Incompressible, inviscid limit of the
compressible Navier-Stokes system.
Nader MASMOUDI
Ceremade - URA CNRS 749
Universit´e Paris IX Dauphine
Place du Mar´echal de Lattre de Tassigny
75775 PARIS Cedex 16 - France
email:[email protected]
January 17, 2000
Abstract. We prove some asymptotic results concerning global (weak) so-lutions of compressible isentropic Navier-Stokes equations. More precisely, we establish the convergence towards solutions of incompressible Euler equa-tions, as the density becomes constant, the Mach number goes to 0 and the Reynolds number goes to infinity.
Limite incompressible et non visqueuse du syst`eme de Navier-Stokes com-pressible
R´esum´eNous prouvons quelques r´esultats asymptotiques concernant des so-lutions (faibles) globales des ´equations de Navier-Stokes (isentropique) com-pressible. Plus pr´ecis´ement, nous ´etablissons la convergence vers une solution des ´equations d’Euler incompressible, lorsque la densit´e devient constante, le nombre de Mach tend vers 0 et le nombre de Reynolds tend vers l’infini.
1
Introduction
From a physical point of view, one can formally derive incompressible models from compressible ones, when the Mach number goes to zero and the density
becomes almost constant. In P.L.Lions and the author [12], this problem is investigated starting form the global solutions of the compressible Navier-Stokes equations constructed by P.L.Lions ([11], see also [15] ). We have shown the convergence towards the incompressible Navier-Stokes equations as well as the convergence towards the incompressible Euler equations (if the viscosity coefficients go to zero and if the initial data are “ well prepared”). These results have been precised and extended in different works (see [13], [2], [1], [14]).
In this paper, we extend the result shown in [12] concerning the conver-gence to the Euler system to the case of more general initial data. In fact if the viscosity goes to zero too, we loose spatial compactness properties. In order to circumvent this difficulty, we use energy arguments. Hence, we have to describe (precisely) the oscillations that take place and include them in the energy estimates. Ideas of this type were introduced by Schochet [18], (see also [4]) and extended to the case of vanishing viscosity coefficients in [17]. Let us now precise the scalings, we are going to use, which are the same as those used in [12]. The unknowns (˜ρ, v) are respectively the density and the velocity of the fluid (gaz). We scale ˜ρ and v (and thus p) in the following way
˜
ρ = ρ(x, ǫt), v = ǫu(x, ǫt) (1) and we assume that the viscosity coefficients ˜µ, ˜ξ ( ˜ξ − ˜µ, ˜µ are called the volumic and dynamical viscosity coefficients) are also small and scale like
˜
µ = ǫµǫ, ˜ξ = ǫξǫ (2)
where ǫ ∈ (0, 1) is a “small parameter ” and the normalized coefficient µǫ, ξǫ
satisfy µǫ> 0, ξǫ+ µǫ > 0. Moreover, we assume that
µǫ→ 0 as ǫ goes to 0+ . (3)
With the preceding scalings, the compressible Navier-Stokes system reads ∂ρ ∂t + div(ρu) = 0 , ρ ≥ 0, ∂ρu
∂t + div(ρu ⊗ u) − µǫ∆u − ξǫ∇divu + a ǫ2∇ρ
γ = 0 .
(4)
One can always assume that a = 1/γ by replacing ε by √aγε. All Throughout this paper the domain Ω will be the the whole space RN or the torus TN
(in this last case, all the functions are defined on RN and assumed to be
periodic with period 2πai in the i-th variable). We recall here that the
inviscid limit, namely the convergence of the Navier-Stokes equations to Euler equations in the case of a domain with boundary is an open problem even in the incompressible case, which seems to be easier (see [16] for a partial result).
