Jiang Xu
email: jiangxu79math@yahoo.com

A LOW-FREQUENCY ASSUMPTION FOR OPTIMAL TIME-DECAY ESTIMATES TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

Keywords: 1991 Mathematics Subject Classification. 76N15, 35Q30, 35L65, 35K65 . Compressible Navier-Stokes system, time-decay estimates, critical Besov spaces

. In this paper, we claim a new low-frequency assumption for barotropic compressible Navier-Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov space Ḃ-σ 1 2,∞ with the regularity σ1 ∈ (1-d/2, 2d/p-d/2] , then a sharp time-weighted inequality including enough time-decay information can be available, where optimal decay exponents for the high frequencies are also exhibited. The proof mainly depends on tricky and non classical Besov estimates. As a by-product, those optimal time-decay rates of L q -L r type are well captured in the critical regularity framework.

Introduction

Consider the following barotropic compressible Navier-Stokes equations (1.1) ∂ t + div ( u) = 0, ∂ t ( u) + div ( u ⊗ u) -div (2µ D(u) + λ div u Id) + ∇Π = 0.

Here, u = u(t, x) ∈ R d (with (t, x) ∈ R + × R d (d ≥ 2)) and = (t, x) stand for the velocity field and the density of fluid, respectively. The barotropic assumption means that the pressure Π P ( ) depends only upon the density and the function P will be taken suitably smooth in all that follows. The notations div and ∇ are the divergence operator and gradient operator with respect to x, respectively. D(u) 1 2 D x u + T D x u stands for the deformation tensor. The Lamé coefficients λ and µ (the bulk and shear viscosities) are density-dependent functions, which are supposed to be smooth enough and to satisfy µ > 0 and ν λ + 2µ > 0.

System (1.1) is supplemented with the initial data

(1.2) ( , u) | t=0 = ( 0 (x), u 0 (x)) , x ∈ R d ,
and in this paper we focus on solutions that are close to some constant state ( ∞ , 0) with ∞ > 0, at infinity. So far there is a huge literature on the existence and large-time behavior of solutions to (1.1). The local existence and uniqueness of smooth solutions were proved by Serrin [START_REF] Serrin | On the uniqueness of compressible fluid motion[END_REF] and Nash [START_REF] Nash | Le problme de Cauchy pour les équations différentielles d'un fluide général[END_REF]. The local existence of strong solutions with Sobolev regularity was constructed by Solonnikov [START_REF] Solonnikov | Estimates of soutions to a nonstationnary Navier-Stokes system[END_REF], Valli [START_REF] Valli | An existence theorem for compressible viscous fluids[END_REF] and Fiszdon-Zajaczkowski [START_REF] Fiszdon | Existence and uniqueness of solutions of the initial boundary value problem for the flow of a baratropic viscous fluid, local in time[END_REF]. Matsumura and Nishida [START_REF] Matsumura | The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids[END_REF][START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat-conductive gases[END_REF] established the first global strong solutions for small perturbations of a linearly stable constant state ( ∞ , 0) in three dimensions. With the additional L 1 assumption of initial data, the definite decay rate was also available:

(1.3) ( -∞ , u)(t) L 2 t -3 4 with t 1 + t 2 ,
which coincides with that of the heat kernel. Indeed, such time-decay rate reveals the dissipative properties of solutions to (1.1)-(1.2). Subsequently, Ponce [START_REF] Ponce | Global existence of small solution to a class of nonlinear evolution equations[END_REF] obtained more general L p decay rates:

(1.4) ∇ k ( -∞ , u)(t) L p t -d 2 (1-1 p )-k 2 , 2 ≤ p ≤ ∞, 0 ≤ k ≤ 2, d = 2, 3.
S. Kawashima [START_REF] Kawashima | Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics[END_REF] exhibited the asymptotic behavior for generally hyperbolic-parabolic composite systems satisfying some dissipative assumption, which is now referred as the "Shizuta-Kawashima" stability condition. There are also some efforts extending Matsumura-Nishida's results to more physical situation where the fluid domain is not R d : for example, the exterior domains were investigated by Kobayashi [START_REF] Kobayashi | Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R 3[END_REF] and Kobayashi-Shibata [START_REF] Kobayashi | Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain of R 3[END_REF], the half space by Kagei & Kobayashi [START_REF] Kagei | On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in R 3[END_REF][START_REF] Kagei | Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space[END_REF]. In 1998, Xin [START_REF] Xin | Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density[END_REF] found a fact that any smooth solution to the Cauchy problem of full compressible Navier-Stokes system without heat conduction (including the present baratropic case) would blow up in finite time if the initial density contains vacuum. For the existence of solutions for arbitrary data, a breakthrough work is due to P.-L. Lions [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF], where he obtained the global existence of weak solutions with finite energy when the adiabatic index is suitably large. Subsequently, some developments were made by Feireisl, Novotny & Petzeltová [START_REF] Feireisl | On the existence of globally defined weak solutions to the Navier-Stokes equations[END_REF] and Jiang-Zhang [START_REF] Jiang | Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids[END_REF].

On the other hand, it is interesting to show the global existence of strong solutions in the critical regularity framework. More precisely, System (1.1) is invariant by the transformation (1.5) (t, x) (l 2 t, lx), u(t, x) u(l 2 t, lx), l > 0 up to a change of the pressure term Π into l 2 Π, hence one employs functional spaces to investigate (1.1), which are endowed with norms enjoying the scaling invariance (1.5). This trick is now classic and has been used by different authors. For instance, recall that the context of the incompressible Navier-Stokes equation, working in critical spaces goes back to the work by Fujita & Kato in [START_REF] Fujita | On the Navier-Stokes initial value problem I[END_REF] (see also results by Kozono & Yamazaki [START_REF] Kozono | Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data[END_REF] and Cannone [START_REF] Cannone | A generalization of a theorem by Kato on Navier-Stokes equations[END_REF]). As observed by Danchin in [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF], one may solve (1.1) globally in critical homogeneous Besov spaces of L 2 type, that is, the initial data belong to ( Ḃ d referred to [START_REF] Charve | A global existence result for the compressible Navier-Stokes equations in the critical L p framework[END_REF][START_REF] Chen | Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity[END_REF][START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fluids[END_REF][START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] or Theorem 1.1 in [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF] directly. In this paper, our main interest is to exhibit (more precisely) the large time behavior of strong solutions constructed in the critical regularity framework. First of all, we would like to mention those efforts in this direction. Okita [START_REF] Okita | Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations[END_REF] studied the optimal time-decay estimates to (1.1)-(1.2) in the L 2 critical framework, thanks to a smart modification of the method of [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equations[END_REF]. However, the result cannot cover the 2D case. In the survey paper [START_REF] Danchin | Fourier Analysis Methods for the Compressible Navier-Stokes Equations[END_REF], Danchin proposed another description of the time decay which allows to handle dimension d ≥ 2. In the joint work with Danchin [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF], we developed that method of [START_REF] Danchin | Fourier Analysis Methods for the Compressible Navier-Stokes Equations[END_REF] so as to establish the optimal decay rates in general L p critical spaces.

Reformulation and main results

Next, in order to state main results of this paper, it is convenient to reformulate system (1.1) as the nonlinear perturbation form of constant equilibrium state ( ∞ , 0), looking at the nonlinearities as source terms. For simplicity, let's normalize the density ∞ , the sound speed c ∞ P ( ∞ ) and the total viscosity ν ∞ λ ∞ + 2µ ∞ (with λ ∞ λ( ∞ ) and µ ∞ µ( ∞ )) to be one. Hence, we have (2.1)

∂ t a + div u = f, ∂ t u -Au + ∇a = g,
with f -div(au), g -u• ∇u-I(a)Au-k(a)∇a+ 1 1+a div(2 µ(a) D(u) + λ(a)div u Id, where

A µ ∞ ∆ + (λ ∞ + µ ∞ ) ∇div such that ν ∞ λ ∞ + 2µ ∞ = 1 and µ ∞ > 0 I(a) a 1+a , k(a) P (1+a) 1+a -1, µ(a) µ(1 + a) -µ(1), λ(a) λ(1 + a) -λ(1).
Note that I(a), k(a), µ(a) and λ(a) are smooth functions satisfying

I(0) = k(0) = µ(0) = λ(0) = 0.
Now, we state our main result as follows. constructed in [START_REF] Charve | A global existence result for the compressible Navier-Stokes equations in the critical L p framework[END_REF][START_REF] Chen | Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity[END_REF][START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fluids[END_REF][START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] (or see Theorem 1.1 in [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF]). Let the real number σ 1 satisfy

(2.3) 1 - d 2 < σ 1 ≤ σ 0 with σ 0 2d p - d 2 .
There exists a positive constant c = c (p, d, λ, µ, P ) such that if in addition the initial data (a 0 0 -1, u 0 ) fulfill

(2.4) D p,0 (a 0 , u 0 ) Ḃ-σ 1 2,∞ ≤ c,
then we have

(2.5) D p (t) (D p,0 + (∇a 0 , u 0 ) h Ḃ d p -1 p,1
) for all t ≥ 0, where the functional D p (t) is defined by

(2.6) D p (t) sup σ∈[ε-σ 1 , d 2 +1] τ σ 1 +σ 2 (a, u) L ∞ t ( Ḃs 2,1 ) + τ α (∇a, u) h L ∞ t ( Ḃ d p -1 p,1 ) + τ α ∇u h L ∞ t ( Ḃ d p p,1 )
with α σ 1 + d 2 + 1 2 -ε for sufficiently small ε > 0. Indeed, the analogue of (2.6) is not sharp in [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF]. Our initial observation allows to gain more decay in time (for example, say τ -β with β ≥ 1) than τ -1 for the improved regularity of velocity. A natural question follows, since bounding the last two norms for high frequencies are absolutely interactional. Which values are optimal for the decay exponents α and β respectively? To explore the optimality, we intend to develop the general low-frequency assumption for compressible fluid equations (1.1) and achieve (2.6) exhibiting optimal decay exponents for the high frequencies of solutions.

The statement of Theorem 2.1 deserves some comments.

(

) 1
The assumption (2.3) enables us to enjoy relatively larger freedom on the choice of σ 1 . For example, the value of σ 1 = σ 0 = d/2 if taking p = 2 is relevant in other contexts like the Boltzmann equation (see [START_REF] Sohinger | The Boltzmann equation, Besov spaces, and optimal time decay rates in R n x[END_REF] by Sohinger and Strain), or hyperbolic systems with dissipation (see the joint works [START_REF] Xu | Frequency-localization Duhamel principle and its application to the optimal decay of dissipative systems in low dimensions[END_REF][START_REF] Xu | The optimal decay estimates on the framework of Besov spaces for generally dissipative systems[END_REF] with Kawashima). We mention those previous works by Matsumura and Nishida [START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat-conductive gases[END_REF], by Li and Zhang [START_REF] Li | Large time behavior of isentropic compressible Navier-Stokes system in R 3[END_REF], by Guo and Wang [START_REF] Guo | Decay of dissipative equations and negative sobolev spaces[END_REF], where different low-frequency assumptions were posed. Noticing that the Sobolev embeddings

L 1 → Ḃ0 1,∞ → Ḃ-d/2 2,∞ , Ḣ-d/2 → Ḃ-d/2 2,∞ ,
we see that (2.4) is exactly less restrictive in contrast with that in [START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat-conductive gases[END_REF][START_REF] Li | Large time behavior of isentropic compressible Navier-Stokes system in R 3[END_REF][START_REF] Guo | Decay of dissipative equations and negative sobolev spaces[END_REF]. (2) If replacing (2.4) by the slightly stronger assumption

(a 0 , u 0 ) Ḃ-σ 1 2,1
≤ c, then we can take ε = 0: that is we can take α = σ 1 + d 2 + 1 2 and change the first term of D p (t) by the slightly stronger norm

sup σ∈[-σ 1 , d 2 +1] τ σ 1 +σ 2 (a, u) L ∞ t ( Ḃσ 2,1 ) .
We believe exponent α = σ 1 + d 2 + 1 2 as well as the upper bound σ ≤ d/2 + 1 for the first term of D p to be optimal (see the very end of the present paper for more explanations). How to guess the lower bound of regularity σ 1 in the lowfrequency assumption? Actually, it may stem from the following elementary inequality:

t 0 t -τ -σ 1 +σ 2 τ -δ dτ t -σ 1 +σ 2 , 0 ≤ σ 1 + σ 2 ≤ δ, δ > 1.
In our case, the minimum value of δ is

σ 1 /2 + d/4 + 1/2, owing to σ ≤ d/2 + 1.
Consequently, the desired lower bound of σ 1 just follows from the requirement

σ 1 /2 + d/4 + 1/2 > 1.
(3) In physical dimensions d = 2, 3, Condition (2.2) allows us to consider the case p > d, so that the regularity exponent d/p-1 for the velocity becomes negative. Our result thus applies to large highly oscillating initial velocities (see [START_REF] Charve | A global existence result for the compressible Navier-Stokes equations in the critical L p framework[END_REF][START_REF] Chen | Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity[END_REF] for more explanation), which is the essentially different ingredient in comparison with related decay works, see [START_REF] Danchin | Fourier Analysis Methods for the Compressible Navier-Stokes Equations[END_REF][START_REF] Guo | Decay of dissipative equations and negative sobolev spaces[END_REF][START_REF] Li | Large time behavior of isentropic compressible Navier-Stokes system in R 3[END_REF][START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat-conductive gases[END_REF][START_REF] Okita | Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations[END_REF] and references therein. (4) Lastly, we would like to mention that (2.4) is of independent interest and thus has an application potential in mathematical analysis of classical Newtonian mechanics or complex fluids.

