Timothée Crin-Barat

Ling-Yun Shou
email: shoulingyun11@gmail.com

Jin Tan

On the overdamping phenomenon for compressible multi-fluid systems with rough coefficients

Keywords: Multi-fluid system, relaxation limit, overdamping, critical regularity, porous media system, Kapila system, Baer-Nunziato System 2020 Mathematics Subject Classification-35Q35, 35B40, 76N10, 76T17

In this paper we investigate two types of relaxation processes quantitatively in the context of small data global-in-time solutions for compressible one-velocity multi-fluid models. First, we justify the pressure-relaxation limit from a one-velocity Baer-Nunziato system to a Kapila model as the pressure-relaxation parameter tends to zero, in a uniform manner with respect to the time-relaxation parameter associated to the friction forces modeled in the equation of the velocity. This uniformity allows us to further consider the time-relaxation limit for the Kapila model. More precisely, we show that the diffusely time-rescaled solution of the Kapila system converges to the solution of a two-phase porous media type system as the time-relaxation parameter tends to zero. For both relaxation limits, we exhibit explicit convergence rates.

.

Introduction

Models and motivations

Multi-fluid systems have been used to simulate a wide range of physical mixing phenomenon, from engineering to biological systems (cf. [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF][START_REF] Bresch | A compressible multifluid system with new physical relaxation terms[END_REF][START_REF] Ishii | Thermo-fluid dynamics of two-phase flow[END_REF][START_REF] Wallis | One-dimensional two-fluid flow[END_REF] and the references therein). In the present paper, we investigate an inviscid compressible one-velocity Baer-Nunziato system with two different pressure laws in the presence of drag forces, which was discussed in the recent work [START_REF] Bresch | Note on the derivation of multicomponent flow systems[END_REF] of Bresch and Hillairet:

                                     α ε,τ + + α ε,τ -= 1, ∂ t α ε,τ + + u ε,τ • ∇α ε,τ + = α ε,τ + α ε,τ - ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) , ∂ t (α ε,τ + ρ ε,τ + ) + div (α ε,τ + ρ ε,τ + u ε,τ ) = 0, ∂ t ρ ε,τ + div (ρ ε,τ u ε,τ ) = 0, ∂ t (ρ ε,τ u ε,τ ) + div (ρ ε,τ u ε,τ ⊗ u ε,τ ) + ∇P ε,τ + ρ ε,τ u ε,τ τ = 0, ρ ε,τ = α ε,τ + ρ ε,τ + + α ε,τ -ρ ε,τ -, P ε,τ = α ε,τ + P + (ρ ε,τ + ) + α ε,τ -P -(ρ ε,τ -), x ∈ R d , t > 0, (BN)
where the unknowns α ε,τ ± = α ε,τ ± (t, x) ∈ [0, 1], ρ ε,τ ± = ρ ε,τ ± (t, x) ≥ 0 and u ε,τ = u ε,τ (t, x) ∈ R d stand for the volume fractions, the densities and the common velocity of two fluids (denoted by + and -), respectively.

The two positive constants ε and τ are (small) relaxation parameters associated to the pressure-relaxation and time-relaxation limits. Finally, the two pressures P + and P -take the gamma-law forms P ± (s) = A ± s γ± with constants A ± > 0, 1 ≤ γ -< γ + .

(1.1)

The Baer-Nunziato terminology refers to the pressure-relaxation mechanism in the equations of the volume fractions. Numerically, such relaxation procedure can simplify its resolution as it reduces the number of constraints by introducing new unknowns: two pressures instead of one. The readers can see [START_REF] Bresch | Note on the derivation of multicomponent flow systems[END_REF] and references therein for more discussions on this pressure-relaxation process. Very recently, the one-dimensional version of System (BN) was rigorously derived by Bresch, Burtea and Lagoutiére in [START_REF] Bresch | Physical relaxation terms for compressible two-phase systems[END_REF].

There is an extensive literature on the mathematical analysis of multi-fluid systems. For example, in the one-velocity case, the global existence of weak solutions has been studied in [START_REF] Bresch | Finite-energy solutions for compressible two-fluid Stokes system[END_REF][START_REF] Li | Global existence of weak solutions to the drift-flux system for general pressure laws[END_REF][START_REF] Novotný | Weak solutions for some compressible multicomponent fluid models[END_REF][START_REF] Vasseur | Global weak solution to the viscous two-fluid model with finite energy[END_REF][START_REF] Wen | On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions[END_REF][START_REF] Yao | Existence and uniqueness of global weak solution to a two-phase flow model with vacuum[END_REF], and the global well-posedness and optimal time-decay rates of strong solutions has been established in the framewok of Sobolev spaces [START_REF] Guo | Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum[END_REF][START_REF] Yao | Existence of asymptotic behavior of global weak solutions to a 2d viscous liquid-gas two-phase flow model[END_REF][START_REF] Zhang | Global existence and optimal convergence rates for the strong solutions in h 2 to the 3d viscous liquid-gas two-phase flow model[END_REF] and critical Besov spaces [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF][START_REF] Hao | Well-posedness for a multidimensional viscous liquid-gas two-phase flow model[END_REF], etc. We also refer to [START_REF] Bresch | Global weak solutions to a generic two-fluid model[END_REF][START_REF] Bresch | A multi-fluid compressible system as the limit of weak solutions of the isentropic compressible Navier-Stokes equations[END_REF][START_REF] Bresch | Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system[END_REF][START_REF] Evje | Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model[END_REF][START_REF] Kracmar | Weak solutions for a bifluid model for a mixture of two compressible noninteracting fluids with general boundary data[END_REF] on the study of multi-fluid systems in the two-velocity case. Complete reviews on multi-fluid systems are presented in [START_REF] Bresch | Multifluid models including compressible fluids[END_REF][START_REF] Wen | Review on mathematical analysis of some two-phase flow models[END_REF]. Concerning the study of relaxation problems associated to systems of conservation laws, it can be traced back to the work [START_REF] Chen | Hyperbolic conservation laws with stiff relaxation terms and entropy[END_REF] by Chen, Levermore and Liu. Recently, Giovangigli and Yong in [START_REF] Giovangigli | Volume viscosity and internal energy relaxation: symmetrization and Chapman-Enskog expansion[END_REF][START_REF] Giovangigli | Volume viscosity and internal energy relaxation: error estimates[END_REF] studied a relaxation problem arising in the dynamics of perfect gases out of thermodynamicalequilibrium.

At the formal level, if the solution (α ε,τ ± , ρ ε,τ ± , u ε,τ ) of System (BN) tends to some limit (α τ ± , ρ τ ± , u τ ) as ε → 0, then (α τ ± , ρ τ ± , u τ ) should satisfy the so-called one-velocity Kapila system (cf. [START_REF] Kapila | Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations[END_REF]):

                             α τ + + α τ -= 1, ∂ t (α τ + ρ τ + ) + div (α τ + ρ τ + u τ ) = 0, ∂ t ρ τ + div (ρ τ u τ ) = 0, ∂ t (ρ τ u τ ) + div (ρ τ u τ ⊗ u τ ) + ∇P τ + ρ τ u τ τ = 0, ρ τ = α τ + ρ τ + + α τ -ρ τ -, P τ = P + (ρ τ + ) = P -(ρ τ -).
(K) System (K) can be understood as a multi-fluid compressible Euler equations with an algebraic closure law (common pressure). For existence of finite energy weak solutions to the related viscous flows, refer to the recent works [START_REF] Bresch | Finite-energy solutions for compressible two-fluid Stokes system[END_REF][START_REF] Li | Global existence of weak solutions to the drift-flux system for general pressure laws[END_REF][START_REF] Novotný | Weak solutions for some compressible multicomponent fluid models[END_REF][START_REF] Wen | On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions[END_REF].

Then, we further investigate the time-relaxation limit of System (K) as τ → 0. Inspired by the works [START_REF] Coulombel | The strong relaxation limit of the multidimensional isothermal Euler equations[END_REF][START_REF] Junca | Strong relaxation of the isothermal euler system to the heat equation[END_REF][START_REF] Xu | Relaxation limit in Besov spaces for compressible Euler equations[END_REF] concerning the relaxation problems for the compressible Euler system with damping, we introduce a large time-scale O( 1 τ ) and define the following charge of variables

(β τ ± , ̺ τ ± , v τ )(s, x) := α τ ± , ρ τ ± , u τ τ s τ , x . (1.2)
Under the diffusive scaling (1.2), System (K) becomes

                           β τ + + β τ -= 1, ∂ s (β τ + ̺ τ + ) + div (β τ ± ̺ τ + v τ ) = 0, ∂ s ̺ τ + div (̺ τ v τ ) = 0, τ 2 ∂ s (̺ τ v τ ) + τ 2 div (̺ τ v τ ⊗ v τ ) + ∇Π τ + ̺ τ v τ = 0, ̺ τ = β τ + ρ τ + + β τ -̺ τ -, Π τ = P + (̺ τ + ) = P -(̺ τ -). (K τ )
As τ → 0, one then expects that (β τ ± , ̺ τ ± , v τ ) converges to some limit (β ± , ̺ ± , v) which is the solution of the following two-phase system:

                           β + + β -= 1, ∂ s (β + ̺ + ) + div (β + ̺ + v) = 0, ∂ s ̺ + div (̺v) = 0, ∇Π + ̺v = 0, ̺ = β + ̺ + + β -̺ -, Π = P + (̺ + ) = P -(̺ -).
(1.3) System (1.3) can be viewed as the Darcy's law (1.3) 4 coupled with a porous media type equations

                     β + + β -= 1, ∂ s β + + v • ∇β + = - (γ + -γ -)β + β - γ + β -+ γ -β + div v, ∂ s Π + v • ∇Π = γ + γ -Π γ + β -+ γ -β + div ∇Π β + ̺ + + β -̺ - , Π = P + (̺ + ) = P -(̺ -).
(PM)

The present paper is a follow-up to the paper [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF] by Burtea, Crin-Barat and Tan where the authors justified the pressure-relaxation limit for the viscous version of System (BN) to System (K) in the context of vanishing viscosity. In [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF], the smallness condition on initial data for the global well-posedness result depends on min{τ, 1 τ } due to the overdamping phenomenon and therefore does not allow to further investigate the limit when τ → 0.

The main goals of this article are to justify the above pressure-relaxation limit rigorously and quantitatively from System (BN) to System (K) as ε → 0 uniformly in τ as well as the time-relaxation limit from System (K τ ) to System (PM) as τ → 0. For both relaxation limits, we will focus on the global-in-time strong solutions close to some constant equilibrium state. In other words, we consider solutions (α ε,τ ± , ρ ε,τ ± , u ε,τ ) to System (BN) (resp. (α τ ± , ρ τ ± , u τ ) to System (K)) with positive densities and volume fractions which, as |x| → ∞, tend to some thermodynamically stable equilibrium state (ᾱ ± , ρ± , 0) fulfilling 0 < ᾱ± < 1, ᾱ+ + ᾱ-= 1, ρ± > 0, P + (ρ + ) = P -(ρ -).

(1.4)

For convenience, we also define the corresponding equilibrium state for the total density and the total pressure as ρ := ᾱ+ ρ+ + ᾱ-ρ-, P := P + (ρ + ) = P -(ρ -).

(

To achieve our goals, we prove the uniform in ε and τ (such that ε ≤ τ ) estimates of global-intime strong solutions for System (BN) under a perturbation framework, which improved the analysis performed in [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF] that did not provide uniform-in-τ estimates. This result can be understood as uniform global well-posedness for a class of non-symmetric partially dissipative hyperbolic systems with rough coefficients in the context of overdamping phenomenon, which is not covered by the very recent lecture of Danchin [START_REF] Danchin | Partially dissipative systems in the critical regularity setting, and strong relaxation limit[END_REF]. Indeed, the proof generalize the techniques developed in [START_REF] Crin-Barat | Global existence for partially dissipative hyperbolic systems in the l p framework, and relaxation limit[END_REF][START_REF] Crin-Barat | Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case[END_REF] which can not be directly applied to System (BN) due to the complex forms of the total pressure and the lack of symmetry.

On the other hand, it is natural to ask what happens for System (BN) as τ tends to 0. To investigate this process, we introduce a diffusive scaling similar to (1.2) as follows

(β ε,τ ± , ̺ ε,τ ± , v ε,τ )(s, x) := α ε,τ ± , ρ ε,τ ± , u ε,τ τ s τ , x . (1.6)
Under such scaling (1.6), System (BN) becomes

                       ∂ s β ε,τ ± + v ε,τ • ∇β ε,τ ± = ± β ε,τ + β ε,τ - ετ P + (̺ ε,τ + ) -P -(̺ ε,τ -) , ∂ s (β ε,τ ± ̺ ε,τ ± ) + div (β ε,τ ± ̺ ε,τ ± v ε,τ ) = 0, τ 2 ∂ s (̺ ε,τ v ε,τ ) + τ 2 div (̺ ε,τ v ε,τ ⊗ v ε,τ ) + ∇Π ε,τ + ̺ ε,τ v ε,τ = 0, ̺ ε,τ = β ε,τ + ̺ ε,τ + + β ε,τ -̺ ε,τ -, Π ε,τ = β ε,τ + P + (̺ ε,τ + ) + β ε,τ -P -(̺ ε,τ -). (BN τ )
The crucial observation is that the parameter τ now also appears under the pressure-relaxation term in the equation of the volume fractions. This reveals that as τ → 0, the two pressures in System (BN τ ) should converge to a common pressure, and the solutions of System (BN) should converge to the solutions of System (K) regardless of ε. Additionally, in the sequel of the paper we are only able to justify the limit in the case ε ≤ τ which corresponds to the situation that the time-scale of the pressure-relaxation is small than the time-scale of the diffusion-relaxation. The condition ε ≤ τ appears in the spectral analysis of the system (see Section 1.3) and is essential for us to close the uniform a-priori estimates in both low and high frequencies (See Subsections 3.1-3.2 for the details). But in a formal way, the condition ε ≤ τ seems not necessary in the limit process τ → 0, so the case ε > τ remains an interesting open problem.

The following Figure 1 summarizes the limit processes that we tackle in this article.

System (BN) System (K) System (K τ ) System (PM) System (BN τ ) ε → 0 + uniformly in τ Change of time-scale (1.2) τ → 0 + Change of time-scale (1.6) τ → 0 + , ε ≤ τ Figure 1: Relaxation limits diagram.
The rest of the paper is organized as follows. In Subsection 1.2, we state our main results. In Subsection 1.3, we first recall a reformulation of System (BN) from [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF] and present an explicit spectral analysis for the associated linear system, then the difficulties and strategies of proof are discussed. Section 2 is devoted to some notations and properties of Besov spaces and Littlewood-Paley decomposition, and the regularity estimates for some linear problems are stated. In Section 3, we establish uniform estimates for the linearized problem. Next, in Section 4, we prove the global existence and uniqueness results of solutions for Systems (BN), (K) and (PM), respectively. The Section 5 is devoted to the justification of the relaxation limits with explicit convergence rates.

Notations. We end this subsection by presenting a few notations. As usual, we denote by C (and sometimes with subscripts) harmless positive constants that may change from line to line, and A B

(A B) means that both A ≤ CB (A ≥ CB), while A ∼ B means that A B and A B. For X a Banach space, p ∈ [1, ∞] and T > 0, the notation L p (0, T ; X) or L p T (X) designates the set of measurable functions f : [0, T ] → X with t → f (t) X in L p (0, T ), endowed with the norm • L p T (X) := • X L p (0,T )
. We agree that C b ([0, T ]; X) denotes the set of continuous and bounded (uniformly in T ) functions from [0, T ] to X. Sometimes, we use the notation L p (X) to designate the space L p (R + ; X) and

• L p (X)
for the associated norm. We will keep the same notations for multi-component functions, namely for

f : [0, T ] → X m with m ∈ N.

Main results

Our first theorem concerns the uniform, in both ε and τ , global well-posedness issue of System (BN) in the critical regularity framework.

(α ±,0 , ρ ±,0 , u 0 ) satisfies (α ±,0 -ᾱ± , ρ ±,0 -ρ± , u 0 ) ∈ Ḃ d 2 -1 ∩ Ḃ d 2 +1
. There exists a positive constant c 0 independent of τ and ε such that if

(α ±,0 -ᾱ± , ρ ±,0 -ρ± , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 ≤ c 0 , (1.7)
then the Cauchy problem of System (BN) with the initial data (α ±,0 , ρ ±,0 , u 0 ) has a unique global solution

(α ε,τ ± , ρ ε,τ ± , u ε,τ ) satisfying                (α ε,τ ± -ᾱ± , ρ ε,τ ± -ρ± , u ε,τ ) ∈ C b (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
),

P + (ρ ε,τ + ) -P -(ρ ε,τ -) ∈ L 1 (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
),

P ε,τ -P ∈ L 1 (R + ; Ḃ d 2 +1 ) ∩ L 2 (R + ; Ḃ d 2 ∩ Ḃ d 2 +1
),

u ε,τ ∈ L 1 (R + ; Ḃ d 2 ∩ Ḃ d 2 +1 ) ∩ L 2 (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
).