1.1
Statement of the results
1.1.1 The whole space case
Let us begin with the case of the whole space. We consider a sequence of global weak solutions (ρǫ, uǫ) of the compressible Navier-Stokes equations (4)
and we assume that ρǫ− 1 ∈ L∞(0, ∞; Lγ2) ∩ C([0, ∞), L p
2) for all 1 ≤ p < γ,
where Lp2 = {f ∈ L1loc, |f|1|f |≥1 ∈ Lp, |f|1|f |≤1 ∈ L2}, uǫ ∈ L2(0, T ; H1) for
all T ∈ (0, ∞) (with a norm which can explode when ε goes to 0), ρǫ|uǫ|2 ∈
L∞(0, ∞; L1) and ρ
ǫuǫ ∈ C([0, ∞) ; L
2γ/(γ+1)
− w) i.e. is continuous with respect to t ≥ 0 with values in L2γ/(γ+1)endowed with its weak topology. We require (4) to hold in the sense of distributions and we impose the following conditions at infinity
ρǫ → 1 as |x| → +∞ , uǫ→ 0 as |x| → +∞ . (5)
Finally, we prescribe initial conditions ρǫ t=0= ρ 0 ǫ , ρǫuǫ t=0= m 0 ǫ (6)
where ρ0ǫ ≥ 0, ρ0ǫ − 1 ∈ Lγ, mǫ0 ∈ L2γ/(γ+1), m0ǫ = 0 a.e. on {ρ0ǫ = 0} and
ρ0 ǫ|u0ǫ| 2 ∈ L1, denoting by u0 ǫ = m0 ǫ ρ0 ǫ on {ρ 0 ǫ > 0}, u0ǫ = 0 on {ρ0ǫ = 0}. The
initial conditions also satisfy the following uniform bounds Z
ρ0ǫ|u0ǫ|2 + 1 ǫ2
Z
(ρ0ǫ)γ− 1 − γ(ρ0ǫ − 1) ≤ C , (7) where, here and below, C denotes various positive constants independent of ǫ. Let us notice that (7) implies in particular that, roughly speaking, ρ0
ǫ is of
Notice that if γ < 2, we cannot deduce any bound for ϕε in L∞(0, T ; L2).
This is why we prefer to work with the following approximation
Φε= 1 ε r 2a γ − 1(ρ γ ǫ − 1 − γ(ρǫ− 1)).
Furthermore, we assume that pρ0
ǫ u0ǫ converges strongly in L2 to some eu0.
Then, we denote by u0 = P ˜u0, where P is the projection on divergence-free
vector fields, we also define Q (the projection on gradient vector fields), hence ˜
u0 = P ˜u0+ Q˜u0. Moreover, we assume that Φ0ε converges strongly in L2 to
some ϕ0. This also implies that ϕ0
ε converges to ϕ0 in L γ 2.
Our last requirement on (ρǫ, uǫ) concerns the total energy : we assume
that we have Eǫ(t) + Z t 0 Dǫ(s)ds ≤ Eǫ0 a.e. t, dEǫ dt + Dǫ ≤ 0 in D ′ (0, ∞) (8) where Eǫ(t) = Z Ω 1 2ρǫ|uǫ| 2(t) + a ǫ2(γ − 1)((ρǫ) γ − 1 − γ(ρε− 1))(t), Dǫ(t) = Z Ω
µǫ|Duǫ|2(t) + ξǫ (divuǫ)2(t) and Eǫ0 =
Z Ω 1 2ρ 0 ǫ|u0ǫ|2+ a ǫ2(γ − 1)((ρ 0 ǫ)γ− 1 − γ(ρ0ǫ − 1)).
We now wish to emphasize the fact that we assume the existence of a sequence of solutions with the above properties, and we shall also assume that γ > N
2. We recall the results of P.L.Lions [11] which yield the existence of such solutions precisely when γ > N
2 if N ≥ 4, γ ≥
9
5 if N = 3 and γ ≥ 3 2
if N = 2. We also refer to [12] for the proof of the uniform bounds.
When ε goes to zero and µε goes to 0, we expect that uε converges to v,
the solution of the Euler system ∂tv + div (v ⊗ v) + ∇π = 0 div v = 0 v|t=0 = u0 (9)
Theorem 1.1 : (The whole space case) We assume that µǫ→ǫ 0 (such that
µǫ+ξǫ> 0 for all ǫ) and that P ˜u0 ∈ Hs for some s > N/2+1, then P (√ρεuǫ)
converges to v in L∞(0, T ; L2) for all T < T∗, where v is the unique solution
of the Euler system in L∞
loc([0, T∗); Hs) and T∗ is the existence time of (9).