Let us present the strategy for proving Theorem 2.1. Since from [START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat-conductive gases[END_REF], a usual approach getting the decay estimates of type (1.3) and (1.4) is to take advantage of L 1 -L 2 decay estimates for the linearized system corresponding to the left-hand side of (2.1), treating the nonlinear right-hand side (f, g) by means of Duhamel formula. In the critical regularity framework however, one cannot afford any loss of regularity for the high frequency part of the solution (and some terms like u•∇a induce a loss of one derivative as one cannot expect any smoothing for a, solution of a transport equation). The sketch of the proof of Theorem 2.1 is divided into three steps according to the three terms of D p . Due to (2.4) in which σ 1 belongs to the whole range (1-d/2, σ 0 ], the low-frequency analysis (in the first step) is quite complicated. More concretely speaking, with aid of low and high frequency decompositions, we split the nonlinear term (f, g) into (f , g ) and (f h , g h ) (see the context below). To handle (f , g ), some non classical product estimates in Besov spaces are well developed, see Lemma 4.2. Secondly, bounding the term (f h , g h ) seems to be more elaborate owing to the regularity of low frequencies. For clarity, Lemmas 4.3-4.4 are devoted to the non oscillation case (2 ≤ p ≤ d), where we deal with cases d/p -d/2 < σ 1 ≤ σ 0 and 1 -d/2 < σ 1 ≤ d/p -d/2 separately. Lemma 4.5 is only responsible for the oscillation case (p > d). It is worth noting that those proofs of Lemmas 4.2-4.5 rely heavily on tricky product Besov estimates with respect to different Sobolev embeddings.

The purpose of the second and third steps is to track the optimal decay exponents for high frequencies with the general assumption (2.4). For convenience of reader, we follow from the fashion of [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF] briefly and highlight new ingredients. Precisely, with the aid of the effective velocity (see for example, [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fluids[END_REF][START_REF] Danchin | Fourier Analysis Methods for the Compressible Navier-Stokes Equations[END_REF])

(2.7) w ∇(-∆) -1 (a -div u),
we get round the technical obstacle that there is a loss of one derivative of density in the second step. The last step is dedicated to establish gain of regularity and decay altogether for the high frequencies of the velocity, which strongly depends on the parabolic maximal regularity for the Lamé semi-group (that are the same as for the heat semi-group, see Remark 3.2). By a by-product, Theorem 2.1 enables us to gain L p decay estimates of Ponce type (the slightly stronger Ḃ0

p,1 norm in fact). Corollary 2.1. The solution ( , u) constructed in Theorem 2.1 fulfills

Λ σ ( -1) L p (D p,0 + (∇a 0 , u 0 ) h Ḃ d p -1 p,1 ) t -σ 1 +σ 2 if -σ 1 < σ ≤ d p , Λ σ u L p (D p,0 + (∇a 0 , u 0 ) h Ḃ d p -1 p,1 ) t -σ 1 +σ 2 if -σ 1 < σ ≤ d p + 1,
where the pseudo differential operator Λ s is defined by

Λ s f F -1 (|ξ| s Ff ).
Furthermore, one can get the optimal decay estimates of L q -L r type, as a consequence of improved Gagliardo-Nirenberg type inequalities which parallel the work of Sohinger and Strain [START_REF] Sohinger | The Boltzmann equation, Besov spaces, and optimal time decay rates in R n x[END_REF] (see also [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2, or [START_REF] Xu | The optimal decay estimates on the framework of Besov spaces for generally dissipative systems[END_REF]).

Corollary 2.2. Let the assumptions of Theorem 2.1 be fulfilled with p = 2. Then the corresponding solution ( , u) satisfies

Λ l ( -1, u) L r (D 2,0 + (∇a 0 , u 0 ) h Ḃ d 2 -1 2,1 ) t -σ 1 2 -d 2 ( 1 2 -1 r )-l 2 for 2 ≤ r ≤ ∞ and l ∈ R satisfying -σ 1 < l + d( 1 2 -1 r ) ≤ d 2 .
Remark 2.1. The proofs of Corollaries 2.1-2.2 follow from essentially the argument as same as [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF], and are thus omitted for brevity. The choice of derivative indices of velocity is somewhat relaxed thanks to the optimal exponents for high frequencies.

Finally, we end the section with an overview of our approach. Up to now, we present the optimal time-weighted estimates in L p critical spaces, for those constructed solutions (close to constant equilibrium) as in [START_REF] Charve | A global existence result for the compressible Navier-Stokes equations in the critical L p framework[END_REF][START_REF] Chen | Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity[END_REF][START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF][START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fluids[END_REF]. As a matter of fact, the vacuum mechanism is ruled out, so the low frequency linear analysis is dominate in our case. What happen, if the vacuum occurs (see e.g., [START_REF] Huang | Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations[END_REF]). The corresponding mathematical theory for viscous fluids is far away well known in critical spaces. In addition, providing an accurate large-time asymptotic description is still out of reach. For instance, one has to resort to detailed pointwise estimates on Green's function if characterizing the wave aspect of solutions. In this research line, we would like to mention the previous work by Zeng [START_REF] Zeng | L 1 Asymptotic behavior of compressible isentropic viscous 1-D flow[END_REF] devoted to the one-dimensional case. Hoff and Zumbrum [START_REF] Hoff | Multidimensional diffusion waves for the Navier-Stokes equations of compressible flow[END_REF] derived the L p decay rates towards diffusion waves in multi-dimensions, which are same as in (1.4). In [START_REF] Liu | The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions[END_REF], Liu and Wang exhibited pointwise estimates of diffusion waves with the optimal time-decay rate in odd dimension, that corresponds to the weak Huygens' principle.

The rest of the paper unfolds as follows: In Section 3, we briefly recall Littlewood-Paley decomposition, Besov spaces and related analysis tools. Section 4 is devoted to the proof of Theorem 2.1. In Section 5 (Appendix), we give some illustrations on the optimality of the regularity and decay exponents in the definition of D p .

Preliminary

Throughout the paper, C > 0 stands for a harmless "constant". For brevity, we sometime write u v instead of u ≤ Cv. The notation u ≈ v means that u v and v u. For any Banach space X and u, v ∈ X, we agree that (u, v) X u X + v X . For all T > 0 and ρ ∈ [1, +∞], we denote by L ρ T (X) L ρ ([0, T ] ; X) the set of measurable functions u : [0, T ] → X such that t → u(t) X is in L ρ (0, T ).

To make the paper self-contained, let us recall Littlewood-Paley decomposition, Besov spaces and related analysis tools. More details may be found for example in Chap. 2 and Chap. 3 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]. Firstly, we introduce a homogeneous Littlewood-Paley decomposition. To do this, we fix some smooth radial non increasing function χ with Supp χ ⊂ B 0, 4 3 and χ ≡ 1 on B 0, 3 4 , then set

ϕ (ξ) = χ (ξ/2) -χ (ξ) so that j∈Z ϕ 2 -j • = 1 in R d \ {0} and Supp ϕ ⊂ ξ ∈ R d : 3/4 ≤ |ξ| ≤ 8/3 .
Then homogeneous dyadic blocks ∆j are defined by ∆j u ϕ(2

-j D)u = F -1 (ϕ(2 -j •)Fu) = 2 jd h(2 j •) u with h F -1 ϕ.
Therefore, one has the following homogeneous decomposition for any tempered distribution u ∈ S (R d )

(3.1) u = j∈Z ∆j u.
As it holds only modulo polynomials, it is convenient to consider the subspace of those tempered distributions u such that (3.2) lim j→-∞ Ṡj u L ∞ = 0, where Ṡj f stands for the low frequency cut-off defined by Ṡj u χ 2 -j D u. Indeed, if equality (3.2) is fulfilled, then (3.1) holds in S R d . For convenience, we denote by S 0 R d the subspace of tempered distributions satisfying (3.2).

With the aid of the Littlewood-Paley decomposition, Besov spaces and related analysis tools will come into play in our context. < +∞ , where

(3.3) u Ḃs p,r (2 js ∆j u L p ) r (Z) .
When studying the evolution PDEs, a class of mixed space-time Besov spaces are also used, which was first introduced by J.-Y. Chemin and N. Lerner [START_REF] Chemin | Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes[END_REF] (see also [START_REF] Chemin | Théorèmes d'unicité pour le systèm de Navier-Stokes tridimensionnel[END_REF] for the particular case of Sobolev spaces).

Definition 3.2. For T > 0, s ∈ R, 1 ≤ r, θ ≤ ∞, the homogeneous Chemin-Lerner space L θ T ( Ḃs p,r
) is defined by

L θ T ( Ḃs p,r ) u ∈ L θ 0, T ; S 0 : u L θ T ( Ḃs p,r ) < +∞ , where (3.4) u L θ T ( Ḃs p,r )
(2 js ∆j u L θ T (L p ) ) r (Z) . For notational simplicity, index T is omitted if T = +∞. We also use the following functional space:

C b (R + ; Ḃs p,r ) u ∈ C(R + ; Ḃs p,r ) s.t u L ∞ ( Ḃs p,r ) < +∞ .
The above norm (3.4) may be compared with those of the standard spaces L θ T ( Ḃs p,r ) by means of Minkowski's inequality.

Remark 3.1. It holds that u L θ T (B s p,r ) ≤ u L θ T (B s p,r ) if r ≥ θ; u L θ T (B s p,r ) ≥ u L θ T (B s p,r ) if r ≤ θ.
Restricting the above norms (3.3) and (3.4) to the low or high frequencies parts of distributions will be fundamental in our approach. For example, let us fix some integer j 0 (the value of which will follow from the proof of the main theorem) and put

1 u Ḃs p,1 j≤j 0 2 js ∆k f L p and u h Ḃs p,1 j≥j 0 -1 2 js ∆j u L p , u L ∞ T ( Ḃs p,1 ) j≤j 0 2 js ∆j u L ∞ T (L p ) and u h L ∞ T ( Ḃs p,1 ) j≥j 0 -1 2 js ∆j u L ∞ T (L p ) .
We often use the following classical properties (see [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]):

• Scaling invariance: For any s ∈ R and (p, r) ∈ [1, ∞] 2 , there exists a constant C = C(s, p, r, d) such that for all λ > 0 and u ∈ Ḃs p,r , we have

C -1 λ s-d p u Ḃs p,r ≤ u(λ•) Ḃs p,r ≤ Cλ s-d p u Ḃs p,r
.

• Completeness: Ḃs p,r is a Banach space whenever s < d p or s ≤ d p and r = 1. • Interpolation: The following inequality is satisfied for 1 ≤ p, r 1 , r 2 , r ≤ ∞, s 1 = s 2 and θ ∈ (0, 1):

u Ḃθs 1 +(1-θ)s 2 p,r u θ Ḃs 1 p,r 1 u 1-θ Ḃs 2 p,r 2 with 1 r = θ r 2 + 1-θ r 2 . • Action of Fourier multipliers: If F is a smooth homogeneous of degree m function on R d \{0} then F (D) : Ḃs p,r → Ḃs-m p,r . Proposition 3.1. (Embedding for Besov spaces on R d ) • For any p ∈ [1, ∞] we have the continuous embedding Ḃ0 p,1 → L p → Ḃ0 p,∞ . • If σ ∈ R, 1 ≤ p 1 ≤ p 2 ≤ ∞ and 1 ≤ r 1 ≤ r 2 ≤ ∞, then Ḃσ p 1 ,r 1 → Ḃσ-d( 1 p 1 -1 p 2 ) p 2 ,r 2 . • The space Ḃ d p
p,1 is continuously embedded in the set of bounded continuous functions (going to zero at infinity if, additionally, p < ∞).

The following product estimates in Besov spaces play a fundamental role in our analysis of the bilinear terms of (2.1).

Proposition 3.2. ([1, 10]) Let s > 0 and 1 ≤ p, r ≤ ∞. Then Ḃs p,r ∩ L ∞ is an algebra and uv Ḃs p,r u L ∞ v Ḃs p,r + v L ∞ u Ḃs p,r
.

Let the real numbers s 1 , s 2 , p 1 and p 2 be such that

s 1 + s 2 > 0, s 1 ≤ d p 1 , s 2 ≤ d p 2 , s 1 ≥ s 2 , 1 p 1 + 1 p 2 ≤ 1.
Then we have

uv Ḃs 2 q,1 u Ḃs 1 p 1 ,1 v Ḃs 2 p 2 ,1 with 1 q = 1 p 1 + 1 p 2 - s 1 d • Additionally, for exponents s > 0 and 1 ≤ p 1 , p 2 , q ≤ ∞ satisfying d p 1 + d p 2 -d ≤ s ≤ min d p 1 , d p 2 and 1 q = 1 p 1 + 1 p 2 - s d , we have uv Ḃ-s q,∞ u Ḃs p 1 ,1 v Ḃ-s p 2 ,∞ . Proposition 3.
∈ R, z Ḃs 2,∞ sup j≤j 0 2 js ∆j z L 2 .
There exists a universal integer N 0 such that for any 2 ≤ p ≤ 4 and s > 0, we have

uv h Ḃ-σ 0 2,∞ ≤ C( u Ḃs p,1 + Ṡj 0 +N 0 u L p * ) v h Ḃ-s p,∞ (3.5) u h v Ḃ-σ 0 2,∞ ≤ C( u h Ḃs p,1 + Ṡj 0 +N 0 u h L p * ) v Ḃ-s p,∞ (3.6
F (0) = 0. For all 1 ≤ p, r ≤ ∞ and s > 0 we have F (u) ∈ Ḃs p,r ∩ L ∞ for u ∈ Ḃs p,r ∩ L ∞ , and F (u) Ḃs p,r ≤ C u Ḃs p,r
with C depending only on u L ∞ , F (and higher derivatives), s, p and d.

In the case s > -min d p , d p then u ∈ Ḃs p,r ∩ Ḃ d p p,1 implies that F (u) ∈ Ḃs p,r ∩ Ḃ d p p,1
, and we have

F (u) Ḃs p,r ≤ C(1 + u Ḃ d p p ,1 ) u Ḃs p,r
.

