Convergence of a low order non-local Navier–Stokes–Korteweg system: The order-parameter model

Abstract

In the present article we consider a capillary compressible system introduced by C. Rohde after works of Bandon, Lin and Rogers, called the order-parameter model, and whose aim is to reduce the numerical difficulties generated by the classical local Korteweg system (involving derivatives of order three) or the non-local system (also introduced by Rohde after works of Van der Waals, and which involves a convolution operator). We prove that this system has a unique global solution for initial data close to an equilibrium and we obtain the convergence of this solution towards the local Korteweg model as well as a convergence rate with respect to the order parameter, in accordance to what conjectured C. Rohde. As a by-product, the a priori estimates we obtain allow to provide global existence results in the $L^{p}$ setting ( $p \neq 2$ ).

Keywords

Compressible Navier–Stokes system local and non-local capillarity Besov spaces Lagrangian change of variables

1. Introduction

1.1. Presentation of the systems

In the mathematical study of liquid-vapour mixture, Gibbs first modelled phase transitions thanks to the minimization of an energy functional with a nonconvex energy density (see [16]). The phases are separated by an hypersurface and there are mainly two ways to describe it: either we consider that the interface behaves like a discontinuity for the fluid parameters (this is the Sharp Interface model), either we consider that between the phases lies a thin region of continuous transition (this is the Diffuse Interface approach, where the phase changes are seen through the variations of the density and which is much simpler numerically). Unfortunately the basic models provide an infinite number of solutions (few of them being physically relevant) and this is why authors tried to penalize the high variations of the density (with capillary terms related to surface tension) in order to select the physically correct solutions.

In the present paper, we are interested in the local and non-local Korteweg systems (in the diffuse interface model). These systems are based upon the compressible Navier–Stokes system with a Van der Waals state law for ideal fluids, and endowed with a capillary tensor.

Let us recall that the local model was introduced by Korteweg and renewed by Dunn and Serrin (see [15]) and the non-local model was introduced by Van der Waals and renewed by F. Coquel, D. Diehl, C. Merkle and C. Rohde. For an in-depth presentation of the capillary models, we refer to [11,17–19,25,28]).

Let ρ and u denote the density and the velocity of a compressible viscous fluid (ρ is a non-negative function and u is a vector-valued function defined on $R^{d}$ ). We denote by $A$ the following diffusion operator $\begin{array}{l} A u = μ Δ u + (λ + μ) \nabla div u, with μ > 0 and ν = λ + 2 μ > 0 . \end{array}$ The Navier–Stokes equations for compressible fluids endowed with internal capillarity read: $\begin{array}{l} \{\begin{matrix} \partial_{t} ρ + div (ρ u) = 0, \\ \partial_{t} (ρ u) + div (ρ u \otimes u) - A u + \nabla (P (ρ)) = κ ρ \nabla D [ρ] . \end{matrix} \end{array}$ The capillary coefficient κ may depend on ρ but in this article it is chosen constant. In the local Korteweg system (NSK), the capillary term $D [ρ]$ is given by (see [15]): $\begin{array}{l} D [ρ] = Δ ρ, \end{array}$ and, in the non-local Korteweg system (NSRW) (see [11,29], and [32]), if ϕ is an interaction potential which satisfies the following conditions $\begin{matrix} (1) & (| \cdot | + | \cdot |^{2}) ϕ (\cdot) \in L^{1} (R^{d}), \int_{R^{d}} ϕ (x) d x = 1, ϕ even and ϕ ⩾ 0, \end{matrix}$ then $D [ρ]$ is the non-local term given by: $\begin{array}{l} D [ρ] = ϕ * ρ - ρ . \end{array}$

Comparing the Fourier transform of the capillary terms, we have $(\hat{ϕ} (ξ) - 1) \hat{ρ} (ξ)$ in the non-local model, and $- | ξ |^{2} \hat{ρ} (ξ)$ in the local model so that a natural question is to study the closedness of the solutions of these models when $\hat{ϕ} (ξ)$ is formally “close” to $1 - | ξ |^{2}$ . For this, we introduced in [7] a specific interaction potential and considered the following non-local system: $\begin{array}{l} (NSR W_{ε}) \{\begin{matrix} \partial_{t} ρ_{ε} + div (ρ_{ε} u_{ε}) = 0, \\ \partial_{t} (ρ_{ε} u_{ε}) + div (ρ_{ε} u \otimes u_{ε}) - A u_{ε} + \nabla (P (ρ_{ε})) = ρ_{ε} \frac{κ}{ε^{2}} \nabla (ϕ_{ε} * ρ_{ε} - ρ_{ε}), \end{matrix} \end{array}$ with $\begin{array}{l} ϕ_{ε} = \frac{1}{ε^{d}} ϕ (\frac{x}{ε}) with ϕ (x) = \frac{1}{{(2 π)}^{d}} e^{- \frac{| x |^{2}}{4}} . \end{array}$ For a fixed ξ the Fourier transform of $ϕ_{ε}$ is $\hat{ϕ_{ε}} (ξ) = e^{- ε^{2} | ξ |^{2}}$ , and when ε is small, $\frac{\hat{ϕ_{ε}} (ξ) - 1}{ε^{2}}$ is close to $- | ξ |^{2}$ .

Using energy methods, we proved in [7] that this system has a unique global strong solution for initial data close to an equilibrium state. The functional setting are classical and hybrid Besov spaces (taylored to the capillary term). We also obtained that when the small parameter ε goes to zero, the solution tends to the corresponding solution of the local Korteweg system and we obtained a rate of convergence in terms of ε. In [9], we provided by Lagrangian methods more precise a priori estimates giving a better understanding of the convergence and the hybrid Besov setting in terms of the linear Fourier structures.

Though, these models are not completely satisfying. On one hand, recalling the results from [2,12] (compressible Navier–Stokes system), [8,14] (local Korteweg model), and [7,19] (non-local Korteweg model), we observe that the density in the local capillary model is far more regular than in the non-local model where it shares the same frequency structure as in the compressible Navier–Stokes model (heat regularization in low frequencies and only a damping in high frequencies).

On the other hand, from a numerical point of view the local model is difficult to handle because the capillary term contains third-order derivatives. The non-local model also presents difficulties in numerical studies: even if the capillary term only contains derivatives of order one, it involves a convolution operator, whose numerical difficulty is comparable.

For this reason, C. Rohde presented in [30] a new model, called the order-parameter model, and inspired by the work of D. Brandon, T. Lin and R.C. Rogers in [5].

This new system consists in introducing in the capillary term $α^{2} \nabla (c - ρ)$ a new variable c called the “order parameter”, which is coupled to the density via the following relation related to the Euler–Lagrange equation from the variational approach (α controls the coupling between ρ and c): $\begin{array}{l} ε^{2} Δ c + α^{2} (ρ - c) = 0 \end{array}$ so that the new system he considers is the following (replacing α by $α / ε$ allows us to fix $ε = 1$ ): $\begin{array}{l} (NSO P_{α}) \{\begin{matrix} \partial_{t} ρ_{α} + div (ρ_{α} u_{α}) = 0, \\ \partial_{t} (ρ_{α} u_{α}) + div (ρ_{α} u_{α} \otimes u_{α}) - A u_{α} + \nabla (P (ρ_{α})) = κ α^{2} ρ_{α} \nabla (c_{α} - ρ_{α}), \\ Δ c_{α} + α^{2} (ρ_{α} - c_{α}) = 0 . \end{matrix} \end{array}$ As emphasized by C. Rohde, from a numerical point of view this system is much more interesting because now we only have one derivative in the capillary tensor (which is local), and the additional equation for the order parameter is a simple linear elliptic equation that can be easily and numerically fast solved at least when the mesh is fixed. Moreover as we will see later, as for the previous non-local capillary model, this system has the same frequency structure as the classical Navier–Stokes model.

Remark 1.
As explained in [30] another numerical interest is that when α is large enough, the momentum equations can be rewritten with the new pressure $\tilde{p} (ρ) = p (ρ) + \frac{α^{2}}{2} ρ^{2}$ , whose derivative at $ρ = 1$ , when α is large enough, becomes positive if the density is bounded from below.

In [30], C. Rohde proves (for $ε = λ = μ = 1$ in the two-dimensional case) that the system has a unique local classical solution:
Theorem 1 ([30]).

Assume that the initial data $(ρ_{0}, u_{0})$ is independent of $α > 0$ with $u_{0} \in H^{4} (R^{2})$ , $ρ_{0} > 0$ and $ρ_{0} - \bar{ρ} \in H^{4} (R^{2})$ for some constant $\bar{ρ}$ . Let $c_{0}$ be the solution of the elliptic problem $- Δ c_{0} + α^{2} c_{0} = α^{2} ρ_{0}$ . There exists a constant $T_{*} > 0$ such that the initial-value problem $(NSO P_{α})$ has a unique solution defined on $[0, T_{*} [$ satisfying: $\begin{array}{l} ρ_{α} - \bar{ρ}, u_{α} \in L_{loc}^{\infty} (0, T_{*}; H^{4} (R^{2})), ρ_{α} > 0, \\ c_{α} - \bar{ρ} \in L_{loc}^{\infty} (0, T_{*}; H^{5} (R^{2})) . \end{array}$ Moreover, for all $t \in [0, T_{*} [$ we have $\begin{array}{l} lim_{α \to \infty} {‖ ρ_{α} (t, \cdot) - c_{α} (t, \cdot) ‖}_{L^{2} (R^{2})} = 0 . \end{array}$

In this paper, C. Rohde also conjectures that when the coupling constant α goes to infinity, these solutions converge to the solution of the local Korteweg model.

1.2. Statement of the results

In the present article, following what we did in the whole space for the non-local system (see [7]) and using Lagrangian methods from [9], we will prove that under smallness conditions, and with less regular initial data, the system has global strong solutions in the following critical spaces (we refer to the Appendix for more details on Besov spaces and hybrid spaces). We also prove the above conjectured convergence and give an explicit rate of convergence with respect to α.

Definition 1.
The space $F_{α}^{s}$ is the set of functions $(q, c, u)$ in $\begin{array}{l} {(C_{b} (R_{+}, {\dot{B}}_{2, 1}^{s - 1} \cap {\dot{B}}_{2, 1}^{s}) \cap L^{1} (R_{+}, {\dot{B}}_{α}^{s + 1, s - 1} \cap {\dot{B}}_{α}^{s + 2, s}))}^{2} \times {(C_{b} (R_{+}, {\dot{B}}_{2, 1}^{s - 1}) \cap L^{1} (R_{+}, {\dot{B}}_{2, 1}^{s + 1}))}^{d} \end{array}$ endowed with the norm $‖ (q, c, u) ‖_{F_{α}^{s}} = ‖ (q, c, u) ‖_{F_{α}^{s} (\infty)}$ where for all t we denote (recall that $ν_{0} = min (μ, ν)$ ) $\begin{array}{l} {‖ (q, c, u) ‖}_{F_{α}^{s} (t)} \overset{def}{=} & ‖ u ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s - 1}} + ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s - 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s}} + ‖ c ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s - 1}} + ν ‖ c ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s}} \\ + ν_{0} ‖ u ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{s + 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 1, s - 1}} \\ (2) & + ν^{2} ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 2, s}} + ν ‖ c ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 1, s - 1}} + ν^{2} ‖ c ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 2, s}} . \end{array}$
Theorem 2.
Let $α > 0$ and assume $min (μ, 2 μ + λ) > 0$ . There exist two positive constants $η_{OP}$ and C only depending on d, μ, λ, κ, and $P^{'} (\overline{ρ})$ (independent of α) such that if $ρ_{0} - \overline{ρ} \in {\dot{B}}_{2, 1}^{\frac{d}{2} - 1} \cap {\dot{B}}_{2, 1}^{\frac{d}{2}}$ , $u_{0} \in {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}$ , $c_{0}$ is defined by $- Δ c_{0} + α^{2} c_{0} = α^{2} ρ_{0}$ and $\begin{array}{l} ‖ ρ_{0} - \overline{ρ} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1} \cap {\dot{B}}_{2, 1}^{\frac{d}{2}}} + ‖ u_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} ⩽ η_{OP} \end{array}$ then system $(NSO P_{α})$ has a unique global solution $(ρ_{α}, c_{α}, u_{α})$ with $(ρ_{α} - \overline{ρ}, c_{α} - \overline{ρ}, u_{α}) \in F_{α}^{\frac{d}{2}}$ such that: $\begin{array}{l} {‖ (ρ_{α} - \overline{ρ}, c_{α} - \overline{ρ}, u_{α}) ‖}_{F_{α}^{\frac{d}{2}}} ⩽ C^{0} \overset{def}{=} C (‖ ρ_{0} - \overline{ρ} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1} \cap {\dot{B}}_{2, 1}^{\frac{d}{2}}} + ‖ u_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}) . \end{array}$ Moreover we have the global in time results: $\begin{array}{l} \{\begin{matrix} ‖ c_{α} - ρ_{α} ‖_{{\tilde{L}}^{\infty} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2} - 1})} + ν ‖ c_{α} - ρ_{α} ‖_{{\tilde{L}}^{\infty} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2}})} \underset{α \to \infty}{⟶} 0, \\ ν ‖ c_{α} - ρ_{α} ‖_{L^{1} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2} - 1})} + ν^{2} ‖ c_{α} - ρ_{α} ‖_{L^{1} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2}})} ⩽ C^{0} α^{- 2} . \end{matrix} \end{array}$

