}_{0}(t),z(t),t)\leq 1/\beta$ and $L^{t}(\tilde{r}_{0}(t),z(t),t)\lesssim 1/\epsilon$

• •

$|\Phi_{*}(x,t)\cdot e_{r}|\geq\beta$ and $\tilde{r}_{0}(t)\geq\beta$

for any $t\in I$, where $(\tilde{r}_{0}(t),z(t))=(\tilde{\Phi}^{-1}\circ\Phi_{*})(x,t)$, and we employ a contradiction argument. By the second assumption: $|\Phi_{*}(x,t)\cdot e_{r}|\geq\beta$, $R$ satisfies the following:

[TABLE]

By the explicit formulas in Section 3, we have the following lemma (these are direct calculations, thus we omit its proof).

Lemma 6.2**.**

For $t=Z^{-1}_{*t}\in I$, we have the following estimates along the axis-length trajectory:

[TABLE]

Moreover, we have

[TABLE]

[TABLE]

[TABLE]

By the above lemma with Remark 3.4, we immediately have $|\Theta^{\prime\prime}|\lesssim 1/\epsilon$ and $\Theta^{\prime\prime\prime}\approx 1/\delta$ (for sufficiently small $\delta$ compare with $\epsilon$) in $t\in I$.

Lemma 6.3**.**

For any $\epsilon>0$, we have

[TABLE]

for sufficiently small $\delta>0$.

Proof.

From Section 4, we see

[TABLE]

in $t\in I$. “remainder” is small compare with the main terms provided by small $\epsilon,\delta>0$. Thus we immediately obtain $|u|^{2}|\partial_{s}\kappa|\gg\kappa D_{t}|u|$ for sufficiently small $\delta>0$. ∎

By Lemma 5.4, we see

[TABLE]

and it is in contradiction, since $\partial_{s}\kappa$ is sufficiently large compare with the other terms.

∎

7. Proof of the main theorem (the vortex breakdown case)

Assume

[TABLE]

and employ a contradiction argument. Recall that, by Remark 3.4, $|\Theta^{\prime}|\approx 1/\epsilon_{1}$, $|\Theta^{\prime\prime}|\lesssim 1/\delta$ and $|\Theta^{\prime\prime\prime}|\lesssim 1/\delta$ in some small time interval. From Section 4, near the axis, we have ($r=R(r_{0},z_{0},t)$)

[TABLE]

Thus near the axis, we have

[TABLE]

Since $\partial_{\bar{r}}D_{t}|u|=3\kappa D_{t}|u|+\partial_{s}\kappa|u|^{2}$ and $\kappa=\partial_{s}\kappa=0$ along the axis, we have $\partial_{\bar{r}}D_{t}|u|=0$ along the axis. By the mean value theorem, we have

[TABLE]

along the axis (note that $\partial_{\bar{r}}\to\partial_{r}$ if the corresponding point approaches the axis). However it is in contradiction, since the right hand side is large, while the left hand side is not large.

8. Appendix: Instantaneous blow-up

In this section we show instantaneous blow-up. Let us consider the Euler equations in the whole space $\mathbb{R}^{3}$:

[TABLE]

