A Multiscale Problem for Viscous Heat-Conducting Fluids in Fast Rotation

Abstract

In the present paper, we study the combined incompressible and fast rotation limits for the full Navier–Stokes–Fourier system with Coriolis, centrifugal and gravitational forces, in the regime of small Mach, Froude and Rossby numbers and for general ill-prepared initial data. We consider both the isotropic scaling (where all the numbers have the same order of magnitude) and the multiscale case (where some effect is predominant with respect to the others). In the case when the Mach number is of higher order than the Rossby number, we prove that the limit dynamics is described by an incompressible Oberbeck–Boussinesq system, where the velocity field is horizontal (according to the Taylor–Proudman theorem), but vertical effects on the temperature equation are not negligible. Instead, when the Mach and Rossby numbers have the same order of magnitude, and in the absence of the centrifugal force, we show convergence to a quasi-geostrophic equation for a stream function of the limit velocity field, coupled with a transport-diffusion equation for a new unknown, which links the target density and temperature profiles. The proof of the convergence is based on a compensated compactness argument. The key point is to identify some compactness properties hidden in the system of acoustic-Poincaré waves. Compared to previous results, our method enables first of all to treat the whole range of parameters in the multiscale problem, and also to consider a low Froude number regime with the somehow critical choice $Fr\,=\,\sqrt{Ma}$, where Ma is the Mach number. This allows us to capture some (low) stratification effects in the limit.

Appendix: A Few Tools from Littlewood–Paley Theory

In this appendix, we present some tools from Littlewood–Paley theory, which we have exploited in our analysis. We refer to Chapter 2 of Bahouri et al. (2011) for details. For simplicity of exposition, we deal with the $\mathbb {R}^d$ case, with $d\ge 1$; however, the whole construction can be adapted also to the d-dimensional torus $\mathbb {T}^d$, and to the “hybrid” case $\mathbb {R}^{d_1}\times \mathbb {T}^{d_2}$.

First of all, we introduce the Littlewood–Paley decomposition. For this, we fix a smooth radial function $\chi$ such that $\mathrm{Supp}\,\chi \subset B(0,2)$, $\chi \equiv 1$ in a neighbourhood of B(0, 1) and the map $r\mapsto \chi (r\,e)$ is non-increasing over $\mathbb {R}_+$ for all unitary vectors $e\in \mathbb {R}^d$. Set $\varphi \left( \xi \right) =\chi \left( \xi \right) -\chi \left( 2\xi \right)$ and $\varphi _j(\xi ):=\varphi (2^{-j}\xi )$ for all $j\ge 0$. The dyadic blocks $(\Delta _j)_{j\in \mathbb {Z}}$ are defined by³

$$\begin{aligned} \Delta _j\,:=\,0\quad \text{ if } \; j\le -2,\qquad \Delta _{-1}\,:=\,\chi (D)\qquad \text{ and } \qquad \Delta _j\,:=\,\varphi \left( 2^{-j}D\right) \quad \text{ if } \; j\ge 0\,. \end{aligned}$$

For any $j\ge 0$ fixed, we also introduce the low frequency cut-off operator

$$\begin{aligned} S_j\,:=\,\chi \left( 2^{-j}D\right) \,=\,\sum _{k\le j-1}\Delta _{k}\,. \end{aligned}$$

(A.1)

Note that $S_j$ is a convolution operator. More precisely, after defining

$$\begin{aligned} K_0\,:=\,\mathcal {F}^{-1}\chi \qquad \qquad \text{ and } \qquad \qquad K_j(x)\,:=\,\mathcal {F}^{-1}\left[ \chi \left( 2^{-j}\cdot \right) \right] (x) = 2^{jd}K_0\left( 2^j x\right) \,, \end{aligned}$$

for all $j\in \mathbb {N}$ and all tempered distributions $u\in \mathcal {S}'$, we have that $S_ju\,=\,K_j\,*\,u$. Thus, the $L^1$ norm of $K_j$ is independent of $j\ge 0$; hence, $S_j$ maps continuously $L^p$ into itself, for any $1 \le p \le +\infty$.

The following property holds true; for any $u\in \mathcal {S}'$, then one has the equality $u=\sum _{j}\Delta _ju$ in the sense of $\mathcal {S}'$. Let us also recall the so-called Bernstein inequalities.

Lemma A.1

Let $0<r<R$. A constant C exists so that, for any nonnegative integer k, any couple (p, q) in $[1,+\infty ]^2$, with $p\le q$, and any function $u\in L^p$, we have, for all $\lambda >0$,

$$\begin{aligned}&{\mathrm{Supp}\,}\, \widehat{u} \subset B(0,\lambda R)\quad \Longrightarrow \quad \Vert \nabla ^k u\Vert _{L^q}\, \le \, C^{k+1}\,\lambda ^{k+d\left( \frac{1}{p}-\frac{1}{q}\right) }\,\Vert u\Vert _{L^p}\;;\\&\quad {\mathrm{Supp}\,}\, \widehat{u} \subset \{\xi \in \mathbb {R}^d\,:\, \lambda r\le |\xi |\le \lambda R\} \quad \Longrightarrow \quad C^{-k-1}\,\lambda ^k\Vert u\Vert _{L^p}\,\\&\quad \le \, \Vert \nabla ^k u\Vert _{L^p}\, \le \, C^{k+1} \, \lambda ^k\Vert u\Vert _{L^p}\,. \end{aligned}$$

By use of Littlewood–Paley decomposition, we can define the class of Besov spaces.

Definition A.2

Let $s\in \mathbb {R}$ and $1\le p,r\le +\infty$. The non-homogeneous Besov space $B^{s}_{p,r}$ is defined as the subset of tempered distributions u for which

$$\begin{aligned} \Vert u\Vert _{B^{s}_{p,r}}\,:=\, \left\| \left( 2^{js}\,\Vert \Delta _ju\Vert _{L^p}\right) _{j\ge -1}\right\| _{\ell ^r}\,<\,+\infty \,. \end{aligned}$$

Besov spaces are interpolation spaces between Sobolev spaces. In fact, for any $k\in \mathbb {N}$ and $p\in [1,+\infty ]$, we have the chain of continuous embeddings $B^k_{p,1}\hookrightarrow W^{k,p}\hookrightarrow B^k_{p,\infty }$, which, when $1<p<+\infty$, can be refined to $B^k_{p, \min (p, 2)}\hookrightarrow W^{k,p}\hookrightarrow B^k_{p, \max (p, 2)}$. In particular, for all $s\in \mathbb {R}$, we deduce that $B^s_{2,2}\equiv H^s$, with equivalence of norms:

$$\begin{aligned} \Vert f\Vert _{H^s}\,\sim \,\left( \sum _{j\ge -1}2^{2 j s}\,\Vert \Delta _jf\Vert ^2_{L^2}\right) ^{\!\!1/2}\,. \end{aligned}$$

(A.2)