The following result deals with the convergence in α: when the initial data are small enough (so that we have global solutions for (NSK) and $(NSO P_{α})$ ) the solution of $(NSO P_{α})$ goes to the solution of (NSK) when α goes to infinity.
Theorem 3.
With the same assumptions as before, if $\begin{array}{l} ‖ ρ_{0} - \overline{ρ} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1} \cap {\dot{B}}_{2, 1}^{\frac{d}{2}}} + ‖ u_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} ⩽ min (η_{K}, η_{OP}), \end{array}$ then systems (NSK) and $(NSO P_{α})$ both have global solutions and $‖ (ρ_{α} - ρ, c_{α} - ρ, u_{α} - u) ‖_{F_{α}^{\frac{d}{2}}}$ goes to zero as α goes to infinity. Moreover, with the same notations as before, there exists a constant $C = C (η, κ, \overline{ρ}, P^{'} (1)) > 0$ such that for all $h \in] 0, 1 [$ (if $d = 2$ ) or $h \in] 0, 1]$ (if $d ⩾ 3$ ) $\begin{array}{l} {‖ (ρ_{α} - ρ, c_{α} - ρ, u_{α} - u) ‖}_{F_{α}^{\frac{d}{2} - h}} ⩽ C α^{- h} . \end{array}$
Remark 2.
As the order parameter $c_{α}$ goes to $ρ_{α}$ , we formally get that when α goes to infinity, the capillary term goes to $κ ρ \nabla Δ ρ$ .
Remark 3.
Thanks to Besov injections into $L^{p}$ spaces and interpolation, the previous convergence rates give for example with $h = 1 / 2$ that if $δ ρ = ρ_{α} - ρ$ and $δ u = u_{α} - u$ : $\begin{array}{l} \{\begin{matrix} ‖ δ u ‖_{L_{t}^{\infty} L^{2}} + ‖ \nabla^{2} δ u ‖_{L_{t}^{1} L^{2}} ⩽ C α^{- 1 / 2}, \\ ‖ δ q ‖_{L_{t}^{\infty} L^{2}} + ‖ \nabla δ q ‖_{L_{t}^{\infty} L^{2}} + ‖ \nabla δ q ‖_{L_{t}^{2} L^{2}} ⩽ C α^{- 1 / 2} (d = 3), \end{matrix} \\ \{\begin{matrix} ‖ δ u ‖_{L_{t}^{4} L^{2}} + ‖ δ u ‖_{L_{t}^{4 / 3} L^{\infty}} + ‖ \nabla δ u ‖_{L_{t}^{4 / 3} L^{2}} ⩽ C α^{- 1 / 2}, \\ ‖ δ q ‖_{L_{t}^{\infty} L^{2}} + ‖ δ q ‖_{L_{t}^{4} L^{2}} ⩽ C α^{- 1 / 2} (d = 2) . \end{matrix} \end{array}$

Let us end this section with the following important remark: Remark 4.
As explained in [9], the high-frequency part in the a priori estimates that we obtain in the present article (see Theorem 4) allow us perform the same method as in [6] and provide global existence results for the order parameter model in the $L^{p}$ setting with $p \neq 2$ .

1.3. Outline of the paper

The article is structured in the following way: Section 2 is devoted to the proof of Theorem 2. We first introduce an interaction potential $ϕ_{α}$ that allows us to rewrite the system into a non-local shape. As we want precise estimates we follow the methods from [9]: we first obtain estimates on the linearized system and then on the advected linear system thanks to a Lagrangian change of variable.

In Section 3 we prove the convergence result from Theorem 3 and in the Appendix, we first recall basic properties of Besov spaces, then we provide estimates for the flow of a smooth vectorfield. The last part of the Appendix is devoted to Bessel functions that are needed for the expression of our new interaction potential.

We wish to emphasize that as we follow our method from [9], we will skip details and only focus on what is different: the difficulty comes from the potential $ϕ_{α}$ (involving Bessel functions). As it has a singularity at the origin and presents weaker decay properties at infinity (compared to the nice rescaled Gaussian function from [9]), the estimates for the commutator of the capillary operator and the Lagrangian change of variable (main point from [9]) will require more work.

2. Proof of Theorem 2

2.1. Interaction potential

As announced in the introduction, we first rewrite the system in a non-local shape. Let us focus on the last equation, we can write that (for more clarity we drop the subscripts with α): $\begin{array}{l} - ε^{2} Δ (ρ - c) + α^{2} (ρ - c) = - ε^{2} Δ ρ \end{array}$ which leads to: $\begin{array}{l} α^{2} (ρ - c) = - α^{2} {(- Δ + \frac{α^{2}}{ε^{2}})}^{- 1} Δ ρ = - {(\frac{- Δ}{α^{2}} + \frac{1}{ε^{2}})}^{- 1} Δ ρ . \end{array}$ So that in Fourier variable: $\begin{array}{l} \hat{α^{2} (ρ - c)} (ξ) = \frac{| ξ |^{2}}{\frac{| ξ |^{2}}{α^{2}} + \frac{1}{ε^{2}}} \hat{ρ} (ξ) = ε^{2} \cdot \frac{α^{2}}{ε^{2}} (1 - \frac{1}{\frac{ε^{2}}{α^{2}} | ξ |^{2} + 1}) \hat{ρ} (ξ) . \end{array}$ Then up to choose $κ = ε^{2}$ and replace α by $α / ε$ , from now on we assume that $ε = 1$ and then if we introduce: $\begin{array}{l} D [ρ] = α^{2} (c - ρ), \end{array}$ we have $\begin{matrix} (3) & \hat{D [ρ]} (ξ) = \frac{- | ξ |^{2}}{\frac{| ξ |^{2}}{α^{2}} + 1} \hat{ρ} (ξ) = α^{2} (\frac{1}{\frac{| ξ |^{2}}{α^{2}} + 1} - 1) \hat{ρ} (ξ) . \end{matrix}$ As a consequence, when α is large, $α^{2} (ρ - c)$ formally goes to $Δ ρ$ as for the non-local capillary term from [9], and the object of this article is to prove that the solutions of this system will go to the solutions of the local Korteweg model. Let us now define the interaction potential $ϕ_{α}$ by: $\begin{matrix} (4) & \hat{ϕ_{α}} (ξ) = \frac{1}{\frac{| ξ |^{2}}{α^{2}} + 1} . \end{matrix}$ We have $\int_{R} ϕ_{α} (x) d x = 1$ and $D [ρ] = α^{2} (ϕ_{α} * ρ - ρ)$ . If we put $ϕ = ϕ_{1}$ then $\begin{matrix} (5) & \hat{ϕ} (ξ) = \frac{1}{| ξ |^{2} + 1}, \hat{ϕ_{α}} = \hat{ϕ} (\cdot / α) and ϕ_{α} = α^{d} ϕ (α \cdot) . \end{matrix}$ In some cases we have explicit expressions for this inverse Fourier transform: for all x, $ϕ (x) = C e^{- | x |}$ when $d = 1$ , $ϕ (x) = C^{'} \frac{e^{- | x |}}{| x |}$ when $d = 3$ (we refer to [31]). In the other cases the expression of ϕ involves Bessel functions. Let us begin by recalling that the Fourier transform of a radial function is also radial, more precisely (see for example [31], p. 213) there exists a constant $C_{d}$ such that if $f (x) = f_{0} (| x |)$ for all $x \in R^{d}$ , then its Fourier transform satisfies for all $ξ \in R^{d}$ , $\hat{f} (ξ) = F_{0} (| ξ |)$ where for all $ρ > 0$ $\begin{array}{l} F_{0} (ρ) = \frac{C_{d}}{ρ^{\frac{d}{2} - 1}} \int_{0}^{\infty} J_{\frac{d}{2} - 1} (ρ r) f_{0} (r) r^{\frac{d}{2}} d r, \end{array}$ where $J_{ν}$ denotes the general Bessel function of real index ν. This formulation is related to the Hankel transform, we refer to the Appendix for more details and properties on Bessel functions. Coming back to our problem, we then obtain that for all $x \in R^{d}$ , $\begin{matrix} (6) & ϕ (x) = \frac{C_{d}}{| x |^{\frac{d}{2} - 1}} \int_{0}^{\infty} J_{\frac{d}{2} - 1} (r | x |) \frac{r^{\frac{d}{2}}}{1 + r^{2}} d r . \end{matrix}$ And thanks to the Hankel–Nicholson integrals (we refer for example to [26], p. 330 or [34], p. 434), under the following assumptions: $\begin{array}{l} a > 0, Re (z) > 0, - 1 < Re (ν) < 2 Re (μ) + \frac{3}{2}, \end{array}$ we have the identity: $\begin{array}{l} \int_{0}^{\infty} \frac{t^{ν + 1} J_{ν} (a t)}{{(t^{2} + z^{2})}^{μ + 1}} d t = \frac{a^{μ} z^{ν - μ}}{2^{μ} Γ (μ + 1)} K_{ν - μ} (a z), \end{array}$ where $K_{ν}$ denotes the modified Bessel function of the second kind and index ν (also called Hankel, Schläfti or Weber function). This allows us to finally write that for all $x \in R^{d}$ provided that $d \in {1, 2, 3, 4}$ (from the previous conditions with $ν = \frac{d}{2} - 1$ , $μ = 0$ , $z = 1$ , $a = | x |$ ), $\begin{matrix} (7) & ϕ (x) = \frac{C_{d}}{| x |^{\frac{d}{2} - 1}} K_{\frac{d}{2} - 1} (| x |) . \end{matrix}$

Remark 5.
Another way to understand the limitation on the dimension consists in observing in the integral (6), that if we roughly approximate the Bessel function by $cos (r) r^{- 1 / 2}$ at infinity, then the integrated function has the following asymptotic expansion at infinity: $cos (r) r^{d / 2 - 5 / 2}$ ( $| x | = 1$ for more simplicity).

In fact (7) is also valid for dimensions $d ⩾ 5$ (like the Fourier transform, the Hankel transform can be generalized for tempered distributions). Let us compute the Fourier transform: for all $ξ \in R^{d}$ $\begin{array}{l} \int_{R^{d}} e^{- i x \cdot ξ} \frac{K_{\frac{d}{2} - 1} (| x |)}{| x |^{\frac{d}{2} - 1}} d x = \int_{0}^{\infty} r^{\frac{d}{2}} K_{\frac{d}{2} - 1} (r) (\int_{S^{d - 1}} e^{- i r ω \cdot ξ} d ω) d r, \end{array}$ and thanks to the radial symmetry: $\begin{array}{l} \int_{S^{d - 1}} e^{- i r ω \cdot ξ} d ω = \int_{S^{d - 1}} e^{- i r | ξ | ω \cdot e_{1}} d ω . \end{array}$ Performing a d-dimensional spherical change of variable we obtain that (with $λ = r | ξ |$ ): $\begin{array}{l} \int_{S^{d - 1}} e^{- i λ ω \cdot e_{1}} d ω & = \int_{0}^{π} \int_{0}^{2 π} \dots \int_{0}^{2 π} e^{- i λ cos θ_{1}} {sin}^{d - 2} θ_{1} {sin}^{d - 3} θ_{2} \dots sin θ_{d - 2} d θ_{1} \dots d θ_{d - 1} \\ (8) & = C_{d} \int_{0}^{π} e^{- i λ cos θ_{1}} {sin}^{d - 2} θ_{1} d θ_{1} . \end{array}$ Using the following integral representation of function $I_{ν}$ (for $Re (ν) > - 1 / 2$ see the Appendix for modified Bessel functions $I_{ν}$ and $K_{ν}$ ) $\begin{array}{l} I_{ν} (z) = \frac{z^{ν}}{2^{ν} π^{\frac{1}{2}} Γ (ν + \frac{1}{2})} \int_{0}^{π} e^{- z cos t} {sin}^{d - 2} t d t . \end{array}$ Then thanks to the following identity (here $a = i | ξ |$ and $b = 1$ ): $\begin{array}{l} \int z I_{ν} (a z) K_{ν} (b z) d z = \frac{z}{a^{2} - b^{2}} (a I_{ν + 1} (a z) K_{ν} (b z) + b I_{ν} (a z) K_{ν + 1} (b z)), \end{array}$ we obtain that: $\begin{array}{l} \int_{R^{d}} e^{- i x \cdot ξ} \frac{K_{\frac{d}{2} - 1} (| x |)}{| x |^{\frac{d}{2} - 1}} d x = \frac{2^{\frac{d}{2} - 1} Γ (\frac{d}{2})}{1 + | ξ |^{2}} \end{array}$ so that in (7), $C_{d} = {(2^{\frac{d}{2} - 1} Γ (\frac{d}{2}))}^{- 1}$ . Considering the asymptotics of function $K_{\frac{d}{2} - 1}$ , ϕ is continuous on $R^{d} ∖ {0}$ . Near 0, and for $d ⩾ 3$ , we have $ϕ (x) \sim C_{d} | x |^{2 - d}$ so that $| x | ϕ (x)$ is a $L^{1}$ function on $R^{d}$ .
2.2. Reformulation of the system

We are now able to write the system into a non-local form: $\begin{array}{l} (NSO P_{α}) \{\begin{matrix} \partial_{t} ρ_{α} + div (ρ_{α} u_{α}) = 0, \\ \partial_{t} (ρ_{α} u_{α}) + div (ρ_{α} u_{α} \otimes u_{α}) - A u_{α} + \nabla (P (ρ_{α})) = ρ_{α} κ α^{2} \nabla (ϕ_{α} * ρ_{α} - ρ_{α}), \end{matrix} \end{array}$ with $\begin{array}{l} ϕ_{α} = α^{d} ϕ (α \cdot) with ϕ (x) = \frac{C_{d}}{| x |^{\frac{d}{2} - 1}} K_{\frac{d}{2} - 1} (| x |) . \end{array}$

Remark 6.
From the previous computations, we immediately get that $\begin{array}{l} c_{α} - ρ_{α} = {(- Δ + α I_{d})}^{- 1} Δ ρ_{α} = ϕ_{α} * ρ_{α} - ρ_{α} \end{array}$ that is $c_{α} = ϕ_{α} * ρ_{α}$ . This is why we cannot choose any initial data for the order parameter and take $c_{0} = ϕ_{α} * ρ_{0}$ .

