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Highly rotating viscous compressible fluids in presence of capillarity effects

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Abstract

We study here a singular limit problem for a Navier–Stokes–Korteweg system with Coriolis force, in the domain \(\mathbb {R}^2\times \,]0,1[\,\) and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter \(\varepsilon \rightarrow 0\), we perform the incompressible and high rotation limits simultaneously; moreover, we consider both the constant and vanishing capillarity regimes. In this last case, the limit problem is identified as a 2-D incompressible Navier–Stokes equation in the variables orthogonal to the rotation axis; if the capillarity is constant, instead, the limit equation slightly changes, keeping however a similar structure, due to the presence of an additional surface tension term. In the vanishing capillarity regime, various rates at which the capillarity coefficient goes to 0 are considered: in general, this produces an anisotropic scaling in the system. The proof of the results is based on suitable applications of the RAGE theorem, combined with microlocal symmetrization arguments.

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Notes

  1. Here, we mean “isolated” in the sense of [20], Chapter III, Paragraph 6.5: it is an isolated point not just of \(\sigma _p\), but of the whole spectrum of the operator.

  2. Throughout we agree that f(D) stands for the pseudo-differential operator \(u\mapsto \mathcal {F}^{-1}(f\,\mathcal {F}u)\).

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Acknowledgments

The author is deeply grateful to I. Gallagher for proposing him the problem and for enlightening discussions about it. The most of the work was completed while he was a post-doc at Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris-Diderot: he wish to express his gratitude also to these institutions. The author wish to thanks also the anonymous referees for their careful reading and relevant remarks, which greatly helped him to improve the final version of the paper. The author was partially supported by the project “Instabilities in Hydrodynamics”, funded by the Paris city hall (program “Émergence”) and the Fondation Sciences Mathématiques de Paris. He is also member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Correspondence to Francesco Fanelli.

Appendix 1: A primer on Littlewood–Paley theory

Appendix 1: A primer on Littlewood–Paley theory

Let us recall here the main ideas of Littlewood–Paley theory, which we exploited in the previous analysis. We refer e.g. to [1] (Chapter 2) and [24] (Chapters 4 and 5) for details.

For simplicity of exposition, let us deal with the \(\mathbb {R}^d\) case; however, the construction can be adapted to the d-dimensional torus \(\mathbb {T}^d\), and then also to the case of \(\mathbb {R}^{d_1}\times \mathbb {T}^{d_2}\).

First of all, let us introduce the so called “Littlewood–Paley decomposition”, based on a non-homogeneous dyadic partition of unity with respect to the Fourier variable.

We, fix a smooth radial function \(\chi \) supported in the ball B(0, 2),  equal to 1 in a neighborhood of B(0, 1) and such that \(r\mapsto \chi (r\,e)\) is nonincreasing 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 byFootnote 2

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

Throughout the paper we will use freely the following classical property: for any \(u\in \mathcal {S}',\) the equality \(u=\sum _{j}\Delta _ju\) holds true in \(\mathcal {S}'\).

Let us also mention the so-called Bernstein’s inequalities, which explain the way derivatives act on spectrally localized functions.

Lemma 7.1

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

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

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

Definition 7.2

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

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

Besov spaces are interpolation spaces between the Sobolev ones. In fact, for any \(k\in \mathbb {N}\) and \(p\in [1,+\infty ]\) we have the following chain of continuous embeddings:

$$\begin{aligned} B^k_{p,1}\hookrightarrow W^{k,p}\hookrightarrow B^k_{p,\infty }, \end{aligned}$$

where \(W^{k,p}\) denotes the classical Sobolev space of \(L^p\) functions with all the derivatives up to the order k in \(L^p\). Moreover, for all \(s\in \mathbb {R}\) we have the equivalence \(B^s_{2,2}\equiv H^s\), with

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

Let us now collect some bounds which are straightforward consequences of Bernstein’s inequalities. The statements are not optimal: we limit to present the properties we used in our analysis.

Lemma 7.3

  1. (i)

    For \(1\le p\le 2\), one has \(\Vert f\Vert _{L^2}\,\le \,C(\Vert f\Vert _{L^p}+\Vert \nabla f\Vert _{L^2})\).

  2. (ii)

    For any \(0<\delta \le 1/2\) and any \(1\le p\le +\infty \), one has

    $$\begin{aligned} \Vert f\Vert _{L^\infty }\,\le \,C\left( \Vert f\Vert _{L^p}+\Vert \nabla f\Vert ^{(1/2)-\delta }_{L^2}\, \left\| \nabla ^2f\right\| ^{(1/2)+\delta }_{L^2}\,\right) . \end{aligned}$$
  3. (iii)

    Let \(1\le p\le 2\) such that \(1/p\,<\,1/d+1/2\). For any \(j\in \mathbb {N}\), there exists a constant \(C_j\), depending just on j, d and p, such that

    $$\begin{aligned} \left\| \left( \mathrm{Id}\,\,-\,S_j\right) f\right\| _{L^2}\,\le \,C_j\,\left\| \nabla f\right\| _{B^0_{p,\infty }}. \end{aligned}$$

