A refined uniqueness result of Leray’s problem in an infinite-long pipe with the Navier-slip boundary

Our main result of the current paper states:

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We review some works related to the solvability of Leray’s problem. Originally, the study of Leray’s problem focuses on the existence, regularity, uniqueness, and asymptotic behavior of Systems (1.1)–(1.3) subject to no-slip boundary condition

as described by Ladyzhenskaya of her works [12, p. 77] and [13, p. 551]. Later in[14], Ladyzhenskaya and Solonnikov provided a comprehensive analysis of the existence, uniqueness, and asymptotic behavior of weak solutions to the Leray’s problem with the help of small-flux conditions. Amick [2, 3] made a significant initial contribution to solving Leray’s problem by reducing the issue to a variational problem, although the uniqueness of the solutions remains an open question. Further details on the decay, far-field asymptotic behavior, and well-posedness of solutions to Leray’s problem under the no-slip boundary condition are available in [4, 10, 22]. For a systematic exploration and analysis of this problem, readers may consult Chapter XIII of [9]. In recent research, Wang and Xie investigated the existence, uniqueness, and uniform structural stability of Poiseuille flows in a pipe, addressing the 3D axially symmetric inhomogeneous Navier-Stokes equations. See [23, 24] and some of their subsequent works.

The Naiver-slip boundary condition (1.2), which was initialed by Navier [20], allows fluid to slip along the boundary with a scale being proportional to its stress tensor. Due to the complexity of the Navier-slip boundary condition, new mathematical strategies are needed to study problems related to it. More and more studies on problems with Navier-slip boundary conditions emerged these years, see [1] and references therein. In [18, 19, 11], authors studied the solvability of the steady Navier-Stokes equations with the total Navier-slip condition ($\alpha=0$). Wang and Xie [25] proved the uniqueness and uniform structural stability of the Poiseuille flow in an infinite straight long pipe with the Navier-slip boundary condition. Li, Pan and Yang [15] gave the characterization of bounded smooth solutions to the axisymmetric Navier-Stokes equations with the total Navier-slip boundary condition and the Navier-Hodge-Lions boundary condition in the infinitely long cylinder.

In the recent paper [17], authors proved the generalized Leray’s problem of stationary 3D Navier-Stokes equations (i.e. existence, uniqueness, together with the higher-order regularity and asymptotic behavior of the stationary 3D Navier-Stokes flow in a distorted infinite pipe with Navier-slip boundary condition) with the flux of flow being small enough, depending on the shape of the pipe and the friction ratio $\alpha$ of the Naiver-slip boundary condition. Meanwhile, the related 2D problem, which is studied in [16], presented an exact formula that measures the smallness of $\Phi$ with respect to $\alpha$ when considering the existence, regularity together with the asymptotic behavior problem:

where $C>0$ is a constant independent with $\alpha$. Estimate (1.8) shows when $\alpha$ is small enough, the smallness restriction of the flux tends weaker, and for the total Navier-slip condition ($\alpha=0$), the solution can be arbitrarily large.

The cross section $\Sigma$ degenerates to an interval in the 2D case, and we assume $\Sigma=]0,1[$ without loss of generality. Direct calculation shows

solves (1.6) (with $\boldsymbol{e_{3}}$ replaced by $\boldsymbol{e_{2}}$). It is clear that

For the 3D case, the gradient bound (1.10) also holds when the cross-section $\Sigma$ is a unit disk. Indeed,

which could be considered as a generalization of the Hagen-Poiseuille flow under the no-slip boundary condition ($\alpha=+\infty$), solves the problem (1.6) provided $\Sigma=\{x_{h}\in\mathbb{R}^{2}:\,|x_{h}|<1\}$. In this case, clearly

follows from direct calculations.

However, in general 3D cases, the topology of the cross section may rather complicated and one can hardly represent the formula of the solution as (1.9) or (1.11). Whether the gradient bound (1.10) still holds for 3D Poiseuille flow in a sufficiently regular pipe is an interesting question. In this paper, we give a positive answer to this question. With the help of this, we prove the Poiseuille flow given in (1.6) is the unique solution to the problem (1.1)–(1.2)–(1.3) provided $\Phi$ is no larger than a constant that is independent with the friction ratio $\alpha$.

Now we outline the idea of proving the main result of the current paper. First and the most important is to prove the upper bound (1.12) holds for Poiseuille flow in the general smooth pipe $\mathcal{D}$. To do this, we first rescale (1.6) to get

To prove $\|\nabla_{h}\varphi_{\alpha}\|_{L^{2}(\Sigma)}\lesssim\alpha$ for $\alpha$ being sufficiently small, we expand $\varphi_{\alpha}$ as:

where $\phi_{0}=|\Sigma|^{-1}$, and $\phi_{n}$ satisfies

The key observation of this decomposition is the sequence (1.14) is convergent when $\alpha$ is small, and the zero-order term $\phi_{0}$ is a constant which makes no sense after taking the gradient. Meanwhile, the uniform boundedness of $\|\nabla_{h}\varphi_{\alpha}\|_{L^{2}(\Sigma)}$ for sufficient large $\alpha$ is concluded by an energy estimate of system (1.13). After showing the estimate

the uniqueness is proved by deriving a Saint-Venant type estimate that was first introduced by Ladyzhenskaya and Solonnikov in [14].