As we consider initial data close to an equilibrium state $(\overline{ρ}, 0)$ we begin with the classical change of function $ρ = \overline{ρ} (1 + q)$ . For simplicity we take $\overline{ρ} = 1$ . The previous system becomes (also denoted by $(NSO P_{α})$ ): $\begin{array}{l} (NSO P_{α}) \{\begin{matrix} \partial_{t} q_{α} + u_{α} . \nabla q_{α} + (1 + q_{α}) div u_{α} = 0, \\ \partial_{t} u_{α} + u_{α} \cdot \nabla u_{α} - A u_{α} + P^{'} (1) \cdot \nabla q_{α} - κ α^{2} \nabla (ϕ_{α} * q_{α} - q_{α}) \\ = K (q_{α}) \cdot \nabla q_{α} - I (q_{α}) A u_{α}, \end{matrix} \end{array}$ where K and I are the real-valued functions defined on $R$ given by: $\begin{array}{l} K (q) = (P^{'} (1) - \frac{P^{'} (1 + q)}{1 + q}) and I (q) = \frac{q}{q + 1} . \end{array}$ The functional spaces we will really use are the following (we refer to the Appendix for definitions of the Besov spaces and hybrid spaces ${\dot{B}}_{α}^{s + 2, s}$ ):
Definition 2.
The space $E_{α}^{s}$ is the set of functions $(q, u)$ in $\begin{array}{l} (C_{b} (R_{+}, {\dot{B}}_{2, 1}^{s - 1} \cap {\dot{B}}_{2, 1}^{s}) \cap L^{1} (R_{+}, {\dot{B}}_{α}^{s + 1, s - 1} \cap {\dot{B}}_{α}^{s + 2, s})) \times {(C_{b} (R_{+}, {\dot{B}}_{2, 1}^{s - 1}) \cap L^{1} (R_{+}, {\dot{B}}_{2, 1}^{s + 1}))}^{d} \end{array}$ endowed with the norm $‖ (q, u) ‖_{E_{α}^{s}} = ‖ (q, u) ‖_{E_{α}^{s} (\infty)}$ where for all t we denote (recall that $ν_{0} = min (μ, ν)$ ) $\begin{array}{l} {‖ (q, u) ‖}_{E_{α}^{s} (t)} \overset{def}{=} & ‖ u ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s - 1}} + ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s - 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s}} \\ (9) & + ν_{0} ‖ u ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{s + 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 1, s - 1}} + ν^{2} ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 2, s}} . \end{array}$
Remark 7.
Due to obvious simplifications we slightly changed the notations for $E_{α}^{s}$ and ${\dot{B}}_{α}^{s + 2, s}$ : with the notations from [9] these spaces would have been respectively denoted by $E_{1 / α}^{s}$ and ${\dot{B}}_{1 / α}^{s + 2, s}$ .

We will now follow the tracks of [7] and [9] to prove the results. Classically in the study in critical spaces of compressible Navier–Stokes-type systems (see [6,12,20]), the proofs of Theorems 2 and 3 (see [7], Section 2) rely on key a priori estimates on the following advected linear system ( $α > 0$ is fixed and for more simplicity we write $(q, u)$ instead of $(q_{α}, u_{α})$ ): $\begin{array}{l} (LO P_{α}) \{\begin{matrix} \partial_{t} q + v \cdot \nabla q + div u = F, \\ \partial_{t} u + v \cdot \nabla u - A u + p \nabla q - κ α^{2} \nabla (ϕ_{α} * q - q) = G, \end{matrix} \end{array}$ where $p = P^{'} (1) > 0$ and $\begin{array}{l} A u = μ Δ u + (λ + μ) \nabla div u . \end{array}$ Although the potential function is different from the Gaussian from [7], we can easily adapt the energy methods and results from this paper. Here we will directly focus on more refined estimates as in [9] and use them in the proof of the last theorem: we can prove that the estimates are similar up to slight changes in the constants:
Theorem 4.
Let $α > 0$ , $- \frac{d}{2} + 1 < s < \frac{d}{2} + 1$ , $I = [0, T [$ or $[0, + \infty [$ and $v \in L^{1} (I, {\dot{B}}_{2, 1}^{\frac{d}{2} + 1}) \cap L^{2} (I, {\dot{B}}_{2, 1}^{\frac{d}{2}})$ . Assume that $(q, u)$ is a solution of system $(LO P_{α})$ defined on I. There exists $α_{0} > 0$ , a constant $C > 0$ depending on d, s such that if $α ⩾ α_{0}$ , for all $t \in I$ (denoting $ν = μ + 2 λ$ and $ν_{0} = min (ν, μ)$ ), $\begin{array}{l} {‖ (q, u) ‖}_{E_{α}^{s} (t)} ⩽ & C_{p, \frac{ν^{2}}{4 κ}} e^{C_{p, \frac{ν^{2}}{4 κ}} C_{visc} \int_{0}^{t} (‖ \nabla v (τ) ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}} + ‖ v (τ) ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}}^{2}) d τ} \\ \times (‖ u_{0} ‖_{{\dot{B}}_{2, 1}^{s - 1}} + ‖ q_{0} ‖_{{\dot{B}}_{2, 1}^{s - 1}} + ν ‖ q_{0} ‖_{{\dot{B}}_{2, 1}^{s}} \\ (10) & + ‖ F ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{s - 1}} + ν ‖ F ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{s}} + ‖ G ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{s - 1}}), \end{array}$ where $\begin{array}{l} \{\begin{matrix} C_{p, \frac{ν^{2}}{4 κ}} = C max (\sqrt{p}, \frac{1}{\sqrt{p}}) max (\frac{4 κ}{ν^{2}}, {(\frac{ν^{2}}{4 κ})}^{2}), \\ C_{visc} = \frac{1 + | λ + μ | + μ + ν}{ν_{0}} + max (1, \frac{1}{ν^{3}}) . \end{matrix} \end{array}$
Remark 8.
The coefficient $C_{visc}$ satisfies: $\begin{array}{l} C_{visc} = \{\begin{matrix} \frac{1 + 2 ν}{μ} + max (1, \frac{1}{ν^{3}}) & if λ + μ > 0, \\ \frac{1 + 2 μ}{ν} + max (1, \frac{1}{ν^{3}}) & if λ + μ ⩽ 0 \end{matrix} \end{array}$ and when both viscosities are small, we simply have $C_{visc} ⩽ max (1, \frac{1}{ν_{0}^{3}})$ .

Up to the necessary changes due to the Fourier expression of the potential, we use the very same methods as is [9] so we will skip details and only state the linear estimates and precise the matrix and eigenvalues for the convenience of the reader.
2.3. Linear estimates

As in [6] and [9] the first step to prove Theorem 4 is to obtain estimates for the following linearized system: $\begin{array}{l} ({OP}_{α}) \{\begin{matrix} \begin{matrix} \partial_{t} q + div u = F, \\ \partial_{t} u - A u + p \nabla q - κ α^{2} \nabla (ϕ_{α} * q - q) = G, \end{matrix} \end{matrix} \end{array}$ with $\begin{array}{l} A u = μ Δ u + (λ + μ) \nabla div u . \end{array}$ In this article, we will use the following frequency-localized estimates:

Proposition 1.
Let $α > 0$ , $s \in R$ , $I = [0, T [$ or $[0, + \infty [$ . Assume that $(q, u)$ is a solution of system $({OP}_{α})$ defined on I. There exists $α_{0} > 0$ , a constant $C > 0$ depending on d, s, $c_{0}$ and $C_{0}$ such that if $α ⩾ α_{0}$ , for all $t \in I$ (we recall that $ν_{0}$ and $C_{p, \frac{ν^{2}}{4 κ}}$ are defined in the previous theorem), and for all $j \in Z$ , $\begin{array}{l} ‖ {\dot{Δ}}_{j} u ‖_{L_{t}^{\infty} L^{2}} + ν_{0} 2^{2 j} ‖ {\dot{Δ}}_{j} u ‖_{L_{t}^{1} L^{2}} + (1 + ν 2^{j}) (‖ {\dot{Δ}}_{j} q ‖_{L_{t}^{\infty} L^{2}} + ν min (α^{2}, 2^{2 j}) ‖ {\dot{Δ}}_{j} q ‖_{L_{t}^{1} L^{2}}) \\ (11) & ⩽ C_{p, \frac{ν^{2}}{4 κ}} ((1 + ν 2^{j}) ‖ {\dot{Δ}}_{j} q_{0} ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} u_{0} ‖_{L^{2}} + (1 + ν 2^{j}) ‖ {\dot{Δ}}_{j} F ‖_{L_{t}^{1} L^{2}} + ‖ {\dot{Δ}}_{j} G ‖_{L_{t}^{1} L^{2}}) . \end{array}$
Proof.
This is an adaptation of the arguments in the proof of Proposition 2 from [9]. We refer the reader to Section 2 therein and we will only point out the differences coming from the eigenvalues and eigenvectors: as in [12] or [6] we first introduce the Helmholtz decomposition of u. Defining the pseudo-differential operator Λ by $Λ f = F^{- 1} (| \cdot | \hat{f} (\cdot))$ , we set: $\begin{matrix} (12) & \{\begin{matrix} v = Λ^{- 1} div u, \\ w = Λ^{- 1} curl u \end{matrix} \end{matrix}$ then $u = - Λ^{- 1} \nabla v + Λ^{- 1} curl w$ and the system turns into: $\begin{array}{l} (L_{α}^{'}) \{\begin{matrix} \partial_{t} q + Λ v = F, \\ \partial_{t} v - ν Δ v - p Λ q + κ α^{2} Λ (ϕ_{α} * q - q) = Λ^{- 1} div G, \\ \partial_{t} w - μ Δ w = Λ^{- 1} curl G . \end{matrix} \end{array}$ The last equation is a decoupled heat equation, easily estimated in Besov spaces (see [2], Chapter 2). As in [9] we can focus on the first two lines and compute the eigenvalues and eigenvectors of the matrix associated to the Fourier transform of the system: $\begin{array}{l} \partial_{t} (\begin{matrix} \hat{q} \\ \hat{v} \end{matrix}) = A (ξ) (\begin{matrix} \hat{q} \\ \hat{v} \end{matrix}) with A (ξ) : = (\begin{matrix} 0 & - | ξ | \\ | ξ | (p + κ \frac{| ξ |^{2}}{\frac{| ξ |^{2}}{α^{2}} + 1}) & - ν | ξ |^{2} \end{matrix}) . \end{array}$ The discriminant of the characteristic polynomial of $A (ξ)$ is: $\begin{array}{l} Δ (ξ) = | ξ |^{2} (ν^{2} | ξ |^{2} - 4 (p + κ \frac{| ξ |^{2}}{\frac{| ξ |^{2}}{α^{2}} + 1})), \end{array}$ and thanks to the variations of function $\begin{array}{l} f_{α} : x \mapsto ν^{2} x - 4 (p + κ \frac{x}{\frac{x}{α^{2}} + 1}) = ν^{2} x - 4 (p + κ α^{2}) + \frac{4 κ α^{2}}{\frac{x}{α^{2}} + 1}, \end{array}$ we obtain the existence of a unique threshold $x_{α} > 0$ such that $\begin{array}{l} Δ (ξ) \{\begin{matrix} < 0 & if | ξ |^{2} < x_{α}, \\ > 0 & if | ξ |^{2} > x_{α} . \end{matrix} \end{array}$ We emphasize that this function has the same variations as in the case of [9]: when $\frac{ν^{2}}{4 K} ⩾ 1$ , $f_{α}$ is an increasing function on $R_{+}$ , and when $\frac{ν^{2}}{4 K} < 1$ , $f_{α}$ is decreasing in $[0, α^{2} (\frac{2 \sqrt{K}}{ν} - 1)]$ and then increasing. □

2.4. Advected linear estimates

The difficulties and methods exposed here are the same as in [9], so we will roughly explain them and focus on what is new.

In order to prove Theorem 4, a natural idea is to use Proposition 1 and put the advection terms as external forces. Unfortunately, there are some obstacles: the main problem is that in $v \cdot \nabla q$ , the term ${\dot{T}}_{v} \nabla q$ can be estimated in ${\dot{B}}_{2, 1}^{s - 1}$ but not in ${\dot{B}}_{2, 1}^{s}$ because it is not enough regular in high frequencies.

A direct use of the linear estimates will be useful only for the low frequencies ( $j ⩽ 0$ ), and in the high frequency regime ( $j > 0$ ), we will perform a Lagrangian change of variable (as in [6,9,13,21,23]) in order to get rid of $v \cdot \nabla q$ . We then aim to use on the new system our linear estimates but we have to be careful with the external force terms introduced by the change of variable. Most of the work in [9] was to provide estimates on the commutator of the non-local operator from the capillarity term and the Lagrangian change of variable.

