Rate of the enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere

In this paper, as a continuation of [32], we consider the incompressible Navier–Stokes equations on the two-dimensional (2D) unit sphere $S^{2}$ in $\mathbb{R}^{3}$:

Here $\mathbf{u}$ is the tangential velocity of the fluid, $p$ is the scalar-valued pressure, and $\mathbf{f}$ is a given tangential 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 $S^{2}$. Here the viscous term is taken to be the twice of the divergence of the deformation tensor $\mathrm{Def}\,\mathbf{u}$:

where $\mathrm{Ric}\equiv 1$ is the Ricci curvature of $S^{2}$. The Navier–Stokes equations on spheres and more general manifolds with this kind of viscous term have been studied by many authors (see e.g. [43, 36, 33, 31, 9, 21, 5, 7, 39, 40, 35, 22, 37, 38]). There are also several works on the Navier–Stokes equations on manifolds in which the viscous term is taken to be $\nu\Delta_{H}\mathbf{u}$ by analogy of the flat domain case (see e.g. [17, 16, 4, 18, 51, 26, 41]). We refer to [12, 1, 11, 44, 6] for the above identity and the choice of the viscous term in the Navier–Stokes equations on manifolds.

Identifying vector fields with one-forms and taking $\mathrm{rot}=\ast d$ in (1.1), where $\ast$ and $d$ are the Hodge star operator and the external derivative, we have the vorticity equation

for the vorticity $\omega=\mathrm{rot}\,\mathbf{u}$ (see [32] for derivation). Here $\nabla_{\mathbf{u}}\omega$ is the directional derivative of $\omega$ along $\mathbf{u}$ and $\Delta$ is the Laplace–Beltrami operator on $S^{2}$ which is invertible on $L_{0}^{2}(S^{2})$, the space of $L^{2}$ functions on $S^{2}$ with zero mean, $\mathbf{n}_{S^{2}}$ is the unit outward normal vector field on $S^{2}$, and $\times$ is the vector product in $\mathbb{R}^{3}$. Note that the vorticity equation (1.2) is equivalent to the Navier–Stokes equations (1.1) since any closed one-form on $S^{2}$ is exact.

For $n\in\mathbb{Z}_{\geq 0}$ and $|m|\leq n$ let $Y_{n}^{m}$ be the spherical harmonics which satisfy $-\Delta Y_{n}^{m}=\lambda_{n}Y_{n}^{m}$ with $\lambda_{n}=n(n+1)$ (see Section 2). The vorticity equation (1.2) with external force $\mathrm{rot}\,\mathbf{f}_{n}^{a}=a\nu(\lambda_{n}-2)Y_{n}^{0}$ has a stationary solution with velocity field

for $n\in\mathbb{N}$ and $a\in\mathbb{R}$, where $\mathbf{x}(\theta,\varphi)$ is the parametrization of $S^{2}$ by the colatitude $\theta$ and the longitude $\varphi$ (see Section 2). The flow (1.3) can be seen as the spherical version of the Kolmogorov flow which is a stationary solution to the Navier–Stokes equations in a 2D flat torus with shear external force (see e.g. [30, 19, 28, 34, 29] for the study of the stability of the plane Kolmogorov flow). Ilyin [18] called (1.3) the generalized Kolmogorov flow and studied its stability for the Navier–Stokes equations on $S^{2}$ with viscous term $\nu\Delta_{H}\mathbf{u}$. Also, Sasaki, Takehiro, and Yamada [39, 40] called (1.3) an $n$-jet zonal flow and investigated its stability for the Navier–Stokes equations on $S^{2}$ with viscous term $\nu(\Delta_{H}\mathbf{u}+2\mathbf{u})$. The stability of (1.3) for the Euler equations on $S^{2}$ was also studied by Taylor [42]. We call (1.3) 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 consider the linear stability of the two-jet Kolmogorov type flow. We substitute $\omega=\omega_{2}^{a}+\tilde{\omega}_{2}$ for (1.2) and omit the nonlinear term with respect to $\tilde{\omega}_{2}$ to get