(1.8)

Moreover, the following uniform estimates holds:

(α ε,τ ± -ᾱ± , ρ ε,τ ± -ρ± , u ε,τ ) L ∞ ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + (∂ t α ε,τ ± , ∂ t ρ ε,τ ± , ∂ t u ε,τ ) L 1 ( Ḃ d 2 ) + 1 ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 1 ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 √ ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 2 ( Ḃ d 2 -1 ∩ Ḃ d 2 +1
)

+ τ P ε,τ -P L 1 ( Ḃ d 2 +1 ) + √ τ P ε,τ -P L 2 ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + u ε,τ L 1 ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + 1 √ τ u ε,τ L 2 ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + ρ ε,τ u ε,τ τ + ∇P ε,τ L 1 ( Ḃ d 2 -1 ∩ Ḃ d 2 ) ≤ C (α ±,0 -ᾱ± , ρ ±,0 -ρ± , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 , (1.9)
where C > 0 is a generic constant.

Remark 1.1. It should be emphasized that the regularity and decay-in-τ properties of the effective flux ρ ε,τ u ε,τ τ + ∇P ε,τ is better than the one verified by the solution (α ε,τ ± , ρ ε,τ ± , u ε,τ ). This is consistent with Darcy's law and plays key role in the justification of the time-relaxation limit.

By classical compactness arguments and the uniform estimates (1.9), we obtain the following global well-posedness theorems for Systems (K) and (PM) in the critical regularity framework.

Theorem 1.2. Let d ≥ 2 and 0 < τ ≤ 1. Given the constants ᾱ± , ρ± verifying (1.4)-(1.5), assume that the initial data (α ±,0 , ρ ±,0 , u 0 ) satisfies (α ±,0 -ᾱ± , ρ ±,0 -ρ± , u 0 ) ∈ Ḃ d 2 -1 ∩ Ḃ d 2 +1
. There exists a positive constant c 1 independent of τ such that if

(α ±,0 -ᾱ± , ρ ±,0 -ρ± , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 ≤ c 1 , (1.10)
then the Cauchy problem of System (K) with the initial data (α ±,0 , ρ ±,0 , u 0 ) admits a unique global

solution (α τ ± , ρ τ ± , u τ ) satisfying        (α τ ± -ᾱ± , ρ τ ± -ρ± , u τ ) ∈ C b (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
),

P τ -P ∈ L 1 (R + ; Ḃ d 2 +1 ) ∩ L 2 (R + ; Ḃ d 2 ∩ Ḃ d 2 +1
),

u τ ∈ L 1 (R + ; Ḃ d 2 ∩ Ḃ d 2 +1 ) ∩ L 2 (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
).

(1.11)

Moreover, the following uniform estimates holds:

(α τ ± -ᾱ± , ρ τ ± -ρ± , u τ ) L ∞ ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + (∂ t α τ ± , ∂ t ρ τ ± , ∂ t u τ ) L 1 ( Ḃ d 2 ) + τ P τ -P L 1 ( Ḃ d 2 +1 ) + √ τ P τ -P L 2 ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + u τ L 1 ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + 1 √ τ u τ L 2 ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + ρ ε,τ u τ τ + ∇P τ L 1 ( Ḃ d 2 -1 ∩ Ḃ d 2 ) ≤ C (α ±,0 -ᾱ± , ρ ±,0 -ρ± , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 , (1.12)
where C > 0 is a uniform constant independent of τ and time.

Theorem 1.3. Let d ≥ 2.
Given the constants ᾱ± , ρ± verifying (1.4)- (1.5), assume that the initial data

(β ±,0 , ̺ ±,0 ) satisfies (β ±,0 -ᾱ± , ̺ ±,0 -ρ± ) ∈ Ḃ d 2 -1 ∩ Ḃ d 2 +1
and

(β ±,0 -ᾱ± , ̺ ±,0 -ρ± ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 ≤ c 2 , (1.13)
for a positive constant c 2 independent of τ , then the Cauchy problem of System (PM) with the initial data (β ±,0 , ̺ ±,0 ) admits a unique global solution (β, ̺), which satisfies

   β ± -ᾱ± ∈ C b (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
),

̺ ± -ρ± ∈ C b (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) ∩ L 1 (R + ; Ḃ d 2 +1
).

(1.14)

Moreover, the following uniform estimates holds:

(β ± -ᾱ± , ̺ ± -ρ± ) L ∞ ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + (∂ t β ± , ∂ t ̺ ± ) L 1 ( Ḃ d 2 ) + ̺ ± -ρ± L 1 ( Ḃ d 2 +1 ∩ Ḃ d 2 +3 ) ≤ C (β ±,0 -ᾱ± , ̺ ±,0 -ρ± ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 , (1.15)
where C > 0 is a uniform constant independent of time.

Finally, we present the rigorous justifications of the pressure-relaxation limit for System (BN) to System (K) as ε → 0, and further the time-relaxation limit for System (K) to System (1.3) as τ → 0, with explicit convergence rates.

Theorem 1.4. Let d ≥ 2 and 0 < ε ≤ τ ≤ 1. Given the constants ᾱ± , ρ± verifying (1.4)-(1.5), let (α ε,τ ± , ρ ε,τ ± , u ε,τ ), (α τ ± , ρ τ ± , u τ
) and (β ± , ̺ ± ) be the global solutions to the Cauchy problems of Systems (BN), (K) and (PM) obtained from Theorems 1.1-1.3 associated to their corresponding initial data (α ε,τ ±,0 , ρ ε,τ ±,0 , u ε,τ 0 ), (α τ ±,0 , ρ τ ±,0 , u τ 0 ) and (β ±,0 , ̺ ±,0 ), respectively.

• Let the initial quantities P ε,τ 0 -P τ 0 and Y ε,τ 0 -Y τ 0 be denoted by (5.1) and (5.3), respectively. If d ≥ 3 and

P + (ρ ε,τ +,0 ) -P -(ρ ε,τ -,0 ), Y ε,τ 0 -Y τ 0 , P ε,τ 0 -P τ 0 , u ε,τ 0 -u τ 0 Ḃ d 2 -2 ∩ Ḃ d 2 -1 ≤ √ ετ , (1.16)
then there exists a universal constant C 1 such that the following estimates hold:

(α ε,τ ± -α τ ± , ρ ε,τ ± -ρ τ ± , u ε,τ -u τ ) L ∞ ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) + √ τ ρ ε,τ ± -ρ τ ± L 2 ( Ḃ d 2 -1 ) + 1 √ τ u ε,τ -u τ L 2 ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) + u ε,τ -u τ L 1 ( Ḃ d 2 -1 ) ≤ C 1 √ ετ .
(1.17)

• Furthermore, define (β τ ± , ̺ τ ± , v τ ) by the diffusive scaling (1.2) and v by Darcy's law (1.3) 4 . Let the initial quantity Z τ 0 -Z 0 be denoted by (5.32). If

Z τ 0 -Z 0 Ḃ d 2 -1 ∩ Ḃ d 2 + ̺ τ ±,0 -̺ ±,0 Ḃ d 2 -1 ≤ τ, (1.18)
then there exists a universal constant C 2 such that the following estimates hold:

(β τ ± -β ± , ̺ τ ± -̺ ± ) L ∞ ( Ḃ d 2 -1 ) + ̺ τ ± -̺ ± L 1 ( Ḃ d 2 +1 ) + v τ -v L 1 ( Ḃ d 2 ) ≤ C 2 τ. (1.19) Corollary 1.1. Let d ≥ 3, 0 < ε ≤ τ ≤ 1
, and the constants ᾱ± , ρ± verify (1.4)- (1.5). Then under the assumptions of Theorem 1.4and let (β ε,τ ± , ̺ ε,τ ± , v ε,τ ) be defined by (1.6), then there is a generic constant C 3 such that

(β ε,τ ± -β ± , ̺ ε,τ ± -̺ ± ) L ∞ ( Ḃ d 2 -1 ) ≤ C 3 ( √ ετ + τ ).

Difficulties and strategies

The first difficulty concerning the study of System (BN) are its lack of dissipativity and symmetrizability. Indeed, the linearization of (BN) admits the eigenvalue 0 and therefore does not satisfy the well-known "Shizuta-Kawashima" stability condition for partially dissipative hyperbolic systems (cf. [START_REF] Shizuta | Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation[END_REF]).

Additionally, System (BN) cannot be written in a conservative form and the entropy naturally associated to (BN) is not positive-definite, therefore the notion of entropic variables does not make sense in this case. Therefore, the first crucial step in our analysis is to partially symmetrize System (BN), by hands.

We refer to [START_REF] Bresch | Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system[END_REF][START_REF] Forestier | Criterion of hyperbolicity for non-conservative quasilinear systems admitting a partially convex conservation law[END_REF] for the treatment of non-conservative systems in similar contexts. In our setting, as explained in [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF], we define the new unknowns

               y ε,τ := α ε,τ + ρ ε,τ + α ε,τ + ρ ε,τ + + α ε,τ -ρ ε,τ - - ᾱ+ ρ+ ᾱ+ ρ+ + ᾱ-ρ- , w ε,τ := α ε,τ + α ε,τ - γ + α ε,τ -+ γ -α ε,τ + P + (ρ ε,τ + ) -P -(ρ ε,τ -) , r ε,τ := P ε,τ -P -(γ + -γ -)w ε,τ , (1.20)
and the corresponding initial data

             y 0 := α +,0 ρ +,0 α +,0 ρ +,0 + α -,0 ρ -,0 - ᾱ+ ρ+ ᾱ+ ρ+ + ᾱ-ρ- , w 0 := α +,0 α -,0 γ + α -,0 + γ -α +,0 P + (ρ +,0 ) -P -(ρ -,0 ) , r 0 := α +,0 P + (ρ +,0 ) + α -,0 P -(ρ -,0 ) -P -(γ + -γ -)w 0 ,
so that the Cauchy problem of System (BN) subject to the initial data (α ±,0 , ρ ±,0 , u 0 ) is reformulated as

                           ∂ t y ε,τ + u ε,τ • ∇y ε,τ = 0, ∂ t w ε,τ + u ε,τ • ∇w ε,τ + ( F1 + G ε,τ 1 )div u ε,τ + ( F2 + G ε,τ 2 ) w ε,τ ε = 0, ∂ t r ε,τ + u ε,τ • ∇r ε,τ + ( F3 + G ε,τ 3 )div u ε,τ = F ε,τ 4 (w ε,τ ) 2 ε , ∂ t u ε,τ + u ε,τ • ∇u ε,τ + u ε,τ τ + ( F0 + G ε,τ 0 )∇r ε,τ + (γ + -γ -)( F0 + G ε,τ 0 )∇w ε,τ = 0, (y ε,τ , w ε,τ , r ε,τ , u ε,τ )(0, x) = (y 0 , w 0 , r 0 , u 0 )(x), (1.21)
where F ε,τ i (i = 0, 1, 2, 3, 4) are the nonlinear terms

                                   F ε,τ 0 := 1 α ε,τ + ρ ε,τ + + α ε,τ -ρ ε,τ - , F ε,τ 1 := (γ + -γ -)α ε,τ + α ε,τ - γ + α ε,τ -+ γ -α ε,τ + ( P + r ε,τ ) + γ 2 + α ε,τ -+ γ 2 -α ε,τ + γ + α ε,τ -+ γ -α ε,τ + w ε,τ , F ε,τ 2 := (γ + α ε,τ -+ γ -α ε,τ + )( P + r ε,τ ) - (γ + -γ 2 + )(α ε,τ -) 2 -(γ --γ 2 -)(α ε,τ + ) 2 α ε,τ + α ε,τ - w ε,τ , F ε,τ 3 := γ + γ - γ + α ε,τ -+ γ -α ε,τ + P ε,τ , F ε,τ 4 := γ + γ - α ε,τ + α ε,τ - (1 -γ + α ε,τ --γ -α ε,τ + ), (1.22)
Fi (i = 0, 1, 2, 3) are the constants

                     F0 := 1 ᾱ+ ρ+ + ᾱ-ρ- > 0, F1 := (γ + -γ -)ᾱ + ᾱ- γ + ᾱ-+ γ -ᾱ+ P > 0, F2 := (γ + ᾱ-+ γ -ᾱ+ ) P > 0, F3 := γ + γ - γ + ᾱ-+ γ -ᾱ+ P > 0, (1.23)
and G ε,τ i (i = 0, 1, 2, 3) are the coefficients

G ε,τ i := F ε,τ i -Fi . (1.24)
In this formulation, the equation (1.21) 1 is purely transport and the linear part of subsystem (1.21) 2 -(1.21) 4 is partially dissipative and satisfies the "Shizuta-Kawashima" stability condition. Thus, we will estimate the undamped unknown y ε,τ and the dissipative components (w ε,τ , r ε,τ , u ε,τ ) separately. We emphasize here that due to the double parameters ε, τ and the lack of time-integrability of G ε,τ i , the dissipative structures of subsystem (1.21) 2 -(1.21) 4 does not fit into the general theorems that can be found in [START_REF] Crin-Barat | Global existence for partially dissipative hyperbolic systems in the l p framework, and relaxation limit[END_REF][START_REF] Crin-Barat | Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case[END_REF][START_REF] Danchin | Partially dissipative systems in the critical regularity setting, and strong relaxation limit[END_REF][START_REF] Shizuta | Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation[END_REF][START_REF] Xu | Global classical solutions for partially dissipative hyperbolic system of balance laws[END_REF][START_REF] Xu | Relaxation limit in Besov spaces for compressible Euler equations[END_REF], and a new analysis is needed to be developed to obtain the uniform estimates with respect to the two relaxation parameters ε, τ .

In order to understand the behaviors of the solution to (1.21) with respect to ε, τ , we perform a spectral analysis of the linearized system for (1.21). For simplicity we set Fi = 1 (i = 0, 2, 3) and F1 = γ + -γ -.

In terms of Hodge decomposition, we denote the compressible part m = Λ -1 div u and the incompressible part Ω = Λ -1 ∇ × u with Λ σ := F -1 (|ξ| σ F (•)) and rewrite the linearized system of (1.21) as

∂ t     w r m     = A     w r m     , A :=     -1 ε 0 -(γ + -γ -)Λ 0 0 -Λ (γ + -γ -)Λ Λ -1 τ     , ∂ t Ω + 1 τ Ω = 0.
The eigenvalues of the matrix A(ξ) satisfy

| A(ξ) -λI 3×3 | = λ 3 + 1 τ + 1 ε λ 2 + 1 ετ + |γ + -γ -| 2 + 1 |ξ| 2 λ + 1 ε |ξ| 2 = 0.
Under the condition 0 < ε << τ , the behaviors of λ i (i = 1, 2, 3) can be analyzed as follows:

• In the low-frequency region |ξ| << 1 τ , by Taylor's expansion near |τ ξ| << 1 as in [START_REF] Matsumura | The Cauchy problem for the equations of motion of compressible viscous and heat-conductive fluids[END_REF], all the eigenvalues are real, and we have

λ 1 = -1 ε + 1 τ O(|τ ξ| 2 ), λ 2 = -τ |ξ| 2 + 1 τ O(|τ ξ| 3 ) and λ 3 = -1 τ + 1 τ O(|τ ξ| 2 ).
• In the medium-frequency region 1 τ << |ξ| << 1 ε , according to Cardano's formula, λ 1 is real and λ i (i = 2, 3) are conjugated complex, and Re λ i -1

τ holds for all i = 1, 2, 3.

• In the high-frequency region |ξ| >> 1 ε , by Taylor's expansion near |εξ| -1 << 1, the real eigenvalue λ 1 and the conjugated complex eigenvalues λ i (i = 2, 3) satisfy λ 1 = -

1 |γ+-γ-| 2 +1 1 ε + 1 ε O 1 |εξ| 2 and λ 2,3 = -1 2τ -|γ+-γ-| 2 |γ+-γ-| 2 +1 1 2ε ± |γ + -γ -| 2 + 1|ξ|i + ( 1 τ + |γ+-γ-| 2 ε )O 1 |εξ| .
The above spectral analysis suggests us to separate the whole frequencies into two parts |ξ| 1 τ and |ξ| 1 τ so as to capture the qualitative properties of solutions for System (1.21). Indeed, the time-decay rates (determined by λ 2 ) achieve the fastest rate in the low-frequency region |ξ| 1 τ . Moreover this region recover the whole frequency-space when τ → 0, as expected from the well-known overdamping phenomenon which will be mentioned below. To this end, the threshold J τ between these two regions is used in the definition of the hybrid Besov spaces in next section.

It should be noted that λ 2 and λ 3 exhibit similar behaviors to the eigenvalues of the compressible Euler equations with damping. Indeed, to study System (1.21), one considers the following simplified system of damped Euler type with rough coefficients:

   ∂ t r τ + (1 + G τ 3 )div u τ = 0, ∂ t u τ + (1 + G τ 0 )∇r τ + u τ τ = 0.
(1.25)

The well-known spectral analysis for the linear Euler part of System (1.25) implies that the frequency space shall be separated into the low-frequency region |ξ| 1 τ and the high-frequency region |ξ| 1 τ to recover the uniform estimates and optimal regularity of solutions. Formally, this implies that as τ → 0, the low-frequency region covers the whole frequency space and is therefore be dominant at the limit. We observe here the classical overdamping phenomenon: as the friction coefficient 1 τ gets larger, the decay rates of r τ do not necessarily increase and on the contrary follow min{τ, 1 τ }, cf. Figure 2. For more discussion on the overdamping phenomenon, see Zuazua's sildes [START_REF] Zuazua | Decay of partially dissipative hyperbolic systems[END_REF]. Recently, in [START_REF] Crin-Barat | Global existence for partially dissipative hyperbolic systems in the l p framework, and relaxation limit[END_REF][START_REF] Crin-Barat | Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case[END_REF], the issue concerning the relaxation limit from compressible Euler system with damping toward the porous media equation has been rigorously justified in critical space

ω * = |ξ| 1 τ = damping ω = decay rate 1 τ * = 2|ξ| ω = 1 2τ ω = 2τ |ξ| 2 1+ √ 1-4τ 2 |ξ| 2
Ḃ d 2 ∩ Ḃ d 2 +1
. The readers also can refer to the work [START_REF] Crin-Barat | The hyperbolic-parabolic chemotaxis system modelling vasculogenesis: global dynamics and relaxation limit[END_REF] about the relaxation limit for a hyperbolic-parabolic chemotaxis system to the Keller-Segel equations. As explained in the notes [START_REF] Danchin | Partially dissipative systems in the critical regularity setting, and strong relaxation limit[END_REF], the regularity index d 2 + 1 is called critical for initial data of general hyperbolic systems since

Ḃ d 2 +1 2,1
is embedded in the set of globally Lipschitz functions.