In addition √ρεuǫ converges to v in Lp(0, T ; L2loc) for all 1 ≤ p < +∞ and
all T < T∗.
1.1.2 The periodic case
Now, we take Ω = TN and consider a sequence of solutions (ρ
ε, uε) of (4),
sat-isfying the same conditions as in the whole space case (the functions are now periodic in space and all the integration are performed over TN). Of course,
the conditions at infinity are removed and the spaces Lp2 can be replaced by Lp. Here, we have to impose more conditions on the oscillating part (acoustic
waves), namely we have to assume that Q˜u0 is more regular than L2. In fact,
in the periodic case, we do not have a dispersion phenomenon as in the case of the whole space and the acoustic waves will not go to infinity, but they are going to interact with each other. This is way, we have to include them in the energy estimates to show our convergence result.
For the next theorem, we assume that Q˜u0, ϕ0 ∈ Hs−1 and that there
exists a nonnegative constant ν such that µǫ + ξǫ ≥ 2ν > 0 for all ε. For
simplicity, we assume that µǫ+ ξǫ converges to 2ν.
Theorem 1.2 : (The periodic case) We assume that µǫ →ǫ 0 (such that
µǫ+ ξǫ→ 2ν > 0) and that P ˜u0 ∈ Hs for some s > N/2 + 1, and Q˜u0, ϕ0 ∈
Hs−1 then P (√ρ
εuǫ) converges to v in L∞(0, T ; L2) for all T < T∗, where
v is the unique solution of the Euler system in L∞
loc(0, T∗; Hs) and T∗ is the
existence time of (9). In addition √ρεuǫconverges weakly to v in L∞(0, T ; L2)
In the above theorem, one can remove the condition 2ν > 0. In that case, we still have the result of theorem 1.2 but only on an interval of time (0, T∗∗)
which is the existence interval for the equation governing the oscillating part, which will be given later on.
Theorem 1.3 : (The periodic case, ν = 0) We assume that µǫ →ǫ 0 (such
that µǫ+ ξǫ→ 0) and that ˜u0∈ Hs for some s > N/2 + 1, and ϕ0 ∈ Hs then
the unique solution of the Euler system in L∞
loc(0, T∗; Hs), T∗ is the existence
time of (9) and T∗∗ the existence time of (32), with ν = 0. In addition √ρ εuǫ
converges weakly to v in L∞(0, T ; L2)
2
The whole space case
We recall that in the case of the whole space, we do not assume any extra condition on the viscosity ξε, neither do we assume any regularity (more than
L2) for the gradient part of the initial data. The proof relies on the dispersion
property of the wave equation ([19], [1]) and the notion of non dissipative solutions for the Euler system ([10], [12]).
First, using the energy bounds, we deduce that ρε− 1 converges to 0 in
L∞(0, T ; Lγ
2) and that there exists some u ∈ L∞(0, T ; L2) and a subsequence
√ρ
εuǫ converging weakly to u. Hence, we also deduce that ρεuǫ converges
weakly to u in L2γ/(γ+1). We next introduce the following group (L(τ), τ ∈ IR) defined by eτ L where L is the operator defined on D′
× (D′)N, by L ϕ v = − divv ∇ϕ . (10)
It is easy to check that eτ L is an isometry on each Hs× (Hs)N
for all s ∈ IR and for all τ . This will allow eτ L
ϕ v = ϕ(τ ) v(τ ) solves ∂ϕ ∂τ = −divv , ∂v ∂τ = −∇ϕ and thus ∂ 2ϕ ∂τ2 − ∆ϕ = 0.
Let (ψε, mε = ∇qε) be the solution of the following system
∂ψε ∂t = − 1 εdivmε ψε(t = 0) = Φ 0 ε ∂mε ∂t = − 1 ε∇ψε mε(t = 0) = Q p ρ0 ǫu0ε (11)
We recall that for all v ∈ Hs, we define Qv = ∇∆−1div v and P v = v − Qv.