where $I$ is the identity operator (see [32] for derivation of the linearized equation). Replacing $\tilde{\omega}_{2}$ and $a_{2}$ by $\omega$ and $a$, we rewrite this equation as

which is considered in $L_{0}^{2}(S^{2})$. Then since $-A$ is nonnegative and self-adjoint in $L_{0}^{2}(S^{2})$ and $\Lambda$ is $A$-compact in $L_{0}^{2}(S^{2})$, the operator $\mathcal{L}^{\nu,a}$ generates an analytic semigroup $\{e^{t\mathcal{L}^{\nu,a}}\}_{t\geq 0}$ in $L_{0}^{2}(S^{2})$ by a perturbation theory of semigroups (see [13]) and thus the solution of (1.4) with initial data $\omega_{0}\in L_{0}^{2}(S^{2})$ is given by $\omega(t)=e^{t\mathcal{L}^{\nu,a}}\omega_{0}$. In [32] the second author of the present paper proved the linear stability of the two-jet Kolmogorov type flow for all $\nu>0$ by getting the exponential decay of $e^{t\mathcal{L}^{\nu,a}}\omega_{0}$ towards a (not orthogonal) projection of $\omega_{0}$ onto the kernel of $\mathcal{L}^{\nu,a}$. Moreover, he showed that $\Lambda$ does not have eigenvalues in $\mathbb{C}\setminus\{0\}$ by making use of the mixing structure of $\Lambda$ expressed by a recurrence relation for $Y_{n}^{m}$, and applied it to find that the enhanced dissipation occurs for the rescaled flow $e^{\frac{t}{\nu}\mathcal{L}^{\nu,a}}\omega_{0}$, which is a solution to $\partial_{t}\omega=A\omega-i\alpha\Lambda\omega$ with $\alpha=a/\nu$, as in the case of an advection-diffusion equation [8, 52, 48]. More precisely, let

which is a closed subspace of $L_{0}^{2}(S^{2})$ invariant under the actions of $A$ and $\Lambda$ (see Section 3.1), and $\mathbb{Q}$ be the orthogonal projection from $\mathcal{X}$ onto the orthogonal complement of the kernel of $\Lambda$ restricted on $\mathcal{X}$. Then it was shown in [32, Theorem 1.4] that

This result in particular gives the convergence of $\mathbb{Q}e^{\frac{t}{\nu}\mathcal{L}^{\nu,a}}\omega_{0}$ to zero in $L^{2}(S^{2})$ as $\nu\to 0$ for each fixed $t>0$ and $a\in\mathbb{R}$, but does not give the actual convergence rate. The purpose of this paper is to give an explicit decay rate of the original flow $\mathbb{Q}e^{t\mathcal{L}^{\nu,a}}\omega_{0}$.

In the plane case, Beck and Wayne [3] numerically conjectured that a perturbation of the plane Kolmogorov flow decays at the rate $O(e^{-\sqrt{\nu}\,t})$ when the viscosity coefficient $\nu$ is sufficiently small. They also verified this enhanced dissipation for a linearized operator without a nonlocal term based on the hypocoercivity method developed by Villani [47]. Lin and Xu [27] proved the enhanced dissipation for a full linearized operator but without an explicit decay rate by using the Hamiltonian structure of a perturbation operator and the RAGE theorem. The decay rate $O(e^{-\sqrt{\nu}\,t})$ was confirmed by Ibrahim, Maekawa, and Masmoudi [15] based on the pseudospectral bound method, by Wei and Zhang [49] based on the hypocoercivity method, and by Wei, Zhang, and Zhao [50] based on the wave operator method.

In this paper we show that $\mathbb{Q}e^{t\mathcal{L}^{\nu,a}}\omega_{0}$ decays at the rate $O(e^{-\sqrt{\nu}\,t})$ as in the plane case. For $m\in\mathbb{Z}$ let $\mathcal{P}_{m}$ be the orthogonal projection from $L^{2}(S^{2})$ onto the space of $L^{2}$ functions on $S^{2}$ of the form $u=U(\theta)e^{im\varphi}$ (see (2.8) for the precise definition). Note that $\mathcal{X}$ is orthogonally decomposed as $\mathcal{X}=\oplus_{m\in\mathbb{Z}\setminus\{0\}}\mathcal{P}_{m}\mathcal{X}$ and

for $u\in\mathcal{X}$ (see Section 3.1). The main result of this paper is as follows.