Nevertheless, the method developed in [START_REF] Crin-Barat | Global existence for partially dissipative hyperbolic systems in the l p framework, and relaxation limit[END_REF][START_REF] Crin-Barat | Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case[END_REF][START_REF] Crin-Barat | The hyperbolic-parabolic chemotaxis system modelling vasculogenesis: global dynamics and relaxation limit[END_REF] is not applicable in the current situation to derive estimates which are uniform with respect to the relaxation parameter τ . This is mainly due to the complex form of the total pressure P ε,τ in the velocity equation (BN) [START_REF] Bresch | Multifluid models including compressible fluids[END_REF] and the fact that one can not expect any time integrability property for the purely transported unknown y ε,τ , which generally leads to a lack of time integrability on

R + for (α ε,τ ± -ᾱ± , ρ ε,τ ± -ρ± ) (see Remarks 3.2-3.
3), and thus for G τ 3 , G τ 0 in System (1.25). In addition, we can not find a a rescaling to reduce the proof to the case τ = 1 and then recover the corresponding uniform estimates with respect to τ thanks to the homogeneity of the Besov norms as in [START_REF] Crin-Barat | Global existence for partially dissipative hyperbolic systems in the l p framework, and relaxation limit[END_REF][START_REF] Crin-Barat | Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case[END_REF]. To overcome this new difficulties, we will keep track of the dependence of ε, τ and perform elaborate energy estimates with mixed L 1 -time and L 2 -time type dissipation. More precisely, we introduce a purely damped mode corresponding to the Darcy's law (1.3) 4 in the low-frequency setting to partially diagonalize the system and derive a priori estimate at a lower regularity compared to [START_REF] Crin-Barat | Global existence for partially dissipative hyperbolic systems in the l p framework, and relaxation limit[END_REF][START_REF] Crin-Barat | Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case[END_REF][START_REF] Crin-Barat | The hyperbolic-parabolic chemotaxis system modelling vasculogenesis: global dynamics and relaxation limit[END_REF].

Due to the lack of symmetry, we need to cancel higher-order terms in the high-frequency setting so as not to lose derivatives. For that, the construction of a Lyapunov functional in the spirit of Beauchard and Zuazua as in [START_REF] Beauchard | Large time asymptotics for partially dissipative hyperbolic systems[END_REF] with additional nonlinear weights allows us to capture the dissipative structures in high frequencies. Moreover, we also establish the uniform L 2 -in-time estimates at Ḃ d 2 +1 -regularity level to recover the necessary bounds of parameters. Applying these ideas, we obtain uniform estimates in terms of the parameters ε, τ satisfying 0 < ε ≤ τ < 1 for the linearized problem (see Proposition 3.1), which is crucial for our later nonlinear analysis.

Let us finally sketch the proof of the justifications of the strong relaxation limits. In fact, to obtain convergence rates, we will not estimate the differences of solutions between systems directly. The reason shares similarities with the proof of global uniform well-posedness. Roughly speaking, since both pressure-relaxation limit and time-relaxation limit are singular limits, there are singular terms which are only uniformly bounded but not necessarily vanishing in the equations satisfied by the difference of solutions. To overcome these difficulties, we discover some auxiliary unknowns associated with the difference systems, which reveal better structures (cancellations), and then perform error estimates on them for each relaxation limit. More details are presented in Sections 5.1 and 5.2.

Tools

In this section, we recall the notations of the Littlewood-Paley decomposition and Besov spaces. The reader can refer to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF][Chapter 2] for a complete overview. Choose a smooth radial non-increasing function χ(ξ) with compact supported in B(0, 4 3 ) and χ(ξ) = 1 in B(0, 3 4 ) such that

ϕ(ξ) := χ (ξ/2) -χ(ξ), j∈Z ϕ(2 -j •) = 1, Supp ϕ ⊂ ξ ∈ R d 3 4 ≤ |ξ| ≤ 8 3 .
For any j ∈ Z, the homogeneous dyadic blocks ∆j and the low-frequency cut-off operator Ṡj are defined by

∆j u := F -1 (ϕ(2 -j •)F u), Ṡj u := F -1 (χ(2 -j •)F u),
where F and F -1 stand for the Fourier transform and its inverse. From now on, we use the shorthand

notation ∆j u = u j . Let S ′ h be the set of tempered distributions on R d such that every u ∈ S ′ h satisfies u ∈ S ′ and lim j→-∞ Ṡj u L ∞ = 0. Then it follows that u = j∈Z u j in S ′ , Ṡj u = j ′ ≤j-1 u j ′ , ∀u ∈ S ′ h ,
With the help of these dyadic blocks, the homogeneous Besov space Ḃs for s ∈ R is defined by

Ḃs := u ∈ S ′ h | u Ḃs := j∈Z 2 js u j L 2 < ∞ .
We denote the Chemin-Lerner type space L ̺ (0, T ; Ḃs ) for s ∈ R and T > 0:

L ̺ (0, T ; Ḃs ) := u ∈ L ̺ (0, T ; S ′ h ) | u L ̺ T ( Ḃs ) := j∈Z 2 js u j L ̺ T (L 2 ) < ∞ .
By the Minkowski inequality, it holds that

u L ̺ T ( Ḃs ) ≤ u L ̺ T ( Ḃs ) , where • L ̺ T ( Ḃs p,r
) is the usual Lebesgue-Besov norm. Moreover, we write

C b (R + ; Ḃs ) := u ∈ C(R + ; Ḃs ) | f L ∞ (R+; Ḃs ) < ∞ .
In order to perform our analysis on the low and high frequencies regions, we set the threshold

J τ := -[log 2 τ ] + k, (2.1)
for suitable negative integer k (to be determined). Denote the following notations for p ∈ [1, ∞] and

s ∈ R: u ℓ Ḃs := j≤Jτ 2 js u j L 2 , u h Ḃs := j≥Jτ -1 2 js u j L ̺ T (L 2 ) , u ℓ L ̺ T ( Ḃs ) := j≤Jτ 2 js u j L 2 , u h L ̺ T ( Ḃs ) := j≥Jτ -1 2 js u j L ̺ T (L 2 ) .
For any u ∈ S ′ h , we also define the low-frequency part u ℓ and the high-frequency part u h by

u ℓ := j≤Jτ -1 u j , u h := u -u ℓ = j≥Jτ u j .
It is easy to check for any

s ′ > 0 that    u ℓ Ḃs ≤ u ℓ Ḃs ≤ 2 Jτ s ′ u ℓ Ḃs-s ′ ≤ 2 s ′ (2 k τ -1 ) s ′ u ℓ Ḃs-s ′ , u h Ḃs ≤ u h Ḃs ≤ 2 -(Jτ -1)s ′ u h Ḃs+s ′ ≤ 2 s ′ (2 -k τ ) s ′ u h Ḃs+s ′ . (2.2)
Next, we state some properties of Besov spaces and related estimates which will be repeatedly used in the rest of paper. The reader can refer to [2, Chapters 2-3] for more details.

The first lemma pertains to the so-called Bernstein's inequalities.

Lemma 2.1. Let 0 < r < R, 1 ≤ p ≤ q ≤ ∞ and k ∈ N. For any function u ∈ L p and λ > 0, it holds    Supp F (u) ⊂ {ξ ∈ R d | |ξ| ≤ λR} ⇒ D k u L q λ k+d( 1 p -1 q ) u L p , Supp F (u) ⊂ {ξ ∈ R d | λr ≤ |ξ| ≤ λR} ⇒ D k u L p ∼ λ k u L p .
The following Morse-type product estimates in Besov spaces play a fundamental role in our analysis of nonlinear terms.

Lemma 2.2. The following statements hold:

• Let s > 0. Then Ḃs ∩ L ∞ is a algebra and uv Ḃs u L ∞ v Ḃs + v L ∞ u Ḃs . (2.3)
• Let s 1 , s 2 satisfy s 1 , s 2 ≤ d 2 and s 1 + s 2 > 0. Then there holds

uv Ḃs 1 +s 2 -d 2 u Ḃs 1 v Ḃs 2 .
(2.4)

The following commutator estimates will used to control some nonlinearities in high frequencies.

Lemma 2.3. Let p ∈ [1, ∞] and -d 2 -1 ≤ s ≤ d 2 + 1. Then it holds j∈Z 2 js [v, ∆j ]∂ i u L 2 ∇v Ḃ d 2 u Ḃs , i = 1, ..., d, (2.5)
for the commutator [A, B] := AB -BA.

We need the following lemma about the continuity for composition of functions. It should be noted that (2.7) will be used to deal with the two-dimensional case in Ḃ0 .

Lemma 2.4. Let m ∈ N, s > 0, and G ∈ C ∞ (R m ) satisfy G(0, ..., 0) = 0. Then for any f i ∈ Ḃs ∩ L ∞ (i = 1, ..., m), there exists a constant C > 0 depending on (f 1 , ..., f m ) L ∞ , F , s, m and d such that G(f 1 , ..., f m ) Ḃs p,r ≤ C (f 1 , ..., f m ) Ḃs p,r .
(2.6)

In the case s > -d 2 and f i ∈ Ḃs ∩ Ḃ d 2 , it holds that G(f 1 , ..., f m ) Ḃs p,r ≤ C(1 + (f 1 , ..., f m ) Ḃ d 2 ) (f 1 , ..., f m ) Ḃs p,r . (2.7)
Proof. The estimate (2.6) can be found in [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF][Pages 387-388]. Then ford 2 < s ≤ d 2 , Taylor's formula implies that there exists a sequence H i (f 1 , ..., f m ) satisfies H i (0, ..., 0) = 0 and

G(f 1 , ..., f m ) = m i=1 ∂ fi G(0, ..., 0) + H i (f 1 , ..., f m ) f i .
This together with the product law (2.4) and the estimate (2.6) yields (2.7).

We recall the optimal regularity estimates for the heat equation as follows (cf. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF][Page 157] for example).

Lemma 2.5. Let c 1 > 0, s ∈ R and 1 ≤ p ≤ ∞. For given time T > 0, assume u 0 ∈ Ḃs and

f ∈ L p (0, T ; Ḃs-2+ 2 p ). If u solves the problem    ∂ t u -c 1 ∆u = f, x ∈ R d , t > 0, u(0, x) = u 0 (x), x ∈ R d ,
then the following estimate is fulfilled:

u L ∞ t ( Ḃs ) + c 1 p 1 1 u L p t ( Ḃs+ 2 p ) ≤ C u 0 Ḃs + c 1 p -1 1 f L p t ( Ḃs-2+ 2 p ) , t ∈ (0, T ),
where C > 0 is a constant independent of T and c 1 .

Finally, we have the optimal regularity estimates of the damped transport equation. Since it can be directly shown by the commutator estimates (2.5) and Grönwall's inequality as in [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF][START_REF] Danchin | Fourier analysis methods for the compressible Navier-Stokes equations[END_REF], we omit the proof for brevity.

Lemma 2.6. Let c 2 ≥ 0, p = 1 or c 2 > 0, 1 ≤ p ≤ ∞. For -d 2 < s ≤ d 2 + 1 and given time T > 0, assume that u 0 ∈ Ḃs , v ∈ L 1 (0, T ; B d 2 +1 ) and f ∈ L p (0, T ; Ḃs ). If u solves the problem    ∂ t u + v • ∇u + c 2 u = f, x ∈ R d , t > 0, u(0, x) = u 0 (x), x ∈ R d , then it holds that u L ∞ t ( Ḃs ) + c 1 p 2 u L p t ( Ḃs ) ≤ Cexp C v L 1 t ( Ḃ d 2
+1
)

u 0 Ḃs + c 1 p -1 2 f L p t ( Ḃs ) , t ∈ (0, T ), where C > 0 is a constant independent of T and c 2 .

Analysis of the linearized system

We now consider the linear problem associated to (1.21), which reads

                       ∂ t y + v • ∇y = 0, ∂ t w + v • ∇w + (h 1 + H 1 )div u + (h 2 + H 2 ) w ε = S 1 , ∂ t r + v • ∇r + (h 3 + H 3 )div u = S 2 , ∂ t u + v • ∇u + u τ + (h 4 + H 4 )∇r + (h 5 + H 5 )∇w = S 3 , (y, w, r, u)(0, x) = (y 0 , w 0 , r 0 , u 0 )(x), (3.1)
where h i (i = 1, ..., 5) are given positive constants and

H i = H i (t, x) (i = 1, ..., 5), S i = S i (t, x) (i = 1, 2, 3)
are given smooth functions.

We first establish the following a-priori estimates for solutions of the linear problem (3.1) uniformly with respect to the parameters ε, τ , which improves the result in [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF] without the uniformity with respect to τ . As explained before, the threshold J τ between low and high frequencies given by (2.1) is the key to our analysis.

Proposition 3.1. Let d ≥ 2, 0 < ε ≤ τ < 1, T > 0,
and the threshold J τ be given by (2.1). Assume

that (w 0 , r 0 , u 0 ) ∈ Ḃ d 2 -1 ∩ Ḃ d 2 +1 , S 1 , S 2 , S 3 ∈ L 1 (0, T ; Ḃ d 2 -1 ∩ Ḃ d 2 +1 ), H i ∈ C([0, T ]; Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) and ∂ t H i ∈ L 1 (0, T ; Ḃ d 2 ) for i = 1, 2, ...5.
There exists a constant c > 0 independent of T , ε and τ such that if

Z(t) : = 5 i=1 H i L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) ≤ c, t ∈ (0, T ), (3.2)
then for t ∈ (0, T ), the solution (y, w, r, u) of the Cauchy problem (3.1) satisfies X (t) := (y, w, r, u)

L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + (∂ t y, ∂ t w, ∂ t r, ∂ t u) L 1 t ( Ḃ d 2 ) + 1 ε w L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 √ ε w L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + τ r ℓ L 1 t ( Ḃ d 2 +1 ∩ Ḃ d 2 +2 ) + r h L 1 t ( Ḃ d 2 +1 ) + τ r L 1 t ( Ḃ d 2 +1 ) + √ τ r L 2 t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + u L 1 t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + 1 √ τ u L 2 t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + 1 τ u + τ (h 4 + H 4 )∇r L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) ≤ C 0 exp C 0 t 0 V(s)ds (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 + (S 1 , S 2 , S 3 ) L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) , (3.3)
where C 0 > 1 is a universal constant, and V(t) is denoted by

V(t) := v(t) Ḃ d 2 ∩ Ḃ d 2 +1 + 5 i=1 ∂ t H i (t) Ḃ d 2 . (3.4)
Proof. First, we deal with the purely transport unknown y. By the regularity estimates in Lemma 2.6

for the transport equation (3.1) 1 , it follows that y

L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) exp t 0 v(s) Ḃ d 2 +1 ds y 0 Ḃ d 2 -1 ∩ Ḃ d 2 +1 . (3.5)
And direct produce law (2.4) for the equation (3.1) 1 gives that

∂ t y L 1 t ( Ḃ d 2 ) t 0 v(s) Ḃ d 2 y(s) Ḃ d 2 +1 ds. (3.6)
Similarly, we also get from (2.4) and (3.1) 3 that

∂ t r L 1 t ( Ḃ d 2 ) t 0 v(s) Ḃ d 2 r(s) Ḃ d 2 +1 ds + (1 + H 3 L ∞ t ( Ḃ d 2 )
) u

L 1 t ( Ḃ d 2 +1 ) + S 2 L 1 t ( Ḃ d 2 ) , (3.7)
and

(∂ t w, ∂ t u) L 1 t ( Ḃ d 2 ) t 0 v(s) Ḃ d 2 (w, u)(s) Ḃ d 2 +1 ds + 1 τ u + τ (h 5 + H 5 )∇r L 1 t ( Ḃ d 2 ) + 1 + 5 i=1 H i L ∞ t ( Ḃ d 2 ) 1 ε w L 1 t ( Ḃ d 2 )
+ u

L 1 t ( Ḃ d 2 +1 ) + (S 1 , S 3 ) L 1 t ( Ḃ d 2 )
.

(3.8)

The conclusion of the proof will follow from Lemmas 3.1-3.3 given and proven in the next three subsections.

Indeed, combining (3.5)-(3.8) and the uniform estimates of (w, r, u) from Lemmas 3.1-3.3 together and taking the constant η > 0 suitable small in Lemma 3.3, we obtain

X (t) (S 1 , S 2 , S 3 ) L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + Z(t) + Z(t) X (t) + t 0 V(s)X (s)ds + (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1
.

Then making use of the Grönwall inequality and the smallness assumption (3.2) of Z(t), we obtain the uniform a-priori estimates (3.3).