For all $j \in Z$ and $t \in I$ , we introduce: $\begin{matrix} (13) & \begin{matrix} U_{j} (t) = & ‖ {\dot{Δ}}_{j} u ‖_{L_{t}^{\infty} L^{2}} + ν_{0} 2^{2 j} ‖ {\dot{Δ}}_{j} u ‖_{L_{t}^{1} L^{2}} \\ + (1 + ν 2^{j}) (‖ {\dot{Δ}}_{j} q ‖_{L_{t}^{\infty} L^{2}} + ν min (α^{2}, 2^{2 j}) ‖ {\dot{Δ}}_{j} q ‖_{L_{t}^{1} L^{2}}) \end{matrix} \end{matrix}$ and $\begin{matrix} (14) & \begin{matrix} U (t) = & ‖ u ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s - 1}} + ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s - 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{s}} \\ + ν_{0} ‖ u ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{s + 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 1, s - 1}} + ν^{2} ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{s + 2, s}} . \end{matrix} \end{matrix}$

2.4.1. Low frequencies

For the low frequencies, we obtain (see [9], Section 3.1 for details) that there exists a nonnegative summable sequence whose sum is 1, denoted by ${(c_{j} (t))}_{j \in Z}$ such that for all $j ⩽ 0$ , $K > 0$ (to be chosen later), and if $α ⩾ N_{1} / log 2$ (we refer to [9], Section 3.1 for this, and to the Appendix for $N_{1}$ which is a constant related to $c_{0}$ and $C_{0}$ in the Littlewood–Paley decomposition, such that if $| j - l | > N_{1}$ then ${\dot{Δ}}_{j} \circ {\dot{Δ}}_{l} = 0$ , in our choice of $c_{0}$ and $C_{0}$ , $N_{1} = 1$ ). $\begin{array}{l} U_{j} (t) ⩽ & C_{p, \frac{ν^{2}}{4 κ}} [U_{j} (0) + (1 + ν 2^{j}) ‖ {\dot{Δ}}_{j} F ‖_{L_{t}^{1} L^{2}} + ‖ {\dot{Δ}}_{j} G ‖_{L_{t}^{1} L^{2}} \\ + \frac{1}{2 K} 2^{- j (s - 1)} \int_{0}^{t} c_{j} (τ) (ν_{0} ‖ u ‖_{{\dot{B}}_{2, 1}^{s + 1}} + ν ‖ q ‖_{{\dot{B}}_{α}^{s + 1, s - 1}} + ν^{2} ‖ q ‖_{{\dot{B}}_{α}^{s + 2, s}}) d τ \\ + C^{2} \frac{K}{2} 2^{- j (s - 1)} \\ (15) & \times \int_{0}^{t} c_{j} (τ) ((max (1, \frac{1}{ν^{3}}) + \frac{1}{ν_{0}}) {‖ v (τ) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2}}}^{2} + C {‖ v (τ) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}) U (τ) d τ] . \end{array}$

Remark 9.
As pointed out in [9] (Remark 29) using the linear estimates for the low frequency case, allows us to get rid of another difficulty introduced by the change of variables: in low frequencies some of the additional external force terms have too much regularity to be absorbed by the left-hand side, and not enough regularity to be controlled with a view to apply the Gronwall lemma. The only way to control them would be to use interpolation arguments that would introduce linear time dependant coefficients, and prevent us to get global in time results.

2.4.2. Lagrangian change of coordinates

As explained, in order to get rid of the advection terms involved in system $(LO P_{α})$ the first step is to consider the following localized equations (as usual we set $f_{j} = {\dot{Δ}}_{j} f$ …): $\begin{array}{l} \{\begin{matrix} \partial_{t} q_{j} + {\dot{S}}_{j - 1} v \cdot \nabla q_{j} + div u_{j} = f_{j}, \\ \partial_{t} u_{j} + {\dot{S}}_{j - 1} v \cdot \nabla u_{j} - A u_{j} + p \nabla q_{j} - κ α^{2} \nabla (ϕ_{α} * q_{j} - q_{j}) = g_{j}, \end{matrix} \end{array}$ where the external force terms are defined by: $\begin{array}{l} f_{j} = F_{j} + ({\dot{S}}_{j - 1} v \cdot \nabla q_{j} - {\dot{Δ}}_{j} (v \cdot \nabla q)) and g_{j} = G_{j} + ({\dot{S}}_{j - 1} v \cdot \nabla u_{j} - {\dot{Δ}}_{j} (v \cdot \nabla u)) . \end{array}$ Both of these terms can be estimated thanks to the following commutator estimate from [13] (we refer to Lemma B.1 from Appendix B):

Lemma 1 ([13]).

There exists a sequence ${(c_{j})}_{j \in Z} \in l^{1} (Z)$ such that $‖ c ‖_{l^{1} (Z)} = 1$ and a constant $C = C (d, σ)$ such that for all $j \in Z$ , $\begin{array}{l} {‖ {\dot{S}}_{j - 1} v \cdot \nabla h_{j} - {\dot{Δ}}_{j} (v \cdot \nabla h) ‖}_{L^{2}} ⩽ C_{j} 2^{- j σ} ‖ \nabla v ‖_{{\dot{B}}_{2, \infty}^{\frac{d}{2}} \cap L^{\infty}} ‖ h ‖_{{\dot{B}}_{2, 1}^{σ}} . \end{array}$

In order to perform the change of variable we define $ψ_{j, t}$ as the flow associated to ${\dot{S}}_{j - 1} v$ : $\begin{matrix} (16) & \{\begin{matrix} \partial_{t} ψ_{j, t} (x) = {\dot{S}}_{j - 1} v (t, ψ_{j, t} (x)), \\ ψ_{j, 0} (x) = x . \end{matrix} \end{matrix}$ we can also write: $\begin{array}{l} ψ_{j, t} (x) = x + \int_{0}^{t} {\dot{S}}_{j - 1} v (τ, ψ_{j, τ} (x)) d τ . \end{array}$ Thanks to Propositions 4 and 5 from the Appendix (we refer to [13] or [6]), there exists a constant C such that: $\begin{matrix} (17) & \begin{matrix} ‖ g \circ ψ_{j, t} ‖_{L^{p}} ⩽ e^{C V} ‖ g ‖_{L^{p}} for all function g in L^{p}, \\ {‖ D ψ_{j, t}^{\pm} ‖}_{L^{\infty}} ⩽ e^{C V}, \\ {‖ D ψ_{j, t}^{\pm} - I_{d} ‖}_{L^{\infty}} ⩽ e^{C V} - 1, \\ {‖ D^{k} ψ_{j, t}^{\pm} ‖}_{L^{\infty}} ⩽ C 2^{(k - 1) j} (e^{C V} - 1) for k ⩾ 2, \end{matrix} \end{matrix}$ where $\begin{matrix} (18) & V (t) \overset{def}{=} \int_{0}^{t} {‖ \nabla v (τ) ‖}_{L^{\infty}} d τ . \end{matrix}$ As explained in [9], considering some of the additional external force terms at the point $ψ_{j, t}^{- 1} (x)$ instead of x will help getting uniform estimates with respect to α, so the Jacobian determinant of the change of variable will play a crucial role in our computations (contrary to the case of Lemma 2.6 from [2] where it produces a term that we are not able to sum here). $\begin{matrix} (19) & \{\begin{matrix} det (D ψ_{j, t} (x)) = e^{\int_{0}^{t} (div {\dot{S}}_{j - 1} v) (τ, ψ_{j, τ (x)}) d τ}, \\ det (D ψ_{j, t}^{- 1} (x)) = e^{- \int_{0}^{t} (div {\dot{S}}_{j - 1} v) (τ, X_{j} (τ, t, x)) d τ} = e^{- \int_{0}^{t} (div {\dot{S}}_{j - 1} v) (τ, ψ_{j, τ} \circ ψ_{j, t}^{- 1} (x)) d τ}, \end{matrix} \end{matrix}$ where $X_{j} (τ, t, x)$ denotes the two parameter flow associated to ${\dot{S}}_{j - 1} v$ (we refer to (75) in the Appendix).

Let us now perform the Lagrangian change of variable, for a function h, we define $\tilde{h} = h \circ ψ_{j, t} = h (t, ψ_{j, t})$ . Then we have $\partial_{t} {\tilde{q}}_{j} (t, x) = (\partial_{t} q_{j} + {\dot{S}}_{j - 1} v \cdot \nabla q_{j}) (t, ψ_{j, t} (x))$ , which provides the following system: $\begin{matrix} (20) & \{\begin{matrix} \begin{matrix} \partial_{t} {\tilde{q}}_{j} + div {\tilde{u}}_{j} = {\tilde{f}}_{j} + R_{j}^{1}, \\ \partial_{t} {\tilde{u}}_{j} - A {\tilde{u}}_{j} + p \nabla {\tilde{q}}_{j} - κ α^{2} \nabla (ϕ_{α} * {\tilde{q}}_{j} - {\tilde{q}}_{j}) = {\tilde{g}}_{j} + R_{j}^{2} + R_{j}^{3} + κ R_{j}, \end{matrix} \end{matrix} \end{matrix}$ where the remainder terms $R_{q}^{1}$ , $R_{q}^{2}$ and $R_{q}^{3}$ are estimated exactly as in [6] and [9]. We refer to [9] for details. As in [9], the only difference with [6] is the following additional remainder term: $\begin{matrix} (21) & R_{j} = α^{2} (ϕ_{α} * \nabla q_{j} - \nabla q_{j}) \circ ψ_{j, t} - α^{2} (ϕ_{α} * \nabla {\tilde{q}}_{j} - \nabla {\tilde{q}}_{j}) . \end{matrix}$ Thanks to the definition of $ϕ_{α}$ and ϕ we can also write that: $\begin{matrix} (22) & L_{α} (f) \overset{def}{=} α^{2} (ϕ_{α} * f - f) = α^{2} \int_{R^{d}} ϕ (z) (f (x - \frac{z}{α}) - f (x)) d z . \end{matrix}$ As in [9] most of the work consists in obtaining bounds in $L_{t}^{1} L^{2}$ that are uniform with respect to α, and go to zero when t is small. Dealing with $R_{j}$ is the object of the rest of this section.

2.4.3. Precisions on the capillary term

As explained, we will skip detail for what is close to the study in [9] and only point out what is different. Let us first go back to the convolution term written in (22): for a function f, $\begin{array}{l} α^{2} \hat{(ϕ_{α} * f - f)} (ξ) = - \frac{| ξ |^{2}}{1 + \frac{| ξ |^{2}}{α^{2}}} \hat{f} (ξ) . \end{array}$ Similarly to [9] we obtain the following equivalence, giving a smooth interpretation of the hybrid norm. Here again, instead of a fixed frequency threshold there is a continuous transition from the parabolically regularized low frequencies and the damped high frequencies:

Proposition 2.
For any suitable function f and any $s \in R$ , we have: $\begin{array}{l} ‖ f ‖_{{\dot{B}}_{α}^{s + 2, s}} & = \sum_{j \in Z} min (α^{2}, 2^{2 j}) 2^{j s} ‖ {\dot{Δ}}_{j} f ‖_{L^{2}} \\ (23) & \sim \sum_{j \in Z} \frac{2^{2 j}}{1 + \frac{2^{2 j}}{α^{2}}} 2^{j s} ‖ {\dot{Δ}}_{j} f ‖_{L^{2}} \sim {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{s}} . \end{array}$
Proof.
Thanks to the monotonicity of function $x \mapsto x / (1 + x)$ , we easily obtain that for all $j \in Z$ : $\begin{array}{l} \frac{1}{2} min (α^{2}, 2^{2 j}) ⩽ \frac{2^{2 j}}{1 + \frac{2^{2 j}}{α^{2}}} ⩽ min (α^{2}, 2^{2 j}) . \end{array}$ We refer to [9] for details. □
Remark 10.
In the $L^{r}$ -setting, we can prove that for all $j \in Z$ : $\begin{array}{l} {‖ α^{2} (ϕ_{α} * {\dot{Δ}}_{j} f - {\dot{Δ}}_{j} f) ‖}_{L^{r}} ⩽ C min (α^{2}, 2^{2 j}) ‖ {\dot{Δ}}_{j} f ‖_{L^{r}} . \end{array}$

2.4.4. Estimates on the capillary remainder $R_{j}$

This section is devoted to giving estimates on the capillary term introduced in (21). As $\begin{array}{l} \nabla {\tilde{q}}_{j} = \nabla (q_{j} \circ ψ_{j, t}) = \nabla q_{j} \circ ψ_{j, t} \times D ψ_{j, t} = \nabla q_{j} \circ ψ_{j, t} \times (D ψ_{j, t} - I_{d}) + \nabla q_{j} \circ ψ_{j, t}, \end{array}$ we obtain the following decomposition: $R_{j} = I_{j} + {II}_{j}$ with $\begin{matrix} (24) & \{\begin{matrix} I_{j} = α^{2} (ϕ_{α} * g_{j} - g_{j}) where g_{j} = \nabla q_{j} \circ ψ_{j, t} \times (I_{d} - D ψ_{j, t}), \\ {II}_{j} = α^{2} (ϕ_{α} * \nabla q_{j} - \nabla q_{j}) \circ ψ_{j, t} - α^{2} (ϕ_{α} * (\nabla q_{j} \circ ψ_{j, t}) - \nabla q_{j} \circ ψ_{j, t}) . \end{matrix} \end{matrix}$ The term $I_{j}$ is dealt exactly as in [9] (see (3.104), Section 3.5), and all the difficulty consists in estimating (locally in frequency) the commutator between the Lagrangian change of variable and the non-local operator $L_{α}$ defined in (22). For a function f, and all $j \in Z$ we set $f_{j} = {\dot{Δ}}_{j} f$ and: $\begin{matrix} (25) & {II}_{j}^{'} = {II}_{j}^{'} (f) = α^{2} (ϕ_{α} * f_{j} - f_{j}) \circ ψ_{j, t} - α^{2} (ϕ_{α} * (f_{j} \circ ψ_{j, t}) - f_{j} \circ ψ_{j, t}) . \end{matrix}$