Let us now recall the following classical Bernstein inequality:

(3.7) D k u L b ≤ C 1+k λ k+d( 1 a -1 b ) u L a that holds for all function u such that Supp Fu ⊂ ξ ∈ R d : |ξ| ≤ Rλ for some R > 0 and λ > 0, if k ∈ N and 1 ≤ a ≤ b ≤ ∞.
More generally, if we assume u to satisfy Supp Fu ⊂ ξ ∈ R d : R 1 λ ≤ |ξ| ≤ R 2 λ for some 0 < R 1 < R 2 and λ > 0, then for any smooth homogeneous of degree m function A on R d \ {0} and 1 ≤ a ≤ ∞, we have (see e.g. Lemma 2.2 in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]):

(3.8) A(D)u L a λ m u L a .
An obvious consequence of (3.7) and (3.8) is that D k u Ḃs p,r ≈ u Ḃs+k p,r for all k ∈ N. A time-dependent version of the following commutator estimate has been used in the second step of the proof of Theorem 2.1.

Proposition 3.5. ([10]) Let 1 ≤ p, p 1 ≤ ∞ and -min d p 1 , d p < s ≤ 1 + min d p , d p 1 with 1 p + 1 p = 1.
There exists a constant C > 0 depending only on s such that for all j ∈ Z and ∈ {1, • • • , d}, we have

[v • ∇, ∂ ∆j ]a L p ≤ Cc j 2 -j(s-1) ∇v Ḃ d p 1 p 1 ,1 ∇a Ḃs-1 p,1
,

where the commutator [•, •] is defined by [f, g] = f g -gf and (c j ) j∈Z denotes a sequence such that (c j ) 1 ≤ 1.
Let us finally recall the following parabolic regularity estimate for the heat equation to end this section.

Proposition 3.6. Let s ∈ R, (p, r) ∈ [1, ∞] 2 and 1 ≤ ρ 2 ≤ ρ 1 ≤ ∞. Let u satisfy ∂ t u -µ∆u = f, u |t=0 = u 0 .
Then for all T > 0 the following a priori estimate is fulfilled:

(3.9) µ 1 ρ 1 u L ρ 1 T ( Ḃs+ 2 ρ 1 p,r ) u 0 Ḃs p,r + µ 1 ρ 2 -1 f L ρ 2 T ( Ḃs-2+ 2 ρ 2 p,r ) .
Remark 3.2. The solutions to the following Lamé system (3.10)

∂ t u -µ∆u -(λ + µ) ∇div u = f, u |t=0 = u 0 ,
where λ and µ are constant coefficients such that µ > 0 and λ + 2µ > 0, also fulfill (3.9) (up to the dependence w.r.t. the viscosity). Indeed, if we denote by P Id + ∇(-∆) -1 div and Q Id -P the orthogonal projectors over divergence-free and potential vector fields, then we see both Pu and Qu satisfy the heat equation, as it can easily be observed by applying P and Q to (3.10).

The proof of time-decay estimates

This section is devoted to the proof of Theorem 2.1 taking for granted the global-intime existence result of Theorem 1.1 in [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF]. We denote the energy norm as follows:

E p (t) (a, u) L ∞ t ( Ḃ d 2 -1 2,1 )
+ (a, u)

L 1 t ( Ḃ d 2 +1
2,1 )

+ (∇a, u) h L ∞ t ( Ḃ d p -1 p,1 ) + (a, ∇u) h L 1 t ( Ḃ d p p,1 ) . (4.1)
In what follows, we shall use repeatedly that for 0 ≤ s 1 ≤ s 2 with s 2 > 1, we have (4.2)

t 0 t -τ -s 1 τ -θ τ θ-s 2 dτ t -s 1 if 0 ≤ θ < 1.
Let us keep in mind that the global solution (a, u) given by Theorem 2.1 satisfies

(4.3) a L ∞ t ( Ḃ d p p,1 )
≤ c 1 for all t ≥ 0.

4.1. First step: Bounds for the low frequencies. Let (G(t)) t≥0 be the semi-group associated with the left-hand side of (2.1). The standard Duhamel principle gives

(4.4) a(t) u(t) = G(t) a 0 u 0 + t 0 G(t -τ ) f (τ ) g(τ ) dτ.
First of all, we state smoothing estimate of the linearized solution (a L , u L ) G(t) (a 0 , u 0 ), which behaves like that of heat kernel.

Lemma 4.1. Let (a L , u L ) be the solution to the linear system

∂ t a L + div u L = 0, ∂ t u L -Au L + ∇a L = 0 with the initial data (a L , u L ) | t=0 = (a 0 , u 0 ) .
Then, for any j 0 ∈ Z, there exists a positive constant c 0 = c 0 (λ ∞ , µ ∞ , j 0 ) such that (a L,j , u L,j ) (t) L 2 e -c 0 2 2j t (a 0,j , u 0,j ) L 2 for t ≥ 0 and j ≤ j 0 , where we set z j = ∆j z for any z ∈ S (R d ).

The reader is referred to [START_REF] Charve | A global existence result for the compressible Navier-Stokes equations in the critical L p framework[END_REF] for the proof of Lemma 4.1. Set U (a, u) and U 0 (a 0 , u 0 ). It is not difficult to deduce that for σ 1 + σ > 0 (see [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF])

sup t≥0 t σ 1 +σ 2 G(t)U 0 Ḃσ 2,1 U 0 Ḃ-σ 1 2,∞
.

Moreover, it is clear that for σ + σ 1 > 0,

G(t)U 0 Ḃσ 2,1 U 0 Ḃ-σ 1 2,∞ j≤j 0 2 j(σ 1 +σ) U 0 Ḃ-σ 1 2,∞
.

Hence, setting t √ 1 + t 2 , we arrive at

(4.5) sup t≥0 t σ 1 +σ 2 G(t)U 0 Ḃσ 2,1 U 0 Ḃ-σ 1 2,∞
.

Furthermore, we get from Duhamel's formula that

t 0 G(t -τ ) (f, g) (τ )dτ Ḃσ 2,1 t 0 t -τ -σ 1 +σ 2 (f, g) (τ ) Ḃ-σ 1 2,∞ dτ.
To bound the above time-weighted integral on the right side, we have the following proposition.

Proposition 4.1. If p satisfies (2.2), then it holds that for all t ≥ 0, (4.6)

t 0 t -τ -σ 1 +σ 2 (f, g) (τ ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t) + E 2 p (t) , provided that -σ 1 < σ ≤ d 2 + 1
, where E p (t) and D p (t) have been defined by (4.1) and (2.6), respectively.

For clarity, we divide the proof of Proposition 4.1 into several lemmas. It is convenient to decompose f and g in terms of low-frequency and high-frequency as follows:

f = f + f h with f -a div u -u • ∇a , f h -a div u h -u • ∇a h and g = g + g h with g -u • ∇u -k(a)∇a + g 3 (a, u ) + g 4 (a, u ), g h -u • ∇u h -k(a)∇a h + g 3 (a, u h ) + g 4 (a, u h ),
where

g 3 (a, v) = 1 1 + a 2 µ (a) div D (v) + λ(a)∇div v -I (a) Av, g 4 (a, v) = 1 1 + a 2 µ (a)D (v) • ∇a + λ (a) div v ∇a and z j<j 0 ∆j z, z h z -z for z = a, u. Lemma 4.2. If p satisfies (2.
2), then it holds that for all t ≥ 0, (4.7)

t 0 t -τ -σ 1 +σ 2 f , g (τ ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t) + E 2 p (t) .
Proof. To bound the six terms of f and g with a or u , we first claim the following two non classical product estimates:

F G Ḃ-σ 1 2,∞ F Ḃ d p p,1 G Ḃ-σ 1 2,1 , (4.8) F G Ḃ d p -d 2 -σ 1 2,∞ F Ḃ d p -d 2 -σ 1 p,1 G Ḃ d p 2,1 . (4.9)
Indeed, if σ 1 = σ 0 , then (4.9) stems from the last term of Proposition 3.2 with s = d p , q = p 1 = 2, p 2 = p. If 1 -d/2 < σ 1 < σ 0 , then (4.9) follows from the second term of Proposition 3.2 with

d p s 1 ≥ s 2 d p - d 2 -σ 1 , q = p 1 =
d p s 1 ≥ s 2 -σ 1 , q = p 2 = 2 and p 1 = p satisfying s 1 + s 2 > 0.
In addition, due to α σ 1 + d 2 + 1 2 -ε ≥ σ 1 2 + d 4 for small enough ε > 0, the embedding in Proposition 3.1 and the definition of D p (t), we obtain (a, u )(τ )

Ḃ d p p,1 τ -σ 1 2 -d 4 τ σ 1 2 + d 4 (a, u)(τ ) Ḃ d 2 2,1 + τ α a(τ ) h Ḃ d p p,1 τ -σ 1 2 -d 4 D p (τ ) (4.10)
and also that, owing to -σ

1 < 1 -σ 1 < 2 -σ 1 < d 2 + 1, (4.11) ∇(a , u )(τ ) Ḃ-σ 1 2,1 τ -1 2 D p (τ ), ∇ 2 u (τ ) Ḃ-σ 1 2,1 τ -1 D p (τ ).
Observe that d p -d 2 -σ 1 < d p -1 and α > 1 for small enough ε > 0, we have by embedding, (4.12) (∇a, u h )(τ )

Ḃ d p -d 2 -σ 1 p,1 ∇a (τ ) Ḃ-σ 1 2,1 + (∇a h , u h )(τ ) Ḃ d p -1 p,1 τ -1 2 D p (τ )
and, thanks to

-σ 1 < d 2 -1 < d 2 ≤ d p + 1 ≤ d 2 + 1, (4.13) ∇(a , u )(τ ) Ḃ d p 2,1 τ - σ 1 2 -d 2p -1 2 D p (τ ).
Now, we begin with prove (4.7). To bound the term with a div u , we write that, owing to (4.8), (4.10) and (4.11),

t 0 t -τ -σ 1 +σ 2 a div u Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃ d p p,1 div u Ḃ-σ 1 2,1 dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 4 -1 2 dτ.
As

σ 1 2 + d 4 + 1 2 > 1 and σ 1 2 + d 4 + 1 2 ≥ σ 1 +σ 2 for σ 1 satisfying (2.3) and σ ≤ d 2 +1, inequality (4.2) implies that t 0 t -τ -σ 1 +σ 2 a div u Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t).
Regarding the term with k(a)∇a , with the aid of Proposition 3.4, (4.2), (4.8) and (4.10)-(4.11), we similarly get

t 0 t -τ -σ 1 +σ 2 k(a)∇a Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃ d p p,1 ∇a Ḃ-σ 1 2,1 dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 4 -1 2 dτ t -σ 1 +σ 2 D 2 p (t).
Regarding the term with g 3 (a, u ), it follows from Proposition 3.4, (4.2), (4.8) and (4.10)-(4.11) that

t 0 t -τ -σ 1 +σ 2 g 3 (a, u ) Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃ d p p,1 ∇ 2 u Ḃ-σ 1 2,1 dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 4 -1 dτ t -σ 1 +σ 2 D 2 p (t). We decompose u • ∇a = u • ∇a + u h • ∇a and u • ∇u = u • ∇u + u h • ∇u .
Bounding u • ∇a and u • ∇u essentially follow from the same procedure as a div u , we omit them. For the term containing u h • ∇a , we note that, owing to (4.9),

t 0 t -τ -σ 1 +σ 2 u h • ∇a Ḃ d p -d 2 -σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 u h Ḃ d p -d 2 -σ 1 p,1 ∇a Ḃ d p 2,1 dτ.
Observe that, as

σ 1 ≤ σ 1 + d 2 -d p (p ≥ 2), we have u h • ∇a Ḃ-σ 1 2,∞ u h • ∇a Ḃ d p -d 2 -σ 1 2,∞ . Using the fact that σ 1 2 + d 2p + 1 ≥ σ 1 2 + d 4 + 1 2 > 1 and σ 1 2 + d 2p + 1 ≥ σ 1 +σ 2 for p ≤ d * and σ ≤ d
2 + 1 as well with (4.12), (4.13) and (4.2), we thus arrive at

t 0 t -τ -σ 1 +σ 2 u h • ∇a Ḃ-σ 1 2,∞ dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p -1 dτ t -σ 1 +σ 2 D 2 p (t).
The term u h • ∇u being completely similar to u h • ∇a . Let us finally handle the term corresponding to g 4 (a, u ). Using (4.9), (4.3) and Proposition 3.4, we have

t 0 t -τ -σ 1 +σ 2 g 4 (a, u ) Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 g 4 (a, u ) Ḃ d p -d 2 -σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 ∇a Ḃ d p -d 2 -σ 1 p,1 ∇u Ḃ d p 2,1
dτ,

where d p -d 2 -σ 1 + 1 ≥ 1 -d p > -d p implies that d p -d 2 -σ 1 +
1 fullfils the regularity requirement in Proposition 3.4). Thanks to (4.12)-(4.13) and (4.2), we end up with

t 0 t -τ -σ 1 +σ 2 g 4 (a, u ) Ḃ-σ 1 2,∞ dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p -1 dτ t -σ 1 +σ 2 D 2 p (t).
Hence, putting all estimates together lead to (4.7).

To bound nonlinear terms in f h and g h , precisely,

a div u h , u • ∇a h , u • ∇u h , k(a)∇a h , g 3 (a, u h ) and g 4 (a, u h ),
we shall proceed those calculations differently depending on whether 2 ≤ p ≤ d or p > d.

