Nonlinear stability of the two-jet Kolmogorov type flow on the unit sphere under a perturbation with nondissipative part

Let $S^{2}$ be the two-dimensional (2D) unit sphere in $\mathbb{R}^{3}$. We consider the Navier–Stokes equations

Here $\mathbf{u}$ is the tangential velocity field of a fluid, $p$ is the pressure, and $\mathbf{f}$ is a given external force. Also, $\nu>0$ is the viscosity coefficient, $\nabla_{\mathbf{u}}\mathbf{u}$ is the covariant derivative of $\mathbf{u}$ along itself, $\Delta_{H}$ is the Hodge Laplacian via identification of vector fields and one-forms, and $\nabla$ and $\mathrm{div}$ are the gradient and the divergence on $S^{2}$. Note that the viscous term in (1.1) contains the zeroth order term since it is twice of the deformation tensor $\mathrm{Def}\,\mathbf{u}$:

Here $\mathrm{Ric}\equiv 1$ is the Ricci curvature of $S^{2}$. There are many works on the Navier–Stokes equations on spheres and manifolds with this kind of viscous term (see e.g. [42, 35, 32, 30, 10, 19, 6, 8, 38, 39, 34, 20, 36, 37]). Also, several authors studied the Navier–Stokes equations on manifolds with viscous term replaced by $\nu\Delta_{H}\mathbf{u}$ in analogy of the flat domain case (see e.g. [16, 15, 45, 5, 17, 51, 24, 40]). We refer to [13, 1, 12, 43, 7] for the identity (1.2) and the choice of the viscous term in the Navier–Stokes equations on manifolds. A crucial difference due to the choice of the viscous term is the presence of nondissipative vector fields. The viscosity always works if one chooses $\nu\Delta_{H}\mathbf{u}$ since $S^{2}$ does not admit nontrivial harmonic forms. On the other hand, if one takes $\nu(\Delta_{H}\mathbf{u}+2\mathbf{u})$, then the viscosity does not work for tangential vector fields of the form $X(x)=\mathbf{a}\times x$, $x\in S^{2}$ with any $\mathbf{a}\in\mathbb{R}^{3}$. Note that this $X$ is a Killing vector field on $S^{2}$, and in general Killing vector fields are nondissipative stationary solutions to the Navier–Stokes equations on a manifold with viscous term given by (1.2). In fact, the problem of the existence and uniqueness of solutions is not so affected by the choice of the viscous term, since the difference of the above two viscous terms is linear and of lower order. However, as we will see below, the long-time behavior of a solution can be different by the choice of the viscous term due to the effect of a nondissipative part of a solution coming from the convection term.

Since $S^{2}$ is 2D and simply connected, the Navier–Stokes equations (1.1) are equivalent to the following equation for the vorticity $\omega=\mathrm{rot}\,\mathbf{u}$:

Here $f=\mathrm{rot}\,\mathbf{f}$ is an external force, $\mathbf{n}_{S^{2}}$ is the unit outward normal vector field of $S^{2}$, and $\Delta$ is Laplace–Beltrami operator on $S^{2}$ with inverse $\Delta^{-1}$ in $L_{0}^{2}(S^{2})$, the space of $L^{2}$ functions on $S^{2}$ with vanishing mean. Also, $\mathbf{a}\cdot\mathbf{b}$ and $\mathbf{a}\times\mathbf{b}$ stand for the inner and vector products of $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$. We refer to [31] for the derivation of (1.3) from (1.1).

For $n\in\mathbb{Z}_{\geq 0}$ and $|m|\leq n$, let $Y_{n}^{m}$ be the spherical harmonics and $\lambda_{n}=n(n+1)$ be the corresponding eigenvalue of $-\Delta$ (see Section 2 for details). Then, for $n\in\mathbb{N}$ and $a\in\mathbb{R}$, the vorticity equation (1.3) with external force $f=a\nu(\lambda_{n}-2)Y_{n}^{0}$ has a stationary solution with corresponding velocity field

where $x(\theta,\varphi)$ is the parametrization of $S^{2}$ by the colatitude $\theta$ and the longitude $\varphi$. The flow (1.4) can be seen as a spherical version of the well-known plane Kolmogorov flow, which is a stationary solution to the Navier–Stokes equations in a 2D flat torus (see e.g. [29, 18, 27, 33, 28] for the study of the stability of the plane Kolmogorov flow). In [17], the flow (1.4) is called the generalized Kolmogorov flow. Also, it is called an $n$-jet zonal flow in [38, 39]. We call (1.4) the $n$-jet Kolmogorov type flow in order to emphasize both the similarity to the plane Kolmogorov flow and the number of jets.