Theorem 5.
Let $σ \in R$ . There exists a constant $C = C_{σ, d}$ such that for all $f \in {\dot{B}}_{α}^{σ + 2, σ}$ , there exists a summable positive sequence ${(c_{j} (f))}_{j \in Z}$ whose sum is 1 such that for all t so small that $\begin{matrix} (26) & e^{2 C V} - 1 ⩽ \frac{1}{2} \end{matrix}$ and for all $j \in Z$ , $\begin{array}{l} {‖ {II}_{j}^{'} (f) ‖}_{L^{2}} ⩽ C e^{C V} (V + e^{2 C V} - 1) c_{j} (f) 2^{- j σ} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}}, \end{array}$ where $V (t) = \int_{0}^{t} ‖ \nabla v (τ) ‖_{L^{\infty}} d τ$ .
Remark 11.
As a by-product we obtain that under the previous assumptions, if t is small enough, $\begin{array}{l} α^{2} {‖ (ϕ_{α} * {\dot{Δ}}_{j} f) \circ ψ_{j, t} - ϕ_{α} * ({\dot{Δ}}_{j} f \circ ψ_{j, t}) ‖}_{L^{2}} \\ ⩽ C e^{C V} (V + e^{2 C V} - 1) c_{j} (f) 2^{- j σ} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} . \end{array}$ Remark that neither of the left-hand side terms are spectrally localized.
Proof of Theorem 5.
We refer to [9] for details and here we will only focus on what changes. The first step is to obtain pointwise estimates and then $L^{2}$ estimates: as in the works of T. Hmidi, S. Keraani, H. Abidi and M. Zerguine ([21–23] and [24]) we wish to retrieve the desired Besov norm thanks to an equivalent expression of ${II}_{j}^{'}$ as an integral formulation involving finite differences of f of order 1 (that is expressions of the type $τ_{- y} f - f$ where $τ_{- y} f (x) = f (x + y)$ ) or order 2. Like in [9] we need to directly consider ${II}_{j}^{'} (ψ_{j, t}^{- 1} (x))$ instead of simply ${II}_{j}^{'} (x)$ and we obtain: $\begin{array}{l} {II}_{j}^{'} (ψ_{j, t}^{- 1} (x)) = α^{2} \int_{R^{d}} ϕ (z) (f_{j} (x - \frac{z}{α}) - f_{j} (x)) \\ (27) & \times [1 - \frac{ϕ (α (ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x - \frac{z}{α})))}{ϕ (z)} e^{- \int_{0}^{t} (div {\dot{S}}_{j - 1} v) (τ, X_{j} (τ, t, x - \frac{z}{α})) d τ}] d z . \end{array}$ Remark 12.
We emphasize here that the previous quotient is well defined near zero and we refer to Section 2.2.

We have to face the same problem as in [9]: as we want to estimate the $L^{2}$ norm of this quantity, that is a Besov norm with integer regularity index $s = 0$ , the finite difference of order 1 will not be sufficient for our need, and we will have to introduce finite differences of order 2 (this is a classical problem for integer indices). Indeed, using the present quantity would only involve a term in $| y | / α$ and when estimating in low frequencies, there would be either an extra multiplicative coefficient α or an extra derivative term $2^{- j}$ (that would prevent any convergence when $- j$ is large). To be able to do a correct estimate we need at least $| y |^{2} / α^{2}$ .

For this, we simply write ${II}_{j}^{'} = \frac{1}{2} ({II}_{j}^{'} + {II}_{j}^{'})$ , and perform the change of variable $z = - y$ in the second integral. If we set: $\begin{matrix} (28) & \{\begin{matrix} R_{-} = \frac{ϕ (α (ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x - \frac{z}{α})))}{ϕ (z)}, \\ R_{+} = \frac{ϕ (α (ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x + \frac{z}{α})))}{ϕ (z)}, \\ B_{-} = \int_{0}^{t} (div {\dot{S}}_{j - 1} v) (τ, X_{j} (τ, t, x - \frac{z}{α})) d τ, \\ B_{+} = \int_{0}^{t} (div {\dot{S}}_{j - 1} v) (τ, X_{j} (τ, t, x + \frac{z}{α})) d τ, \end{matrix} \end{matrix}$ then $\begin{matrix} (29) & \begin{matrix} {II}_{j}^{'} (ψ_{j, t}^{- 1} (x)) = & \frac{α^{2}}{2} \int_{R^{d}} ϕ (z) (f_{j} (x - \frac{z}{α}) - f_{j} (x)) [1 - R_{-} e^{- B_{-}}] d z \\ + \frac{α^{2}}{2} \int_{R^{d}} ϕ (z) (f_{j} (x + \frac{z}{α}) - f_{j} (x)) [1 - R_{+} e^{- B_{+}}] d z \\ = & II I_{j} (x) + {IV}_{j} (x), \end{matrix} \end{matrix}$ where $\begin{matrix} (30) & \{\begin{matrix} {III}_{j} (x) = \frac{α^{2}}{2} \int_{R^{d}} ϕ (z) (f_{j} (x - \frac{z}{α}) + f_{j} (x + \frac{z}{α}) - 2 f_{j} (x)) [1 - R_{-} e^{- B_{-}}] d z, \\ {IV}_{j} (x) = \frac{α^{2}}{2} \int_{R^{d}} ϕ (z) (f_{j} (x + \frac{z}{α}) - f_{j} (x)) [R_{-} e^{- B_{-}} - R_{+} e^{- B_{+}}] d z . \end{matrix} \end{matrix}$ Taking $L^{2}$ norms we have: $\begin{matrix} (31) & {‖ {II}_{j}^{'} \circ ψ_{j, t}^{- 1} ‖}_{L^{2}} ⩽ ‖ {III}_{j} ‖_{L^{2}} + ‖ {IV}_{j} ‖_{L^{2}}, \end{matrix}$ and thanks to estimates on the Jacobian determinant of the flow (see (17)), Theorem 5 is immediately implied by the following proposition: Proposition 3.
Under the previous assumptions, there exist a positive constant $C = C_{σ, d}$ and a nonnegative sequence ${(c_{j} = c_{j} (f))}_{j \in Z}$ whose sum is 1, such that if t is so small that $e^{2 C V (t)} - 1 ⩽ \frac{1}{2}$ , we have: $\begin{array}{l} ‖ {III}_{j} ‖_{L^{2}} + ‖ {IV}_{j} ‖_{L^{2}} ⩽ C (V + e^{2 C V} - 1) e^{C V} 2^{- j σ} c_{j} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} . \end{array}$

To prove this result we will successively prove the following lemmas: Lemma 2.
There exists a constant C only depending on the dimension d such that for all $j \in Z$ , f, and all t is so small that $e^{2 C V (t)} - 1 ⩽ \frac{1}{2}$ , $\begin{array}{l} ‖ {III}_{j} ‖_{L^{2}} ⩽ & C e^{C V} (V + e^{2 C V} - 1) \\ (32) & \times α^{2} \int_{R^{d}} e^{- \frac{| z |}{8}} \frac{1 + | z |^{- \frac{1}{2}}}{| z |^{d - \frac{3}{2}}} {‖ f_{j} (\cdot - \frac{z}{α}) + f_{j} (\cdot + \frac{z}{α}) - 2 f_{j} (\cdot) ‖}_{L^{2}} d z \end{array}$ and $\begin{array}{l} ‖ {IV}_{j} ‖_{L^{2}} ⩽ & C e^{C V} (V + e^{2 C V} - 1) \\ (33) & \times α^{2} \int_{R^{d}} e^{- \frac{| z |}{8}} \frac{1 + | z |^{- \frac{1}{2}}}{| z |^{d - \frac{3}{2}}} min (1, \frac{2^{j} | z |}{α}) {‖ f_{j} (\cdot + \frac{z}{α}) - f_{j} (\cdot) ‖}_{L^{2}} d z, \end{array}$ where V is defined in ( 18 ).
Lemma 3.
For all $σ \in R$ there exists a constant $C_{σ, d}$ such that for any $f \in {\dot{B}}_{2, 1}^{σ}$ , there exists a nonnegative summable sequence ${(c_{j} (f))}_{j \in Z}$ with $‖ c_{j} (f) ‖_{l^{1} (Z)} = 1$ , such that $\begin{array}{l} α^{2} \int_{R^{d}} e^{- \frac{| z |}{8}} \frac{1 + | z |^{- \frac{1}{2}}}{| z |^{d - \frac{3}{2}}} {‖ f_{j} (\cdot - \frac{z}{α}) + f_{j} (\cdot + \frac{z}{α}) - 2 f_{j} (\cdot) ‖}_{L^{2}} d z \\ + α^{2} \int_{R^{d}} e^{- \frac{| z |}{8}} \frac{1 + | z |^{- \frac{1}{2}}}{| z |^{d - \frac{3}{2}}} min (1, \frac{2^{j} | z |}{α}) {‖ f_{j} (\cdot + \frac{z}{α}) - f_{j} (\cdot) ‖}_{L^{2}} d z \\ (34) & ⩽ C_{σ, d} 2^{- j σ} c_{j} (f) {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} . \end{array}$
Proof of Lemma 2.
A direct estimate gives: $\begin{array}{l} ‖ {III}_{j} ‖_{L^{2}} ⩽ \frac{α^{2}}{2} \int_{R^{d}} ϕ (z) {‖ f_{j} (\cdot - \frac{z}{α}) + f_{j} (\cdot + \frac{z}{α}) - 2 f_{j} (\cdot) ‖}_{L_{x}^{2}} {‖ 1 - R_{-} e^{- B_{-}} ‖}_{L_{x}^{\infty}} d z \end{array}$ and $\begin{array}{l} ‖ {IV}_{j} ‖_{L^{2}} ⩽ \frac{α^{2}}{2} \int_{R^{d}} ϕ (z) {‖ f_{j} (\cdot + \frac{z}{α}) - f_{j} (\cdot) ‖}_{L_{x}^{2}} ‖ R_{-} e^{- B_{-}} - R_{+} e^{- B_{+}} ‖_{L_{x}^{\infty}} d z . \end{array}$ So that we first have to focus on the $L^{\infty}$ norms: Lemma 4.
Under the same assumptions, for all $η \in] 0, 1 [$ , there exists three constants C, $C_{d}$ and $C_{d, η}$ such that for all $j \in Z$ , all f, and all $t > 0$ so small that $e^{2 C V (t)} - 1 ⩽ \frac{1}{2}$ , we have: $\begin{array}{l} {‖ 1 - R_{-} e^{- B_{-}} ‖}_{L_{x}^{\infty}} ⩽ C_{d, η} e^{C_{d} V (t)} (e^{2 C V} - 1 + V) e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}), \\ {‖ R_{-} e^{- B_{-}} - R_{+} e^{- B_{+}} ‖}_{L_{x}^{\infty}} \\ (35) & ⩽ C_{d, η} e^{C_{d} V (t)} (e^{2 C V} - 1 + V) min (1, \frac{2^{j} | z |}{α}) e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}) . \end{array}$
Proof.
As in [9] every term is close to 1 when t is small and we obtain the result by carefully estimating the differences between these terms. For this we write: $\begin{matrix} (36) & \{\begin{matrix} Q_{1} = 1 - R_{-} e^{- B_{-}} = 1 - R_{-} + R_{-} (1 - e^{- B_{-}}), \\ Q_{2} = R_{-} e^{- B_{-}} - R_{+} e^{- B_{+}} = (R_{-} - R_{+}) e^{- B_{-}} + R_{+} (e^{- B_{-}} - e^{- B_{+}}) . \end{matrix} \end{matrix}$ Thanks to the following elementary consequence of the mean-value theorem, we obtain (we refer to [9] for details) that there exists a constant $C > 0$ such that (we refer to Theorem 5 for the definition of V): $\begin{matrix} (37) & \{\begin{matrix} | 1 - e^{- B_{-}} | ⩽ C V (t) e^{C V (t)}, \\ | e^{- B_{-}} | ⩽ e^{C V (t)}, \\ | e^{- B_{-}} - e^{- B_{+}} | ⩽ 2 C V (t) e^{2 C V (t)} min (1, \frac{| z | 2^{j}}{α}) . \end{matrix} \end{matrix}$ We now turn to the estimates involving $R_{\pm}$ . As obtained in (7), the interaction potential can be expressed thanks to the modified Bessel function of second kind $K_{\frac{d}{2} - 1}$ : $\begin{array}{l} ϕ (x) = \frac{C_{d}}{| x |^{\frac{d}{2} - 1}} K_{\frac{d}{2} - 1} (| x |), \end{array}$ so that if we denote by $Y_{\pm}$ the variation ratio of the flow: $\begin{matrix} (38) & Y_{\pm} \overset{def}{=} \frac{| ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |}{\frac{| z |}{α}} \end{matrix}$ we have the expression: $\begin{matrix} (39) & R_{\pm} = {(\frac{1}{Y_{\pm}})}^{\frac{d}{2} - 1} \frac{K_{\frac{d}{2} - 1} (α | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |)}{K_{\frac{d}{2} - 1} (| z |)} = {(\frac{1}{Y_{\pm}})}^{\frac{d}{2} - 1} \frac{K_{\frac{d}{2} - 1} (| z | Y_{\pm})}{K_{\frac{d}{2} - 1} (| z |)} . \end{matrix}$ We bound the first factor thanks to the following estimates on the variation ratio: Lemma 5 ([9], Section 3.4).