Here we note that the exponent of $|m|$ in (1.5) is $2/3$ in the sphere case, while it is $1/2$ in the plane case (see [15, Corollary 3.14]). This difference comes from the factor $|m|^{-1/2}$ appearing in the estimate for the $L^{\infty}(S^{2})$-norm of $u=U(\theta)e^{im\varphi}$ by the $L^{2}(S^{2})$-norm of $\nabla u$ (see Lemma 2.3) which yields a different coercive estimate for $\mathbb{Q}_{m}(\mu-\Lambda_{m})$ with $\mu\in\mathbb{R}$, $\mathbb{Q}_{m}=\mathbb{Q}|_{\mathcal{P}_{m}\mathcal{X}}$, and $\Lambda_{m}=m^{-1}\Lambda|_{\mathcal{P}_{m}\mathcal{X}}$ (see Lemma 3.8).

To prove Theorem 1.1 we follow the argument of the work by Ibrahim, Maekawa, and Masmoudi [15] which verified the enhanced dissipation for the plane Kolmogorov flow by the pseudospectral bound method. By rescaling time as $t\mapsto\nu t$ in (1.4) and taking the Fourier series with respect to the longitude $\varphi$, we consider the equation

in $\mathcal{X}_{m}=\mathcal{P}_{m}\mathcal{X}$ for each $m\in\mathbb{Z}\setminus\{0\}$, where $\alpha=a/\nu\in\mathbb{R}$ and $M_{\cos\theta}$ is the multiplication operator by $\cos\theta$. By the definition of $\mathcal{X}_{m}$, the operator $-A_{m}$ is self-adjoint and positive in $\mathcal{X}_{m}$. Also, the kernel of $\Lambda_{m}$ is $\{cY_{2}^{m}\mid c\in\mathbb{C}\}$ when $|m|=1,2$ and trivial when $|m|\geq 3$. We intend to estimate the semigroup $e^{tL_{\alpha,m}}$ generated by $L_{\alpha,m}$, especially $\mathbb{Q}_{m}e^{tL_{\alpha,m}}$ or equivalently $e^{t\mathbb{Q}_{m}L_{\alpha,m}}$, where $\mathbb{Q}_{m}$ is the orthogonal projection from $\mathcal{X}_{m}$ onto $\mathcal{Y}_{m}$, the orthogonal complement of the kernel of $\Lambda_{m}$. To this end, we apply abstract results given in Section 4 to $L_{\alpha,m}$ to derive an estimate for the quantity