In this paper we focus on the case $n=2$ and consider the stability of the two-jet Kolmogorov type flow for the vorticity equation (1.3) (see Remark 1.3 for the one-jet case). Our particular interest is in studying the effect of a nondissipative part of a perturbation in the stability problem. Since $-\Delta Y_{1}^{m}=2Y_{1}^{m}$ for $|m|=0,1$, the viscosity does not work for functions in $\mathrm{span}\{Y_{1}^{0},Y_{1}^{\pm 1}\}$, for which the corresponding velocity fields are Killing vector fields on $S^{2}$. If a perturbation contains such a nondissipative part, then it seems to be natural to ask how the nondissipative part affects the long-time behavior of the perturbation through the convection term. Our aim is to give an explicit answer to this problem.

The stability of the Kolmogorov type flows was studied by Ilyin [17] and Sasaki, Takehiro, and Yamada [38, 39] for the Navier–Stokes equations on $S^{2}$ and by Taylor [41] for the Euler equations on $S^{2}$ (and these authors dealt with the case of a rotating sphere). For the viscous case, Ilyin [17] investigated the linear stability and showed that the $n$-jet Kolmogorov type flow is globally asymptotically stable for all $\nu>0$ when $n=1,2$ but it becomes unstable for small $\nu>0$ when $n\geq 3$. In that paper, however, the viscous term in the Navier–Stokes equations was taken as $\nu\Delta_{H}\mathbf{u}$, which becomes $\nu\Delta\omega$ in the vorticity form. Hence a perturbation does not contain a nondissipative part in the setting of [17]. For the case of the viscous term $\nu(\Delta\omega+2\omega)$, Sasaki, Takehiro, and Yamada studied the linear and nonlinear stability of the Kolmogorov type flows in [38] and [39], respectively, and obtained the same results as in [17]. However, they considered a perturbation in the orthogonal complement of $\mathrm{span}\{Y_{1}^{0},Y_{1}^{\pm 1}\}$, so the effect of a nondissipative part was not taken into account.

In our previous work [31], we studied the linear stability of the two-jet Kolmogorov type flow under a perturbation which contains a nondissipative part. The two-jet Kolmogorov type flow is of the form

Then the linearized equation for the vorticity equation (1.3) around (1.5) is

where $I$ is the identity operator. We refer to [31] for the derivation of (1.6) from (1.3). In [31] we proved that a solution $\omega(t)$ to (1.6) with initial data $\omega_{0}\in L_{0}^{2}(S^{2})$ satisfies

where the equilibrium $\omega_{\infty,\mathrm{lin}}$ is given by

and $C>0$ is a constant independent of $t$, $\nu$, $a_{2}$, and $\omega_{0}$. Note that $\omega_{\infty,\mathrm{lin}}$ contains the nonzero $Y_{2}^{\pm 1}$-components when $\omega_{0,1}^{\pm 1}\neq 0$, although the functions $Y_{2}^{\pm 1}$ themselves are dissipative in (1.6). This result shows that the nondissipative part indeed affects the long-time behavior of the perturbation through the interaction between the viscosity and convection even in the linear stability case. In [31] and the companion paper [26] we also observed that a part of a solution to (1.6) decays at a rate faster than the usual viscous rate $O(e^{-\nu t})$ when $\nu>0$ is small. Such a phenomenon is called the enhanced dissipation, and it has been attracting interest of many researchers in recent years (see e.g. [9, 52, 48] and [3, 25, 14, 49, 50] for the study of the enhanced dissipation for advection-diffusion equations and for the plane Kolmogorov flow, respectively).