Lemma 4.3. Let 2 ≤ p ≤ d and d/p -d/2 < σ 1 ≤ σ 0 . If p satisfies (2.
2), then it holds that for all t ≥ 0, (4.14)

t 0 t -τ -σ 1 +σ 2 f h , g h (τ ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t) + E 2 p (t) .
Proof. We first deal with the case

d p -d 2 < σ 1 ≤ σ 0 if 2 ≤ p ≤ d. Owing to the embedding L p 2 → Ḃ-σ 0
2,∞ and Hölder inequality, one has

(4.15) F G Ḃ-σ 1 2,∞ F G Ḃ-σ 0 2,∞ F G L p 2 F L p G L p .
By Sobolev embedding properties, we obtain

(a, u) L p ≤ (a, u) L p + (a, u) h L p (a, u) Ḃ d 2 -d p 2,1 + (a, u) h Ḃ d p -1 p,1 . Remembering that -σ 1 < d 2 -d p < d 2 + 1 and α ≥ σ 1 2 + d 4 -d 2p
for small enough ε > 0 and using the definition of D p (t), we get

(4.16) (a, u)(τ ) L p τ - σ 1 2 -d 4 + d 2p D p (τ ), ∇a h (τ ) L p τ -α D p (τ ) for all τ ≥ 0.
For the term with u • ∇a h , we use (4.15) and (4.16), and get

t 0 t -τ -σ 1 +σ 2 u • ∇a h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 u L p ∇a h L p dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 4 + d 2p -α dτ.
The relation σ ≤ d 2 + 1 implies that α ≥ σ 1 +σ 2 for small enough ε > 0. Consequently, we have

σ 1 2 + d 4 - d 2p + α ≥ σ 1 + σ 2 .
Additionally, it is not hard to check that σ 1 2 + d 4 -d 2p + α > 1. Therefore, inequality (4.2) implies that

t 0 t -τ -σ 1 +σ 2 u • ∇a h Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t).
To handle the term with k(a)∇a h , thanks to (4.15)-(4.16), we similarly arrive at

t 0 t -τ -σ 1 +σ 2 k(a)∇a h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a L p ∇a h L p dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 4 + d 2p -α dτ t -σ 1 +σ 2 D 2 p (t).
For the term with a div u h , we consider cases t ≥ 2 and t ≤ 2 separately. Then we use that, owing to (4.15), if t ≥ 2,

t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a L p div u h L p dτ = 1 0 + t 1 (• • • ) dτ I 11 + I 12 .
Note that d p -1 ≥ 0, it follows from the definitions of E p (t) and D p (t) that

I 11 t -σ 1 +σ 2 sup 0≤τ ≤1 a L p 1 0 u h Ḃ d p p,1 dτ t -σ 1 +σ 2 D p (1)E p (1).
On the other hand, using (4.2), (4.16) and the fact that τ ≈ τ as τ ≥ 1, one has for all -σ 1 < σ ≤ d 2 + 1,

I 12 = t 1 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 4 + d 2p -α τ σ 1 2 + d 4 -d 2p a L p τ α div u h L p dτ sup 1≤τ ≤t τ σ 1 2 + d 4 -d 2p a L p sup 1≤τ ≤t τ α u h Ḃ d p p,1 × t 1 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 4 + d 2p -α dτ t -σ 1 +σ 2 D 2 p (t).
Therefore, we deduce that

t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D p (t)E p (t) + D 2 p (t) for all t ≥ 2.
In the trivial case t ≤ 2, we have t ≈ 1 and t -τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, and

t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ a L ∞ t (L p ) u h L 1 t ( Ḃ d p p,1 )
D p (t)E p (t).

Those terms with u • ∇u h and g 3 (a, u h ) may be treated along the same lines.

Finally, in order to bound the term with g 4 (a, u h ), we use (4.15) and get if t ≥ 2,

t 0 t -τ -σ 1 +σ 2 g 4 (a, u h ) Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 ∇a L p ∇u h L p dτ = 1 0 + t 1 (• • • ) dτ J 11 + J 12 . Note that -σ 1 < d 2 -d p + 1 < d 2 + 1 and α ≥ σ 1 2 + d 4 -d 2p + 1 2 > 0 for small enough ε > 0, we have ∇a(τ ) L p a(τ ) Ḃ d 2 -d p +1 2,1 + a h Ḃ d p p,1 τ - σ 1 2 -d 4 + d 2p -1 2 D p (τ ) for all τ ≥ 0.
Hence using the embedding and the definitions of E p (t) and D p (t) yields

J 11 t -σ 1 +σ 2 sup 0≤τ ≤1 ∇a L p 1 0 u h Ḃ d p p,1 dτ t -σ 1 +σ 2 D p (1)E p (1).
Thanks to (4.2), we conclude that

J 12 sup 1≤τ ≤t τ σ 1 2 + d 4 -d 2p + 1 2 ∇a L p sup 1≤τ ≤t τ α u h Ḃ d p p,1 × t 1 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 4 + d 2p -1 2 -α dτ t -σ 1 +σ 2 D 2 p (t).
Therefore, we end up with

t 0 t -τ -σ 1 +σ 2 g 4 (a, u h ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D p (t)E p (t) + D 2 p (t) for all t ≥ 2.
The case t ≤ 2 implies that t ≈ 1 and t -τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, so we get

t 0 t -τ -σ 1 +σ 2 g 4 (a, u h ) Ḃ-σ 1 2,∞ dτ ∇a L ∞ t (L p ) u h L 1 t ( Ḃ d p p,1 )
D p (t)E p (t).

In summary, the time-weighted inequality (4.14) is followed.

Lemma 4.4. Let 2 ≤ p ≤ d and 1 -d/2 < σ 1 ≤ d/p -d/2. If p satisfies (2.
2), then it holds that for all t ≥ 0, (4.17)

t 0 t -τ -σ 1 +σ 2 f h , g h (τ ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t) + E 2 p (t) .
Proof. We handle another case 1

-d 2 < σ 1 ≤ d p -d 2 if 2 ≤ p ≤ d.
By Sobolev embedding properties and Hölder inequality, we have

F G h Ḃ-σ 1 2,∞ F G h Ḃ0 2,∞ F G h L 2 F L p * G h L p F Ḃσ 0 p,1 G h Ḃ d p -1 p,1 , (4.18) F G h Ḃ-σ 1 2,∞ F G h Ḃ0 2,∞ F G h L 2 F L d G h L d * F Ḃ d p -1 p,1 G h Ḃ d p p,1 , (4.19)
where 1/p + 1/p * = 1/d + 1/d * = 1/2. In addition, we also get, due to the embedding

Ḃ d p 2,1 → Ḃσ 0 p,1 , (4.20) (a, u )(τ ) Ḃσ 0 p,1 (a, u)(τ ) Ḃ d p 2,1 + a(τ ) h Ḃ d p p,1 τ - σ 1 2 -d 2p D p (τ ),
where we noticed the fact that -σ

1 < d 2 -1 ≤ d p < d 2 +1 for p ≤ d * , and that α ≥ σ 1 2 + d 2p
for small enough ε > 0.

To bound the term with a div u h , we take advantage of (4.18) and get, if t ≥ 2,

t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃσ 0 p,1 div u h Ḃ d p -1 p,1 dτ = 1 0 + t 1 (• • • ) dτ I 21 + I 22 .
With the aid of (4.20), one can obtain

I 21 t -σ 1 +σ 2 sup 0≤τ ≤1 a Ḃσ 0 p,1 1 0 u h Ḃ d p p,1 dτ t -σ 1 +σ 2 D p (1)E p (1)
.

By using (4.2) and (4.20), we conclude that for all -σ 1 < σ ≤ d 2 + 1,

I 22 = t 1 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p -α τ σ 1 2 + d 2p a Ḃσ 0 p,1 τ α div u h Ḃ d p -1 p,1 dτ sup 1≤τ ≤t τ σ 1 2 + d 2p a Ḃσ 0 p,1 sup 1≤τ ≤t τ α u h Ḃ d p p,1 × t 1 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p -α dτ t -σ 1 +σ 2 D 2 p (t),
where we noticed that

σ 1 2 + d 2p + α > 1 and that σ 1 2 + d 2p + α ≥ σ 1 +σ 2
for small enough ε > 0 and σ ≤ d 2 + 1. Therefore, we obtain (4.21)

t 0 t-τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D p (t)E p (t) + D 2 p (t) for all t ≥ 2.
In the case t ≤ 2, we have t ≈ 1 and t -τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, and

(4.22) t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ a L ∞ t ( Ḃσ 0 p,1 ) u h L 1 t ( Ḃ d p p,1 ) D p (t)E p (t).
Regarding the term with u • ∇a h , we write that u • ∇a h = u • ∇a h + u h • ∇a h . To handle the term containing u • ∇a h , we use the following inequality, owing to the relation

∇a(τ ) h Ḃ d p -1 p,1 τ -α D p (τ ),
together with (4.18), (4.20) and (4.2), we thus get

t 0 t -τ -σ 1 +σ 2 u • ∇a h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 u Ḃσ 0 p,1 ∇a h Ḃ d p -1 p,1 dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p -α dτ t -σ 1 +σ 2 D 2 p (t).
To bound the term with u h • ∇a h , we have by using (4.18) and the relation

σ 0 ≤ d p , t 0 t -τ -σ 1 +σ 2 u h • ∇a h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 u h Ḃ d p p,1 ∇a h Ḃ d p -1 p,1 dτ = 1 0 + t 1 (• • • )dτ J 21 + J 22 .
It is clear that

J 21 t -σ 1 +σ 2 E 2 p (1)
and that, owing to (4.2) and (4.23), if t ≥ 1,

J 22 = t 1 t -τ -σ 1 +σ 2 τ -2α τ α u h Ḃ d p p,1 τ α ∇a h Ḃ d p -1 p,1 dτ sup 1≤τ ≤t τ α u h Ḃ d p p,1 sup 1≤τ ≤t τ α a h Ḃ d p p,1 t 1 t -τ -σ 1 +σ 2 τ -2α dτ t -σ 1 +σ 2 D 2 p (t), since α ≥ σ 1 2 + d 2p
for small enough ε > 0. For the term with k(a)∇a h , we use (4.18), (4.3) and Proposition 3.4 with σ 0 > -d p , and get

(4.24) k(a)∇a h Ḃ-σ 1 2,∞ k(a) Ḃσ 0 p,1 ∇a h Ḃ d p -1 p,1 a Ḃσ 0 p,1 a h Ḃ d p p,1
.

Furthermore, by using (4.20), (4.23) and (4.2), we end up with

t 0 t -τ -σ 1 +σ 2 k(a)∇a h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃσ 0 p,1 a h Ḃ d p p,1 dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p -α dτ t -σ 1 +σ 2 D 2 p (t).
For the term with g 3 (a, u h ), by repeating the procedure leading to (4.21) and (4.22), we conclude that

t 0 t-τ -σ 1 +σ 2 g 3 (a, u h ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 (D p (t)E p (t)+D 2 p (t)) for -σ 1 < σ ≤ d 2 +1.
To handle the term involving u • ∇u h , we write that, thanks to (4.19)

t 0 t -τ -σ 1 +σ 2 u • ∇u h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 u Ḃ d p -1 p,1 ∇u h Ḃ d p p,1 dτ = 1 0 + t 1 (• • • ) dτ K 21 + K 22 .
It following from the definition of E(t) that

K 21 t -σ 1 +σ 2 E 2 p (1). Using the relation -σ 1 < d 2 -1 < d 2 + 1 and α ≥ σ 1 2 + d 4 -1 2 for small enough ε > 0 indicates that u Ḃ d p -1 p,1 u Ḃ d 2 -1 2,1 + u h Ḃ d p -1 p,1 τ -σ 1 2 -d 4 + 1 2 D p (τ )
together with (4.2), we thus get, if t ≥ 1,

K 22 t 1 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 4 + 1 2 -α τ σ 1 2 + d 4 -1 2 u Ḃ d p -1 p,1 τ α ∇u h Ḃ d p p,1 dτ sup 1≤τ ≤t τ σ 1 2 + d 4 -1 2 u Ḃ d p -1 p,1 sup 1≤τ ≤t τ α ∇u h Ḃ d p p,1 × t 1 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 4 + 1 2 -α dτ t -σ 1 +σ 2 D 2 p (t), since the relations σ 1 > 1 -d 2 and α > 1 lead to σ 1 2 + d 4 -1 2 + α > 1 and σ 1 2 + d 4 -1 2 + α ≥ σ 1 +σ 2
for σ ≤ d 2 + 1. Let us finally look at the term with g 4 (a, u h ). We observe that, owing to (4.19) and Proposition 3.4,

t 0 t -τ -σ 1 +σ 2 g 4 (a, u h ) Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃ d p p,1 ∇u h Ḃ d p p,1 dτ = 1 0 + t 1 (• • • ) dτ L 21 + L 22 .

It is clear that

L 21 t -σ 1 +σ 2 E 2 p (1)
and that, owing to (4.2) and (4.10), if t ≥ 1,

L 22 = t 1 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 2 -α τ σ 1 2 + d 4 a Ḃ d p p,1 τ α ∇u h Ḃ d p p,1 dτ sup 1≤τ ≤t τ σ 1 2 + d 4 a Ḃ d p p,1 sup 1≤τ ≤t τ α u h Ḃ d p +1 p,1 × t 1 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 4 -α dτ t -σ 1 +σ 2 D 2 p (t).
Therefore, the proof of Lemma 4.4 is complete.

t 0 t -τ -σ 1 +σ 2 f h , g h (τ ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t) + E 2 p (t) .
Proof. Let us end the first step in the case where (2.2) is fulfilled and p > d. Using the fact that σ 1 ≤ σ 0 and applying (3.5) with s = 1

-d p yield (4.26) F G h Ḃ-σ 1 2,∞ F G h Ḃ-σ 0 2,∞ F Ḃ1-d p p,1 + Ṡj 0 +N 0 F L p * G h Ḃ d p -1 p,1 with 1 p * 1 2 -1 p and thus, because Ḃ d p 2,1 → L p * , (4.27) F G h Ḃ-σ 1 2,∞ F G h Ḃ-σ 0 2,∞ F Ḃ d p 2,1 + F Ḃ1-d p p,1 G h Ḃ d p -1 p,1 .
On the other hand, we claim that (4.28) (a, u )(τ )

Ḃ1-d p p,1 τ -σ 1 +1-σ 0 2 D p (τ )
and also that (4.29) (a, u) (τ )

Ḃ d p 2,1 τ - σ 1 2 -d 2p D p (τ ) for all τ ≥ 0.
Indeed, owing to the embedding and the definition of D p (t), we have

(a, u) (τ ) Ḃ1-d p p,1 (a, u) (τ ) Ḃ1-σ 0 2,1 τ -σ 1 +1-σ 0 2 D p (τ ).
Here, we used the relation p > d which indicates that -σ

1 < d 2 -1 < 1 -σ 0 < d 2 + 1. As 1 -d p < d p , it holds that a h (τ ) Ḃ1-d p p,1 a h (τ ) Ḃ d p p,1 τ -α D p (τ ),
and (4.28) is thus fulfilled by a h due to

σ 1 +1-σ 0 2 < σ 1 2 + d 4 + 1 2 ≤ α for small enough ε > 0. Thanks to -σ 1 < d 2 -1 ≤ d p < d 2 + 1 for p ≤ d *
, the inequality (4.29) is easily followed by the definition of D p (t).