which was introduced in [14] and called the pseudospectral bound of $\mathbb{Q}_{m}L_{\alpha,m}$ in [15]. Then we get an estimate for the semigroup generated by $\mathbb{Q}_{m}L_{\alpha,m}$ by combining the estimate for the pseudospectral bound of $\mathbb{Q}_{m}L_{\alpha,m}$ with an abstract theorem for an $m$-accretive operator on a weighted Hilbert space which is a version of the Gearhart–Prüss type theorem shown by Wei [48]. In order to use the abstract results, we need to confirm several assumptions for $A_{m}$ and $\Lambda_{m}$. The main effort is to verify a coercive estimate for $\mathbb{Q}_{m}(\mu-\Lambda_{m})$ on $\mathcal{Y}_{m}$ with $\mu\in\mathbb{R}$ (see Section 3.2). This coercive estimate involves a loss of derivative, but thanks to a small factor in front it can be controlled by the smoothing effect of $-A_{m}$ in the estimate for the pseudospectral bound of $\mathbb{Q}_{m}L_{\alpha,m}$. When $|\mu|\geq 1$, we easily get the coercive estimate by taking the $L^{2}(S^{2})$-inner product of $\mathbb{Q}_{m}(\mu-\Lambda_{m})u$ with $(I+6\Delta^{-1})u$ for $u\in\mathcal{Y}_{m}$ and using the coerciveness of $I+6\Delta^{-1}$ on $\mathcal{Y}_{m}$ (see Lemmas 3.7 and 3.8). On the other hand, the proof is more involved when $|\mu|<1$ (see Lemma 3.9). In this case, noting that $u\in\mathcal{Y}_{m}$ is of the form $u=U(\theta)e^{im\varphi}$, we analyze an ordinary differential equation (ODE) with variable $\theta$ corresponding to $\mathbb{Q}_{m}(\mu-\Lambda_{m})$ and prove the coercive estimate by a contradiction argument after giving auxiliary statements. The main difficulty comes from the nonlocalities of the orthogonal projection $\mathbb{Q}_{m}$ which is nontrivial when $|m|=1,2$ and of the inverse operator $\Delta^{-1}$ appearing in $\Lambda_{m}$. These nonlocalities affect the analysis for different values of $\mu$. The nonlocality of $\mathbb{Q}_{m}$ with $|m|=1,2$ is relevant to the case where $\mu$ is closed to zero, which is the eigenvalue of $\Lambda_{m}$. On the other hand, the nonlocality of $\Delta^{-1}$ causes a difficulty when $\mu$ is close to $\pm 1$, which are the critical values of $\cos\theta$ appearing in $\Lambda_{m}$, i.e. the derivative $(\cos\theta)^{\prime}=-\sin\theta$ vanishes when $\cos\theta=\pm 1$. Hence we can deal with these difficulties separately by using a contradiction argument. Moreover, since $I-\mathbb{Q}_{m}$ is the orthogonal projection onto the kernel of $\Lambda_{m}$ which is spanned by the smooth function $Y_{2}^{m}$, we can handle the nonlocality of $\mathbb{Q}_{m}$ by introducing a suitable auxiliary function which involves $Y_{2}^{m}$. Also, when $\mu$ is close to $\pm 1$, the use of a contradiction argument enables us to focus on analysis of functions concentrating around the critical points of $\cos\theta$, i.e. $\theta=0,\pi$, for which the nonlocal term consisting of $\Delta^{-1}$ essentially becomes a small order by the smoothing effect of $\Delta^{-1}$. By these facts we can verify the coercive estimate for $|\mu|<1$, but the actual proof requires very long and careful calculations.

The coercive estimate for $\mathbb{Q}_{m}(\mu-\Lambda_{m})$ given in this paper is basically the same as the one in the plane case [15]. In the sphere case, the surface measure on $S^{2}$ is of the form $\sin\theta\,d\theta\,d\varphi$ under the spherical coordinate system. The weight function $\sin\theta$ vanishes at the critical points $\theta=0,\pi$ of $\cos\theta$ appearing in $\Lambda_{m}$, so one may expect to have a better coercive estimate than the one in the plane case when $|\mu|<1$ and $\mu$ is close to $\pm 1$, but we cannot get such an estimate. In fact, for $\theta_{\mu}=\arccos\mu$ with $\mu$ close to $\pm 1$, a small parameter $\varepsilon>0$, and a function on $S^{2}$ of the form $u=U(\theta)e^{im\varphi}$, we have a better bound $\varepsilon\sin\theta_{\mu}$ due to the weight function $\sin\theta$ when we estimate the $L^{2}$-norm of $u$ on a narrow band on $S^{2}$ corresponding to $\theta\in(\theta_{\mu}-\varepsilon,\theta_{\mu}+\varepsilon)$ by the $L^{\infty}$-norm of $U$ on $(\theta_{\mu}-\varepsilon,\theta_{\mu}+\varepsilon)$. However, we also have a factor $1/\sin\theta_{\mu}$ when we use an interpolation type inequality which bounds the $L^{\infty}$-norm $U$ on $(\theta_{\mu}-\varepsilon,\theta_{\mu}+\varepsilon)$ by the $L^{2}$- and $H^{1}$-norms of $u$ on $S^{2}$ (see Lemma 2.6), so the resulting coercive estimate is the same as in the plane case.