Under the general previous assumptions, for all $x, z \in R^{d}$ with $z \neq 0$ , we have: $\begin{array}{l} e^{- C V (t)} ⩽ Y_{\pm} ⩽ e^{C V (t)} \end{array}$ and $\begin{array}{l} | Y_{\pm} - 1 | ⩽ e^{2 C V (t)} - 1, | \frac{1}{Y_{\pm}} - 1 | ⩽ e^{2 C V (t)} - 1 . \end{array}$

Using the upper and lower bounds for ϕ given by Proposition 7, we can write that for a fixed $η \in] 0, 1 [$ (to be precised later) there exists a constant $C_{d, η}$ such that: $\begin{array}{l} R_{\pm} ⩽ C_{d, η} e^{C (\frac{d}{2} - 1) V (t)} \frac{e^{- (1 - η) α | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |}}{{(α | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |)}^{\frac{d - 1}{2}}} e^{| z |} \cdot \{\begin{matrix} | z |^{\frac{d}{2} - 1} & if d > 2, \\ (1 + | z |^{\frac{1}{2}}) & if d = 2 \end{matrix} \end{array}$ that is, with the previous notations: $\begin{array}{l} R_{\pm} ⩽ C_{d, η} e^{C (\frac{d}{2} - 1) V (t)} e^{| z | (1 - (1 - η) Y_{\pm})} {(\frac{1}{Y_{\pm}})}^{\frac{d - 1}{2}} \frac{1}{| z |^{\frac{d - 1}{2}}} \cdot \{\begin{matrix} | z |^{\frac{d}{2} - 1} & if d > 2, \\ (1 + | z |^{\frac{1}{2}}) & if d = 2, \end{matrix} \end{array}$ and then $\begin{array}{l} R_{\pm} ⩽ C_{d, η} e^{C (d - \frac{3}{2}) V (t)} e^{η | z |} e^{(1 - η) | z | (1 - Y_{\pm})} (1 + | z |^{- \frac{1}{2}}) . \end{array}$ Thanks again to Lemma 5, we finally obtain: $\begin{matrix} (40) & R_{\pm} ⩽ C_{d, η} e^{C_{d} V (t)} e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}) . \end{matrix}$ There are two terms left to estimate: $1 - R_{-}$ and $R_{-} - R_{+}$ . Again, we simply write that: $\begin{array}{l} 1 - R_{\pm} = P_{1} + P_{2} \\ = [1 - {(\frac{1}{Y_{\pm}})}^{\frac{d}{2} - 1}] + {(\frac{1}{Y_{\pm}})}^{\frac{d}{2} - 1} [1 - \frac{K_{\frac{d}{2} - 1} (α | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |)}{K_{\frac{d}{2} - 1} (| z |)}] . \end{array}$ An elementary use of the mean-value theorem to the function $h (y) = y^{\frac{d}{2} - 1}$ gives that (we recall that in Lemma 5, we proved $e^{- C V (t)} ⩽ Y_{\pm} ⩽ e^{C V (t)}$ ): $\begin{matrix} (41) & \begin{matrix} | h (1) - h (\frac{1}{Y_{\pm}}) | & ⩽ | 1 - \frac{1}{Y_{\pm}} | (\frac{d}{2} - 1) sup_{y \in [e^{- C V (t)}, e^{C V (t)}]} y^{\frac{d}{2} - 2} \\ ⩽ C_{d} | 1 - \frac{1}{Y_{\pm}} | e^{C | \frac{d}{2} - 2 | V (t)}, \end{matrix} \end{matrix}$ that is: $\begin{matrix} (42) & | P_{1} | ⩽ C_{d} (e^{2 C V (t)} - 1) e^{C_{d} V (t)} . \end{matrix}$ Moreover, the second term is bounded by: $\begin{array}{l} | P_{2} | ⩽ \frac{e^{C_{d} V (t)}}{K_{\frac{d}{2} - 1} (| z |)} | K_{\frac{d}{2} - 1} (| z |) - K_{\frac{d}{2} - 1} (α | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |) | . \end{array}$ Similarly, we can write that: $\begin{array}{l} | K_{\frac{d}{2} - 1} (| z |) - K_{\frac{d}{2} - 1} (α | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |) | \\ ⩽ | z | | Y_{\pm} - 1 | \int_{0}^{1} | K_{\frac{d}{2} - 1}^{'} (| z | (1 + u (| Y_{\pm} - 1 |)) | d u \\ (43) & ⩽ (e^{2 C V (t)} - 1) | z | \cdot sup_{y \in [| z | (1 - (e^{2 C V (t)} - 1)), | z | (1 + (e^{2 C V (t)} - 1))]} | K_{\frac{d}{2} - 1}^{'} (y) | . \end{array}$ If t is so small that $e^{2 C V (t)} - 1 ⩽ \frac{1}{2}$ , then thanks to the bound for $K_{\frac{d}{2} - 1}^{'}$ from Proposition 7 we obtain that: $\begin{array}{l} | K_{\frac{d}{2} - 1} (| z |) - K_{\frac{d}{2} - 1} (α | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x \pm \frac{z}{α}) |) | \\ ⩽ (e^{2 C V (t)} - 1) | z | \cdot sup_{y \in [\frac{1}{2} | z |, \frac{3}{2} | z |]} C_{d, η} \frac{e^{- (1 - η) y}}{y^{\frac{d + 1}{2}}} \\ (44) & ⩽ C_{d, η} (e^{2 C V (t)} - 1) \frac{e^{- (1 - η) | z | (1 - (e^{2 C V (t)} - 1))}}{| z |^{\frac{d - 1}{2}}} . \end{array}$ Gathering this with the lower bound for $K_{\frac{d}{2} - 1}$ from Proposition 7, we finally bound $P_{2}$ : $\begin{array}{l} | P_{2} | ⩽ e^{C_{d} V (t)} C_{d, η} (e^{2 C V (t)} - 1) \frac{e^{- (1 - η) | z | (1 - (e^{2 C V (t)} - 1))}}{| z |^{\frac{d - 1}{2}}} \cdot e^{| z |} \cdot \{\begin{matrix} | z |^{\frac{d}{2} - 1} & if d > 2, \\ (1 + | z |^{\frac{1}{2}}) & if d = 2 \end{matrix} \\ (45) & ⩽ C_{d, η} e^{C_{d} V (t)} (e^{2 C V (t)} - 1) e^{η | z |} e^{(1 - η) | z | (e^{2 C V (t)} - 1)} (1 + | z |^{- \frac{1}{2}}) . \end{array}$ Finally together with (42), we finally obtain the estimate: $\begin{matrix} (46) & | 1 - R_{\pm} | ⩽ C_{d, η} e^{C_{d} V (t)} (e^{2 C V (t)} - 1) e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}) . \end{matrix}$ Let us finally turn to the last term: $R_{-} - R_{+}$ . Using the notations, we can write: $\begin{array}{l} R_{-} - R_{+} & = {(\frac{1}{Y_{-}})}^{\frac{d}{2} - 1} \frac{K_{\frac{d}{2} - 1} (| z | Y_{-})}{K_{\frac{d}{2} - 1} (| z |)} - {(\frac{1}{Y_{+}})}^{\frac{d}{2} - 1} \frac{K_{\frac{d}{2} - 1} (| z | Y_{+})}{K_{\frac{d}{2} - 1} (| z |)} \\ = [{(\frac{1}{Y_{-}})}^{\frac{d}{2} - 1} - {(\frac{1}{Y_{+}})}^{\frac{d}{2} - 1}] \frac{K_{\frac{d}{2} - 1} (| z | Y_{-})}{K_{\frac{d}{2} - 1} (| z |)} \\ + {(\frac{1}{Y_{+}})}^{\frac{d}{2} - 1} \frac{K_{\frac{d}{2} - 1} (| z | Y_{-}) - K_{\frac{d}{2} - 1} (| z | Y_{+})}{K_{\frac{d}{2} - 1} (| z |)} \\ (47) & = P_{1}^{'} + P_{2}^{'} . \end{array}$ As we did in (41), if $k (y) = y^{- (\frac{d}{2} - 1)}$ , we can write: $\begin{array}{l} | k (\frac{1}{Y_{-}}) - k (\frac{1}{Y_{+}}) | ⩽ C_{d} | Y_{-} - Y_{+} | e^{C_{d} V (t)} . \end{array}$ The other term in $P_{1}^{'}$ has already been estimated when estimating $R_{\pm}$ : $\begin{array}{l} \frac{K_{\frac{d}{2} - 1} (| z | Y_{-})}{K_{\frac{d}{2} - 1} (| z |)} ⩽ C_{d, η} e^{C_{d} V (t)} e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}), \end{array}$ and we get that: $\begin{matrix} (48) & | P_{1}^{'} | ⩽ C_{d, η} e^{C_{d} V (t)} | Y_{-} - Y_{+} | e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}) . \end{matrix}$ For $P_{2}^{'}$ , as in (2.4.4), we can write: $\begin{array}{l} | K_{\frac{d}{2} - 1} (| z | Y_{-}) - K_{\frac{d}{2} - 1} (| z | Y_{+}) | \\ ⩽ | z | | Y_{-} - Y_{+} | \int_{0}^{1} | K_{\frac{d}{2} - 1}^{'} (| z | ((1 - u) Y_{+} + u Y_{-})) | d u \\ ⩽ | z | | Y_{-} - Y_{+} | \int_{0}^{1} | K_{\frac{d}{2} - 1}^{'} (| z | + | z | ((1 - u) (Y_{+} - 1) + u (Y_{-} - 1))) | d u \\ ⩽ | z | | Y_{-} - Y_{+} | \cdot sup_{y \in [| z | (1 - (e^{2 C V (t)} - 1)), | z | (1 + (e^{2 C V (t)} - 1))]} | K_{\frac{d}{2} - 1}^{'} (y) | \\ (49) & ⩽ C_{d, η} | Y_{-} - Y_{+} | \frac{e^{- (1 - η) | z | (1 - (e^{2 C V (t)} - 1))}}{| z |^{\frac{d - 1}{2}}} . \end{array}$ Using once again the lower bound for $K_{\frac{d}{2} - 1}$ implies that: $\begin{array}{l} | P_{2}^{'} | ⩽ C_{d, η} e^{C_{d} V (t)} | Y_{-} - Y_{+} | e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}) \end{array}$ and together with (48), this implies that: $\begin{matrix} (50) & | R_{-} - R_{+} | ⩽ C_{d, η} e^{C_{d} V (t)} | Y_{-} - Y_{+} | e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}) . \end{matrix}$ Thanks to Lemma 5, we get: $\begin{matrix} (51) & | Y_{-} - Y_{+} | ⩽ e^{C V (t)} (e^{2 C V (t)} - 1) . \end{matrix}$ Unfortunately, as explained, this will not be sufficient: this estimate is useful for high frequencies j, but after integration in z, the result is not summable when j goes to $- \infty$ . This is why, as in [9], we need a much more precise estimate on $| Y_{-} - Y_{+} |$ : $\begin{array}{l} | Y_{-} - Y_{+} | & = \frac{α}{| z |} (| ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x - \frac{z}{α}) | - | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x + \frac{z}{α}) |) \\ = \frac{α}{| z |} \frac{| ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x - \frac{z}{α}) |^{2} - | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x + \frac{z}{α}) |^{2}}{| ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x - \frac{z}{α}) | + | ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x + \frac{z}{α}) |} \\ (52) & = \frac{α}{| z |} \frac{N_{j}}{D_{j}} . \end{array}$ As in the proof of Lemma 8 from [9], we simply use the identity $| a |^{2} - | b |^{2} = (a + b ∣ a - b)$ and write: $\begin{array}{l} N_{j} & = (2 ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x - \frac{z}{α}) - ψ_{j, t}^{- 1} (x + \frac{z}{α}) | ψ_{j, t}^{- 1} (x - \frac{z}{α}) - ψ_{j, t}^{- 1} (x + \frac{z}{α})) \\ (53) & ⩽ | 2 ψ_{j, t}^{- 1} (x) - ψ_{j, t}^{- 1} (x - \frac{z}{α}) - ψ_{j, t}^{- 1} (x + \frac{z}{α}) | \cdot | ψ_{j, t}^{- 1} (x - \frac{z}{α}) - ψ_{j, t}^{- 1} (x + \frac{z}{α}) | . \end{array}$ Thanks to the mean value theorem (used twice for the first factor and once for the second), we can write: $\begin{array}{l} N_{j} ⩽ {(\frac{| z |}{α})}^{2} {‖ D^{2} ψ_{j, t}^{- 1} ‖}_{L^{\infty}} \times (\frac{| z |}{α}) {‖ D ψ_{j, t}^{- 1} ‖}_{L^{\infty}}, \end{array}$ and using the estimates for the flow (see (17)), we obtain that $\begin{array}{l} N_{j} ⩽ e^{C V} (e^{C V} - 1) \frac{2^{j} | z |^{3}}{α^{3}} . \end{array}$ On the other hand, $\begin{array}{l} D_{j} = \frac{| z |}{α} (Y_{+} + Y_{-}) ⩾ 2 e^{- C V (t)} \frac{| z |}{α} . \end{array}$ From these estimates, we easily conclude that: $\begin{array}{l} | Y_{-} - Y_{+} | ⩽ \frac{α}{| z |} \times e^{C V} (e^{C V} - 1) \frac{2^{j} | z |^{3}}{α^{3}} \times \frac{1}{2} e^{C V} \frac{α}{| z |} ⩽ e^{2 C V} (e^{C V} - 1) \frac{2^{j} | z |}{α} . \end{array}$ Then, combined with (51), we obtain: $\begin{matrix} (54) & | Y_{-} - Y_{+} | ⩽ e^{2 C V} (e^{2 C V} - 1) min (1, \frac{2^{j} | z |}{α}) \end{matrix}$ and then, thanks to (50), we obtain that: $\begin{matrix} (55) & | R_{-} - R_{+} | ⩽ C_{d, η} e^{C_{d} V (t)} (e^{2 C V} - 1) min (1, \frac{2^{j} | z |}{α}) e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} (1 + | z |^{- \frac{1}{2}}) . \end{matrix}$ Gathering (37), (40), (46) and (55), we obtain the result from Lemma 4. □