Note also that if p > d and p fulfills (2.2), then we have by interpolation

u h (τ ) Ḃ1-d p p,1 u h (τ ) d p Ḃ d p -1 p,1 u h (τ ) 1-d p Ḃ d p +1 p,1 .
Consequently, it follows from the definition of D p (t) that (4.30)

u h (τ ) Ḃ1-d p p,1 τ -(1-d p )α τ -d p α D p (τ ) for all τ ≥ 0.
Here, it is easy to check that 0

< 1 - d p α < 1 - d p 2d p + 1 2 = 1 - d 2p -2 d p - 1 2 2 < 1.
Taking F = u and G h = ∇a h in (4.27) and using (4.28), (4.29) and (4.30), we get

t 0 t -τ -σ 1 +σ 2 u • ∇a h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 u Ḃ d p 2,1 + u Ḃ1-d p p,1 + u h Ḃ1-d p p,1 ∇a h Ḃ d p -1 p,1 dτ D 2 p (t) t 0 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p + τ -σ 1 +1-σ 0 2 + τ -(1-d p )α τ -d p α τ -α dτ. Keeping in mind α > 1 > d p and α ≥ σ 1 2 + d 4 for small enough ε > 0, it is shown that σ 1 + σ 2 ≤ σ 1 2 + 1 2 + d 4 < σ 1 2 + d 2p + α < 2α, σ 1 + σ 2 ≤ σ 1 + 1 -σ 0 2 + α, and σ 1 2 + d 2p + α > 1 2 - d 4 + d 2p + α > 1, σ 1 + 1 -σ 0 2 + α > 1, since p ≤ d * implies that d/p ≥ d/2 -1.
Therefore, we deduce (thanks to (4.2)) that

t 0 t -τ -σ 1 +σ 2 u • ∇a h Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D 2 p (t).
To bound the term with k(a)∇a h , we see that, using the composition inequality in Lebesgue spaces and the embeddings

Ḃ d p 2,1 → Ḃσ 0 p,1 → L p * , we get k(a) L p * a L p * a Ḃ d p 2,1 + a h Ḃσ 0 p,1 a Ḃ d p 2,1 + a h Ḃ d p p,1
.

Hence, taking advantage of (4.26) and Proposition 3.4,

(4.31) k(a)∇a h Ḃ-σ 1 2,∞ a Ḃ1-d p p,1 + a Ḃ d p 2,1 + a h Ḃ d p p,1 ∇a h Ḃ d p -1 p,1
, and one can thus bound the term corresponding to k(a)∇a h as u • ∇a h . To bound the term with a div u h , we use (4.27), and get if t ≥ 2,

t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃ d p 2,1 + a Ḃ1-d p p,1 div u h Ḃ d p -1 p,1 dτ = 1 0 + t 1 (• • • ) dτ I 1 + I 2 .
Remembering the definitions of E p (t) and D p (t) and using (4.28) and (4.29), we obtain

I 1 t -σ 1 +σ 2 sup 0≤τ ≤1 a Ḃ d p 2,1 + sup 0≤τ ≤1 a Ḃ1-d p p,1 1 0 u h Ḃ d p p,1 dτ t -σ 1 +σ 2 D p (1)E p (1),
Using (4.2) and the fact that τ ≈ τ when τ ≥ 1, one has for all -σ 1 < σ ≤ d 2 + 1,

I 2 sup 1≤τ ≤t τ σ 1 2 + d 2p a Ḃ d p 2,1 + sup 1≤τ ≤t τ σ 1 +1-σ 0 2 a Ḃ1-d p p,1 sup 1≤τ ≤t τ α u h Ḃ d p p,1 × t 1 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p + τ -σ 1 +1-σ 0 2 τ -α dτ t -σ 1 +σ 2 D 2 p (t).
Therefore, we arrive at

t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D p (t)E p (t) + D 2 p (t) for all t ≥ 2. Owing to d p -1 < 1 -d p and d p ≥ d 2 -1, we discover that (4.32) (a, u) Ḃ d p 2,1 (a, u) Ḃ d 2 -1 2,1 , (a, u) Ḃ1-d p p,1 (a, u) Ḃ d p -1 p,1 (a, u) Ḃ d 2 -1 2,1 .
In the case t ≤ 2, we have t ≈ 1 and t -τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, and

t 0 t -τ -σ 1 +σ 2 a div u h Ḃ-σ 1 2,∞ dτ a L ∞ t ( Ḃ d 2 -1 2,1 ) + a h L ∞ t ( Ḃ d p p,1 ) u h L 1 t ( Ḃ d p p,1 ) E 2 p (t).
In order to bound the term with g 3 (a, u h ), we mimic the proof of (4.31) and get

g 3 (a, u h ) Ḃ-σ 1 2,∞ a Ḃ1-d p p,1 + a Ḃ d p 2,1 + a h Ḃ d p p,1 ∇ 2 u h Ḃ d p -1 p,1
, and one can conclude that exactly as the previous term a div u h that

t 0 t -τ -σ 1 +σ 2 g 3 (a, u h ) Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D p (t)E p (t) + D 2 p (t) + E 2 p (t) .
To handle the term with u • ∇u h , we note that, thanks to (4.27) and interpolation,

u • ∇u h Ḃ-σ 1 2,∞ u Ḃ d p 2,1 + u Ḃ1-d p p,1 + u h Ḃ d p -1 p,1 ∇u h Ḃ d p p,1 . If t ≥ 2, then we write that t 0 t -τ -σ 1 +σ 2 u • ∇u h Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 u Ḃ d p 2,1 + u Ḃ1-d p p,1 + u h Ḃ d p -1 p,1 ∇u h Ḃ d p p,1 dτ = 1 0 + t 1 (• • • )dτ J 1 + J 2 .
It just follows from (4.28), (4.29) and the definitions of D p (t) and E p (t) that

J 1 t -σ 1 +σ 2 D p (1)E p (1) + E 2 p (1)
. and also that

J 2 D 2 p (t) t 1 t -τ -σ 1 +σ 2 τ - σ 1 2 -d 2p + τ -σ 1 +1-σ 0 2 + τ -α τ -α dτ t -σ 1 +σ 2 D 2 p (t).
Consequently, for all t ≥ 2, we have

t 0 t -τ -σ 1 +σ 2 u • ∇u h Ḃ-σ 1 2,∞ dτ t -σ 1 +σ 2 D p (t)E p (t) + D 2 p (t) + E 2 p (t) .
In the case t ≤ 2, we have t ≈ 1 and t -τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2. Furthermore, it follows from (4.32) that

t 0 t -τ -σ 1 +σ 2 u • ∇u h Ḃ-σ 1 2,∞ dτ u L ∞ t ( Ḃ d 2 -1 2,1 ) + u h L ∞ t ( Ḃ d p -1 p,1 ) u h L 1 t ( Ḃ d p +1 p,1 ) E 2 p (t).
Lastly, we consider the term with g 4 (a, u h ). Applying (3.6) with s = 1 -d p yields for any smooth function K vanishing at 0,

∇K(a) ⊗ ∇u h Ḃ-σ 0 2,∞   ∇u h Ḃ1-d p p,1 + j 0 +N 0 -1 j=j 0 ∆j ∇u h L p *   ∇K(a) Ḃ d p -1 p,1 .
As p * ≥ p, we get from Bernstein inequality that for j 0 ≤ j < j 0 + N 0 , ∆j ∇u h L p * ∆j ∇u h L p . Hence, using Proposition 3.4, the fact that σ 1 ≤ σ 0 and that 1 -d p < d p , we have

g 4 (a, u h ) Ḃ-σ 1 2,∞ g 4 (a, u h ) Ḃ-σ 0 2,∞ a Ḃ d p p,1 ∇u h Ḃ1-d p p,1 a Ḃ d p p,1 ∇u h Ḃ d p p,1 . Therefore, t 0 t -τ -σ 1 +σ 2 g 4 (a, u h ) Ḃ-σ 1 2,∞ dτ t 0 t -τ -σ 1 +σ 2 a Ḃ d p p,1 ∇u h Ḃ d p p,1 dτ = 1 0 + t 1 (• • • )dτ K 1 + K 2 .
For K 1 , it is clear that

K 1 t -σ 1 +σ 2 E 2 p (1
) and that, owing to the definition of D p (t) and (4.2), (4.10), if t ≥ 2,

K 2 sup 1≤τ ≤t τ σ 1 2 + d 4 a Ḃ d p p,1 sup 1≤τ ≤t τ α ∇u h Ḃ d p p,1 × t 1 t -τ -σ 1 +σ 2 τ -σ 1 2 -d 4 -α dτ t -σ 1 +σ 2 D 2 p (t).
Hence, the proof of Lemma 4.5 is complete.

Combining those estimates in Lemmas 4.2-4.5, we achieve the inequality (4.6). Consequently, the proof of Proposition 4.1 is finished. Furthermore, together with (4.5) for bounding the term of (4.4) pertaining to the data, we deduce that (4.33) t

σ 1 +σ 2 (a, u)(t) Ḃσ 2,1 D p,0 + D 2 p (t) + E 2 p (t) for all t ≥ 0, provided that -σ 1 < σ ≤ d 2 + 1. 4.2.
Second step: Decay estimates for the high frequencies of (∇a, u). This section is devoted to bound the second term of D p (t), which follows from the elementary energy approach of L p type. For the end, we first present a time-weighted product estimate, which can be regarded as the generalization of Proposition 3.2. Lemma 4.6. Let s > 0 and 1 ≤ p, r ≤ ∞. For T > 0 and β = β 1 + β 2 with β 1 , β 2 ≥ 0, one has

t β uv L θ T ( Ḃs p,r ) t β 1 u L θ 1 T (L ∞ ) t β 2 v L θ 2 T ( Ḃs p,r ) + t β 1 v L θ 1 T (L ∞ ) t β 2 u L θ 2 T ( Ḃs p,r ) , where 1/θ = 1/θ 1 + 1/θ 2 .
Proof. In terms of Bony's decomposition (see the definition in Chap. 2 of [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]), we have

t β uv = T t β 1 u (t β 2 v) + T t β 1 v (t β 2 u) + R(t β 1 u, t β 2 v).
According to those results of continuity for the paraproduct and the remainder (see [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]), it is not difficult to get

T t β 1 u (t β 2 v) L θ T ( Ḃs p,r ) t β 1 u L θ 1 T (L ∞ ) t β 2 v L θ 2 T ( Ḃs p,r ) , T t β 1 v (t β 2 u) L θ T ( Ḃs p,r ) t β 1 v L θ 1 T (L ∞ ) t β 2 u L θ 2 T ( Ḃs p,r ) , R(t β 1 u, t β 2 v) L θ T ( Ḃs p,r ) t β 1 u L θ 1 T ( Ḃ0 ∞,∞ ) t β 2 v L θ 2
T ( Ḃs p,r ) . Hence, Remark 3.1 and the embedding L ∞ → Ḃ0 ∞,∞ leads to the desired conclusion. Proposition 4.2. If p satisfies (2.2), then it holds that for all T ≥ 0, (4.34)

t α (∇a, u) h L ∞ T ( Ḃ d p -1 p,1 ) (∇a 0 , u 0 ) h Ḃ d p -1 p,1 + E 2 p (T ) + D 2 p (T )
with α = σ 1 + d 2 + 1 2 -ε for sufficiently small ε > 0, where E p (T ) and D p (T ) are defined by (4.1) and (2.6), respectively.

Proof. The proof of Proposition 4.2 is to exhibit the optimal decay exponents for the high frequencies of (a, u) with the assumption of (2.3)-(2.4). Let P Id+∇ (-∆) -1 div be the Leray projector onto divergence-free vector-fields. Then Pu satisfies a heat equation

∂ t Pu -µ ∞ ∆Pu = Pg.
Let us introduce the effective velocity w (see (2.7)) as in [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fluids[END_REF], which was originated from Hoff's viscous effective flux in [START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF]. It is easy to check that w fulfills some constant coefficient heat equation, while a satisfies a damped transport equation. Precisely, one has

∂ t a + a = f -div w, ∂ t w -∆w = ∇(-∆) -1 (f -div g) + w -(-∆) -1 ∇a.
Then, by performing the L p energy method in [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF], we can end up with

t α (∇a, u) h L ∞ T ( Ḃ d p -1 p,1 ) (∇a 0 , u 0 ) h Ḃ d p -1 p,1 + j≥j 0 -1 sup 0≤t≤T t α t 0 e -c 0 (t-τ ) 2 j( d p -1) Z j (τ )dτ . (4.35) with Z j Z 1 j + • • • + Z 5 j and Z 1 j ∆j (au) L p , Z 2 j g j L p , Z 3 j ∇ ∆j (a div u) L p , Z 4 j R j L p , Z 5 j div u L ∞ ∇a j L p ,
where a j ∆j a, g j ∆j g and R j [u • ∇, ∇ ∆j ]a. Firstly, we observe that

j≥j 0 -1 sup 0≤t≤2 t α t 0 e -c 0 (t-τ ) 2 j( d p -1) Z j (τ )dτ 2 0 j≥j 0 -1 2 j( d p -1) Z j (τ )dτ.
Furthermore, it follows from Propositions 3.2 and 3.5 that (4.36)

2 0 j≥j 0 -1 2 j( d p -1) Z j (τ )dτ 2 0 (au, g) h Ḃ d p -1 p,1 + ∇u Ḃ d p p,1 a Ḃ d p p,1 dτ.
It is obvious that the last term of the r.h.s. of (4.36) may be bounded by CE 2 p (2) and thanks to Proposition 3.2, we arrive at

au h L 1 t ( Ḃ d p -1 p,1 )
au

L 1 t ( Ḃ d p p,1 ) a L 2 t ( Ḃ d p p,1 ) u L 2 t ( Ḃ d p p,1 )
.