In this paper we study the nonlinear stability of the two-jet Kolmogorov type flow under a perturbation with nondissipative part. The nonlinear stability problem for the vorticity equation (1.3) around the two-jet Kolmogorov type flow (1.5) is

where $\omega_{0}\in L_{0}^{2}(S^{2})$ is an initial perturbation (see [31] for the derivation of the perturbation operator). To state our main result, we fix some notations. For $u\in L_{0}^{2}(S^{2})$, we write

Note that $u_{n}^{-m}=(-1)^{m}\overline{u_{n}^{m}}$ if $u$ is real-valued, since $Y_{n}^{-m}=(-1)^{m}\overline{Y_{n}^{m}}$ (see Section 2). In what follows, we consider real-valued initial data and solutions to (1.8), although we take the inner product of (1.8) with the complex spherical harmonics $Y_{n}^{m}$ in order to make some expressions simple. For a real-valued initial data $\omega_{0}\in L_{0}^{2}(S^{2})$, we set

Then we define $\omega_{\infty}=\sum_{m=-2}^{2}\omega_{2,\infty}^{m}Y_{2}^{m}\in L_{0}^{2}(S^{2})$ by

Also, we set $H_{0}^{1}(S^{2})=L_{0}^{2}(S^{2})\cap H^{1}(S^{2})$ and write $H_{0}^{-1}(S^{2})$ for the dual space of $H_{0}^{1}(S^{2})$ with duality product $\langle\cdot,\cdot\rangle_{H_{0}^{1}}$. The main result of this paper is as follows.

Note that the last term of (1.12) is well-defined since

for $\omega\in H_{0}^{1}(S^{2})$ by the Sobolev embedding (see e.g. [2]). Also, a test function $\psi$ in (1.12) is assumed to be real-valued, but in fact we may take a complex-valued $\psi$ by considering its real and imaginary parts separately.

It is expected that we can show that the enhanced dissipation occurs for a part of a solution to (1.8) as in the linear stability case [31, 26], but we need to analyze carefully the interaction between various components of a solution with different longitudinal wave numbers through the convection term. The study of the enhanced dissipation for the nonlinear stability problem will be done in another paper.

The result of Theorem 1.1 can be extended to the case of a rotating sphere. In that case, we consider the vorticity equation with Coriolis force (see e.g. [17, 38, 39])

where $\Omega\in\mathbb{R}$ is the rotation speed of a sphere. For each $n\in\mathbb{N}$ and $a\in\mathbb{R}$, the Kolmogorov type flow (1.4) is still a stationary solution to (1.16) with $f=a\nu(\lambda_{n}-2)Y_{n}^{0}$ since it is independent of the longitude $\varphi$. When $n=2$, the nonlinear stability problem for (1.16) around the two-jet Kolmogorov type flow (1.5) is

Then we easily find by direct calculations (see Section 6 for the outline) that solutions $\omega$ to (1.8) and $\zeta$ to (1.17) are related by

in spherical coordinates. Thus, for a given $\zeta_{0}\in L_{0}^{2}(S^{2})$, we apply Theorem 1.1 to

substitute (1.18) for (1.13) and (1.14), make the change of variable $\varphi\mapsto\varphi+\Omega t$ which does not change the $L^{2}(S^{2})$-norm, and use $Y_{n}^{m}(\theta,\varphi+\Omega t)=e^{im\Omega t}Y_{n}^{m}(\theta,\varphi)$ to obtain the following result.

Let us explain the outline of the proof of Theorem 1.1. Since $S^{2}$ is 2D, we can show the global existence and uniqueness of a weak solution to (1.8) by the Galerkin method with basis functions $Y_{n}^{m}$ and the energy method as in the case of the Navier–Stokes equations in 2D flat domains (see e.g. [44, 4]). Hence we omit details in this paper. To prove (1.13) and (1.14), we first take the inner product of (1.8) with $Y_{1}^{m}$, $|m|=0,1$. Then we find that $\frac{d}{dt}\omega_{1}^{m}(t)=0$ and thus $\omega_{=1}(t)=\omega_{0,=1}$ by using $-\Delta Y_{n}^{m}=\lambda_{n}Y_{n}^{m}$ with $\lambda_{1}=2$ and $\lambda_{2}=6$, a recurrence relation for $\cos\theta\,Y_{n}^{m}$ (see (2.7)), and the fact that a vector field of the form

is a Killing vector field on $S^{2}$. Here the last fact is essential to show that the inner product of the convection term with $Y_{1}^{m}$ vanishes (see Lemma 2.2). To prove that result, we apply an identity for a Killing vector field on a general manifold given in Lemma 2.1, which seems to have its own interest.