Low-frequency analysis

Motivated by Darcy's law (1.3) 4 , we introduce the following effective flux

z := u + τ (h 4 + H 4 )∇r, (3.9)
which undergoes a purely damped effect in the low-frequency region |ξ| ≤ 2 k τ and allows us to diagonalize the subsystem (3.1) 2 -(3.1) 4 up to some higher-order terms that can be absorbed. Indeed, substituting (3.9) into (3.1), we obtain

                 ∂ t w + h 2 ε w = L 1 + R 1 + S 1 , ∂ t r -h 3 h 4 τ ∆r = L 2 + R 2 + S 2 , ∂ t z + z τ = L 3 + R 3 + S 3 , (w, r, z)(0, x) = (w 0 , r 0 , z 0 )(x), (3.10)
where the higher-order linear terms L i (i = 1, 2, 3) are denoted as

       L 1 := h 1 h 4 τ ∆r -div z , L 2 := -h 3 div z, L 3 := h 3 h 4 τ ∇ h 4 τ ∆r -div z -h 5 ∇w, (3.11)
and the nonlinear terms R i (i = 1, 2, 3) are defined by

         R 1 := -v • ∇w -H 1 div u + h 1 τ div (H 4 ∇r) - 1 ε H 2 w, R 2 := -v • ∇r -H 3 div u + h 3 τ div (H 4 ∇r), R 3 := -v • ∇u -H 5 ∇w -τ ∂ t (H 4 ∇r) + h 4 τ ∇R 2 .
(3.12)

Now, to establish the

Ḃ d 2 -1 ∩ Ḃ d 2 -
estimates in low frequencies to the solutions of System (3.1) uniformly with respect to both ε and τ , we understand the equations in (3.10) are decoupled. More precisely, we will treat the equations of w and z as damped equations and r as a heat equation, respectively. This viewpoint plays a key role in the proof of the following lemma. Lemma 3.1. Let T > 0, and the threshold J τ be given by (2.1). Then for t ∈ (0, T ), the solution (w, r, u)

to the linear problem (3.1) 2 -(3.1) 4 satisfies (w, r, u) ℓ L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 ε w ℓ L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 √ ε w ℓ L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + τ r ℓ L 1 t ( Ḃ d 2 +1 ∩ Ḃ d 2 +2 ) + √ τ r ℓ L 2 t ( Ḃ d 2 ∩ Ḃ d 2
+1
)

+ u ℓ L 1 t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + 1 √ τ u ℓ L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 τ z ℓ L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) (w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 + (S 1 , S 2 , S 3 ) ℓ L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + Z(t)X (t) + t 0 V(s)X (s)ds, (3.13)
where Z(t), X (t), V(t) and z are defined by (3.2), (3.3), (3.4) and (3.9), respectively.

Remark 3.1. In [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF], the authors obtained the low-frequency estimates by constructing a related Lyapunov functional. However, that method does not lead to the desired estimates which uniform with respect to τ .

Moreover, it should be noted that the effective unknown z given by (3.9) enables us to capture the heat-like behavior of the unknown r in low frequencies directly, which is consistent with the parabolic nature of the limiting porous media equations.

The Ḃ d 2 -estimates

We first perform Ḃ d 2 -estimates in low frequencies for the heat equation (3.10) 2 . It follows from the regularity estimates in Lemma 2.5 that

r ℓ L ∞ t ( Ḃ d 2 ) + τ r ℓ L 1 t ( Ḃ d 2 +2 ) r 0 ℓ Ḃ d 2 + L 2 ℓ L 1 t ( Ḃ d 2 ) + R 2 ℓ L 1 t ( Ḃ d 2 ) + S 2 ℓ L 1 t ( Ḃ d 2 ) r 0 ℓ Ḃ d 2 + 2 Jτ z ℓ L 1 t ( Ḃ d 2 ) + R 2 ℓ L 1 t ( Ḃ d 2 ) + S 2 ℓ L 1 t ( Ḃ d 2 )
.

(3.14)

Applying Lemma 2.6 to the damped equation (3.10) 1 , we get

w ℓ L ∞ t ( Ḃ d 2 ) + 1 ε w ℓ L 1 t ( Ḃ d 2 ) w 0 ℓ Ḃ d 2 + L 1 ℓ L 1 t ( Ḃ d 2 ) + R 1 ℓ L 1 t ( Ḃ d 2 ) + S 1 ℓ L 1 t ( Ḃ d 2 ) w 0 ℓ Ḃ d 2 + τ r ℓ L 1 t ( Ḃ d 2 +2 ) + 2 Jτ z ℓ L 1 t ( Ḃ d 2 ) + R 1 ℓ L 1 t ( Ḃ d 2 ) + S 1 ℓ L 1 t ( Ḃ d 2 ) (w 0 , r 0 ) ℓ Ḃ d 2 + 2 Jτ z ℓ L 1 t ( Ḃ d 2 ) + (R 1 , R 2 ) ℓ L 1 t ( Ḃ d 2 ) + (S 1 , S 2 ) ℓ L 1 t ( Ḃ d 2 ) (3.15)
where we used inequality (3.14) to control terms involving r in equation (3.10) 1 .

Similarly, by virtue of inequality (3.14) and Lemmas 2.5-2.6, we have for equation (3.10) 3 that

z ℓ L ∞ t ( Ḃ d 2 ) + 1 τ z ℓ L 1 t ( Ḃ d 2 ) z 0 ℓ Ḃ d 2 + L 3 ℓ L 1 t ( Ḃ d 2 ) + (R 3 , S 3 ) ℓ L 1 t ( Ḃ d 2 ) z 0 ℓ Ḃ d 2 + 2 Jτ w ℓ L 1 t ( Ḃ d 2 ) + τ 2 2 Jτ r ℓ L 1 t ( Ḃ d 2 +2 ) + τ 2 2Jτ z ℓ L 1 t ( Ḃ d 2
+2
)

+ (R 3 , S 3 ) ℓ L 1 t ( Ḃ d 2 )
.

(3.16)

Since the threshold J τ satisfies condition (2.1), thus τ 2 Jτ ∼ 2 k << 1 for suitable negative integer k. Due to the condiiton ε ≤ τ so that 2 Jτ w ℓ

L 1 t ( Ḃ d 2 ) ≤ 2 k+1 ε w ℓ L 1 t ( Ḃ d 2 )
, we have by the inequalities (3.14

)-(3.16) that (w, r, z) ℓ L ∞ t ( Ḃ d 2 ) + τ r ℓ L 1 t ( Ḃ d 2 +2 ) + 1 ε w ℓ L 1 t ( Ḃ d 2 ) + 1 τ z ℓ L 1 t ( Ḃ d 2 ) (w 0 , r 0 , z 0 ) ℓ Ḃ d 2 + (R 1 , R 2 , R 3 ) ℓ L 1 t ( Ḃ d 2 ) + (S 1 , S 2 , S 3 ) ℓ L 1 t ( Ḃ d 2 )
.

(3.17)

The terms on the right-hand side of (3.17) can be estimated as follows. First, one derives from inequality (2.2) and product law (2.4) and the composition estimate (2.6) that

z 0 ℓ Ḃ d 2 (r 0 , u 0 ) ℓ Ḃ d 2 + H 4 (0) Ḃ d 2 r 0 Ḃ d 2 (r 0 , u 0 ) Ḃ d 2 . (3.18)
By the product law (2.4) again, we also get

               v • ∇w ℓ L 1 t ( Ḃ d 2 ) t 0 v(s) Ḃ d 2 w(s) Ḃ d 2 +1 ds, H 1 div u ℓ L 1 t ( Ḃ d 2 ) H 1 L ∞ t ( Ḃ d 2 )
u

L 1 t ( Ḃ d 2 +1
)

,

1 ε H 2 w ℓ L 1 t ( Ḃ d 2 ) H 2 L ∞ t ( Ḃ d 2 )
1 ε w

L 1 t ( Ḃ d 2 )
.

(3.19)
According to (2.2) and (2.3)-(2.4), the tricky nonlinear term H 4 ∇r in (3.12) can be estimated as

τ div (H 4 ∇r) ℓ L 1 t ( Ḃ d 2 ) τ H 4 ∇r ℓ ℓ L 1 t ( Ḃ d 2 +1 ) + τ H 4 ∇r h ℓ L 1 t ( Ḃ d 2 +1 ) τ H 4 ∇r ℓ ℓ L 1 t ( Ḃ d 2
+1
)

+ H 4 ∇r h ℓ L 1 t ( Ḃ d 2 ) H 4 L ∞ t ( Ḃ d 2 ) τ r ℓ L 1 t ( Ḃ d 2 +2 ) + H 4 L ∞ t ( Ḃ d 2 +1 ) τ r ℓ L 1 t ( Ḃ d 2
+1
)

+ H 4 L ∞ t ( Ḃ d 2 ) r h L 1 t ( Ḃ d 2 +1
)

.

( -estimates in the both low and high frequencies in the later Subsection 3.3. Indeed, in the low-frequency setting, the uniform

L ∞ t ( Ḃ d 2 )-norm is not enough to produce the uniform L ∞ t ( Ḃ d 2 +1
)-estimates required in (3.20) due to the inclusion (2.2). Now, one derives from inequalities (3.18)-(3.20) that

R 1 ℓ L 1 t ( Ḃ d 2 ) v • ∇w ℓ L 1 t ( Ḃ d 2 ) + H 1 div u ℓ L 1 t ( Ḃ d 2 ) + τ H 4 ∇r ℓ L 1 t ( Ḃ d 2 +1 ) + 1 ε H 2 w ℓ L 1 t ( Ḃ d 2 ) Z(t)X (t) + t 0 V(s)X (s)ds. (3.21)
Similarly, we have

R 2 ℓ L 1 t ( Ḃ d 2 ) v • ∇r ℓ L 1 t ( Ḃ d 2 ) + H 3 div u ℓ L 1 t ( Ḃ d 2 ) + τ H 4 ∇r ℓ L 1 t ( Ḃ d 2
+1
)

Z(t)X (t) + t 0 V(s)X (s)ds. (3.22)
To estimate R 3 , we notice that (2.2) together with (2.4) implies

τ ∂ t (H 4 ∇r) ℓ L 1 t ( Ḃ d 2 ) ∂ t (H 4 ∇r) ℓ L 1 t ( Ḃ d 2 -1 ) t 0 ∂ t H 4 (s) Ḃ d 2 r(s) Ḃ d 2 ds + H 4 L ∞ t ( Ḃ d 2 ) ∂ t r L 1 t ( Ḃ d 2 ) , (3.23)
and

H 5 ∇w ℓ L 1 t ( Ḃ d 2 ) 1 τ H 5 ∇w ℓ L 1 t ( Ḃ d 2 -1 )
H 5

L ∞ t ( Ḃ d 2 2,1 ) 1 ε w L 1 t ( Ḃ d 2 )
, where we used the assumption ε ≤ τ . Thus, it holds that

R 3 ℓ L 1 t ( Ḃ d 2 ) t 0 v(s) Ḃ d 2 u(s) Ḃ d 2 +1 ds + τ H 4 ∇r ℓ L 1 t ( Ḃ d 2
+1
)

+ H 5 ∇w ℓ L 1 t ( Ḃ d 2 ) + τ ∂ t (H 4 ∇r) ℓ L 1 t ( Ḃ d 2 -1 ) + R 2 ℓ L 1 t ( Ḃ d 2 -1 ) Z(t)X (t) + t 0 V (
(w, r, z) ℓ L ∞ t ( Ḃ d 2 ) + τ r ℓ L 1 t ( Ḃ d 2 +2 ) + √ τ r ℓ L 2 t ( Ḃ d 2 +1 ) + 1 ε w ℓ L 1 t ( Ḃ d 2 ) + 1 √ ε w ℓ L 2 t ( Ḃ d 2 ) + 1 τ z ℓ L 1 t ( Ḃ d 2 ) (w 0 , r 0 , u 0 ) Ḃ d 2 + (S 1 , S 2 , S 3 ) ℓ L 1 t ( Ḃ d 2 ) + Z(t)X (t) + t 0 V(s)X (s)ds. (3.25)
Thence, we rewrite the form (3.9) and use inequalities (2.2) and (3.20) to obtain the L 1 t ( Ḃ d 2 )-estimate of u as follows:

u ℓ L 1 t ( Ḃ d 2 +1 ) z ℓ L 1 t ( Ḃ d 2 +1 ) + τ ∇r ℓ L 1 t ( Ḃ d 2 +1 ) + τ H 4 ∇r ℓ L 1 t ( Ḃ d 2 +1 ) 1 τ z ℓ L 1 t ( Ḃ d 2 ) + τ r ℓ L 1 t ( Ḃ d 2 +2 ) + H 4 L ∞ t ( Ḃ d 2 ) τ r ℓ L 1 t ( Ḃ d 2 +2 ) + H 4 L ∞ t ( Ḃ d 2 +1 ) τ r ℓ L 1 t ( Ḃ d 2
+1
)

+ H 4 L ∞ t ( Ḃ d 2 ) r h L 1 t ( Ḃ d 2 +1
)

.

Similarly, we have

u ℓ L ∞ t ( Ḃ d 2 ) (z, r) ℓ L ∞ t ( Ḃ d 2 ) + H 4 L ∞ t ( Ḃ d 2 )
r

L ∞ t ( Ḃ d 2 )
, and

1 √ τ u ℓ L 2 t ( Ḃ d 2 ) 1 √ τ z ℓ L 2 t ( Ḃ d 2 ) + √ τ r ℓ L 2 t ( Ḃ d 2 +1 ) + H 4 L ∞ t ( Ḃ d 2 ) √ τ r L 2 t ( Ḃ d 2 +1
)

.

We thus obtain from inequality (3.25) that

u ℓ L ∞ t ( Ḃ d 2 ) + 1 √ τ u ℓ L 2 t ( Ḃ d 2 ) + u ℓ L 1 t ( Ḃ d 2 +1 ) (w 0 , r 0 , u 0 ) Ḃ d 2 + (S 1 , S 2 , S 3 ) ℓ L 1 t ( Ḃ d 2 )
+ Z(t)X (t) + t 0 V(s)X (s)ds.

(3.26)

3.1.2 The Ḃ d 2 -1 -estimates
We perform the Ḃ d 2 -1 -estimates so as to control τ r ℓ

L 1 t ( Ḃ d 2
+1
)

, as explained in Remark 3.2. Arguing similarly as for inequalities (3.14)-(3.17), we have

(w, r, z) ℓ L ∞ t ( Ḃ d 2 -1 ) + τ r ℓ L 1 t ( Ḃ d 2 +1 ) + 1 ε w ℓ L 1 t ( Ḃ d 2 -1 ) + 1 τ z ℓ L 1 t ( Ḃ d 2 -1 ) (w 0 , r 0 , u 0 ) Ḃ d 2 -1 + (R 1 , R 2 , R 3 ) ℓ L 1 t ( Ḃ d 2 -1 ) + (S 1 , S 2 , S 3 ) ℓ L 1 t ( Ḃ d 2 -1 )
.

(3.27)

Direct calculations give (R 1 , R 2 ) ℓ L 1 t ( Ḃ d 2 -1 ) t 0 v(s) Ḃ d 2 (w, r)(s) Ḃ d 2 ds + (H 1 , H 3 ) L ∞ t ( Ḃ d 2 ) u L 1 t ( Ḃ d 2 ) + H 4 L ∞ t ( Ḃ d 2 )
τ r

L 1 t ( Ḃ d 2 +1 ) + H 2 L ∞ t ( Ḃ d 2 ) 1 ε w L 1 t ( Ḃ d 2 -1 ) Z(t)X (t) + t 0 V(s)X (s)ds. (3.28)
By inequalities (3.23), (3.28) and product law (2.4) for d ≥ 2, the term R 3 can be bounded by

R 3 ℓ L 1 t ( Ḃ d 2 -1 ) t 0 v(s) Ḃ d 2 u(s) Ḃ d 2 ds + H 5 L ∞ t ( Ḃ d 2 2,1 ) 1 ε w L 1 t ( Ḃ d 2 ) + τ ∂ t (H 4 ∇r) ℓ L 1 t ( Ḃ d 2 -1 ) + R 2 ℓ L 1 t ( Ḃ d 2 -1 ) Z(t)X (t) + t 0 V(s)X (s)ds. (3.29)
Inserting (3.28) and (3.29) into (3.27) and taking advantage of interpolation, we obtain

(w, r, z) ℓ L ∞ t ( Ḃ d 2 -1 ) + τ r ℓ L 1 t ( Ḃ d 2 +1 ) + √ τ r ℓ L 2 t ( Ḃ d 2 ) + 1 ε w ℓ L 1 t ( Ḃ d 2 -1 ) + 1 √ ε w ℓ L 2 t ( Ḃ d 2 -1 ) + 1 τ z ℓ L 1 t ( Ḃ d 2 -1 ) (w 0 , r 0 , u 0 ) Ḃ d 2 -1 + (S 1 , S 2 , S 3 ) ℓ L 1 t ( Ḃ d 2 -1 ) + Z(t)X (t) + t 0 V(s)X (s)ds. (3.30)
This together with inequality (2.2) and the fact that u = z -τ (h 4 + H 4 )∇r leads to

u ℓ L ∞ t ( Ḃ d 2 -1 ) (z, r) ℓ L ∞ t ( Ḃ d 2 -1 ) + H 4 L ∞ t ( Ḃ d 2 ) r L ∞ t ( Ḃ d 2 -1 )
.

Similarly, one gets

1 √ τ u ℓ L 2 t ( Ḃ d 2 -1 ) 1 √ τ z ℓ L 2 t ( Ḃ d 2 -1 ) + √ τ r ℓ L 2 t ( Ḃ d 2 ) + H 4 L ∞ t ( Ḃ d 2 ) √ τ r L 2 t ( Ḃ d 2 )
, and

u ℓ L 1 t ( Ḃ d 2 ) 1 τ z ℓ L 1 t ( Ḃ d 2 -1 ) + τ r ℓ L 1 t ( Ḃ d 2 +1 ) + H 4 L ∞ t ( Ḃ d 2 )
τ r

L 1 t ( Ḃ d 2 +1
) .