End of the proof of Lemma 2 . Thanks to Lemma 4, and Proposition 7 (with the same η), all that remains is to estimate the following function:

\begin{array}{l} ϕ (z) e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} & ⩽ C_{d, η} \frac{e^{- (1 - η) | z |}}{| z |^{d - \frac{3}{2}}} e^{| z | [η + (1 - η) (e^{2 C V (t)} - 1)]} \\ (56) & ⩽ C_{d, η} \frac{1}{| z |^{d - \frac{3}{2}}} e^{| z | [- 1 + 2 η + (1 - η) (e^{2 C V (t)} - 1)]} . \end{array}

Using that

e^{2 C V (t)} - 1 ⩽ \frac{1}{2}

, we have

\begin{array}{l} - 1 + 2 η + (1 - η) (e^{2 C V (t)} - 1) ⩽ - \frac{1}{2} + \frac{3}{2} η ⩽ - \frac{1}{8} \end{array}

if we choose

η = \frac{1}{4}

(in fact we need

η < \frac{1}{3}

). □

Proof of Lemma 3.
As $f_{j} = {\dot{Δ}}_{j} f$ , we can write that for all x and z, $\begin{array}{l} f_{j} (x - \frac{z}{α}) + f_{j} (x + \frac{z}{α}) - 2 f_{j} (x) = (τ_{- \frac{z}{α}} {\dot{Δ}}_{j} f + τ_{\frac{z}{α}} {\dot{Δ}}_{j} f - 2 {\dot{Δ}}_{j} f) (x) \end{array}$ and we refer to [9] (proof of Lemma 5) for the following result which is adapted from [2] (Theorems 2.36 and 2.37): Lemma 6.
Under the same assumptions, there exists a nonnegative summable sequence of summation 1, that we will also denote by ${(c_{j})}_{j \in Z}$ , such that: $\begin{array}{l} ‖ τ_{- \frac{z}{α}} {\dot{Δ}}_{j} f + τ_{\frac{z}{α}} {\dot{Δ}}_{j} f - 2 {\dot{Δ}}_{j} f ‖_{L^{2}} \\ (57) & ⩽ C_{σ} 2^{- j σ} c_{j} max (α^{- 2}, 2^{- 2 j}) {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} min (1, \frac{2^{2 j} | z |^{2}}{α^{2}}), \\ ‖ τ_{\frac{z}{α}} {\dot{Δ}}_{j} f - {\dot{Δ}}_{j} f ‖_{L^{2}} \\ (58) & ⩽ C_{σ} 2^{- j σ} c_{j} max (α^{- 2}, 2^{- 2 j}) {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} min (1, \frac{2^{j} | z |}{α}) . \end{array}$

It appears then clearly that both integrals are bounded by: $\begin{array}{l} C_{σ} 2^{- j σ} c_{j} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} max (α^{- 2}, 2^{- 2 j}) α^{2} \times I \\ = C_{σ} 2^{- j σ} c_{j} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} max (1, \frac{α^{2}}{2^{2 j}}) \times I, \end{array}$ where we define: $\begin{array}{l} I \overset{def}{=} & \int_{R^{d}} e^{- \frac{| z |}{8}} \frac{1 + | z |^{- \frac{1}{2}}}{| z |^{d - \frac{3}{2}}} min (1, \frac{2^{2 j} | z |^{2}}{α^{2}}) d z \\ = & C_{d} \int_{0}^{\infty} e^{- \frac{r}{8}} \frac{1 + r^{- \frac{1}{2}}}{r^{d - \frac{3}{2}}} min (1, \frac{2^{2 j} r^{2}}{α^{2}}) r^{d - 1} d r \\ (59) & = & C_{d} \int_{0}^{\infty} e^{- \frac{r}{8}} (1 + r^{\frac{1}{2}}) min (1, \frac{2^{2 j} r^{2}}{α^{2}}) d r = C_{d} (I_{1} + I_{2}), \end{array}$ where we have splitted the integral into: $\begin{array}{l} I_{1} = \frac{2^{2 j}}{α^{2}} \int_{0}^{\frac{α}{2^{j}}} e^{- \frac{r}{8}} r^{2} (1 + r^{\frac{1}{2}}) d r and I_{2} = \int_{\frac{α}{2^{j}}}^{\infty} e^{- \frac{r}{8}} (1 + r^{\frac{1}{2}}) d r . \end{array}$ As in [9], the second integral is easily bounded: as there exists a constant $C > 0$ such that for all $x > 0$ , $(1 + r^{\frac{1}{2}}) e^{- \frac{r}{8}} ⩽ C e^{- \frac{r}{16}}$ , we have $\begin{array}{l} I_{2} ⩽ C \int_{\frac{α}{2^{j}}}^{\infty} e^{- \frac{r}{16}} d r ⩽ 16 C e^{- \frac{1}{16} \frac{α}{2^{j}}} . \end{array}$ For the second integral, if we use the same argument, we get the estimate: $\begin{matrix} (60) & I_{1} ⩽ C \frac{2^{2 j}}{α^{2}} \int_{0}^{\frac{α}{2^{j}}} e^{- \frac{r}{16}} d r ⩽ 16 C \frac{2^{2 j}}{α^{2}} min (1, \frac{α}{2^{j}}) \end{matrix}$ which, multiplied by $max (1, \frac{α^{2}}{2^{2 j}})$ gives a resulting term in $max (1, \frac{2^{j}}{α})$ that is not summable for high frequencies. To overcome this difficulty we simply write that there is a constant $C^{'} > 0$ such that for all $r \in [0, \frac{α}{2^{j}}]$ , $\begin{array}{l} r^{2} (1 + r^{\frac{1}{2}}) e^{- \frac{r}{8}} ⩽ \frac{α}{2^{j}} r (1 + r^{\frac{1}{2}}) e^{- \frac{r}{8}} ⩽ C^{'} \frac{α}{2^{j}} e^{- \frac{r}{16}} \end{array}$ and then we obtain: $\begin{array}{l} I_{1} ⩽ C^{'} \frac{2^{2 j}}{α^{2}} \frac{α}{2^{j}} \int_{0}^{\frac{α}{2^{j}}} e^{- \frac{r}{16}} d r, \end{array}$ which is obviously interesting only when $\frac{α}{2^{j}} ⩽ 1$ that is for high frequencies: in this case it cancels the diverging $max (1, \frac{2^{j}}{α})$ . But in low frequencies this new term is much bigger that the one from (60), so that we have to combine both estimates and finally get that: $\begin{array}{l} I_{1} ⩽ C \frac{2^{2 j}}{α^{2}} min (1, \frac{α}{2^{j}}) \int_{0}^{\frac{α}{2^{j}}} e^{- \frac{r}{16}} d r \\ ⩽ 16 C \frac{2^{2 j}}{α^{2}} min (1, \frac{α^{2}}{2^{2 j}}) . \end{array}$ We are now able to estimate $I$ : $\begin{array}{l} I ⩽ C_{d} (\frac{2^{2 j}}{α^{2}} min (1, \frac{α^{2}}{2^{2 j}}) + e^{- \frac{1}{16} \frac{α}{2^{j}}}), \end{array}$ and then $\begin{array}{l} C_{σ} 2^{- j σ} c_{j} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} max (1, \frac{α^{2}}{2^{2 j}}) \times I \\ ⩽ C_{σ} 2^{- j σ} c_{j} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}} (max (1, \frac{α^{2}}{2^{2 j}}) \frac{2^{2 j}}{α^{2}} min (1, \frac{α^{2}}{2^{2 j}}) \\ + max (e^{- \frac{1}{16} \frac{α}{2^{j}}}, \frac{α^{2}}{2^{2 j}} e^{- \frac{1}{16} \frac{α}{2^{j}}})) \\ (61) & ⩽ C_{σ} 2^{- j σ} c_{j} {‖ α^{2} (ϕ_{α} * f - f) ‖}_{{\dot{B}}_{2, 1}^{σ}}, \end{array}$ which concludes the proof of Lemma 3. □

2.4.5. End of the proof of Theorem 4

As this part is strictly the same as in [9] (Section 3.5) we will not give details (in particular we refer to [6] or [9] for estimates on $R_{j}^{1 (2, 3)}$ and to [33]). Going back to system (20), we use the linear estimates from Proposition 1. Combining this with the previous estimates on the commutator a final use of the Gronwall lemma allows us to obtain that if: $\begin{matrix} (62) & \begin{matrix} V (t) \overset{def}{=} & \int_{0}^{t} {‖ \nabla v (τ) ‖}_{L^{\infty}} d τ ⩽ W (t) \\ \overset{def}{=} & \int_{0}^{t} ({‖ \nabla v (τ) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2}}} + {‖ v (τ) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2}}}^{2}) d τ, \end{matrix} \end{matrix}$ then for t small enough, $\begin{array}{l} U (t) ⩽ & 2 C_{p, \frac{ν^{2}}{4 κ}} (U (0) + ‖ F ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{s - 1}} + ν ‖ F ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{s}} + ‖ G ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{s - 1}}) \\ (63) & \times e^{2 C_{p, \frac{ν^{2}}{4 κ}} (\frac{1 + | λ + μ | + μ + ν}{ν_{0}} + max (1, \frac{1}{ν^{3}})) W (t)} . \end{array}$ Then we globalize the result as in [9].

2.5. Proof of Theorem 2

2.5.1. Existence and uniqueness, uniform estimates

As explained in the Introduction, once we have defined the interaction potential $ϕ_{α}$ , we can follow the very same methods as in [7,9] to obtain the existence of a solution. To obtain uniqueness for a fixed α, using (10) the computations are close to those for the compressible Navier–Stokes system. As in [2] we need to separate the cases $d ⩾ 3$ and $d = 2$ (the case $d = 2$ is more difficult because of endpoints for the remainder estimates in the Littlewood–Paley paradecomposition).

Let us just recall some details for the global existence: using estimate (10) with $s = \frac{d}{2}$ , we obtain that for all $t \in R_{+}$ (we refer to [7] for the expressions of the external force terms F and G): $\begin{array}{l} g (t) = & {‖ (q_{α}, u_{α}) ‖}_{E_{α}^{\frac{d}{2}} (t)} \\ ⩽ & C_{p, \frac{ν^{2}}{4 κ}} e^{C_{p, \frac{ν^{2}}{4 κ}} C_{visc} \int_{0}^{t} (‖ \nabla u_{α} (τ) ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}} + ‖ u_{α} (τ) ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}}^{2}) d τ} \\ (64) & \times (h (0) + ‖ F ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + ν ‖ F ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2}}} + ‖ G ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}) . \end{array}$ Let $η > 0$ be small (it will be fixed later) and assume that: $\begin{array}{l} g (0) \overset{def}{=} {‖ u (0) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + {‖ q (0) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + ν {‖ q (0) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2}}} ⩽ η . \end{array}$ Let us now define $\begin{array}{l} T = sup {t \in R_{+}, g (t) ⩽ 2 C_{p, \frac{ν^{2}}{4 κ}} g (0)} . \end{array}$ As $g (0) ⩽ η$ , we have $T > 0$ ( $C > 1$ ) and the aim is to prove by contradiction that $T = \infty$ . Assume that $T < \infty$ , then we obtain that for all $t ⩽ T$ , denoting by $C = C_{p, \frac{ν^{2}}{4 κ}}^{'}$ and $C_{visc}^{'} = C_{visc} / ν_{0}$ $\begin{array}{l} g (t) ⩽ C e^{2 C^{2} C_{visc}^{'} η (1 + 2 C η + e^{2 C^{2} C_{visc}^{'} η (1 + 2 C η)})} g (0) . \end{array}$ So that if $η = η (p, \frac{ν^{2}}{4 κ}, C_{visc} / ν_{0}) > 0$ is small enough then for all $t ⩽ T$ , $g (t) < 2 C_{p, \frac{ν^{2}}{4 κ}} g (0)$ which contradicts the fact that T is maximal. It then implies that $T = \infty$ .

Remark 13.
When for example $λ = 0$ we have $μ = ν = ν_{0}$ and for a small ν, $C_{visc}^{'} \sim ν^{- 4}$ , so that the previous condition simply implies that $r = η / ν^{4} ⩽ 1$ must be small enough so that $e^{2 C^{2} r (1 + 2 C + e^{2 C^{2}})} < 2$ . A sufficient condition is that r satisfies $e^{6 r C^{2} e^{2 C^{2}}} < 2$ .