It follows from Propositions 3.2 and 3.4 that

g h L 1 t ( Ḃ d p -1 p,1 )
u

L ∞ t ( Ḃ d p -1 p,1 )
∇u

L 1 t ( Ḃ d p p,1 )
+ a

L ∞ t ( Ḃ d p p,1 )
∇u

L 1 t ( Ḃ d p p,1 )
+ a

L 2 t ( Ḃ d p p,1 )
∇a

L 2 t ( Ḃ d p -1 p,1 )
.

Combining interpolation and the definition of E p (t), we deduce that a

L 2 t ( Ḃ d p p,1 ) + u L 2 t ( Ḃ d p p,1 ) E p (t).
Furthermore, we can conclude that the first term in the right-hand side of (4.36) may be bounded by E 2 p (2). We end up with (4.37)

j≥j 0 -1 sup 0≤t≤2 t α t 0 e -c 0 (t-τ ) 2 j( d p -1) Z j (τ )dτ E 2 p (2).
Secondly, let us handle the supremum for 2 ≤ t ≤ T in the last term of (4.35). To do this, one can split the integral on [0, t] into integrals [0, 1] and [1, t]. The [0, 1] part of the integral may be bounded by E 2 p (1). Indeed, owing to e -c 0 (t-τ ) ≤ e -c 0 t 2 for 2 ≤ t ≤ T and 0 ≤ τ ≤ 1, we obtain

j≥j 0 -1 sup 2≤t≤T t α 1 0 e -c 0 (t-τ ) 2 j( d p -1) Z j (τ )dτ j≥j 0 -1 sup 2≤t≤T t α e -c 0 2 t 1 0 2 j( d p -1) Z j (τ )dτ 1 0 j≥j 0 -1 2 j( d p -1) Z j (τ )dτ.
Hence, following from the procedure leading to (4.37), we arrive at (4.38)

j≥j 0 -1 sup 2≤t≤T t α 1 0 e -c 0 (t-τ ) 2 j( d p -1) Z j (τ )dτ E 2 p (1)
.

In order to handle the [1, t] part of the integral for 2 ≤ t ≤ T , we start from (4.39)

j≥j 0 -1 sup 2≤t≤T t α t 1 e -c 0 (t-τ ) 2 j( d p -1) Z j (τ )dτ j≥j 0 -1 2 j( d p -1) sup 1≤t≤T t α Z j (t).
To bound the contribution of Z 1 j and Z 2 j in (4.35), we use the fact that (4.40)

j≥j 0 -1 2 j( d p -1) sup 1≤t≤T t α Z 1 j (t) + Z 2 j (t) t α (au, g) h L ∞ T ( Ḃ d p -1 p,1 )
.

For the term au, we decompose it as follows au = au h + a h u + a u .

Product laws in Proposition 3.2 adapted to tilde spaces ensure that

t α au h h L ∞ T ( Ḃ d p -1 p,1 ) a L ∞ T ( Ḃ d p p,1 ) t α u h L ∞ T ( Ḃ d p -1 p,1 ) E p (T )D p (T ), t α a h u h L ∞ T ( Ḃ d p -1 p,1 ) t α a h L ∞ T ( Ḃ d p p,1 ) u L ∞ T ( Ḃ d p -1 p,1 )
D p (T )E p (T ).

Bernstein inequality and embedding imply that

t α a u h L ∞ T ( Ḃ d p -1 p,1 ) t α a u L ∞ T ( Ḃ d 2 +1 2,1 )
, and thus it follows from Lemma 4.6 (noticing that Ḃ

d 2 2,1 → L ∞ ) that t α a u h L ∞ T ( Ḃ d p -1 p,1 ) t σ 1 2 + d 4 a L ∞ T ( Ḃ d 2 2,1 ) t σ 1 2 + d 4 + 1 2 -ε u L ∞ T ( Ḃ d 2 +1
2,1 )

+ t σ 1 2 + d 4 u L ∞ T ( Ḃ d 2 2,1 ) t σ 1 2 + d 4 + 1 2 -ε a L ∞ T ( Ḃ d 2 +1 2,1 )
.

Owing to the definition of D p (t) and Remark 3.1, we have

t σ 1 2 + d 4 + 1 2 -ε (a, u) (t) L ∞ T ( Ḃ d 2 +1 2,1 ) t σ 1 2 + d 4 + 1 2 -ε (a, u) (t) L ∞ T ( Ḃ d 2 +1-2ε 2,1 ) D p (T ), whence t α a u h L ∞ T ( Ḃ d p -1 p,1 ) D 2 p (T ).
In what follows, we shall use repeatedly the two inequalities for sufficiently small ε > 0:

t σ 1 2 + d 4 + m 2 -ε 2 (∇ m a , ∇ m u )(τ ) L ∞ T ( Ḃ d p p,1 )
D p (T ), (4.41)

t σ 1 2 + d 4 + m 2 -ε 2 (∇a h , u h )(τ ) L ∞ T ( Ḃ d p -1 p,1 )
D p (T ) for m = 0, 1. (4.42) Indeed, it follows from the embedding, the definition of D p (t) and tilde norms that

t σ 1 2 + d 4 + m 2 -ε 2 (∇ m a , ∇ m u ) L ∞ T ( Ḃ d p p,1 ) t σ 1 2 + d 4 + m 2 -ε 2 (∇ m a , ∇ m u ) L ∞ T ( Ḃ d 2 2,1 ) t σ 1 2 + d 4 + m 2 -ε 2 (a, u) L ∞ T ( Ḃ d 2 +m-ε 2,1 ) D p (T )
and also that, due to α

≥ σ 1 2 + d 4 + m 2 -ε 2 for small enough ε > 0, t σ 1 2 + d 4 + m 2 -ε 2 (∇a h , u h ) L ∞ T ( Ḃ d p -1 p,1 ) t α (∇a h , u h )(τ ) L ∞ T ( Ḃ d p -1 p,1 )
D p (T ) for m = 0, 1.

Next, let us pay attention to (4.40) corresponding to the nonlinear term g, which consists of u • ∇u, k(a)∇a, g 3 (a, u) and g 4 (a, u).

Regarding the convection term u • ∇u, we write

u • ∇u = u • ∇u h + u h • ∇u + u • ∇u .
Then Proposition 3.2 adapted to tilde spaces and (4.41)-(4.42) ensure that

t α (u • ∇u h ) h L ∞ T ( Ḃ d p -1 p,1 ) u L ∞ T ( Ḃ d p -1 p,1 ) t α ∇u h L ∞ T ( Ḃ d p p,1 ) E p (T )D p (T ), t α (u h • ∇u ) h L ∞ T ( Ḃ d p -1 p,1 ) t σ 1 2 + d 4 -ε 2 u h L ∞ T ( Ḃ d p -1 p,1 ) t σ 1 2 + d 4 + 1 2 -ε 2 ∇u L ∞ T ( Ḃ d p p,1 ) D 2 p (T )
and

t α (u • ∇u ) h L ∞ T ( Ḃ d p -1 p,1 ) t α (u • ∇u ) L ∞ T ( Ḃ d p p,1 ) t σ 1 2 + d 4 -ε 2 u L ∞ T ( Ḃ d p p,1 ) t σ 1 2 + d 4 + 1 2 -ε 2 ∇u L ∞ T ( Ḃ d p p,1 ) D 2 p (T ).
To bound the term containing k(a)∇a, we use that

t α k(a)∇a h L ∞ T ( Ḃ d p -1 p,1 ) t σ 1 2 + d 4 -ε 2 a L ∞ T ( Ḃ d p p,1 ) t σ 1 2 + d 4 + 1 2 -ε 2 ∇a L ∞ T ( Ḃ d p p,1 ) D 2 p (T ), t α k(a)∇a h h L ∞ T ( Ḃ d p -1 ) a L ∞ T ( Ḃ d p p,1 ) t α ∇a h L ∞ T ( Ḃ d p -1 p,1 ) E p (T )D p (T ).
The term g 3 (a, u) is of the type H(a)∇ 2 u with H(0) = 0, and the term g 4 (a, u) is of the type ∇K(a) ⊗ ∇u with K(0) = 0. Consequently, we derive that thanks to Propositions 3.2 and 3.4 as well as (4.41)-(4.42), The term Z 3 j is similar to g 3 (a, u) and g 4 (a, u) of g. Indeed, we have D 2 p (T ) + E p (T )D p (T ). For the term Z 4 j , we employ a small modification of the commutator estimate as in Proposition 3.5 (just include t α in the definition of the commutator, follow the proof treating the time variable as a parameter, and take the supremum on [0, T ] at the end) and obtain j≥j 0 -1 The global existence result (see Theorem 1.1 in [START_REF] Danchin | Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L p framework[END_REF]) ensures that E p (t) E p,0 1 and as (a 0 , u 0 )

t α H(a)∇ 2 u h L ∞ T ( Ḃ d p -1 p,1 ) + t α ∇K(a) ⊗ ∇u h L ∞ T ( Ḃ d p -1 p,1 ) t σ 1 2 + d 4 -ε 2 a L ∞ T ( Ḃ d p p,1 ) t σ 1 2 + d 4 + 1 2 -ε 2 ∇u L ∞ T ( Ḃ d p p,1 ) D 2 p (T ), t α H(a)∇ 2 u h h L ∞ T ( Ḃ d p -1 p,1 ) + t α ∇K(a) ⊗ ∇u h h L ∞ T ( Ḃ d p -1 p,1 )
Ḃ d 2 -1 2,1 D p,0 1,
one can conclude that (2.5) is satisfied for all time if D p,0 and E p,0 are small enough. This completes the proof of Theorem 2.1.

Appendix

We end the paper with some heuristics concerning the optimality of the regularity and decay exponents in the definition of D p (t). Let us first explain why the regularity exponent σ in the first term D p (t) has to satisfy σ ≤ d 2 + 1. The general fact (based on Inequality (4.2)) that we used repeatedly is that the time decay exponent δ for (f, g) Ḃ-σ 1 2,∞ must satisfy δ ≥ σ 1 +σ 2 • Now, if we look at the term a div u , then a necessary condition for having

a div u Ḃ-σ 1 2,∞ a Ḃσ 2 2,∞ div u Ḃσ 3 2,∞ is that σ 2 + σ 3 ≤ d 2 -σ 1 .
As the decay exponent of the right-hand side is σ 1 + σ

Theorem 2 . 1 .

 21 Let d ≥ 2 and p fulfill (2.2) 2 ≤ p ≤ min(4, d * ) and, additionally, p = 4 if d = 2, where d * 2d/(d -2). Let ( , u) be the corresponding global solution of System (1.1)

Definition 3 . 1 .

 31 For s ∈ R and 1 ≤ p, r ≤ ∞, the homogeneous Besov spaces Ḃs p,r is defined by Ḃs p,r u ∈ S 0 : u Ḃs p,r

  2 are not enough to bound the possible case p > d in the proof of Theorem 2.1, so we have the following non-classical product estimates. Proposition 3.3. ([10]) Let j 0 ∈ Z, and denote z Ṡj 0 z, z h z -z and, for any s

) with σ 0 2d p -d 2 and 1 p * 1 2 - 1 p

 111 , and C depending only on j 0 , d and s. System (2.1) also involves compositions of functions (through I(a), k(a), λ(a) and µ(a)) and they are bounded according to the following conclusion. Proposition 3.4. Let F : R → R be smooth with

d p - 1 ≥

 1 0 and the definition of D p (t),(4.23)

Lemma 4 . 5 .

 45 Let p satisfy (2.2) and p > d. It holds that for all t ≥ 0,(4.25)

  ) + E p (T )D p (T ).

Furthermore, reverting to ( 4 . 40 ), we end up with j≥j 0 - 1 2

 4401 ) + E p (T )D p (T ).