Combining the above three estimates, we are led to

u ℓ L ∞ t ( Ḃ d 2 -1 ) + 1 √ τ u ℓ L 2 t ( Ḃ d 2 -1 ) + u ℓ L 1 t ( Ḃ d 2 ) (w 0 , r 0 , u 0 ) Ḃ d 2 -1 + (S 1 , S 2 , S 3 ) ℓ L 1 t ( Ḃ d 2 -1 )
+ Z(t)X (t) + t 0 V(s)X (s)ds.

High-frequency analysis

In this subsection, we establish some uniform high-frequency estimates of solutions to the linear problem (3.1) in terms of the Lyapunov functional. More precisely, we establish the

L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 )
estimates, and furthermore obtain the higher-order

L 1 t ( Ḃ d 2 ∩ Ḃ d 2 +1
)-norm estimates.

Lemma 3.2. Let T > 0, and the threshold J τ be given by (2.1). Then for any t ∈ (0, T ), the solution

(w, r, u) to the linear problem (3.1) 2 -(3.1) 4 satisfies (w, r, u) h L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + (w, r, u) h L 1 t ( Ḃ d 2 +1 ) + 1 ε w h L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 √ ε w h L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + √ τ r h L 2 t ( Ḃ d 2 +1 ) + 1 τ z h L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 )
(w 0 , r 0 , u 0 ) h

Ḃ d 2 +1 + (S 1 , S 2 , S 3 ) h L 1 t ( Ḃ d 2
+1
)

+ Z(t)X (t) + t 0 V(s)X (s)ds.

(3.32)

Proof. To prove of Lemma 3.2, we localize in frequencies for the equations (3.1) 2 -(3.1)

4 as            ∂ t w j + v • ∇w j + (h 1 + H 1 )div u j + (h 2 + H 2 ) w j ε = ∆j S 1 + T 1 j , ∂ t r j + v • ∇r j + (h 3 + H 3 )div u j = ∆j S 2 + T 2 j , ∂ t u j + v • ∇u j + u j τ + (h 4 + H 4 )∇r j + (h 5 + H 5 )∇w j = ∆j S 3 + T 3 j , (3.33)
with the commutator terms

         T 1 j := [v, ∆j ]∇w + [H 1 , ∆j ]div u + 1 ε [H 2 , ∆j ]w, T 2 j := [v, ∆j ]∇r + [H 3 , ∆j ]div u, T 3 j := [v, ∆j ]∇u + [H 4 , ∆j ]∇r + [H 5 , ∆j ]∇w.
(3.34)

Multiplying (3.33) 3 by u j and integrating the resulting equation by parts, we get

d dt R d 1 2 |u j | 2 dx + R d 1 τ |u j | 2 dx - R d (h 4 + H 4 )r j div u j + (h 5 + H 5 )w j div u j dx div v L ∞ u j 2 L 2 + ∆j S 3 L 2 + T 3 j L 2 u j L 2 + ∇H 4 L ∞ (w j , r j ) L 2 u j L 2 . (3.35)
Thence, we multiply (3.33) 1 by h5+H5 h1+H1 w j and integrate the resulting equation by parts to show

d dt R d 1 2 h 5 + H 5 h 1 + H 1 |w j | 2 dx + R d (h 5 + H 5 )w j div u j + (h 2 + H 2 )(h 5 + H 5 ) ε(h 1 + H 1 ) |w j | 2 dx ∂ t h 5 + H 5 h 1 + H 1 L ∞ + h 5 + H 5 h 1 + H 1 L ∞ div v L ∞ + ∇ h 5 + H 5 h 1 + H 1 L ∞ v L ∞ w j 2 L 2 + h 5 + H 5 h 1 + H 1 L ∞ ∆j S 1 L 2 + T 1 j L 2 w j L 2 .
(3.36)

Similarly, direct computations on (3.33) 2 yield

d dt R d 1 2 h 4 + H 4 h 3 + H 3 |r j | 2 dx + R d (h 4 + H 4 )r j div u j dx ≤ ∂ t h 4 + H 4 h 3 + H 3 L ∞ + h 4 + H 4 h 3 + H 3 L ∞ div v L ∞ + ∇ h 4 + H 4 h 3 + H 3 L ∞ v L ∞ r j 2 + h 4 + H 4 h 3 + H 3 L ∞ ∆j S 2 L 2 + T 2 j L 2 r j L 2 .
(3.37)

To derive the cross estimate and capture the dissipative property of r j , we gain by taking the L 2 -inner product of (3.33) 3 with ∇r j that

R d ∂ t u j • ∇r j dx + R d (h 4 + H 4 )|∇r j | 2 dx + R d (h 5 + H 5 )∇w j • ∇r j + 1 τ u j • ∇r j dx v L ∞ ∇u j L 2 + ∆j S 3 L 2 + T 3 j L 2 ∇r j L 2 , (3.38)
and taking the L 2 -inner product of (3.33) 2 with div u j that

R d u j • ∇∂ t r j dx - R d (h 3 + H 3 )|div u j | 2 dx v L ∞ ∇r j L 2 + ∆j S 2 L 2 + T 2 j L 2 div u j L 2 .
(3.39)

In the spirit of the work [START_REF] Beauchard | Large time asymptotics for partially dissipative hyperbolic systems[END_REF] by Beauchard and Zuazua, for a small constant η * > 0 to be determined, we define the following Lyapunov functional with nonlinear weights as

L j (t) := R d 1 2 h 5 + H 5 h 1 + H 1 |w j | 2 + h 4 + H 4 h 3 + H 3 |r j | 2 + |u j | 2 dx + η * τ 2 -2j R d u j • ∇r j dx,
and its dissipation rate

H j (t) : = R d 1 τ |u j | 2 + (h 2 + H 2 )(h 5 + H 5 ) ε(h 1 + H 1 ) |w j | 2 dx + η * τ 2 -2j R d (h 4 + H 4 )|∇r j | 2 + (h 5 + H 5 )∇w j • ∇r j + 1 τ u j • ∇r j dx.
One derives from assumption (3.2) and the embedding

Ḃ d 2 ֒→ L ∞ that H i L ∞ t (L ∞ ) + ∇H i L ∞ t (L ∞ ) H i L ∞ t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) c << 1, (3.40)
which together with estimates (3.35)-(3.39) and the fact that 2 -j τ ≤ 1 for any j ≥ J τ -1 yields the following Lyapunov inequality:

d dt L j (t) + H j (t) ( div v L ∞ + v L ∞ + 5 i=1 ∂ t H i L ∞ ) (r j , w j , u j ) 2 L 2 + 5 i=1 ∂ t H i L ∞ + ∆j (S 1 , S 2 , S 3 ) L 2 + (T 1 j , T 2 j , T 3 j ) L 2 (r j , w j , u j ) L 2 .
(3.41)

It follows from the smallness condition (3.40), Bernstein's inequality in Lemma 2.1 and the fact 2

-j τ that (1 -η * ) (w j , r j , u j ) 2 L 2 L j (t) (1 + η * ) (w j , r j , u j ) 2 L 2
, and

H j (t) 1 τ u j 2 L 2 + 1 ε w j 2 L 2 + η * τ 2 -2j ∇r j L 2 -∇w j 2 L 2 - 1 τ 2 u j L 2 1 τ (1 -η * ) u j 2 L 2 + 1 ε (1 -η * ) w j 2 L 2 + η * τ r j L 2 .
where one has used the condition ε ≤ τ . Thus, we can choose a sufficiently small constant η * > 0 independent of ε and τ so that

L j (t) ∼ (w j , r j , u j ) 2 L 2 , H j (t) 1 τ (w j , r j , u j ) 2 L 2 1 τ L j (t). (3.42)
Dividing the two sides of (3.41) by L j (t) + η for any η > 0, we have

d dt L j (t) + η + 1 τ L j (t) + η div v L ∞ + v L ∞ + 5 i=1 ∂ t H i L ∞ (r j , w j , u j ) L 2 + ∆j (S 1 , S 2 , S 3 ) L 2 + (T 1 j , T 2 j , T 3 j ) L 2 ,
which together with (3.42) and the embedding

Ḃ d 2 ֒→ L ∞ gives rise to τ (w, r, u) h L ∞ t ( Ḃ d 2 +1 ) + (w, r, u) h L 1 t ( Ḃ d 2 +1 ) τ (w 0 , r 0 , u 0 ) h Ḃ d 2 +1 + t 0 v(s) Ḃ d 2 ∩ Ḃ d 2 +1 + 5 i=1 ∂ t H i (s) Ḃ d 2 τ (w, r, u)(s) h Ḃ d 2 +1 ds + τ j≥Jτ -1 2 j( d 2 +1) (T 1 j , T 2 j , T 3 j ) L 1 t (L 2 ) + τ (S 1 , S 2 , S 3 ) h L 1 t ( Ḃ d 2 +1
) .

(3.43)

According to the commutator estimate (2.5), it follows that

τ j≥Jτ -1 2 j( d 2 +1) (T 1 j , T 2 j , T 3 j ) L 1 t (L 2 ) t 0 v(s) Ḃ d 2 +1 (w, r, u)(s) Ḃ d 2 +1 ds + 4 i=1 H i L ∞ t ( Ḃ d 2 +1 ) 1 ε w L 1 t ( Ḃ d 2 )
+ τ r

L 1 t ( Ḃ d 2 +1 ) + u L 1 t ( Ḃ d 2 +1 ) Z(t)X (t).
This with inequality (3.43) leads to

τ (w, r, u) h L ∞ t ( Ḃ d 2 +1 ) + (w, r, u) h L 1 t ( Ḃ d 2 +1 ) τ (w 0 , r 0 , u 0 ) h Ḃ d 2 +1 + τ (S 1 , S 2 , S 3 ) h L 1 t ( Ḃ d 2 +1 ) + Z(t)X (t) + t 0 V(s)X (s)ds. (3.44)
On the other hand, for any η > 0, we deduce from inequality (3.36) that

d dt w j 2 L 2 + η + 1 ε w j 2 L 2 + η 2 j u j L 2 + (∂ t H 1 , ∂ t H 5 ) L ∞ w j L 2 + div v L ∞ w j L 2 + v L ∞ w j L 2 + ∆j S 1 L 2 + T 1 j L 2 ,
which together with (3.44) implies

1 ε w h L 1 t ( Ḃ d 2 ) w 0 h Ḃ d 2 + u h L 1 t ( Ḃ d 2 +1 ) + S 1 h L 1 t ( Ḃ d 2 ) + τ j≥Jτ -1 2 j( d 2 +1) T 1 j L 1 t (L 2 ) + t 0 (∂ t H 1 , ∂ t H 5 )(s) Ḃ d 2 + v(s) Ḃ d 2 ∩ Ḃ d 2 +1 w(s) Ḃ d 2 ds τ (w 0 , r 0 , u 0 ) h Ḃ d 2 +1 + τ (S 1 , S 2 , S 3 ) h L 1 t ( Ḃ d 2 +1 ) + Z(t)X (t) + t 0 V(s)X (s)ds.
Thanks to inequality (2.2), one has

(w, r, u) h L ∞ t ( Ḃ d 2 -1 ) τ (w, r, u) h L ∞ t ( Ḃ d 2 ) τ (w, r, u) h L ∞ t ( Ḃ d 2
+1
)

.

(3.45)

Finally, the remain estimates in (3.32) can be achieved similarly to (3.45). We omit the details here and complete the proof of Lemma 3.2.

Recovering the Ḃ d 2 +1 -estimates

As explained in Remark 3.3, we need to establish the uniform

L ∞ t ( Ḃ d 2 +1
)-norm estimate of (w, r, u) which in fact leads to the uniform

L 2 t ( Ḃ d 2 +1 )-norm estimates of ( 1 √ ε w, 1 √ τ u)
for both low and high frequencies as a byproduct. Lemma 3.3. Let T > 0, and the threshold J τ be given by (2.1). Then for any t ∈ (0, T ), the solution (w, r, u) to the linear problem (3.1) 2 -(3.1) 4 satisfies (w, r, u)

L ∞ t ( Ḃ d 2 +1 ) + 1 √ ε w L 2 t ( Ḃ d 2 +1 ) + 1 √ τ u L 2 t ( Ḃ d 2 +1 ) (w 0 , r 0 , u 0 ) Ḃ d 2 +1 + (S 1 , S 2 , S 3 ) L 1 t ( Ḃ d 2
+1
)

+ (η + Z(t))X (t) + 1 η t 0 V(s)X (s)ds, (3.46)
where η ∈ (0, 1) is a constant to be chosen, and Z(t), X (t) and V(t) are defined by (3.2), (3.3) and (3.4), respectively.

Proof. We perform a L 2 -in-time type estimates and make use of the decay-in-τ estimates of u. In fact, for any j ∈ Z, by combining inequalities (3.35)-(3.36) together, we get

d dt R d 1 2 h 5 + H 5 h 1 + H 1 |w j | 2 + h 4 + H 4 h 3 + H 3 |r j | 2 + |u j | 2 dx + R d 1 τ |u j | 2 + (h 2 + H 2 )(h 5 + H 5 ) ε(h 1 + H 1 ) |w j | 2 dx div v L ∞ (w j , r j , u j ) 2 L 2 + 5 i=1 ∂ t H i L ∞ + div v L ∞ + v L ∞ (w j , r j ) 2 L 2 + ∇H 4 L ∞ (w j , r j ) L 2 u j L 2 + T 1 j L 2 w j L 2 + T 2 j L 2 r j L 2 + T 3 j L 2 u j L 2 + ∆j (S 1 , S 2 , S 3 ) L 2 (w j , r j , u j ) L 2 .
(3.47) Furthermore, from (3.47) we have (w, r, u)

L ∞ t ( Ḃ d 2 +1 ) + 1 √ ε w L 2 t ( Ḃ d 2 +1 ) + 1 √ τ u L 2 t ( Ḃ d 2 +1 ) (w 0 , r 0 , u 0 ) Ḃ d 2 +1 + v 1 2 L 1 t ( Ḃ d 2 +1 ) (w, r, u) L ∞ t ( Ḃ d 2 +1 ) + t 0 5 i=1 ∂ t H i (s) Ḃ d 2 + v(s) Ḃ d 2 ∩ Ḃ d 2 +1 (w, r)(s) Ḃ d 2 +1 ds 1 2
(w, r)

1 2 L ∞ t ( Ḃ d 2
+1
)

+ H 4 L ∞ t ( Ḃ d 2 +1 ) 1 √ τ u L 2 t ( Ḃ d 2 +1 ) √ τ (w, r) L 2 t ( Ḃ d 2
+1
)

1 2 + j∈Z 2 j( d 2 +1) t 0 T 1 j L 2 w j L 2 + T 2 j L 2 r j L 2 + T 3 j L 2 u j L 2 ds 1 2 + (S 1 , S 2 , S 3 ) L 1 t ( Ḃ d 2 +1 ) (w, r, u) L ∞ t ( Ḃ d 2 +1 ) 1 2 .
(3.48)

The right-hand side of inequality (3.48) can be estimated as follows. By the commutator estimate (2.5), we have

j∈Z 2 j( d 2 +1) t 0 T 1 j L 2 w j L 2 ds w L ∞ t ( Ḃ d 2 +1 ) t 0 v(s) Ḃ d 2 +1 w(s) Ḃ d 2 +1 ds + H 1 L ∞ t ( Ḃ d 2 +1 ) w L ∞ t ( Ḃ d 2 +1 ) u L 1 t ( Ḃ d 2 +1 ) + H 2 L ∞ t ( Ḃ d 2 +1 ) 1 ε w 2 L 2 t ( Ḃ d 2 +1 ) Z(t)X 2 (t) + X (t) t 0 V(s)X (s)ds.
Similarly, it holds

j∈Z 2 j( d 2 +1) t 0 T 2 j L 2 r j L 2 ds r L ∞ t ( Ḃ d 2 +1 ) t 0 v(s) Ḃ d 2 +1 r(s) Ḃ d 2 +1 ds + H 3 L ∞ t ( Ḃ d 2 +1 ) r L ∞ t ( Ḃ d 2 +1 ) u L 1 t ( Ḃ d 2 +1 ) Z(t)X 2 (t) + X (t) t 0 V(s)X (s)ds,
and j∈Z 2 j( d 2 +1) t 0 T 3 j L 2 u j L 2 ds u L ∞ t ( Ḃ d 2 +1 ) t 0 v(s) Ḃ d 2 +1 u(s) Ḃ d 2 +1 ds + (H 4 , H 5 ) L ∞ t ( Ḃ d 2 +1 ) r L ∞ t ( Ḃ d 2 +1 ) u L 1 t ( Ḃ d 2 +1 ) Z(t)X 2 (t) + X (t) t 0 V(s)X (s)ds.
We conclude from the above estimates that (w, r, u)

L ∞ t ( Ḃ d 2 +1 ) + 1 √ ε w L 2 t ( Ḃ d 2 +1 ) + 1 √ τ u L 2 t ( Ḃ d 2 +1 ) (w 0 , r 0 , u 0 ) Ḃ d 2 +1 + (S 1 , S 2 , S 3 ) L 1 t ( Ḃ d 2 +1 ) + Z(t)X (t) + t 0 V(s)X (s)ds 1 2

X (t).

Applying Hölder's inequality to above estimate leads to inequality (3.46).