Lastly, let us explain difference from the plane case [15] in the proof of the coercive estimate. The authors of [15] derived coercive estimates for $\mu-\Lambda_{m}$ on $\mathcal{X}_{m}$ with $\mu\in\mathbb{R}$ away from zero and for $\mathbb{Q}_{m}(\mu-\Lambda_{m})$ on $\mathcal{Y}_{m}$ with $\mu\in\mathbb{R}$ close to zero in order to avoid the nonlocality of $\mathbb{Q}_{m}$ when $\mu$ is away from zero. In this paper, however, we deal with $\mathbb{Q}_{m}(\mu-\Lambda_{m})$ for all $\mu\in\mathbb{R}$ since the nonlocality of $\mathbb{Q}_{m}$ can be handled by taking the inner product in $\mathcal{Y}_{m}$ when $|\mu|\geq 1$ and by introducing an auxiliary function when $|\mu|<1$. Also, we encounter an additional difficulty due to the larger coefficient of the nonlocal operator $\Delta^{-1}$ in $\Lambda_{m}$. To prove the coercive estimate when $|\mu|<1$ and $\mu$ is close to $1$ (or $-1$), we analyze an auxiliary function $w$ on $S^{2}$ of the form $w=W(\theta)e^{im\varphi}$ and derive an estimate for the $L^{2}$-norm of $w$ on a subset of $S^{2}$ corresponding to $\theta\in(\theta_{\mu},\pi)$ (or $\theta\in(0,\theta_{\mu})$) with $\theta_{\mu}=\arccos\mu$ by using an ODE for $W$ (see Lemma 3.17). In the proof of the estimate, we need to show $J+K\geq C\int_{\theta_{\mu}}^{\pi}|W(\theta)|^{2}\sin\theta\,d\theta$ with a constant $C>0$, where

Here $J$ is always nonnegative but $K$ may become negative by the presence of $\cos\theta$. Moreover, the coefficient $5$ of $K$ comes from the coefficient $6$ of $\Delta^{-1}$ in $\Lambda_{m}$. In the plane case, one also needs to deal with integrals similar to $J$ and $K$, but the coefficient of $K$ becomes $1/2$ since the coefficient of $\Delta^{-1}$ in $\Lambda_{m}$ is $1$. Thus one can easily get an estimate just by using the zeroth order term of $W$ in $J$. In our case, however, the use of only the zeroth order term does not work for $|m|=1$ since the coefficient $5$ of $K$ is too large. To overcome this difficulty, we take into account the first order term of $W$ in $J$. In fact, it is natural to use both the zeroth and first order terms since

for the gradient of $w=W(\theta)e^{im\varphi}$ on $S^{2}$. In the actual proof, we split $K$ into the integrals $K_{1}$ and $K_{2}$ over $(\theta_{\mu},\Theta)$ and $(\Theta,\pi)$ with $\Theta\in(\theta_{\mu},\pi/2)$ and estimate them separately. To $K_{1}$ we just use $\cos\theta\geq\cos\Theta$ for $\theta\in(\theta_{\mu},\Theta)$. Also, we carry out integration by parts for $K_{2}$ and apply Young’s inequality to get the integral of a function involving the term $|\nabla w|^{2}$. Then, taking an appropriate $\Theta$ and estimating the integrand, we obtain an estimate for $K_{2}$ which gives the lower bound of $J+K$ when combined with the estimate for $K_{1}$.

The rest of this paper is organized as follows. In Section 2 we fix notations and give auxiliary inequalities. Section 3 is devoted to the study of the linearized operator for the two-jet Kolmogorov type flow. We provide settings and basic results in Section 3.1, verify coercive estimates for $\mathbb{Q}_{m}(\mu-\Lambda_{m})$ in Section 3.2, and derive estimates for the semigroup generated by $L_{\alpha,m}$ and prove Theorem 1.1 in Section 3.3. Section 4 gives abstract results used in Section 3. In Section 5 we show basic formulas of differential geometry on $S^{2}$.

In this section we fix notations and give auxiliary inequalities. We also refer to Section 5 for some notations and basic formulas of differential geometry.