The rest of this paper is organized as follows. Section 2 contains the preliminary work of the proof, in which some useful lemmas are presented. Section 3 is devoted to deriving the key estimate of the Poiseuille flow (1.6). In Section 4, we finish the proof of the uniqueness of the solution, and prove the critical flux $\Phi_{0}$ in no smaller than $\frac{\pi}{16}$ if the cross section of the pipe is a unit disk.

We finish this section with a list of notations that will appear throughout the paper.

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  $C_{a,b,...}$ denotes a positive constant depending on $a,\,b,\,...$ which may be different from line to line. $A\lesssim B$ means $A\leq CB$, and $A\lesssim_{a,b,...}B$ denotes $A\leq C_{a,b,...}B$. Meanwhile, $A\simeq B$ means both $A\lesssim B$ and $B\lesssim A$, in other words, $A$ and $B$ can be controlled by each other.

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  In the standard Euclidean coordinates framework $\boldsymbol{e_{1}},\ \boldsymbol{e_{2}}$ and $\boldsymbol{e_{3}}$, for a 3D vector field $\boldsymbol{w}$, we denote $\boldsymbol{w}=w_{1}\boldsymbol{e_{1}}+w_{2}\boldsymbol{e_{2}}+w_{3}\boldsymbol{e_{3}}$.

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  We set $\boldsymbol{n}=(n_{1},n_{2},0)$ is the unit outer normal vector on $\partial\mathcal{D}=\partial\Sigma\times\mathbb{R}$, where $\bar{\boldsymbol{n}}=(n_{1},n_{2})$ is the unit outer normal vector on $\partial\Sigma$. Meanwhile, $\boldsymbol{\tau}=(\tau_{1},\tau_{2},0)$ is the unit tangent vector on $\partial\Sigma\times\mathbb{R}$ that is orthogonal to $\boldsymbol{e_{3}}$, and $(\boldsymbol{\tau},\boldsymbol{e_{3}},\boldsymbol{n})$ form a right-hand coordinate system. See Figure 1.

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  $\mathfrak{H}$ stands for a multi-index such that $\mathfrak{H}=(h_{1},h_{2},h_{3})$ where $h_{1},h_{2},h_{3}\in\mathbb{N}\cup\{0\}$ and $|\mathfrak{H}|=h_{1}+h_{2}+h_{3}$, $\nabla^{\mathfrak{H}}=\partial_{x_{1}}^{h_{1}}\partial_{x_{2}}^{h_{2}}\partial_{x_{3}}^{h_{3}}$.

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  We use standard notations for Lebesgue and Sobolev functional spaces in $\mathbb{R}^{3}$: For $1\leq p\leq\infty$ and $k\in\mathbb{N}$, $L^{p}$ denotes the Lebesgue space with norm

  $$\|f\|_{L^{p}}:=\left\{\begin{aligned} &\left(\int_{\mathbb{R}^{3}}|f(x)|^{p}dx\right)^{1/p},\quad 1\leq p<\infty,\\ &\mathop{esssup}_{x\in\mathbb{R}^{3}}|f(x)|,\quad\quad\quad\quad p=\infty.\\ \end{aligned}\right.$$

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  $H^{m}$ denotes the $L^{2}$-based Sobolev space with its norm

  $$\begin{split}\|f\|_{H^{m}}:=&\sum_{0\leq|\mathfrak{H}|\leq m}\|\nabla^{\mathfrak{H}}f\|_{L^{2}}\,.\\ \end{split}$$

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  For any $\zeta>1$, we define

  $$\mathcal{D}_{\zeta}:=\left\{x\in\mathcal{D}:-\zeta<x_{3}<\zeta\right\}$$

  the truncated pipe with the length equals to $2\zeta$. Meanwhile, $\Omega^{\pm}_{\zeta}$ are defined by

  $$\Omega_{\zeta}^{+}:=\left(\mathcal{D}_{\zeta}-\mathcal{D}_{\zeta-1}\right)\cap\left\{x\in\mathcal{D}:x_{3}>0\right\},\quad\Omega_{\zeta}^{-}:=\left(\mathcal{D}_{\zeta}-\mathcal{D}_{\zeta-1}\right)\cap\left\{x\in\mathcal{D}:x_{3}<0\right\}\,,$$

  respectively.

Below, the proposition shows an $L^{2}$ estimate related to the pressure in the truncated pipe $\Omega_{Z}^{+}$ or $\Omega_{Z}^{-}$ could be bounded by $L^{2}$ norm of $\nabla\boldsymbol{u}$.

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Noticing $\boldsymbol{P}_{\Phi}$ is an exact solution of (1.1), by subtracting the equation of $\boldsymbol{P}_{\Phi}$ from the equation of ${\boldsymbol{v}}$, one finds

Multiplying $\boldsymbol{w}$ on both sides of (4.29), and integrating on $\mathcal{D}_{\zeta}$, one derives

Using the divergence-free property and the Navier-slip boundary condition of $\boldsymbol{v}$ and ${\boldsymbol{P}_{\Phi}}$, one deduces

Here $\boldsymbol{n}=(n_{1},n_{2},n_{3})$ is the unit outer normal vector on $\partial\mathcal{D}$.

On the other hand, using integration by parts, we may derive alternatively:

where