    Moreover, denoting \(\beta \,:=\,1\,-\,d(1/p\,-\,1/2)>0\), we have the explicit formula

    $$\begin{aligned} C_j=\left( \frac{1}{1-2^{-2\beta }}\right) ^{\!\!1/2}\;2^{-\beta (j-1)}. \end{aligned}$$

    In particular, if \(\nabla f=\nabla f_1 + \nabla f_2\), with \(\nabla f_1\in B^0_{2,\infty }\) and \(\nabla f_2\in B^0_{p,\infty }\), then

    $$\begin{aligned} \left\| \left( \mathrm{Id}\,\,-\,S_j\right) f\right\| _{L^2}\,\le \,\widetilde{C}_j\,\left( \left\| \nabla f_1\right\| _{B^0_{2,\infty }}+ \left\| \nabla f_2\right\| _{B^0_{p,\infty }}\right) , \end{aligned}$$

    for a new constant \(\widetilde{C}_j\) still going to 0 for \(j\rightarrow +\infty \).

Proof

For the first inequality, it is enough to write \(f=\Delta _{-1}f+(\mathrm{Id}\,-\Delta _{-1})f\). The former term can be controlled by \(\Vert f\Vert _{L^p}\) by Bernstein’s inequalities; for the latter, instead we can write

$$\begin{aligned} \Vert (\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^2}\le & {} \sum _{k\ge 0}\Vert \Delta _k(\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^2} \\\le & {} C\,\sum _{k\ge 0}2^{-k}\,\Vert \Delta _k(\mathrm{Id}\,-\Delta _{-1})\nabla f\Vert _{L^2}\;\le \;C\,\Vert \nabla f\Vert _{L^2}, \end{aligned}$$

where we used again Bernstein’s inequalities and the characterization \(L^2\equiv B^0_{2,2}\).

In order to prove the second estimate, we proceed exactly as before. Again, Bernstein’s inequalities allow us to bound low frequencies by \(\Vert f\Vert _{L^p}\). Next we have:

$$\begin{aligned} \Vert (\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^\infty }\le & {} C\,\sum _{k\ge 0}2^{3k/2}\,\Vert \Delta _k(\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^2} \\\le & {} C\,\sum _{k\ge 0}2^{-\delta k}\,\left\| |D|^{\delta +3/2}\Delta _k(\mathrm{Id}\,-\Delta _{-1})f\right\| _{L^2} \end{aligned}$$

(for any ), where we denoted |D| the Fourier multiplier having symbol equal to \(|\xi |\). By interpolation we can write

$$\begin{aligned}&\left\| |D|^{\delta +3/2}\Delta _k(\mathrm{Id}\,-\Delta _{-1})f\right\| _{L^2}\,\le \,C\,\left\| \Delta _k(\mathrm{Id}\,-\Delta _{-1})\nabla f\right\| ^{\sigma }_{L^2}\,\\&\quad \left\| \Delta _k(\mathrm{Id}\,-\Delta _{-1})\nabla ^2f\right\| ^{1-\sigma }_{L^2}, \end{aligned}$$

for \(\sigma \,\in \,]0,1[\,\) (actually, \(\sigma =(1/2)-\delta \)), and this immediately gives the conclusion.

Let us finally prove the third claim. By spectral localization we can write

$$\begin{aligned} \left\| \left( \mathrm{Id}\,-S_j\right) f\right\| ^2_{L^2}\le & {} \sum _{k\ge j-1}\Vert \Delta _kf\Vert ^2_{L^2}\;\le \; \sum _{k\ge j-1}2^{-2k}\Vert \nabla \Delta _kf\Vert ^2_{L^2} \\\le & {} \sum _{k\ge j-1}2^{2kd\left( 1/p\,-\,1/2\right) }\,2^{-2k}\,\Vert \nabla \Delta _kf\Vert ^2_{L^p}. \end{aligned}$$

Keeping in mind that, by hypothesis, \(d\left( 1/p\,-\,1/2\right) -1=-\beta <0\), we infer the desired inequality and the explicit expression for \(C_j\). \(\square \)

Finally, let us recall that one can rather work with homogeneous dyadic blocks \((\dot{\Delta }_j)_{j\in \mathbb {Z}}\), with

$$\begin{aligned} \dot{\Delta }_j:=\varphi (2^{-j}D)\quad \text{ for } \text{ all } \quad j\in \mathbb {Z}, \end{aligned}$$

and introduce the homogeneous Besov spaces \(\dot{B}^s_{p,r}\), defined by the condition

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

We do not enter into the details here; we just limit ourselves to recall refined embeddings of homogeneous Besov spaces into Lebesgue spaces (see Theorem 2.40 of [1]).

Proposition 7.4

For any \(2\le p<+\infty \), one has the continuous embeddings \(\dot{B}^0_{p,2}\,\hookrightarrow \,L^p\) and \(L^{p'}\,\hookrightarrow \,\dot{B}^0_{p',2}\).

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Fanelli, F. Highly rotating viscous compressible fluids in presence of capillarity effects. Math. Ann. 366, 981–1033 (2016). https://doi.org/10.1007/s00208-015-1358-x

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