4 Global existence and uniqueness for the nonlinear problems In this section, we prove (uniform in ε, τ ) global existence and uniqueness of solutions to the Cauchy problem for (BN) subject to the initial data (α ±,0 , ρ ±,0 , u 0 ) (for simplicity, we omit the dependence on the parameters ε, τ ), i.e. Theorem 1.1.

Proof of Theorem 1.1: Let (y 0 , w 0 , r 0 , u 0 ) satisfy the smallness condition (1.7) and set (y 0 , w 0 , r 0 , u 0 ) := (y 0 , w 0 , r 0 , u 0 ). For any n ≥ 0, we consider the approximate scheme for (1.21) as follows:

                           ∂ t y n+1 + u n • ∇y n+1 = 0, ∂ t w n+1 + u n • ∇w n+1 + ( F1 + G n 1 )div u n+1 + ( F2 + G n 2 ) w n+1 ε = 0, ∂ t r n+1 + u n • ∇r n+1 + ( F3 + G n 3 )div u n+1 = F n 4 (w n ) 2 ε , ∂ t u n+1 + u n • ∇u n+1 + u n+1 τ + ( F0 + G n 0 )∇r n+1 + (γ + -γ -)( F0 + G n 0 )∇w n+1 = 0, (y n+1 , w n+1 , r n+1 , u n+1 )(0, x) = ( Ṡn y 0 , Ṡn w 0 , Ṡn r 0 , Ṡn u 0 )(x), (4.1)
with (1.23) and (1.24), respectively.

F n i = F ε,τ i (y n , w n , r n ), F i and G n i = G ε,τ i (y n , w n , r n ) defined in (1.22),
We define E t the functional space associated to the following norm:

(y, w, r, u) Et : = (y, w, r, u)

L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + (∂ t y, ∂ t w, ∂ t r, ∂ t u) L 1 t ( Ḃ d 2 ) + 1 ε w L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 √ ε w L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + τ r ℓ L 1 t ( Ḃ d 2 +1 ∩ Ḃ d 2 +2 ) + √ τ r ℓ L 2 t ( Ḃ d 2 ∩ Ḃ d 2 +3 ) + r h L 1 t ( Ḃ d 2 +1 ) + r h L 2 t ( Ḃ d 2 +1 ) + u L 1 t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + 1 √ τ u L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + 1 τ u + τ ( F0 + G 0 )∇r L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) + 1 √ τ u + τ ( F0 + G 0 )∇r L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1
)

.

For any fixed n ≥ 1, we assume that (y n , w n , r n , u n ) satisfies

(y n , w n , r n , u n ) Et ≤ 2C 0 (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 , t > 0, (4.2)
where the constant C 1 > 1 is given by (3.3). Then by virtue of classical theorems for linear transport and parabolic equations (cf. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF][START_REF] Danchin | Fourier analysis methods for the compressible Navier-Stokes equations[END_REF]), there exists a unique global solution

(y n+1 , w n+1 , r n+1 , u n+1 ) ∈ C b (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
). Our goal is to show that (y n+1 , w n+1 , r n+1 , u n+1 ) also satisfies the estimate (4.2) uniformly in n ≥ 0. To this end, it follows from (1.7), (4.2) and the composition estimates in Lemma 2.4 that

4 i=0 G n i L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) (y n , r n , w n ) L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 ,
and similarly,

4 i=0 ∂ t G n i L 1 t ( Ḃ d 2 ) (1 + (y n , r n , w n ) L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1
)

) (∂ t y n , ∂ t w n , ∂ t r n ) L 1 t ( Ḃ d 2 ) (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 .
Thus, (3.2) follows when (y 0 , w 0 , r 0 , u 0 )

Ḃ d 2 -1 ∩ Ḃ d 2 +1
is sufficiently small, and we are able to employ the uniform a-priori estimates established in Proposition 3.1, (4.2) and the product laws (2.3)-(2.4) to obtain

(y n+1 , w n+1 , r n+1 , u n+1 ) Et ≤ C 1 exp C 1 t 0 u n (s) Ḃ d 2 ∩ Ḃ d 2 +1 + 4 i=0 ∂ t G n i (s) Ḃ d 2 ds × (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 + C ε w n 2 L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) ≤ 2C 1 (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 , t > 0, (4.3)
as long as (y 0 , w 0 , r 0 , u 0 )

Ḃ d 2 -1 ∩ Ḃ d 2 +1
is suitably small. Thus, the uniform estimates (4.2) hold for any n ≥ 0.

Then, for any given time T > 0 and χ ∈ C c (R d × (0, T )), in view of the Aubin-Lions lemma and the cantor diagonal process, there is a limit (y ε,τ , w ε,τ , r ε,τ , u ε,τ ) such that as n → ∞, up to a subsequence,

(y n , w n , r n , u n ) converges strongly to (y ε,τ , w ε,τ , r ε,τ , u ε,τ ) in C([0, T ]; Ḃs ) (s < d 2 + 1
). In addition, taking the advantage of Fatou property, we have

(y ε,τ , w ε,τ , r ε,τ , u ε,τ ) Et (y 0 , w 0 , r 0 , u 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1
, and as in [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF], one can obtain

(y ε,τ , w ε,τ , r ε,τ , u ε,τ ) ∈ C b (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
). Therefore, the limit (y ε,τ , w ε,τ , r ε,τ , u ε,τ ) is indeed a global strong solution to the Cauchy problem (1.21). The uniqueness can be shown by repeating same arguments as in [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF]. Finally, applying the inverse function theorem, we can see that once α ε,τ ± and ρ ε,τ ± are determined by (1.20), then

(α ε,τ ± , ρ ε,τ ± , u ε,τ ) ∈ C b (R + ; Ḃ d 2 -1 ∩ Ḃ d 2 +1
) is the unique global strong solution to the original Cauchy problem of System (BN) with the initial data (α ±,0 , ρ ±,0 , u 0 ) satisfying the properties (1.8)-(1.9), which concludes the proof of Theorem 1.1.

The Cauchy problem of System (K)

We provide a brief explanation on the proof of the global existence and uniqueness for System (K).

The uniformity of the estimates (1.9) for System (BN) essentially ensures us to construct solutions of System (K) by passing to the limit on the relaxation parameter ε.

Proof of Theorem 1.2: Let the initial data (α τ ±,0 , ρ τ ±,0 , u τ 0 ) satisfy the assumption (1.10). For any ε ∈ (0, 1), we consider the approximate problem, namely, System (BN) and the regularized initial data

(α ε,τ ±,0 , ρ ε,τ ±,0 , u ε,τ 0 )(x) := Ṡ[ 1 ε ] (α τ ±,0 , ρ τ ±,0 , u τ 0 )(x). (4.4) It is clear that the sequence (α ε,τ ±,0 , ρ ε,τ ±,0 , u ε,τ 0 ) converges to (α τ ±,0 , ρ τ ±,0 , u τ 0 ) strongly in Ḃ d 2 -1 ∩ Ḃ d 2 +1
and uniformly in τ as ε → 0. By virtue of Theorem 1.1, System (BN) with initial data (α ε,τ ±,0 , ρ ε,τ ±,0 , u ε,τ 0 ) has a unique global solution (α ε,τ ± , ρ ε,τ ± , u ε,τ ) satisfying the uniform estimates (1.9). In view of the Aubin-Lions lemma and the cantor diagonal process, there exists a limit (α τ ± , ρ τ ± , u τ ) such that as ε → 0, up to a subsequence, χ(α ε,τ ± , ρ ε,τ ± , u ε,τ ) converges to χ(α τ ± , ρ τ ± , u τ ) strongly (and uniformly in τ ) in C([0, T ]; Ḃs ) (s < d 2 + 1) for any given time

T > 0 and χ ∈ C ∞ c (R d × [0, T ]
). Thus, we can check that (α τ ± , ρ τ ± , u τ ) satisfies System (K) with the initial data (α τ ±,0 , ρ τ ±,0 , u τ 0 ) in the sense of distributions. Taking the advantage of the Fatou property, one can show that (α τ ± , ρ τ ± , u τ ) satisfies properties (1.11)-(1.12). For brevity and since it can follow the exact same arguments as in [START_REF] Burtea | Relaxation limit for a damped one-velocity Baer-Nunziato model to a Kappila model[END_REF], we omit here the details of proof of uniqueness.

The Cauchy problem of System (PM)

The following lemma states the uniform estimates of the solutions of System (K τ ), which are rescaled from estimates (1.12) in Theorem 1.2 for System (K). Lemma 4.1. Let (α τ ± , ρ τ ± , u τ ) be the global solution to the Cauchy problem of System (K) subject to the initial data (α τ ±,0 , ρ τ ±,0 , u τ 0 ) given by Theorem 1.2 and (β τ ± , ̺ τ ± , v τ ) be defined by the diffusive scaling (1.2), then it holds that

(β τ ± -ᾱ± , ̺ τ ± -ρ± ) L ∞ ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + (Π τ -P , ̺ τ ± -ρ± ) L 1 ( Ḃ d 2
+1
)

+ (Π τ -P , ̺ τ ± -ρ) L 2 ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + v τ L 1 ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + v τ L 2 ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + 1 τ z τ L 1 ( Ḃ d 2 -1 ∩ Ḃ d 2 ) ≤ C (α τ ±,0 -ᾱ± , ρ τ ±,0 -ρ± , u τ 0 ) Ḃ d 2 -1 ∩ Ḃ d 2 +1 , (4.5)
with z τ := v τ + 1 ̺ τ ∇Π τ , and C > 0 a universal constant.

Proof of Theorem 1.3: Assume that the initial data (β ±,0 , ̺ ±,0 ) satisfies (1.13). For any τ ∈ (0, 1),

we define the regularized data as

(α τ ±,0 , ρ τ ±,0 )(x) := Ṡ[ 1 τ ] (β ±,0 , ̺ ±,0 )(x) and u τ 0 (x) := 0.
Hence, by employing Theorem 1.2 we can obtain a sequence (α τ ± , ρ τ ± , u τ ), which is the global solution to System (K) subject to the initial data (α τ ±,0 , ρ τ ±,0 , u τ 0 ). Taking the diffusive scaling (1.2), one has that (β τ ± , ̺ τ ± , v τ ) is the global solution to System (K τ ) subject to the initial data (α ε ±,0 , ρ ε ±,0 , u τ 0 /τ ). In view of the uniform estimates (4.5) established in Lemma 4.1, the Aubin-Lions lemma and the cantor diagonal process, there exists a limit (β ± , ̺ ± ) such that as τ → 0, up to a subsequence,

χ(β τ ± , ̺ τ ± ) converges to χ(β ± , ̺ ± ) in C([0, T ]; Ḃs ) (s < d 2 + 1) strongly for any given time T > 0 and χ ∈ C ∞ c (R d × [0, T ]
). Thus, we can check that (β ± , ̺ ± ) solves System (PM) in the sense of distributions. Furthermore, taking the advantage of the Fatou property and the optimal regularity estimates in Lemma 2.5 for the equation of Π, we can conclude (1.14). Finally, the uniqueness can be obtained in a simple fashion. The interested reader may also refer to [START_REF] Crin-Barat | Global existence for partially dissipative hyperbolic systems in the l p framework, and relaxation limit[END_REF] for more details.

5 Relaxation limits with convergence rates 5.1 Pressure-relaxation limit: System (BN) to System (K)

In this subsection, we prove Theorem 1.4 related to the convergence rate of the relaxation process between System (BN) and System (K). Let (α ε,τ ± , ρ ε,τ ± , u ε,τ ) and (α τ ± , ρ τ ± , u τ ) be the global solutions to System (BN) with the initial data (α ε,τ ±,0 , ρ ε,τ ±,0 , u ε,τ 0 ) and System (K) with the initial data (α ε,τ ±,0 , ρ ε,τ ±,0 , u ε,τ 0 ) given by Theorems 1.1 and 1.2, respectively. Denote the error variables

(δα ± , δρ ± , δρ, δP ± , δP, δu) := (α ε,τ ± -α τ ± , ρ ε,τ ± -ρ τ ± , ρ ε,τ -ρ τ , P ± (ρ ε,τ ± ) -P ± (ρ τ ± ), P ε,τ -P τ , u ε,τ -u τ ),
and the initial data of δP δP (x, 0) = P ε,τ

0 -P τ 0 , P ε 0 := α ε,τ +,0 P + (ρ ε,τ +,0 ) + α ε,τ -,0 P -(ρ ε,τ -,0 ), P τ 0 := P + (ρ τ +,0 ). (5.1)
First, to avoid dealing with difficult nonlinearities in the equation of δα ± , we work with the following purely transported variable instead of δα ± :

δY := α ε,τ + ρ ε,τ + ρ ε,τ - α τ + ρ τ + ρ τ . (5.2)
with the initial data

δY (x, 0) = Y ε,τ 0 (x) -Y τ 0 (x), Y ε,τ 0 := α ε,τ +,0 ρ ε,τ +,0 α ε,τ +,0 ρ ε,τ +,0 + α ε,τ -,0 ρ ε,τ -,0 , Y τ 0 := α τ +,0 ρ τ +,0 α τ +,0 ρ τ +,0 + α τ -,0 ρ τ -,0 . (5.3)
Lemma 5.1. For d ≥ 3, under the assumption (1.16), δY satisfies the following estimate:

δY L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) √ ετ + o(1) δu L 1 t ( Ḃ d 2 -1 )
.

(5.4)

Proof. Since the equation of δY reads

∂ t δY + u ε,τ • ∇δY = -δu • ∇ α τ + ρ τ + ρ τ ,
Lemma 2.6 and the product law (2.4) for d ≥ 3 gives δY

L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) exp u ε,τ L 1 t ( Ḃ d 2 +1 ) √ ετ + δu L 1 t ( Ḃ d 2 -1 ) ∇ α τ + ρ τ + ρ τ L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 )
.

This together with the uniform estimate (1.9) leads to (5.4).

We are now ready to estimate (δα ± , δρ ± , δP ± , δP ). It is easy to verify that P ε,τ satisfies

∂ t P ε,τ + u ε,τ • ∇P ε,τ = -γ + α ε,τ + P + (ρ ε,τ + ) + γ -α ε,τ -P -(ρ ε,τ -) div u ε,τ -α ε,τ + α ε,τ -(γ + -1)P + (ρ ε,τ + ) -(γ --1)P -(ρ ε,τ -) P + (ρ ε,τ + ) -P -(ρ ε,τ -) ε .
(5.5)

And the equation of P τ reads

∂ t P τ + u τ • ∇P τ + γ + γ -P τ γ + α τ -+ γ -α τ + div u τ = 0. (5.6)
However it is not suitable to estimate δP directly from (5.5)-(5.6) as the decay rate of P + (ρ ε,τ + )-P -(ρ ε,τ -) can not be faster than ε in view of (1.9). To overcome this difficulty, we introduce an auxiliary unknown

Q ε,τ := P ε,τ -Γ ε,τ 1 P + (ρ ε,τ + ) -P -(ρ ε,τ -) which verifies ∂ t Q ε,τ + u ε,τ • ∇Q ε,τ + Γ ε,τ 2 div u ε,τ = -Γ ε,τ 3 (P + (ρ ε,τ + ) -P -(ρ ε,τ -) div u ε,τ + (∂ t Γ ε,τ 1 + u ε,τ • ∇Γ ε,τ 1 ) P + (ρ ε,τ + ) -P -(ρ ε,τ -) (5.7) with                    Γ ε,τ 1 := α ε,τ + α ε,τ -(γ + -1)P + (ρ ε,τ + ) -(γ --1)P -(ρ ε,τ -) γ + α ε,τ -P + (ρ ε,τ + ) + γ -α ε,τ + P -(ρ ε,τ -) , Γ ε,τ 2 := γ + γ -P + (ρ ε,τ + )P -(ρ ε,τ -) γ + α -P + (ρ ε,τ + ) + γ -α ε,τ + P -(ρ ε,τ -) , Γ ε,τ 3 := α ε,τ + α ε,τ -γ + P + (ρ ε,τ + ) -γ -P -(ρ ε,τ -) γ + α ε,τ -P + (ρ ε,τ + ) + γ -α ε,τ + P -(ρ ε,τ -)
.

With this formulation, it will be possible to derive the O(ε) bounds for the last term on the right-hand side of (5.7). Define

δQ := P ε,τ -P τ -Γ ε,τ 1 P + (ρ ε,τ + ) -P -(ρ ε,τ -) . (5.8)
The next lemma implies that to estimate (δα ± , δρ ± , δP ± , δP ), it is sufficient to control (δY, δQ, P + (ρ ε,τ + )-P -(ρ ε,τ -)).

Lemma 5.2. For d ≥ 3, under the assumption (1.16), the following estimates follow:

     (δα ± , δρ ± , δρ) L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 )
δY, δQ, P + (ρ ε,τ

+ ) -P -(ρ ε,τ -) L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) , δρ ± L 2 t ( Ḃ d 2 -1 ) δQ, P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 2 t ( Ḃ d 2 -1 )
.