Let $S^{2}$ be the unit sphere in $\mathbb{R}^{3}$ equipped with the Riemannian metric induced by the Euclidean metric of $\mathbb{R}^{3}$. We denote by $\theta$ and $\varphi$ the colatitude and the longitude so that $S^{2}$ is parametrized by $\mathbf{x}(\theta,\varphi)=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$ for $\theta\in[0,\pi]$ and $\varphi\in[0,2\pi)$. For a (complex-valued) function $u$ on $S^{2}$, we sometimes abuse the notation

when no confusion may occur. Thus the integral of $u$ over $S^{2}$ is given by

where $\mathcal{H}^{k}$ is the Hausdorff measure of dimension $k\in\mathbb{N}$. As usual, we set

where $\bar{v}$ is the complex conjugate of $v$, and write $H^{k}(S^{2})$, $k\in\mathbb{Z}_{\geq 0}$ for the Sobolev spaces of $L^{2}$ functions on $S^{2}$ with $H^{0}(S^{2})=L^{2}(S^{2})$ (see [2]).

Let $\nabla$ and $\Delta$ be the gradient and Laplace–Beltrami operators on $S^{2}$. It is well known (see e.g. [46, 45]) that $\lambda_{n}=n(n+1)$ is an eigenvalue of $-\Delta$ with multiplicity $2n+1$ for each $n\in\mathbb{Z}_{\geq 0}$ and the corresponding eigenvectors are the spherical harmonics

Here $P_{n}^{0}$, $n\in\mathbb{Z}_{\geq 0}$ are the Legendre polynomials defined as

and the associated Legendre functions $P_{n}^{m}$, $n\in\mathbb{Z}_{\geq 0}$, $|m|\leq n$ are given by

so that $Y_{n}^{-m}=(-1)^{m}\overline{Y_{n}^{m}}$ (see [23, 10]). Moreover, the set of all $Y_{n}^{m}$ forms an orthonormal basis of $L^{2}(S^{2})$, i.e. $u=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}(u,Y_{n}^{m})_{L^{2}(S^{2})}Y_{n}^{m}$ for each $u\in L^{2}(S^{2})$. It is also known that the recurrence relation

holds (see [23, (7.12.12)]) and thus (see also [46, Section 5.7])

for $n\in\mathbb{Z}_{\geq 0}$ and $|m|\leq n$, where we consider $Y_{|m|-1}^{m}\equiv 0$. Note that the superscript $m$ of $a_{n}^{m}$ just corresponds to that of $Y_{n}^{m}$ and does not mean the $m$-th power.

Let $L_{0}^{2}(S^{2})$ be the space of $L^{2}$ functions on $S^{2}$ with vanishing mean, i.e.

Then $\Delta$ is invertible and self-adjoint as a linear operator

Also, for $s\in\mathbb{R}$, the operator $(-\Delta)^{s}$ is defined on $L_{0}^{2}(S^{2})$ by

We easily observe by a density argument and integration by parts that

By this relation and Poincaré’s inequality (see e.g. [2, Corollary 4.3])

we have the norm equivalence

Let $u$ be a function on $S^{2}$. We write $u=U(\theta)e^{im\varphi}$ if $u$ is of the form

with some function $U$ of the colatitude $\theta$ and $m\in\mathbb{Z}$. In this case,

by (5.1), where $U^{\prime}=\frac{dU}{d\theta}$ and $\nabla^{2}u$ is the covariant Hessian of $u$, and

If $u=U(\theta)e^{im\varphi}\in L^{2}(S^{2})$, then $(u,Y_{n}^{m^{\prime}})_{L^{2}(S^{2})}=0$ for $m^{\prime}\neq m$. In particular, if $m\neq 0$, then $u\in L_{0}^{2}(S^{2})$ and we can apply (2.4)–(2.6) to $u$. Also, if $u=U(\theta)e^{im\varphi}\in H^{1}(S^{2})$ for $m\in\mathbb{Z}$, then $U\in H_{loc}^{1}(0,\pi)\subset C(0,\pi)$ by (2.7). We use these facts without mention.

For a function $u$ on $S^{2}$ and $m\in\mathbb{Z}$ we define a function $\mathcal{P}_{m}u$ on $S^{2}$ by

Note that $\mathcal{P}_{m}u=u$ and $\mathcal{P}_{m^{\prime}}u=0$ for $m^{\prime}\neq m$ if $u=U(\theta)e^{im\varphi}$.

The following results are shown in [32, Section 2].