  R j (t) L p D 2 p (T ) + E p (T )D p (T ).The term Z 5 j is clearly bounded by the r.h.s. of(4.43). Putting all the above estimates together, we thus have(4.44) Z j (t) E p (T )D p (T ) + D 2 p (T). Plugging (4.44) in (4.39), and remembering (4.35), (4.37) and (4.38), we eventually conclude that (4.34). Hence, the proof of Proposition 4.2 is complete. 4.3. Third step: Decay and gain of regularity for the high frequency of u. In the last step, our aim is to bound the third norm in the functional D p (t) and close the time-weighted energy inequality. Proposition 4.3. If p satisfies (2.2), then it holds that for all t ≥ 0, with α = σ 1 + d 2 + 1 2 -ε for sufficiently small ε > 0, where E p (t) and D p (t) are defined by (4.1) and (2.6), respectively. Proof. The proof lies in a new observation, which enables us to obtain more decay for the improved regularity of velocity. Precisely, it follows from the velocity equation in (2.1) that (4.46) ∂ t u -Au = g -∇a. To obtain desired estimates, we reformulate (4.46) as follows ∂ t (t α Au) -A (t α Au) = t α Ag + αt α-1 Au -t α A∇a, t α Au| t=0 = 0. Taking advantage of the smoothing property for Lamé semi-group in Remark 3.2, we thus get τ α ∇ 2 u is exactly same as the second step, and one thus arrive at (4.45) readily. Finally, adding up (4.45) to (4.33) and (4.34) yields for all T ≥ 0, D p (T ) D p,0 + (∇a 0 , u 0 ) h

		h L ∞ t (	Ḃ d p -1 p,1 )		τ α g h L ∞ t (	Ḃ d p -1 p,1 )	+ τ α-1 u	h L ∞ t (	Ḃ d p -1 p,1 )	+ τ α ∇a h L ∞ t (	Ḃ d p,1 ) p -1	.
	Due to the fact α > 1, we see that
	τ α-1 u	h L ∞ t (	Ḃ d p -1 p,1 )		τ α u h L ∞ t (	Ḃ d p -1 p,1 )	,	τ α ∇a h L ∞ t (	Ḃ d p -1 p,1 )	τ α a h L ∞ t (	Ḃ d p,1 ) p	.
	Furthermore, we conclude that
	(4.47)	τ α ∇u h L ∞ t (	Ḃ d p p,1 )		τ α g h L ∞ t (		Ḃ d p -1 p,1 )	+ τ α (∇a, u) h L ∞ t (	Ḃ d p,1 ) p -1	.
	2 j( d p -1) sup 0≤t≤T In light of (4.34), the second norm can be bounded by t α R j (t) L p t σ 1 2 + d 4 + 1 2 -ε 2 ∇u (∇a 0 , u 0 ) h p,1 Ḃ d p -1 + E 2 p (t) + D 2 p (t). L ∞ T (	Ḃ d p p,1 )	t	σ 1 2 + d 4 -ε 2 a	L ∞ T (	Ḃ d p p,1 )
	(4.43) Bounding the norm τ α g h L ∞ t (	Ḃ d p -1 p,1 )		+ t α ∇u h	L ∞ T (	Ḃ d p p,1 )	a	L ∞ T (	Ḃ d p p,1 )	.
	Hence, taking advantage of (4.41) and (4.42) gives
				2 j( d p -1) sup
		j≥j 0 -1								p,1 Ḃ d p -1	+ E 2 p (T ) + D 2 p (T ).
				2	j( d p -1) sup
				j≥j 0 -1
	(4.45)			τ α ∇u h L ∞ t (	Ḃ d p p,1 )		(∇a 0 , u 0 ) h Ḃ d p,1 p -1	+ E 2 p (t) + D 2 p (t)

0≤t≤T t α 1≤t≤T t α

  2 +σ 3 +1 2 , we deduce that δ ≤ d 4 + σ 1 2 + 1 2 • Hence we must have σ ≤ d 2 + 1. To see that the decay rate for the second term in D p (t) cannot be more than σ 1 + d 2 + 1 2 , one can observe that, owing to σ ≤ d 2 + 1, the term a ∇a (which at most has the same regularity as ∇a ) cannot be estimated in a space with higher regularity than Ḃ d , it follows from the first term in D p (t) that the right-hand side has decay exponent σ 1 + d 2 + 1 2 • A similar argument shows that the decay rate for the third term in D p (t) is optimal.

						2 2,1 . As
	the corresponding estimate reads
	a ∇a	Ḃ d 2 2,1	a	Ḃ d 2 2,1	∇a	2 Ḃ d 2,1

Note that for technical reasons, we need a small overlap between low and high frequencies.

Acknowledgments

The author is partially supported by the National Natural Science Foundation of China (11471158) and the Fundamental Research Funds for the Central Universities (NE2015005). Last but not least, he would like to thank Dr. Weixuan Shi for some communications.


Sho Fujimura

Shuji Shiraishi

On the number of perfect matchings of line graphs II

Keywords: perfect matching, line graph, 2-chain

For any general graphs, we provide a formula for the number of perfect matchings in line graphs.

We have shown a method for counting perfect matchings in line graphs and a formula for the number of perfect matchings in L(G) for graphs G of maximum degree at most 3 by [START_REF] Fujimura | On the number of perfect matchings of line graphs[END_REF]. In this paper, we generalize this result for any general graph G. Let T be an extended spanning tree S and v a vertex in T .

Let n v be the number of edges (v, v i ) such that f (v i ) is even, where f is a descendants function of T . We remark that a set of 2-chain covers of G does not depends on the choice of spanning trees of G. Then we have the following theorem:

Theorem. Let G = (V, E) be a graph.
Then the number NPM of perfect matchings of L(G) is

NPM =    ∑ T ∏ f (v) even (n v -1)!! ∏ f (v) odd n v !! if |E| is even 0 if |E| is odd
where the sum runs over all extended spanning trees T.

Proof:

The following holds for each vertex v in T :

(i) If f (v) is even, there are (n v -1)!! choices for 2-chains of G. (ii) If f (v) is odd, there are n v !! choices for 2-chains of G. where (-1)!!=1. (i) Since f (v) is even, there is an even number n v = 2m of children {v i1 , • • • , v i2m } such that each f (v ij ) is even. So we can partition the edges of T v = {{v, v i1 }, • • • , {v, v i2m }} into m 2-
( 2m 2 )( 2m-2 2 ) • • • m! = n v ! 2 nv/2 • nv 2 ! = (n v -1)!!. (i) is obtained. (ii) If f (v)
is odd, then we apply a 2-chain partition to the set

T v = {{v p , v}, {v, v i1 }, • • • , {v, v i2m-1 }}, where v p is the parent vertex of v and each f (v ij ) is even. In the same way, the number of such 2-chain partitions of T v is n v !!
There are 2 k choices for an extended spanning tree, where k = |E| -|V | + 1 is the number of non-tree edges in G and each such tree generates different 2-chain covers. Since a 2-chain cover of G corresponds to a perfect matching of L(G), the result now follows.

2

When the maximum dergee of G is at most 3, NPM is 2 |E|-|V |+1 if |E| is even. This follows automat- ically from n v ≤ 2. Clearly, if n v > 2 for some v in G then n v = 3, v is a root of T and |E| is odd.

Example

Fig. 1 represents the graph G whose vertex set V is {a, b, c, d, e, f, g}, and whose edge set E is {1 = ad, 2 = ab, 3 = ae, 4 = de, 5 = be, 6 = bc, 7 = ef, 8 = cf, 9 = ce, 10 = f g}. The red lines represent a spanning tree S with root a.

f : f (a) = 10, f (b) = 4, f (c) = 2, f (d) = 9, f (e) = 8, f (f ) = 1, f (g) = 0, f (a ′ ) = 0, f (a ′′ ) = 0, f (c ′ ) = 0, f (f ′ ) = 0, where a ′ , a ′′ , c ′ , f ′ are vertices obtained by cutting off vertices a, c, f . The values f (a), f (b), f (c), f (e), f (g), f (a ′ ), f (a ′′ ), f (c ′ ), f (f ′ ) are even, so n a = (0 -1)!! = 1, n b = (2 -1)!! = 1, n c = (0 -1)!! = 1, n e = (4 -1)!! = 3, n g = (0 -1)!! = 1, n a ′ = (0 -1)!! = 1, n a ′′ = (0 -1)!! = 1, n c ′ = (0 -1)!! = 1, n f ′′ = (0 -1)!! = 1.
The values f (d), f (f ) are odd, so

n d = 1!! = 1, n f = 1!! = 1.
Thus the number of 2-chain covers for the extended spanning tree T 12 , denoted by NPM T12 , is 3:

NPM T12 = 3.
In the same way, NPM T1 = 1,

NPM T2 = 3, NPM T3 = 1, NPM T4 = 1, NPM T5 = 1, NPM T6 = 1, NPM T7 = 1, NPM T8 = 1, NPM T9 = 1, NPM T10 = 3, NPM T11 = 3, NPM T12 = 3, NPM T13 = 1, NPM T14 = 1, NPM T15 =

Figure 2 .

 2 shows all distinct 16 extended spanning trees based on S. The descendants functions values of extented spanning trees are underlined. The n v values are placed in parentheses. For Fig. 2 (Tree10) , The values of the descendants function

Fig. 1 :Fig. 2 :

 12 Fig. 1: Spanning tree S of the graph G


Gyula Simon

Miklós Molnár

László Gönczy
email: gonczy@mit.bme.hu

Bernard Cousin
email: bcousin@irisa.fr

Dependable k-coverage algorithms for sensor networks

Keywords: sensor network, k-coverage, dependable, sleepscheduling

Redundant sensing capabilities are often required in sensor network applications due to various reasons, e.g. robustness, fault tolerance, or increased accuracy. At the same time high sensor redundancy offers the possibility of increasing network lifetime by scheduling sleep intervals for some sensors and still providing continuous service with help of the remaining active sensors. In this paper centralized and distributed algorithms are proposed to solve the k-coverage sensing problem and maximize network lifetime. When physically possible, the proposed robust Controlled Greedy Sleep Algorithm provides guaranteed service independently of node and communication errors in the network. The performance of the algorithm is illustrated and compared to results of a random solution by simulation examples.

I. INTRODUCTION

Wireless sensor networks are constructed from small, autonomous sensors and are utilized for various measurement purposes. The individual power capacity of the sensors is very limited (using batteries it may be only some days) but the expected useful lifetime of the network is required to be in the range of weeks or months, depending on the application. To achieve network longevity low duty cycle operation is utilized. In continuously operating sensor networks redundant sensors are deployed from which only a little subset is active at a time; the major part of sensors is turned off and thus saves energy. In redundant dense sensor networks various scheduling algorithms are used to control energy conservation.

In sensor networks used to austerely monitor an area in space it may also be a requirement that multiple sensors be able to provide measurements from each point in space. This property may either be necessary because of the applied measurement technology, safety or performance reasons or to satisfy accuracy requirements with relatively low-quality sensors. High redundancy present in the network is necessary to achieve this goal. In general this class of problems can be treated as the k-coverage problem, where coverage means the ability of a sensor to perform measurements over a certain area. While in a dense sensor network there may be several equally good solutions to the general k-coverage problem, the energy conservation criterion narrows the range of the acceptable solutions.

In many applications a required coverage is satisfactory as well, so larger coverage does not provide higher performance. Thus finding an 'economical' solution to the k-coverage problem with small number of participating nodes at the same time results energy conservation and thus longer network lifetime as well. Effective scheduling algorithm is required to organize the alternation of active (awake) and sleeping sensor sets to provide continuous service of the network.

In this paper new robust algorithms are proposed to provide dependable k-coverage and prolonged network lifetime. In Section II the main previous results are summarized. Section III introduces a centralized algorithm and its fully distributed variant (Controlled Greedy Sleep Algorithm) to provide robust k-coverage while minimizing the number of awaken sensors at the same time. In Section IV new quality of service metrics are introduced and simulation results are presented to illustrate the capabilities of the proposed algorithms.

II. PREVIOUS RESULTS

Because of their fragility and power deprivation the dependability of sensor networks is an important and hot research topic. There are several propositions to ensure fault tolerance at different levels [START_REF] Koushanfar | Fault Tolerance in Wireless Ad Hoc Sensor Networks[END_REF]. Since the sensors are performing both sensing and communication tasks the main problems of sensor networks are associated to these two activities [START_REF] Ahmed | The holes problem in wireless sensor networks: a survey[END_REF]. The sensor network should be capable of taking measurement in the observed area and transmitting the measured values to sink nodes. The k-coverage problem is associated to the measurement functionality [START_REF] Huang | The coverage problem in a wireless sensor network[END_REF]: every point of the target area must be covered by at least k sensors (k is determined by the application). It also has several implications to connectivity issues [START_REF] Ammari | Coverage, connectivity, and fault tolerance measures of wireless sensor networks[END_REF].

The life-time of the network is generally prolonged by scheduling sleep intervals for some sensors, meanwhile the continuous service is provided by the active sensors (see examples in [START_REF] Krashinsky | Minimizing energy for wireless web access with bounded slowdown[END_REF], [START_REF] Brakmo | Sleep: a technique for reducing energy consumption in handheld devices[END_REF], [START_REF] Xing | Integrated coverage and connectivity configuration for energy conservation in sensor networks[END_REF]). The lifetime longevity and the network operability require efficient trade-offs, realized by different scheduling algorithms, which can mainly be divided into two main groups: random and coordinated scheduling algorithms [START_REF] Hsin | Network coverage using low duty-cycled sensors: random & coordinated sleep algorithms[END_REF]. A distributed, random sleeping algorithm was proposed in [START_REF] Chen | Span: An Energy-Efficient Coordination Algorithm for Topology Maintenance in Ad Hoc Wireless Networks[END_REF] where nodes make local decisions on whether to sleep or to join a forwarding backbone, to ensure communications. Each node bases its decision on an estimate of how many of its neighbours will benefit from its being awake and the amount of energy available to it.

In [START_REF] Kumar | On k-coverage in a mostly sleeping sensor network[END_REF] the authors propose a randomized, simple scheduling for dense and mostly sleeping sensor networks. They suppose that there are many redundant sensors in the target area and one can compute the required (identical) duty cycle for individual sensors. In the proposed Randomized Independent Sleeping algorithm, time is divided into periods. At the beginning of each period, each sensor decides whether to go to sleep (with probability p computed from the duty cycle) or not, thus the lifetime of the network is increased by a factor close to 1/p. This solution is very simple and does not require communication between sensors. The drawback of the proposition is that there is no guarantee for coverage nor for network connectivity. Furthermore, since the sleeping factor is the same for all sensors, this solution cannot adapt to inhomogeneous or mobile sensor setups.

To handle the basic coverage problem the authors in [START_REF] Hsin | Network coverage using low duty-cycled sensors: random & coordinated sleep algorithms[END_REF] propose a Role-Alternating, Coverage-preserving, Coordinated Sleep Algorithm (RACP). Each sensor sends a message periodically to its neighbourhood containing its location, residual energy and other control information. An explicit acknowledgment-based election algorithm permits to decide the sleep/awake status. The coordinated sleep is more robust and reduces the duty cycle of sensors compared to the random sleep algorithm, and it guarantees 1-coverage in the network. In this solution the topology can affect the behaviour; thus the sensors can adapt their sleeping to the needs. The price of the performance is the significant communication overhead increasing power consumption.