(5.9)

Proof. Due to (5.2) and

δρ = (ρ ε,τ + -ρ ε,τ -)δα + + α τ + δρ + + α τ -δρ -, (5.10)
it holds that

δY = 1 ρ ε,τ ρ τ ρ ε,τ + ρ τ δα ε,τ + + α τ + ρ τ δρ + -α τ + ρ τ + δρ = 1 ρ ε,τ ρ τ (α τ -ρ ε,τ + ρ τ -+ α τ + ρ τ + ρ ε,τ -)δα + + α τ + α τ -ρ τ -δρ + -α τ + α τ -ρ τ + δρ -.
This implies

δα + = 1 α τ -ρ ε,τ + ρ τ -+ α τ + ρ τ + ρ ε,τ - ρ ε,τ ρ τ δY -α τ + α τ -ρ τ -δρ + + α τ + α τ -ρ τ + δρ -. (5.11)
Inserting (5.11) into (5.10), we have

δρ = ρ ε,τ + -ρ ε,τ - α τ -ρ ε,τ + ρ τ -+ α τ + ρ τ + ρ ε,τ - ρ ε,τ ρ τ δY -α τ + α τ -ρ τ -δρ + + α τ + α τ -ρ τ + δρ -+ α τ + δρ + + α τ -δρ -.
(5.12)

Moreover, we have

δP ± = δρ ± 1 0 P ′ ± (θρ ε,τ ± + (1 -θ)ρ τ ± )dθ and δP = α ε,τ + (P ε,τ + -P ε,τ -) + δP -. (5.13)
Using the previous uniform estimates (1.9) and (4.5), the product laws (2.3)-(2.4) and the composition estimates (2.6)-(2.7), for some constant states Γi > 0 (i = 1, 2, 3), we have

3 i=1 ( Γ i -Γi L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 +1 ) + ∂ t Γ i L 1 t ( Ḃ d 2 )
) << 1.

(5.14)

Therefore, (5.9) follows from (5.8), (5.11)-(5.14), the product laws (2.3)-(2.4) and the fact δα + = -δα + .

The next lemma pertains to O( √ ετ ) bounds for P + (ρ ε,τ + ) -P -(ρ ε,τ -), which leads to the convergence rate √ ετ .

Lemma 5.3. For d ≥ 3, under the assumption (1.16), the following estimate is valid:

P + (ρ ε,τ + ) -P -(ρ ε,τ -) L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) + 1 √ ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 -1 )
√ ετ .

(5.15)

Proof. It is easy to verify from (BN) that P + (ρ ε,τ + ) -P -(ρ ε,τ -) satisfies the damped equation

∂ t P + (ρ ε,τ + ) -P -(ρ ε,τ -) + u ε,τ • ∇ P + (ρ ε,τ + ) -P -(ρ ε,τ -) + c * ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) = ((γ + α ε,τ -P + (ρ ε,τ + ) + γ -α ε,τ + P -(ρ ε,τ -)) -c * ) 1 ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) -γ + P + (ρ ε,τ + ) -γ -P -(ρ ε,τ -) div u ε,τ =: W 1 + W 2 .
(5.16) with c * := (γ + ᾱ-+ γ -ᾱ+ ) P . Thence the L 2 -in-time type estimates in Lemma 2.6 for the damped transport equation (5.16) lead to

P + (ρ ε,τ + ) -P -(ρ ε,τ -) L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) + 1 √ ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 2 t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) exp u ε,τ L 1 t ( Ḃ d 2 +1 ) √ ετ + √ ε (W 1 , W 2 ) L 2 t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 )
.

By (1.9) and (2.4), there holds

√ ε W 1 L 2 t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) (α ε,τ ± -ᾱ± , ρ ε,τ ± -ρ± ) L ∞ t ( Ḃ d 2 ) 1 √ ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 2 t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 )
o(1)

1 √ ε P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 2 t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 )
.

and √ ε W 2 L 2 t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) √ ε u ε,τ L 2 t ( Ḃ d 2 -1 ∩ Ḃ d 2 )
√ ετ .

Therefore, we gain (5.15).

We are going to estimate (δQ, δu). By virtue of (BN), (K) and (5.6)-(5.7), (δQ, δu) satisfies the following equations of damped Euler type with rough coefficients:

     ∂ t δQ + u ε,τ • ∇δQ + Γ ε,τ 2 div δu = δF 1 , ∂ t δu + u ε,τ • ∇δu + 1 ρ ∇δQ + ( 1 ρ ε,τ - 1 ρ τ )∇P τ + δu τ = δF 2 , (5.17)
with the nonlinear terms

             δF 1 = -δu • ∇P τ -Γ ε,τ 2 - γ + γ -P τ γ + α τ -+ γ -α τ + div u τ -Γ ε,τ 3 P + (ρ ε,τ + ) -P -(ρ ε,τ -) div u ε,τ + (∂ t Γ ε,τ 1 + u ε,τ • ∇Γ ε,τ 1 ) P + (ρ ε,τ + ) -P -(ρ ε,τ -) , δF 2 := -δu • ∇u τ - 1 ρ ε,τ ∇ Γ ε,τ 1 P + (ρ ε,τ + ) -P -(ρ ε,τ -) .
In order to establish the uniform-in-τ convergence estimates, we follow the ideas in Section 3 to overcome the issue caused by the overdamping phenomenon.

Lemma 5.4. Let d ≥ 3, 0 < ε ≤ τ ≤ 1, and the threshold J τ be given by (2.1). Then under the assumption (1.16), there holds (δQ, δu)

L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) + τ δQ ℓ L 1 t ( Ḃ d 2 ) + δQ h L 1 t ( Ḃ d 2 -1 ) + √ τ δQ L 2 t ( Ḃ d 2 -1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 )
+ δu

L 1 t ( Ḃ d 2 -1 ) √ ετ + o(1) δY L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 )
.

(5.18)

Proof. As in Section 3, we split the proof into three parts:

• Step 1: Ḃ d 2 -2 -

estimates in low frequencies

We introduce the new damped mode (effective flux)

δz := δu + τ ρ ε,τ ∇δQ + τ ( 1 ρ ε,τ - 1 ρ τ )∇P τ , so that (5.17) is rewritten as        ∂ t δQ - Γ2 τ ρ ∆δQ = -Γ2 div z + δF 3 , ∂ t δz + δz τ = τ ρ ∇( Γ2 τ ρ ∆δQ -Γ2 div z) + δF 4 , (5.19)
where Γ2 > 0 is the constant state of Γ ε,τ 2 , and δF i (i = 3, 4) is defined by

       δF 3 := -u ε,τ • ∇δQ -(Γ ε,τ 2 -Γ2 )div δu + Γ2 τ div ( 1 ρ ε,τ - 1 ρ )∇δQ + ( 1 ρ ε,τ - 1 ρ τ )∇P τ + δF 1 , δF 4 := -u ε,τ • ∇δu + τ ρ ∇δF 3 + τ ( 1 ρ ε,τ - 1 ρ )∇∂ t δQ -τ ∂ t ( 1 ρ ε,τ )∇δQ + τ ∂ t ( 1 ρ ε,τ - 1 ρ τ )∇P τ + δF 2 .
Then by similar arguments used to get (3.14)-(3.17), we deduce from (5.19) and the choice (2.1) of the threshold J τ that

(δQ, δz) ℓ L ∞ t ( Ḃ d 2 -2 ) + τ δQ ℓ L 1 t ( Ḃ d 2 ) + τ ∂ t δQ ℓ L 1 t ( Ḃ d 2 -2 ) + √ τ δQ ℓ L 2 t ( Ḃ d 2 -1 ) + 1 τ δz ℓ L 1 t ( Ḃ d 2 -2 ) √ ετ + (δF 3 , δF 4 ) ℓ L 1 t ( Ḃ d 2 -2 )
.

(5.20)

We first estimate δF 3 . From (1.9), (5.14) and the product map

Ḃ d 2 -2 × Ḃ d 2 → Ḃ d 2 -2 for d ≥ 3, one obtains u ε,τ • ∇δQ + (Γ ε,τ 2 -Γ2 )div δu L 1 t ( Ḃ d 2 -2 ) 1 √ τ u ε,τ L 2 t ( Ḃ d 2 ) √ τ δQ L 2 t ( Ḃ d 2 -1 ) + Γ ε,τ 2 -Γ2 L ∞ t ( Ḃ d 2 ) δu L 1 t ( Ḃ d 2 -1 ) o(1)( √ τ δQ L 2 t ( Ḃ d 2 -1 ) + δu L 1 t ( Ḃ d 2 -1 )
).

(5.21)

By virtue of (1.9), (2.2), (5.9), (5.15) and (2.4), we also have

τ div ( 1 ρ ε,τ - 1 ρ τ )∇P τ ℓ L 1 t ( Ḃ d 2 -2 ) δρ L ∞ t ( Ḃ d 2 -1 ) τ P τ -P L 1 t ( Ḃ d 2 +1 ) o(1) (δY, δQ) L ∞ t ( Ḃ d 2 -1 ) + √ ετ .
As in the previous analysis (3.20), the tricky nonlinear term can be estimated as

τ div ( 1 ρ ε,τ - 1 ρ )∇δQ ℓ L 1 t ( Ḃ d 2 -2 ) τ ( 1 ρ ε,τ - 1 ρ )∇δQ ℓ ℓ L 1 t ( Ḃ d 2 -1 ) + ( 1 ρ ε,τ - 1 ρ )∇δQ h ℓ L 1 t ( Ḃ d 2 -2 ) o(1)(τ δQ ℓ L 1 t ( Ḃ d 2 ) + δQ h L 1 t ( Ḃ d 2 -1 )
).

Similarly, one can show

δF 1 L 1 t ( Ḃ d 2 -2 ) 1 √ τ δu L 2 t ( Ḃ d 2 -2 ) √ τ Q τ L 2 t ( Ḃ d 2 ) + Γ ε,τ 2 - γ + γ -P τ γ + α τ -+ γ -α τ + L ∞ t ( Ḃ d 2 -2 ) u τ L 1 t ( Ḃ d 2 +1
)

+ P + (ρ ε,τ + ) -P -(ρ ε,τ -) L ∞ t ( Ḃ d 2 -2 ) u ε,τ L 1 t ( Ḃ d 2 +1 ) + ( ∂ t Γ ε,τ 1 L 1 t ( Ḃ d 2 ) + u ε,τ L 1 t ( Ḃ d 2 ) ∇Γ ε,τ 1 L ∞ t ( Ḃ d 2
)

) P + (ρ ε,τ + ) -P -(ρ ε,τ -) L ∞ t ( Ḃ d 2 -2 ) o(1) (δY, δQ) L ∞ t ( Ḃ d 2 -2 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -2 ) + √ ετ . (5.22)
Therefore, we have

δF 3 ℓ L 1 t ( Ḃ d 2 -2 ) o(1) (δY, δQ) L ∞ t ( Ḃ d 2 -2 ) + δQ L 1 t ( Ḃ d 2 ) + √ τ δQ L 2 t ( Ḃ d 2 -1 2,1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -2 ) + δu L 1 t ( Ḃ d 2 -1 ) + √ ετ . (5.23)
We turn to the estimate of δF 4 . Similar calculations give

-u ε,τ • ∇δu + τ ρ ε,τ ∇δF 3 L 1 t ( Ḃ d 2 -2 )
o(1) δu

L 1 t ( Ḃ d 2 -1 2,1 ) + δF 3 L 1 t ( Ḃ d 2 -2 )
, and

τ ∂ t ( 1 ρ ε,τ )∇δQ L 1 t ( Ḃ d 2 -2 ) ∂ t ρ ε,τ L 1 t ( Ḃ d 2 ) δQ L ∞ t ( Ḃ d 2 -1 )
o(1) δQ

L ∞ t ( Ḃ d 2 -1 )
.

For the third difficult term in δF 4 , we apply (2.2), the product law (2.4) for d ≥ 3 and the fact τ

( 1 ρ ε,τ - 1 ρ )∇∂ t δQ = τ ∇ ( 1 ρ ε,τ -1 ρ )∂ t δQ -τ ∇( 1 ρ ε,τ -1 ρ )∂ t δQ to have τ ( 1 ρ ε,τ - 1 ρ )∇∂ t δQ ℓ L 1 t ( Ḃ d 2 -2 ) τ ∇ ( 1 ρ ε,τ - 1 ρ )∂ t δQ ℓ L 1 t ( Ḃ d 2 -2 ) + τ ∇( 1 ρ ε,τ - 1 ρ )∂ t δQ ℓ L 1 t ( Ḃ d 2 -2 ) ( 1 ρ ε,τ - 1 ρ )∂ t δQ ℓ L 1 t ( Ḃ d 2 -2 ) + ∇( 1 ρ ε,τ - 1 ρ )∂ t δQ ℓ L 1 t ( Ḃ d 2 -2 ) ρ ε,τ -ρ L ∞ t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) ∂ t δQ L 1 t ( Ḃ d 2 -2 ) o(1) ∂ t δQ L 1 t ( Ḃ d 2 -2 )
.

Similarly, the term δF 2 can be easily estimated as follows:

δF 2 L 1 t ( Ḃ d 2 -2 ) δu L 1 t ( Ḃ d 2 -1 ) u τ L ∞ t ( Ḃ d 2 ) + P + (ρ ε,τ + ) -P -(ρ ε,τ -) L 1 t ( Ḃ d 2 -1 )
o(1) δu

L 1 t ( Ḃ d 2 -1 ) + √ ετ + ε o(1) δu L 1 t ( Ḃ d 2 -1 ) + √ ετ .
To bound the term τ ∂ t ( 1 ρ ε,τ -1 ρ τ )∇P τ , since ∂ t δρ = -div δρu ε,τ + ρ τ δu) follows, we use (1.9), (5.9), (5.15) and (2.6)-(2.7) that

τ ∂ t ( 1 ρ ε,τ - 1 ρ τ )∇P τ L 1 t ( Ḃ d 2 -2 ) ∂ t δρ L 1 t ( Ḃ d 2 -2 ) ∇P τ L ∞ t ( Ḃ d 2 )
+ δρ

L ∞ t ( Ḃ d 2 -2 ) τ ∇P τ L 1 t ( Ḃ d 2 )
o(1)( (δY, δQ)

L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) + δu L 1 t ( Ḃ d 2 -1 )
) + √ ετ .

We thence get

δF 3 ℓ L 1 t ( Ḃ d 2 -2 ) o(1) (δY, δQ) L ∞ t ( Ḃ d 2 -2 ) + δQ L 1 t ( Ḃ d 2 ) + √ τ δQ L 2 t ( Ḃ d 2 -1 2,1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -2 ) + δu L 1 t ( Ḃ d 2 -1 ) + √ ετ .
(5.24)

Substituting the above estimates (5.23)-(5.24) into (5.20) and taking the advantage of δu = δz -

τ ρ ε,τ ∇δQ -τ ( 1 ρ ε,τ -1 ρ τ )∇P τ , we obtain (δQ, δz) ℓ L ∞ t ( Ḃ d 2 -2 ) + τ δQ ℓ L 1 t ( Ḃ d 2 ) + √ τ δQ ℓ L 2 t ( Ḃ d 2 -1 ) + 1 τ δz ℓ L 1 t ( Ḃ d 2 -2 ) + δu ℓ L ∞ t ( Ḃ d 2 -2 ) + 1 √ τ δu ℓ L 2 t ( Ḃ d 2 -2 ) + δu ℓ L 1 t ( Ḃ d 2 -1 ) √ ετ + o(1)( (δY, δQ) L ∞ t ( Ḃ d 2 -2 ∩ Ḃ d 2 -1 ) + τ δQ ℓ L 1 t ( Ḃ d 2 ) + δQ h L 1 t ( Ḃ d 2 -1 ) + √ τ δQ L 2 t ( Ḃ d 2 -1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -2 ) + δu L 1 t ( Ḃ d 2 -1 ) + ∂ t δQ L 1 t ( Ḃ d 2 -2 )
).

(5.25)

•

Step 2: Ḃ d 2 -2 -estimates of (δQ, δu) in high frequencies

Applying ∆ j to (5.17), one gets

     ∂ t ∆j δQ + u ε,τ • ∇ ∆j δQ + Γ ε,τ 2 div ∆j δu = ∆j δF 1 + δR 1,j , ∂ t ∆j δu + u ε,τ • ∇ ∆j δu + 1 ρ ε,τ ∇ ∆j δQ + ∆j δu τ = -∆j ( 1 ρ ε,τ - 1 ρ τ )∇P τ + ∆j δF 2 + δR 2,j + δR 3,j , with δR 1,j := [u ε,τ , ∆j ]∇δQ + [Γ ε,τ 2
, ∆j ]div ∆j δu, δR 2,j := [u ε,τ , ∆j ]∇δu and δR 3,j := [ 1 ρ ε,τ , ∆j ]∇δQ. Similarly to the high-frequency analysis in Subsection 3.2, one gains

d dt ( 1 ρ ε,τ | ∆j δQ| 2 + Γ ε,τ 2 | ∆j δu| 2 )dx + 1 τ ∆j δu 2 L 2 div u ε,τ L ∞ + ∇Γ ε,τ 2
L ∞ + ∇ 1 ρ ε,τ L ∞ + ∂ t 1 ρ ε,τ L ∞ + ∂ t Γ ε,τ 2 L ∞ ∆j δQ L 2 ∆j δu L 2 + ∆j ( 1 ρ ε,τ - 1 ρ τ )∇P τ L 2 ∆j δu L 2 + ∆j (δF 1 , δF 2 ) L 2 ∆j (δQ, δu) L 2 + δR 1,j L 2 ∆j δQ L 2 + (δR 2,j , δR 3,j ) L 2 ∆j δu L 2 , (5.26)
and the cross term

d dt ∆j δu • ∇ ∆j ∇δP dx + ( 1 ρ ε,τ |∇ ∆j δQ| 2 -Γ ε,τ 2 |div ∆j δu j | 2 + 1 τ ∆j δu • ∇ ∆j ∇δP )dx ( u ε,τ L ∞ ∇ ∆j δu L 2 + ∆j ( 1 ρ ε,τ - 1 ρ τ )∇P τ L 2 + ( ∆j δF 2 , δR 2,j , δR 3,j ) L 2 ) ∇ ∆j ∇δP L 2 + ( u ε,τ L ∞ ∇ ∆j δQ L 2 + ∆j δF 1 L 2 + δR 1,j L 2 ) δu L 2 .
(5.27) For all j ≥ J τ , multiplying (5.27) by a suitable small constant and adding the resulting inequality and

τ (δQ, δu) h L ∞ t ( Ḃ d 2 -1 ) + (δQ, δu) h L 1 t ( Ḃ d 2 -1 ) √ τ ε + ( u ε,τ L 1 t ( Ḃ d 2 ∩ Ḃ d 2 +1 ) + (∂ t ρ ε,τ , ∂ t Γ ε,τ 2 ) L 1 t ( Ḃ d 2 ) )τ (δQ, δu) h L ∞ t ( Ḃ d 2 -1 ) + (ρ ε,τ -ρ, Γ ε,τ 2 -Γ2 ) L ∞ t ( Ḃ d 2 +1 ) (δQ, δu) h L 1 t ( Ḃ d 2 -1 ) + τ ( 1 ρ ε,τ - 1 ρ τ )∇P τ h L 1 t ( Ḃ d 2 -1 ) + τ j≥Jτ -1 2 ( d 2 -1)j (δR 1,j , δR 2,j , δR 3,j ) L 2 + τ (δF 1 , δF 2 ) h L 1 t ( Ḃ d 2 -1 )
.