2.6. Order parameter estimates

As already explained (see Remark 6), we have $c_{α} = ϕ_{α} * ρ_{α}$ , so that $c_{α} - 1 = ϕ_{α} * q_{α}$ and for all $s \in R$ , $‖ c_{α} - 1 ‖_{{\dot{B}}_{2, 1}^{s}} ⩽ ‖ q_{α} ‖_{{\dot{B}}_{2, 1}^{s}}$ . Moreover: $\begin{array}{l} ‖ c_{α} - ρ_{α} ‖_{{\dot{B}}_{2, 1}^{s}} = ‖ ϕ_{α} * q_{α} - q_{α} ‖_{{\dot{B}}_{2, 1}^{s}} = \frac{1}{α^{2}} ‖ q_{α} ‖_{{\dot{B}}_{α}^{s + 2, s}} = \sum_{l \in Z} 2^{l s} min (1, \frac{2^{2 l}}{α^{2}}) ‖ {\dot{Δ}}_{l} q_{α} ‖_{L^{2}} . \end{array}$ On one hand, thanks to the Lebesgue theorem (for series), for $s \in {\frac{d}{2} - 1, \frac{d}{2}}$ this involves that: $\begin{array}{l} ‖ c_{α} - ρ_{α} ‖_{{\tilde{L}}^{\infty} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2} - 1})} + ν ‖ c_{α} - ρ_{α} ‖_{{\tilde{L}}^{\infty} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2}})} \underset{α \to \infty}{⟶} 0, \end{array}$ and on the other hand, thanks again to the energy estimates, we end up with: $\begin{array}{l} ν ‖ c_{α} - ρ_{α} ‖_{L^{1} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2} - 1})} + ν^{2} ‖ c_{α} - ρ_{α} ‖_{L^{1} (R_{+}, {\dot{B}}_{2, 1}^{\frac{d}{2}})} ⩽ \frac{C^{0}}{α^{2}} . \end{array}$ This concludes the proof of Theorem 2.

3. Rate of convergence (Theorem 3)

In this section we prove that the solution of $(NSO P_{α})$ goes to the solution of (K), and we give estimates of the rate of convergence as α goes to infinity. Here we follow what we did in [7]: everything relies on Theorem 4. As already explained, if the initial data satisfy $\begin{array}{l} ‖ q_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + ν ‖ q_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}} + ‖ u_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} ⩽ η ⩽ min (η_{K}, η_{OP}), \end{array}$ then systems (K) and $(NSO P_{α})$ both have global solutions $(q, u)$ and $(q_{α}, u_{α})$ , and with the same notations as before, denoting $C = C_{p, \frac{ν^{2}}{4 κ}}$ then for all $t \in R$ we have, $\begin{array}{l} g_{α}^{\frac{d}{2}} (t) \overset{def}{=} & {‖ (q_{α}, u_{α}) ‖}_{E_{α}^{\frac{d}{2}} (t)} \overset{def}{=} ‖ u ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{\frac{d}{2}}} \\ + ν_{0} ‖ u ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} + 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{\frac{d}{2} + 1, \frac{d}{2} - 1}} + ν^{2} ‖ q ‖_{{\tilde{L}}_{t}^{1} {\dot{B}}_{α}^{\frac{d}{2} + 2, \frac{d}{2}}} \\ (65) & ⩽ & C η \end{array}$ and $\begin{array}{l} g^{\frac{d}{2}} (t) \overset{def}{=} & ‖ u ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + ν ‖ q ‖_{{\tilde{L}}_{t}^{\infty} {\dot{B}}_{2, 1}^{\frac{d}{2}}} + ν_{0} ‖ u ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} + 1}} \\ + ν ‖ q ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} + 1}} + ν^{2} ‖ q ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} + 2}} \\ (66) & ⩽ & C η . \end{array}$ As in [7], up to an additional forcing term, let us rewrite system (K) with a capillary term as in system $(NS O_{α})$ : $\begin{array}{l} (K) \{\begin{matrix} \partial_{t} q + u \cdot \nabla q + (1 + q) div u = 0, \\ \partial_{t} u + u \cdot \nabla u - A u + P^{'} (1) \cdot \nabla q - κ α^{2} \nabla (ϕ_{α} * q - q) = K (q) \cdot \nabla q - I (q) A u + R_{α}, \end{matrix} \end{array}$ where the remainder $R_{α} \overset{def}{=} κ \nabla (Δ q - α^{2} (ϕ_{α} * q - q))$ and K and I are defined in the introduction. Let us now write the system satisfied by the difference $(δ q, δ u) = (q_{α} - q, u_{α} - u)$ : $\begin{matrix} (67) & \{\begin{matrix} \partial_{t} δ q + u_{α} \cdot \nabla δ q + div δ u = δ F, \\ \partial_{t} δ u + u_{α} \cdot \nabla δ u - A δ u + P^{'} (1) \cdot \nabla δ q - κ α^{2} \nabla (ϕ_{α} * δ q - δ q) = δ G - R_{α}, \end{matrix} \end{matrix}$ where $\begin{array}{l} \{\begin{matrix} δ F \overset{def}{=} \sum_{i = 1}^{3} δ F_{i} \\ δ G \overset{def}{=} \sum_{i = 1}^{5} δ G_{i} \end{matrix} with \{\begin{matrix} δ F_{1} = - δ u \cdot \nabla q \\ δ F_{2} = - δ q \cdot div u_{α} \\ δ F_{3} = - q \cdot div δ u \end{matrix} and \{\begin{matrix} δ G_{1} = - δ u \cdot \nabla u \\ δ G_{2} = (K (q_{α}) - K (q)) \cdot \nabla q_{α} \\ δ G_{3} = K (q) \cdot \nabla δ q \\ δ G_{4} = (I (q_{α}) - I (q)) A u_{α} \\ δ G_{5} = - I (q) A δ u . \end{matrix} \end{array}$ Except $R_{α}$ , all these additional terms are exactly the same as in [7], so we refer to this article for details on their estimates and we will only focus on what changes. As η is small, we can additionally assume $η ⩽ 1$ , let us denote once again by $C$ a constant only depending on $\frac{ν^{2}}{4 κ}$ and d (that may change from line to line). If we introduce for $h \in [0, 1 [$ $\begin{array}{l} f_{α} (t) \overset{def}{=} {‖ (δ q, δ u) ‖}_{E_{α}^{\frac{d}{2} - h} (t)}, \end{array}$ then using (10), we obtain that for all $t \in R$ (see [7] for details), $\begin{matrix} (68) & \begin{matrix} f_{α} (t) ⩽ C e^{2 C \frac{C_{visc}}{ν_{0}} η} \\ \times [\int_{0}^{t} f_{α} (τ) F (τ) d τ \\ + \int_{0}^{t} ‖ δ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - h + 1}} (‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} + (1 + ν) ‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}}) d τ + ‖ R_{α} ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} - h - 1}}] \\ ⩽ C e^{2 C \frac{C_{visc}}{ν_{0}} η} [\int_{0}^{t} f_{α} (τ) F (τ) d τ + η \frac{1}{ν_{0}} (1 + \frac{1}{ν}) f_{α} (t) + ‖ R_{α} ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} - h - 1}}], \end{matrix} \end{matrix}$ where $\begin{array}{l} F (t) = ‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}} + ν ‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 2}} + ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}} + (1 + \frac{1}{ν}) ‖ u_{α} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}} . \end{array}$

Remark 14.

We recall that, due to endpoints in the Littlewood–Paley remainder term estimates, as in [7] we have the condition $h < d - 1$ , which is why we impose $h < 1$ in order to work for any dimension $d ⩾ 2$ .

If η is so small that $η C e^{2 C \frac{C_{visc}}{ν_{0}} η} max (1, \frac{1}{ν_{0}}) (1 + \frac{1}{ν}) ⩽ \frac{1}{2}$ , then we obtain thanks to the Gronwall lemma, $\begin{array}{l} f_{α} (t) ⩽ C e^{2 C \frac{C_{visc}}{ν_{0}} η} ‖ R_{α} ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} - h - 1}} e^{C e^{2 C \frac{C_{visc}}{ν_{0}} η} \int_{0}^{t} F (τ) d τ} . \end{array}$ Thanks to (65) and (66) we have, $\begin{array}{l} \int_{0}^{t} F (τ) d τ ⩽ (\frac{1}{ν} + \frac{1}{ν_{0}} (2 + \frac{1}{ν})) C η \end{array}$ so that using the condition on η, we end up with: $\begin{matrix} (69) & f_{α} (t) ⩽ C (\frac{C_{visc}}{ν_{0}}, \frac{ν^{2}}{4 κ}, d) ‖ R_{α} ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} - h - 1}} . \end{matrix}$

Remark 15.

For small viscosities, in the case $λ = 0$ , the previous condition on η is roughly $C \frac{η}{ν^{2}} e^{C \frac{η}{ν^{4}}} ⩽ \frac{1}{2}$ which is obviously implied by the condition required in the existence result (see Remark 13).

To estimate the remainder in the case $h \in] 0, 1 [$ , we simply write that: $\begin{array}{l} \hat{R_{α}} (ξ) = - i κ ξ \frac{| ξ |^{4}}{α^{2} + | ξ |^{2}} \hat{q} (ξ), \end{array}$ which allows us to write: $\begin{array}{l} ‖ R_{α} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - h - 1}} = κ \sum_{j \in Z} 2^{j (\frac{d}{2} - h - 1)} \frac{2^{5 j}}{α^{2} + 2^{2 j}} ‖ q_{j} ‖_{L^{2}} = κ \sum_{j \in Z} 2^{j (\frac{d}{2} + 2)} ‖ q_{j} ‖_{L^{2}} \frac{2^{j (2 - h)}}{α^{2} + 2^{2 j}} . \end{array}$ We now consider two cases:

If $2^{j} ⩾ α$ then $\begin{array}{l} \frac{2^{j (2 - h)}}{α^{2} + 2^{2 j}} = 2^{- j h} \frac{2^{2 j}}{α^{2} + 2^{2 j}} ⩽ α^{- h} . \end{array}$

If $2^{j} ⩽ α$ then as $h \in [0, 1 [$ $\begin{array}{l} \frac{2^{j (2 - h)}}{α^{2} + 2^{2 j}} = \frac{2^{j (2 - a)}}{α^{2}} ⩽ \frac{α^{(2 - a)}}{α^{2}} ⩽ α^{- h} . \end{array}$

We conclude that:

\begin{array}{l} ‖ R_{α} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - h - 1}} ⩽ α^{- h} ‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 2}}, \end{array}

and then

\begin{array}{l} f_{α} (t) ⩽ C (\frac{C_{visc}}{ν_{0}}, \frac{ν^{2}}{4 κ}, d) ‖ q ‖_{L_{t}^{1} {\dot{B}}_{2, 1}^{\frac{d}{2} + 2}} α^{- h} ⩽ C (\frac{C_{visc}}{ν_{0}}, \frac{ν^{2}}{4 κ}, d) α^{- h} . \end{array}

To complete the proof of the theorem, all that remains is to estimate

\begin{array}{l} c_{α} - ρ = ϕ_{α} * ρ_{α} - ρ = ϕ_{α} * (ρ_{α} - ρ) + ϕ_{α} * ρ - ρ = ϕ_{α} * (q_{α} - q) + ϕ_{α} * q - q . \end{array}

Thanks to what precedes only the last term has to be estimated, and similarly we have

\begin{array}{l} ‖ ϕ_{α} * q - q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - h - 1}} & = \sum_{j \in Z} 2^{j (\frac{d}{2} - h - 1)} \frac{2^{2 j}}{α^{2} + 2^{2 j}} ‖ q_{j} ‖_{L^{2}} \\ = \sum_{j \in Z} 2^{j (\frac{d}{2} - 1)} ‖ q_{j} ‖_{L^{2}} \frac{2^{j (2 - h)}}{α^{2} + 2^{2 j}} \\ ⩽ α^{- h} ‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} \end{array}

and

\begin{array}{l} ‖ ϕ_{α} * q - q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - h}} ⩽ α^{- h} ‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}} . \end{array}

For the hybrid norms, we do the same:

\begin{array}{l} ‖ ϕ_{α} * q - q ‖_{{\dot{B}}_{α}^{\frac{d}{2} - h + 1, \frac{d}{2} - h - 1}} & ⩽ \sum_{j \in Z} 2^{j (\frac{d}{2} - h - 1)} min (α^{2}, 2^{2 j}) min (1, \frac{2^{2 j}}{α^{2}}) ‖ q_{j} ‖_{L^{2}} \\ (70) & = \sum_{j \in Z} 2^{j (\frac{d}{2} + 1)} ‖ q_{j} ‖_{L^{2}} α^{2} 2^{- j (h + 2)} min {(1, \frac{2^{2 j}}{α^{2}})}^{2} ⩽ α^{- h} ‖ q ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}} \end{array}

so that we finally obtain that if

h \in] 0, 1 [

\begin{array}{l} {‖ (ρ_{α} - ρ, c_{α} - ρ, u_{α} - u) ‖}_{F_{α}^{\frac{d}{2} - h}} ⩽ C (\frac{C_{visc}}{ν_{0}}, \frac{ν^{2}}{4 κ}, d) α^{- h} . \end{array}

Coming back to (69) in the case

h = 0

, we simply write that:

\begin{array}{l} ‖ R_{α} ‖_{{\dot{B}}_{2, 1}^{s}} ⩽ κ \sum_{j \in Z} 2^{j s} 2^{3 j} min (1, \frac{2^{2 j}}{α^{2}}) ‖ q_{j} ‖_{L^{2}}, \end{array}

and thanks to the Lebesgue theorem for series we obtain that the norm

‖ (ρ_{α} - ρ, c_{α} - ρ, u_{α} - u) ‖_{F_{α}^{\frac{d}{2} - h}}

goes to zero when α goes to infinity, which ends the proof of the theorem.

Footnotes

Acknowledgements

The author wishes to thank Raphaël Danchin, Nicolas Fournier, and François Vigneron for useful discussions.

The first part is devoted to a quick presentation of the Littlewood–Paley theory and specific properties for hybrid Besov norms used in this paper. The second section to general considerations on flows.

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