Next we derive an equation for $\omega_{\geq 2}(t)=\omega(t)-\omega_{0,=1}$ from (1.8) and take the inner product of that equation with $(I+6\Delta^{-1})\omega_{\geq 2}(t)$. Then we obtain a differential inequality which yields the estimate (1.13) for $\omega_{\geq 3}(t)$ by applying $\lambda_{2}=6$ and $\lambda_{n}\geq 12$ for $n\geq 3$, integration by parts, and the identity

given in Lemma 2.3, where $X$ is given by

Here we do not have an estimate for $\omega_{\geq 2}(t)=\omega_{=2}(t)+\omega_{\geq 3}(t)$ since $(I+6\Delta^{-1})\omega_{=2}(t)=0$ by $\lambda_{2}=6$. Moreover, the identity (1.19) is used to show

where the term $X\cdot\nabla(I+2\Delta^{-1})\omega_{\geq 2}(t)$ comes from the interaction between the nondissipative part $\omega_{0,=1}$ and the dissipative part $\omega_{\geq 2}(t)$ through the convection term. We further note that, since the above $X$ is of the form $X(x)=\mathbf{a}\times x$, $x\in S^{2}$ with some $\mathbf{a}\in\mathbb{R}^{3}$ expressed in terms of $\omega_{0,1}^{0}$ and $\omega_{0,1}^{1}$, it is a Killing vector field on $S^{2}$ and thus we can apply again the identity for a Killing vector field on a general manifold given in Lemma 2.1 to get the identity (1.19).

Lastly, we derive an equation for $\omega_{=2}(t)=\omega_{\geq 2}(t)-\omega_{\geq 3}(t)$, prove the estimate (1.14), and determine $\omega_{\infty}=\sum_{m=-2}^{2}\omega_{2,\infty}^{m}Y_{2}^{m}$. It turns out that the evolution of $\omega_{=2}(t)$ is described by a system of linear ordinary differential equations (ODEs) of the form

where $\bm{I}_{5}$ is the $5\times 5$ identity matrix, $\bm{A}$ is a constant self-adjoint matrix, $\bm{M}(t)$ and $\bm{f}_{\geq 3}(t)$ are a matrix-valued function and a vector field decaying exponentially in time, and $\bm{c}$ is a constant vector coming from the nondissipative part $\omega_{0,=1}$ through the perturbation operator (see (3.15) and (3.17) for the precise definitions). Then, noting that $4\nu\bm{I}_{5}+i\bm{A}$ is invertible since $\bm{A}$ is self-adjoint, we set

and use (1.21) to derive an estimate for $\bm{\omega}(t)-\bm{\omega}_{\infty}$ which corresponds to (1.14). Moreover, using the explicit forms of $\bm{A}$ and $\bm{c}$, we can solve $(4\nu\bm{I}_{5}+i\bm{A})\bm{\omega}_{\infty}=\bm{c}$ to determine $\bm{\omega}_{\infty}$. Here the matrix $\bm{A}$ comes from the interaction between the nondissipative part $\omega_{0,=1}$ and the dissipative part $\omega_{=2}(t)$ through the convection term. Indeed, each entry of $\bm{A}$ is given by the $Y_{2}^{m}$-component of $X\cdot\nabla\omega_{=2}$ with $X$ given by (1.20). Also, we can express $X\cdot\nabla Y_{2}^{m}$ as a linear combination of $Y_{2}^{0}$, $Y_{2}^{\pm 1}$, and $Y_{2}^{\pm 2}$ for each $|m|=0,1,2$ (see Lemma 2.4), so we can write $\bm{A}$ explicitly in terms of $\omega_{0,1}^{0}$ and $\omega_{0,1}^{1}$ and find that $\bm{A}$ is self-adjoint.

The rest of this paper is organized as follows. In Section 2 we fix notations and give auxiliary results on calculus on a manifold and $S^{2}$. The main part of this paper is Section 3, which is devoted to the proof of Theorem 1.1. In Section 4 we give the proof of Lemma 2.4 which consists of elementary but slightly long calculations. As appendices, we briefly explain the behavior of a perturbation for the one-jet Kolmogorov type flow in Section 5 and observe that solutions to (1.8) and (1.17) are related by (1.18) in Section 6.

We fix notations and give auxiliary results on calculus on a manifold and $S^{2}$.