By (1.9), (5.9), (5.15), the product laws (2.4) and the commutator estimates (2.5), it is easy to show

τ ( 1 ρ ε,τ - 1 ρ τ )∇P τ h L 1 t ( Ḃ d 2 -1 ) δρ L ∞ t ( Ḃ d 2 -1 ) τ P τ -P L 1 t ( Ḃ d 2 +1 ) o(1) (δY, δQ) L ∞ t ( Ḃ d 2 -1 ) + √ ετ , and
τ j≥Jτ -1 2 ( d 2 -1)j (δR 1,j , δR 2,j ) L 2 ∇u ε,τ L 1 t ( Ḃ d 2 )
(δQ, δu)

L ∞ t ( Ḃ d 2 -1 ) o(1) (δQ, δu) L ∞ t ( Ḃ d 2 -1 )
.

For the tricky commutator term R 3,j , we have

τ j≥Jτ -1 2 ( d 2 -1)j R 3,j L 1 t (L 2 ) τ j≥Jτ -1 2 d 2 j [ 1 ρ ε,τ , ∆j ]∇δQ ℓ L 1 t (L 2 ) + j≥Jτ -1 2 ( d 2 -1)j [ 1 ρ ε,τ , ∆j ]∇δQ h L 1 t (L 2 ) ∇ρ ε,τ L ∞ t ( Ḃ d 2 ) (τ δQ ℓ L 1 t ( Ḃ d 2 ) + δQ h L 1 t ( Ḃ d 2 -1 ) ) o(1)(τ δQ ℓ L 1 t ( Ḃ d 2 ) + δQ h L 1 t ( Ḃ d 2 -1 )
).

For δF 1 and δF 2 , similar computations give rise to (δF 1 , δF 2 )

L 1 t ( Ḃ d 2 -1 )
δu

L 1 t ( Ḃ d 2 -1 ) (P τ -P , u τ ) L ∞ t ( Ḃ d 2 +1 ) + Γ ε,τ 2 - γ + γ -P τ γ + α τ -+ γ -α τ + L ∞ t ( Ḃ d 2 -1 ) u τ L 1 t ( Ḃ d 2
+1
)

+ ( u ε,τ L 1 t ( Ḃ d 2
+1
)

+ ∂ t Γ ε,τ 1 + u ε,τ • ∇Γ ε,τ 1 L 1 t ( Ḃ d 2
)

) P + (ρ ε,τ + ) -P -(ρ ε,τ -)

L ∞ t ( Ḃ d 2 -1 )
+ P + (ρ ε,τ + ) -P -(ρ ε,τ -)

L 1 t ( Ḃ d 2 -1 ∩ Ḃ d 2 )
o(1)( (δY, δQ)

L ∞ t ( Ḃ d 2 -1 )
+ δu

L 1 t ( Ḃ d 2 -1 )
) + √ ετ + ε.

( + τ δu

L 1 t ( Ḃ d 2 -1 )
).

(5.29)

• Step 3:

Ḃ d 2 -1 -

estimates of (δQ, δu) in all frequencies

We need to further establish the uniform Ḃ d 2 -1 -bounds. To this end, owing to (5.26), we obtain the L 2 -in-time estimates (δQ, δu)

L ∞ t ( Ḃ d 2 -1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -1 ) √ ετ + u ε,τ L 1 t ( Ḃ d 2 )
+ (∂ t ρ ε,τ , ∂ t Γ ε,τ 2 )

L 1 t ( Ḃ d 2 ) 1 2 (δQ, δu) h L ∞ t ( Ḃ d 2 -1 )
+ (ρ ε,τ -ρ, Γ ε,τ 2 -Γ2 )

1 2 L ∞ t ( Ḃ d 2 +1 ) √ τ δQ h L 2 t ( Ḃ d 2 -1 ) 1 2 1 √ τ δu h L 2 t ( Ḃ d 2 -1 ) 1 2
+ (δF 1 , δF 2 )

1 2 L 1 t ( Ḃ d 2 -1 )
(δQ, δu)

1 2 L ∞ t ( Ḃ d 2 -1 )
+ ( 1 ρ ε,τ - One has

1 ρ τ )∇P τ 1 2 L 2 t ( Ḃ d 2 -1 )
( 1 ρ ε,τ - 1 ρ τ )∇P τ 1 2 L 2 t ( Ḃ d 2 -1 ) δu 1 2 L 2 t ( Ḃ d 2 -1 )
δρ

L ∞ t ( Ḃ d 2 -1 ) √ τ P τ -P L 2 t ( Ḃ d 2 +1
)

1 2 1 √ τ δu L 2 t ( Ḃ d 2 -1 ) 1 2
o(1) (δY, δQ)

L ∞ t ( Ḃ d 2 -1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -1 ) + √ ετ .
Concerning the commutator terms, we have δQ

L ∞ t ( Ḃ d 2 -1 )
√ τ δQ

L 2 t ( Ḃ d 2 -1 ) 1 2 + 1 √ τ δu L 2 t ( Ḃ d 2 -1 ) ∇Γ ε,τ 2 L ∞ t ( Ḃ d 2 +1 ) √ τ δQ L 2 t ( Ḃ d 2 -1 ) 1 2 + ∇u ε,τ L 1 t ( Ḃ d 2 )
δu 2

L ∞ t ( Ḃ d 2 -1 ) 1 2 + ∇ρ ε,τ L ∞ t ( Ḃ d 2 )
√ τ δQ

L ∞ t ( Ḃ d 2 -1 ) 1 √ τ δu L 2 t ( Ḃ d 2 -1 ) 1 2
o(1) (δQ, δu)

L ∞ t ( Ḃ d 2 -1 ) + √ τ δQ L 2 t ( Ḃ d 2 -1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -1 )
.

Gathering (5.28) and the above three estimates, we have (δQ, δu)

L ∞ t ( Ḃ d 2 -1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -1 )
√ ετ + o(1) (δY, δQ, δu)

L ∞ t ( Ḃ d 2 -1 )
+ δQ

L 2 t ( Ḃ d 2 -1 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -1 )
+ √ ετ .

(5.30)

• Step 4: Proof of convergence rate Finally, we make use of the equation (5.17) 1 , (5.21) and (5.22) to get

∂ t δQ L 1 t ( Ḃ d 2 -2 ) u ε,τ • ∇δQ + (Γ ε,τ 2 -Γ2 )div δu L 1 t ( Ḃ d 2 -2 )
+ div δu

L 1 t ( Ḃ d 2 -2 ) + δF 1 L 1 t ( Ḃ d 2 -2 )
(δY, δQ)

L ∞ t ( Ḃ d 2 -2 ) + 1 √ τ δu L 2 t ( Ḃ d 2 -2 )
+ δu

L 1 t ( Ḃ d 2 -1 )
+ √ ετ .

(5.31)

Combining (5.4), (5.25), (5.29), (5.30) and (5.31) together, we end up with (5.18) which completes the proof of Lemma 5.4.

5.2 Time-relaxation limit: System (K τ ) to System (PM)

This subsection is devoted to the proof of (1.19) in Theorem 1.4. Define the error variables (δβ ± , δ̺ ± , δ̺, δΠ, δv) := (β τ ± -β ± , ̺ τ ± -̺ ± , ̺ τ -̺, Π τ -Π, v τ -v).

First, similarly to (5.1), we need to estimate the variable δZ := , Z 0 := β +,0 ̺ +,0 β +,0 ̺ +,0 + β -,0 ̺ -,0 .

β τ + ̺ τ + ̺ τ
(5.32) Indeed, arguing similarly as in Lemma 5.2, we obtain from (K τ ) and (PM) that

                 δβ + = 1 β τ -̺ τ + ̺ -+ β + ̺ + ̺ τ - ̺ τ ̺δZ -β + β -̺ -δ̺ + + β τ + β -̺ + δ̺ -, δ̺ τ = ̺ τ + -̺ τ - β -̺ τ + ̺ -+ β + ̺ + ̺ τ - ̺ τ ̺δZ -β + β -̺ -δ̺ + + β + β -̺ + δ̺ -+ β + δ̺ + + β -δ̺ τ -, δΠ = δ̺ + 1 0 P ′ + (θ̺ τ + + (1 -θ)̺ + )dθ = δ̺ - 1 0 P ′ -(θ̺ τ -+ (1 -θ)̺ -)dθ, (5.33)
which leads to

     δ̺ ± L ∞ t ( Ḃ d 2 -1 ) + δ̺ ± L 1 t ( Ḃ d 2 +1 )
∼ δΠ

L ∞ t ( Ḃ d 2 -1 )
+ δΠ

L 1 t ( Ḃ d 2 +1 ) , δβ ± L ∞ t ( Ḃ d 2 -1 )
(δZ, δΠ)

L ∞ t ( Ḃ d 2 -1 )
.

(5.34)

It is therefore sufficient to estimate (δΠ, δv, δZ) to recover the information on all the error unknowns.

Next, note that δZ satisfies the transport equation

∂ t δZ + v τ • ∇δZ = -δv • ∇ β + ̺ + ̺ . (5.35)
Using Lemma 2.6, (4.5) and the product law (2.4), we get δZ

L ∞ t ( Ḃ d 2 -1 ∩ Ḃ d 2 ) exp v τ L 1 t ( Ḃ d 2
+1
) δv

L 1 t ( Ḃ d 2 ) ∇ β + ̺ + ̺ L ∞ t ( Ḃ d 2 ∩ Ḃ d 2 +1 )
o(1) δv

L 1 t ( Ḃ d 2 )
.

(5.36)

Then, we perform the key estimates of δΠ. From (K τ ), it is easy to see

∂ t Π τ + v τ • ∇Π τ = γ + γ -Π τ γ + β τ -+ γ -β τ + div ( ∇Π τ ̺ τ ) - γ + γ -Π τ γ + β τ -+ γ -β τ + div z τ , z τ := v τ + ∇Π τ ̺ τ .
Thence by the above equation and (PM), δΠ satisfies

∂ t δΠ -c∆δΠ = -v τ • ∇δΠ -δv • ∇Π + γ + γ -Π τ γ + β τ + + γ -β τ - - γ + γ -Π γ + β + + γ -β - div ( 1 ̺ τ ∇Π τ ) + γ + γ -Π γ + β + + γ -β - div ( 1 ̺ τ - 1 ̺ )∇Π τ - γ + γ -Π τ γ + β τ -+ γ -β τ + div z τ , (5.37)
with the constant c := γ+γ-P+( ρ+) (γ+ ᾱ-+γ-ᾱ+) ρ > 0. We mention that the convergence rate τ is O(τ ) bound comes from the uniform estimates (4.5) of the effective unknown z τ . Indeed, by Lemma 2.5, the uniform estimates (4.5), the smallness of the initial data (1.7), the product laws (2.3) and the composition estimates + δΠ

L 1 t ( Ḃ d 2 +1 ) τ + v τ L 2 t ( Ḃ d 2 )
δΠ

L 2 t ( Ḃ d 2 -1 )
+ δv

L 1 t ( Ḃ d 2 )
Π -P

L ∞ t ( Ḃ d 2 )
+ δΠ

L 2 t ( Ḃ d 2 )
Π τ -P

L 2 t ( Ḃ d 2
+1
)

+ ( 1 ̺ τ - 1 ̺ )∇Π τ L 1 t ( Ḃ d 2 ) + z τ L 1 t ( Ḃ d 2 )
o(1)( δZ

L ∞ t ( Ḃ d 2 )
+ (δ̺ ± , δΠ)

L ∞ t ( Ḃ d 2 -1 )
+ (δ̺ ± , δΠ)

L 1 t ( Ḃ d 2 +1
)

) + τ, (5.38) where we have used the key fact

( 1 ̺ τ - 1 ̺ )∇Π τ L 1 t ( Ḃ d 2 )
( δZ

L ∞ t ( Ḃ d 2 )
Π τ -P

L 1 t ( Ḃ d 2
+1
)

+ δ̺ ± L 2 t ( Ḃ d 2 )
Π τ -P

L 2 t ( Ḃ d 2
+1
)

o(1)( δZ

L ∞ t ( Ḃ d 2 ) + δ̺ ± L 2 t ( Ḃ d 2 )
),

(5.39) derived from (2.4), (4.5) and (5.33). Gathering (5.34) and (5.38) together, we get δΠ

L ∞ t ( Ḃ d 2 -1 )
+ δΠ

L 1 t ( Ḃ d 2 +1
) o(1) δZ

L ∞ t ( Ḃ d 2 )
+ δΠ

L 1 t ( Ḃ d 2 +1
)

) + τ.

(5.40)

For the error unknown δv, in view of (4.5), (5.39) and δv = ( 1 ̺ τ -1 ̺ )∇Π τ -1 ̺ ∇δΠ + z τ , it can be bounded by δv

L 1 t ( Ḃ d 2 )
δΠ

L 1 t ( Ḃ d 2
+1
)

+ ( 1 ̺ τ - 1 ̺ )∇Π τ L 1 t ( Ḃ d 2 ) + z τ L 1 t ( Ḃ d 2 )
δZ

L ∞ t ( Ḃ d 2 )
+ δΠ

L ∞ t ( Ḃ d 2 -1 )
+ δΠ

L 1 t ( Ḃ d 2 +1
) + τ.

(5.41)

The combination of estimate (5.34) and inequalities (5.36)- (5.40) gives rise to estimate (1.19), which completes the proof of Theorem 1.4.

Theorem 1 . 1 .

 11 Let d ≥ 2 and 0 < ε ≤ τ ≤ 1. Given the constants ᾱ± , ρ± verifying (1.4)-(1.5), assume that the initial data

Figure 2 :

 2 Figure 2: A graph of overdamping phenomenon for System (1.25).

  s)X (s)ds. (3.24) We substitute inequalities (3.18), (3.21)-(3.22) and (3.24) into inequality (3.17) and use standard interpolation to get

(3. 31 )

 31 Putting the estimates (3.13), (3.25), (3.26) and (3.30), (3.31) together, we complete the proof of Lemma 3.1.

4. 1

 1 The Cauchy problem of System (BN)

( 5 .

 5 26) together, we can derive the Lyapunov inequality similar to (3.35)-(3.42) and then show the following L 1 -in-time type estimates:

  j L 2 ∆j δQ L 2 + (δR 2,j , δR 3,j ) L 2 ∆j u L 2 ds 1 2 .

j∈Z 2 ( d 2 - 1 )j t 0 δR 1 ,j L 2

 221012 ∆j δQ L 2 + δR 2,j L 2 ∆j u L 2 ds

  δβ, where the initial data of δz isδZ(x, 0) = Z τ 0 (x) -Z 0 (x), Z τ

  .20) Remark 3.2. The above estimate (3.20) for H 4 ∇r arising from two pressures implies that one needs uniform Ḃ d 2 -1 -estimates for low frequencies. Indeed, as H 4 does not have the either L 1 -in-time or L 2 -in-time integrability property, the product law (2.3) in

	which can not be obtained from the	Ḃ d 2 +1 indicates us to discover the control of τ r ℓ L 1 t ( 2 -estimates in this subsection. Ḃ d	Ḃ d 2	+1 )	,
	Remark 3.3. It is also one of the reasons why we need to perform the	Ḃ d 2 +1

  .28)

	We thus get
	(δQ, δu) h L ∞ t (	Ḃ d 2	-2 )	+ τ (δQ, δu) h L ∞ t (	Ḃ d 2	-1 )
	+ (δQ, δu) h L 1 t (	Ḃ d 2	-1 )	+	√ τ δQ h L 2 t (	Ḃ d 2	-1 )	+	1 √ τ	u h L 2 t (	Ḃ d 2	-2 )
							L ∞ t (	Ḃ d 2	-1 )	+ τ δQ ℓ L 1 t (	Ḃ d 2 )	+ τ δQ h L 1 t (	Ḃ d 2	-1 )

√

ετ + o(1)( (δQ, δu)

Acknowledgments TCB is partially supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon). J. Tan is supported by the ANR project BORDS (ANR-16-CE40-0027-01) and by the Labex MME-DII.

Data availability statement. Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of interest. On behalf of all authors, the corresponding author states that there is no conflict of interest.