The first existence results for (8) were proved in the framework of Hölder spaces by Gyunter [18], Lichtenstein [23] and Wolibner [33]. More refined results were obtained subsequently by Kato [22], Swann [31], Bardos and Frisch [1], Ebin [13], Chemin [10], Constantin [12] and Majda and Bertozzi [26] among others. On the other hand, Bardos and Titi [2] found examples of solutions in Hölder spaces $C^{\alpha}$ and the Zygmund space $B^{1}_{\infty,\infty}$ which exhibit an instantaneous loss of smoothness in the spatial variable for any $0<\alpha<1$ (see also [11, 27]). Similar examples in logarithmic Lipschitz spaces $\mathrm{logLip}^{\alpha}$ were given by the authors in [27]. In another direction Cheskidov and Shvydkoy [11] constructed periodic solutions that are discontinuous in time at $t=0$ in the Besov spaces $B^{s}_{p,\infty}$ where $s>0$ and $2<p\leq\infty$. After their work, in a series of papers Bourgain and Li [3, 4] constructed smooth solutions which exhibit instantaneous blowup in borderline spaces such as $W^{n/p+1,p}$ for any $1\leq p<\infty$ and $B^{n/p+1}_{p,q}$ for any $1\leq p<\infty$ and $1<q\leq\infty$ as well as in the standard spaces $C^{k}$ and $C^{k-1,1}$ for any integer $k\geq 1$; see also Elgindi and Masmoudi [14] and [28]. As observed in [4] the cases $C^{k}$ and $C^{k-1,1}$ are particularly intriguing in view of the classical existence and uniqueness results mentioned above.

In [29] (see also [28]), they revisited the picture of local well-posedness in the sense of Hadamard for the Euler equations in Hölder spaces. They present a simple example based on a DiPerna-Majda type shear flow which shows that in general the data-to-solution map of (1.5) is not continuous into the space $L^{\infty}([0,T),C^{1,\alpha})$ for any $0<\alpha<1$. On the other hand, continuity of this map is restored (in the strong sense) if the Cauchy problem is restricted to the so called little Hölder space $c^{1,\alpha}$.

Remark 8.1*.*

For $u_{0}\in c^{2,\alpha}$, we can also show that there exists a unique solution $u$ which is in (see [26, Section 4.4] and [29] for example)

[TABLE]

Therefore, if the solution $u$ is axi-symmetric, then the corresponding components $v_{r}$ and $v_{z}$ satisfy

[TABLE]

[TABLE]

and

[TABLE]

In this appendix, we show that even if the solution to the Euler equations is wellposed, such as, in $c^{2,\alpha}$, it may blows up (in some norm) instantaneously.

Theorem 8.2**.**

There is an axisymmetric initial data $u_{0}\in c^{2,\alpha}(\mathbb{R}^{3})$ such that the corresponding unique solution $u$ is not in $C^{1}([0,T]:C^{2}(\mathbb{R}^{3}))$ for any $T>0$. More precisely, we choose an axisymmetric initial data as the following: there is sufficiently small $\beta>0$ such that for any $\{\epsilon_{j}\}_{j}$ $(\epsilon_{j}\to 0)$ and $\{z_{j}\}_{j}$ $(z_{j}\to z)$, there is $\{\delta_{j}\}_{j}$ ($\delta_{j}\to 0$ as $j\to\infty$) such that

[TABLE]

Then we have

[TABLE]

Proof.

The proof is similar to the “vortex breakdown” case. By Remark 3.4, we can figure out that $\Theta^{\prime\prime\prime}|_{r=0}$ is large for some small time interval. The same argument holds true that $\Theta^{\prime}|_{r=0}$ and $\Theta^{\prime\prime}|_{r=0}$ are not large. Due to Remark 8.1, we see $R^{\prime}$, $R^{\prime\prime}$ and $R^{\prime\prime\prime}$ are all small. Let $r=R(r_{0},z_{0},t)$. By Lemma 5.3, near the axis, we see

[TABLE]

Thus near the axis, we have

[TABLE]

By the same argument as in the previous section, we have

[TABLE]

along the axis. This estimate tells us that $|\partial_{t}\partial_{r}^{2}v_{r}(0,z_{j},0)|\approx 1/\delta_{j}$.

∎

Acknowledgments. The author would like to thank Professor Norikazu Saito for letting me know the book [17], Professor Hiroshi Suito for letting me know “Womersley number”, and also Doctor Kento Yamada for letting me know the articles [5, 16, 19, 30]. The author was partially supported by Grant-in-Aid for Young Scientists A (17H04825), Japan Society for the Promotion of Science (JSPS), and also partially supported by JST CREST.