In [START_REF] Liu | A study of the coverage of large-scale sensor networks[END_REF] the asymptotic behavior of coverage in large-scale sensor networks is studied. For the k-coverage problem, formulated as a decision problem, polynomial-time algorithms (in terms of the number of sensors) are presented in [START_REF] Huang | The coverage problem in a wireless sensor network[END_REF]. A comprehensive study on both coverage and connectivity issues can be found in [START_REF] Ghosh | Coverage and connectivity issues in wireless sensor networks[END_REF].

In this paper a new coordinated algorithm is proposed that is able to guarantee k-coverage in the network where it is physically possible and at the same time can provide prolonged network lifetime. The proposed algorithm takes into account both the power status and the sensing assignment of the sensors. First, a centralized solution will be proposed, which will be approximated by a distributed algorithm. The latter solution is more feasible in practice due to its low communication overhead.

III. SCHEDULING ALGORITHM

A. Background

The sensing assignment in a sensor network can be represented by a bipartite graph ( )

E S R G , ∪
, where the two disjoints sets of vertices represent the nodes S and geographical regions R (see Fig. 1). The regions are defined by the subset of sensors that can monitor them. Generally, they cover the whole measured area and are disjoint. In G there is an edge e between region R r ∈ and sensor S s ∈ if and only if s (completely) covers region r.

The simple k-coverage problem is to find a sub-graph

( ) E S R G ′ ′ ′ , ∪
where S S ⊆ ′ so that for all vertices R in G′ the degree is at least k.

The minimal k-coverage problem is to find a nonredundant sub-graph G′ that solves the k-coverage problem. A graph G′ is non-redundant if there exists no ( )

E S R G ′ ′ ′ ′ ′ ′ , ∪ and S S ′ ⊂ ′ ′
that solves the simple kcoverage problem.

B. Assumptions

For the sake of simplicity we assume that the coverage of each sensor can be modeled by a sensing disk and a corresponding sensing radius (we use a constant sensing radius but it's not essential). Within its sensing radius a sensor is able to perform measurements, while outside the sensing circle the sensing performance may degrade (but not necessarily).

It's also assumed that the communication radius is at least twice of the sensing radius. In most practical cases it's a sensible assumption and it automatically provides networkwide communication if 1-coverage in sensing is provided [START_REF] Xing | Integrated coverage and connectivity configuration for energy conservation in sensor networks[END_REF]. Generally, from the assumption it also follows that network connectivity is higher than k when sensing k-coverage is provided [START_REF] Ammari | Coverage, connectivity, and fault tolerance measures of wireless sensor networks[END_REF].

s 2 s 3 s 4 R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R 10 R 11 s 2 s 1 s 1 s 3 R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R 10 R 11 s 4

C. The centralized k-coverage algorithm

In this version a central scheduler unit is charged with the control of awake/doze states of sensors. The scheduler collects the sensor state information periodically, calculates a special drowsiness factor for each sensor and sends to sleep a subset of sensors, depending on their drowsiness factor. The algorithm is the following:

1. Run the network for a period of T 2. Wake up all sensors and collect state information 3. Compute drowsiness factor for each node. 4. Select the node with the largest positive drowsiness factor. Send this node to sleep. 5. Repeat Steps 3-4 while possible (i.e. there is at least one node with positive drowsiness factor). The drowsiness factor of a node s with current energy E s is defined as

( )      - ∀ > Φ Φ = ∑ ∈ ∀ otherwise 1 0, if , 1 r s r E D r R r r s s δ α , (1)
where ( )

   = otherwise 0 and between in edge an is there if 1 , r s G s r δ (2)
And α is a positive constant (e.g.

2 = α
), and r Φ is the coverage ratio of region r defined as follows:

     - > - = Φ otherwise 1 if 1 K c K c r r r (3)
where c r is the degree of node r in G. The coverage ratio r Φ is positive if the region is over-covered, i.e. more than k sensors cover region r. r Φ is negative if region r is not overcovered: in this case the operation of all sensors covering is essential.

The drowsiness factor D s takes into account the energy of sensor s: the smaller the energy of a sensor the larger its drowsiness. Negative drowsiness indicates that the sensor is not allowed to sleep.

A sensor participating in many regions that have low overcoverage is likely to participate in more possible solutions than sensors covering regions also covered by many other sensors. Thus a heuristic property is included in D s to increase the lifetime of the network: sensors participating in regions only slightly over-covered have larger drowsiness. The drowsiness factor for each sensor includes the sum of the coverage ratios of the regions the sensor is able to observe. This property enforces the sensors in critical positions to go to sleep whenever it is possible, to conserve their energy for times when their participation will become inevitable.

Although the centralized algorithm solves the k-coverage problem and conserves the energy of the network at the same time, it assumes network-wide information distribution. In a large network it would mean excessive amount of messages transfer and thus the solution would impose an important communication overhead. Instead, an approximation of the algorithm is proposed that uses information locally available in the neighborhood of a node.

D. The distributed k-coverage algorithm

The following robust, fault tolerant, distributed algorithm solves the k-coverage problem using locally available information only and thus its communication overhead is low. The algorithm is based on the following observations: To perform approximately the same scheduling as it was shown in the centralized algorithm, a sensor s can go to sleep if its neighbors with larger drowsiness factor decided their state for the next period and s has no critical (not over covered) region to monitor. For this, each sensor should know the drowsiness factor of the neighbors and the decision of neighbors with larger factor. To minimize the local communication, a communication delay (STD) can be associated with each sensor. This delay is inversely proportional with the drowsiness factor. So the sensors with large factor broadcast their decision earlier. Only the awake state decision should be broadcasted explicitly, in this way the communication overhead can be minimal.

Controlled Greedy Sleep (CGS) Algorithm

1. Run the network for a period of T 2. Wake up all sensors 3. Nodes with energy enough for at least one more period broadcast local Hello messages containing node locations. Based on received Hello messages node s builds up its local set of alive neighbor nodes (S s ) with their locations. 4. Each node s calculates its drowsiness factor D s (see Eq. 1). Instead of the global graph G use the locally known subgraph ( )

s s s s E S R G , ∪
. R s and E s are defined in Note 1. 5. Based on D s each node selects a Shout Time Delay (STD). Small drowsiness means large STD, large drowsiness means small STD. 6. Each node s broadcasts its STD s and starts collecting other nodes' Awake Messages (AMs). From the received AMs each node builds a List of Awake Nodes (LAN). 7. After STD s each node s makes a decision based upon the received AMs:

-if the k-coverage problem can be solved using only nodes present in the LAN and nodes with STD larger than STD s then go to sleep -otherwise stay awake and broadcast an AM to inform other nodes. Note 1: For each node the covered regions are represented by a set of squares, as illustrated in Fig. 2. (In the example there are 24 regions, which can be increased to provide better approximation). Let's denote the node's location by (x s , y s ), and the center of region i by ( )

i i y x ∆ ∆ ,
in the node's local coordinate system. As a pessimistic approximation, in G s there is an edge between node w placed at location (x w , y w ) and region i if

( ) ( ) 2 2 2 w i s w i s y y y x x x R - ∆ + + - ∆ + > ∆ - (4)
The square model may seem a rude approximation of the sensing disk, but since the sensing disk model is inherently a rather imperfect estimation of the real sensing area of a sensor, there is no point to put great effort to accurately approximate it. The proposed solution is pessimistic, i.e. certain areas may be unnecessarily over-covered. Note 2: Each node goes to sleep in a greedy manner: if the coverage problem can be solved with the already known awake neighbors (with higher drowsiness factor) in the LAN (these nodes already have decided on their sleeping status) and some of the neighbors with lower drowsiness factor (these nodes will decide their sleep/awake status later) then the node greedily elects to sleep.

Note 3: The nodes make their decision based upon received Hello, STD, and Awake messages. Thus the algorithm is insensitive to lost messages in the sense that the coverage is always provided (if and where it is possible). Naturally, communication problems may cause nodes to stay awake unnecessarily and thus they shorten the lifetime of the network but do not affect the quality of service.

Note 4: The election algorithm tolerates node failures as well. The only situation a failing node can cause problem is when the node dies right after transmitting its STD message. In this case nodes with higher drowsiness factor may incorrectly rely on the presence of the failed node.

Note 5: The communication overhead of the algorithm is low. In each cycle every node broadcasts only at most three messages (two if it the node will go to sleep, three otherwise). In addition to this, nodes must stay awake in order to complete the election process. During this extra T e time nodes consume energy. The communication and awake-time overhead can be neglected if T is significantly longer than T e , which is true in most practical cases.

IV. RESULTS

A. Quality of Service Metrics:

The first (functional) requirement is coverage: the network must maintain k-coverage at the largest possible part of the area. The second (non-functional) requirement is the long life-time of the network. The performance of the network can be characterized with the size of the fully covered regions. A possible metric k Θ can be the k-coverage-ratio defined as

A A k k = Θ , (5)
where k A is the area of the k-covered regions and A is the total area of the target space. If the regions have approximately the same size then another similar metric k Θ′ can be defined to approximate k Θ :

N N k k = Θ′ , (6)
where N k is the number of the k-covered regions while N is the total number of regions in the target space. In a critical application the k-coverage must be maintained as long as possible and with k Θ as high as possible. If in a degrading network k Θ is not satisfactory any more it may still be important to maintain a high value for

1 - Θ k , 2 - Θ k , etc, e.

B. Simulation results

The proposed CGS algorithm and the random k-coverage algorithm [START_REF] Kumar | On k-coverage in a mostly sleeping sensor network[END_REF] were simulated in Prowler, a probabilistic sensor network simulator [START_REF] Simon | Simulation-based optimization of communication protocols for large-scale wireless sensor networks[END_REF]. The simulator parameters were set to model the Berkeley MICA motes' MAC layer [START_REF] Hill | Mica: A Wireless Platform for Deeply Embedded Networks[END_REF]. The radio propagation model includes realistic effects, e.g. fading, collisions and lost messages.

The tests were performed with a well controlled setup containing 100 nodes placed uniformly on a grid, as can be seen in Fig. 3, showing Prowler's main display. The distance of adjacent nodes on the grid was 10 m, the sensing radius was 15 m, and the communication radius was approximately 40 m (see the parameter settings in Fig. 4). In the simulation the initial energy of all sensors was set to 20 units and in each period awake sensors consumed 1 unit of energy. The period T was set to 1 hour and the required coverage was 3.

In Fig. 5 the performance of the random k-coverage and the CGS algorithms can be compared. The plots show kcoverage ratios k Θ for k = 1,2,3, as a function of time. In the experiments the random algorithm's sleeping probability was set to p sleep = 0.4 and p sleep = 0.25. In the first case the lifetime of the random and CGS networks were approximately the same, but while the CGS algorithm managed to provide the required coverage, the random algorithm's 3-coverage ratio was only around 90%. During the service degradation phase (after time instant 26) the CGS still provided much better coverage. When p sleep was set to 0.25 the random algorithm improved its performance (approx. 95%) but the degradation of the network was much more abrupt: at time instant 34 all sensors were completely drained ( 0

1 2 3 = Θ = Θ = Θ
), while at this instant for the proposed algorithm % The number of awake sensors is shown, as a function of time, in Fig. 6. With p sleep = 0.25 much more sensors were awake in the random network than in the CGS network. To provide similar energy savings similar to the CGS algorithm, the p sleep = 0.4 setting was appropriate for the random algorithm. With this setting, however, the coverage properties were much worse, according to Fig. 5. CGS, however, provides constant good quality service and long network lifetime at the same time. Algorithms were proposed that solve the k-coverage problem and can provide prolonged network lifetime. We showed that the proposed periodic rescheduling of the sleeping and awake nodes saves energy in the network and extends overall network lifetime. The controlled schedule algorithm guarantees k-coverage in the whole network whenever the topology of the network permits it.

The centralized version of the algorithm requires networkwide communication and thus it is not feasible in large networks. A distributed approximation was proposed that uses only locally available information. The Controlled Greedy Sleep Algorithm requires only a few messages to be broadcasted from every node in each period, thus the energy wasted on the communication overhead is small, compared to the gain in the total energy saving in the network, supposing the period of the scheduling is sufficiently large.

The CGS algorithm is robust and fault tolerant: the algorithm provides the required coverage network-wide (if possible) independently of node failures or even high amount of lost messages. The algorithm was compared to the random k-coverage algorithm and was proved to be superior in two senses: while it is possible, the CGS algorithm guarantees the required coverage all over the network. Also, the degradation curve is much gentler, thus the network service is provided for a longer time.

Fig. 1 .

 1 Fig. 1. An example of the target area covered by four sensors (s1, …, s4). The sensing disks and the sensing regions (R1, …, R11) are also show, along with the corresponding bipartite graph

Fig. 2 .

 2 Fig. 2. Possible approximation of the sensing disk of a sensor s by a set of regions Rs

  g. in order to provide full connectivity in the remaining network

Fig. 3 .

 3 Fig.3. Prowler simulating the 10x10-grid network. Node 5 and 87 (big dots) are transmitting AM messages. Small LEDs in boxes show nodes that have already transmitted AM and will stay awake during the next period. The numbers indicate the actual energy reserve of the nodes.

Fig. 4 .

 4 Fig.4. Simulation parameters in Prowler. The plot illustrates a possible (random) signal power vs. distance function

Fig. 5 .

 5 Fig. 5. Degradation of QoS characteristics of the CGS (a) and the random algorithm with psleep = 0.4 (b) and psleep = 0.25 (c), for the sensor network shown in Fig. 3.

Fig. 6 .

 6 Fig. 6. The number of awake sensors as a function of time for the random and CGS algorithms

ACKNOWLEDGMENT

This research was partially supported by "Dependable, intelligent networks" (F-30/05) project of the Hungarian-French Intergovernmental S&T Cooperation Program and by the Hungarian Government under contract NKFP2-00018/2005.