∇qε∗ χδ, where χ ∈ C0∞(RN) such that
R
χ = 1 and χδ(x) = δ1Nχ(xδ). Since
(11) is linear it is easy to see that (ψε,δ, ∇qε,δ) is a solution of the same system
with regularized initial data. Using (as in [1]) Strichartz type inequality, we get ψε,δ ∇qε,δ Lp(IR;Ws,q(IRN))) ≤ Cε 1/p Φ0 ε Q(pρ0 ǫu0ǫ) ∗ χδ Hs+σ (12)
for all p, q > 2 and σ > 0 such that 2 q = (N − 1)( 1 2 − 1 p), σ = N + 1 N − 1.
This yields that for all fixed δ and for all s ∈ IR, we have as ε goes to 0 (ψε,δ, ∇qε,δ) → 0 in Lp(IR; Ws,q(IRN))). (13)
Now, we turn to the energy estimates. Let us rewrite the energy inequality for almost all t
1 2 Z Ω ρε|uε|2(t) + Φ2ε(t) + Z t 0 Z Ω µε|Duε|2+ ξε(div uε)2 ≤ 1 2 Z Ω ρ0ε|u0ε|2+ Φ2ε(0). (14) Then, the conservation of energy for v reads
Z Ω 1 2|v| 2(t) = Z Ω 1 2|u0| 2 (15)
and using the fact that L is an isometry on L2, we obtain for all t
Z Ω 1 2ψ 2 ε,δ(t) + 1 2|∇qε,δ| 2(t) = Z Ω 1 2ψ 2 ε,δ(0) + 1 2|∇qε,δ| 2(0) (16)
Next, the weak formulation of the conservation of mass yields for almost all t Z Ω ψε,δϕε(t) + 1 ε Z t 0 Z Ω div(ρǫuǫ)ψε,δ+ div(∇qε,δ)ϕǫ = Z Ω ψε,δϕε(0) (17)
while the weak formulation of (4) implies that we have for almost all t Z Ω ρεuε.v(t) + Z t 0 Z Ω ρεuε.(v.∇v + ∇p)+ − Z t 0 Z Ω ρεuεuε.∇v + µε Z t 0 Z Ω∇u ε.∇v = Z Ω ρ0εu0ε.u0. (18)
Z Ω ρεuε.∇qε,δ(t) + Z t 0 Z Ω ρεuε.( 1 ε∇ψε,δ) − Z t 0 Z Ω ρεuεuε.∇mε,δ+ + Z t 0 Z Ω
µε∇uε.∇mε,δ+ ξεdiv(uε)div(mε,δ)
− Z t 0 Z Ω (1 εϕε+ γ − 1 2 Φ 2 ε)div(∇qε,δ) = Z Ω ρ0εu0ε.m0ε,δ. (19)
Summing up (14), (15), (16) and subtracting (17), (18), (19), we deduce from straightforward computations the following inequality
1 2 Z Ω| √ ρεuε− v − mε,δ|2(t) + (Φε− ψε,δ)2(t) + Z t 0 Z Ω µǫ|∇uε|2+ ξǫ|div(uε)|2 ≤ − Z Ω (Φε− ϕε)ψε,δ(t) + Z Ω (Φε− ϕε)ψε,δ(0) + Z Ω (√ρε− 1)√ρεuε.(v + mε,δ)(t) − ( p ρ0 ε− 1) p ρ0 εu0ε.(v0+ m0ε,δ) − Z t 0 Z Ω µǫ|∇uε|2+ ξǫ|div(uε)|2 + Z t 0 Z Ω
µǫ∇uε.∇(v + mε,δ) + ξǫdiv(uε)div(mε,δ)
+ Z t 0 Z Ω ρεuε.(v.∇v + ∇p) − ρεuε(uε.∇v) (∗) − Z t 0 Z Ω ρεuε(uε.∇mε,δ) − (γ − 1 2 Φ 2 ε)div(∇qε,δ) +1 2 Z Ω| p ρ0 εu0ε− v0− m0ε,δ|2+ (Φε− ψε,δ)2(0) (20)
Only the term marked by (∗) in the second hand side of (20) needs some special treatment, we are going to compute it below. In the sequel, we denote by wε,δ = √ρεuε− v − mε,δ. Then, we have easily
Z t 0 Z Ω ρεuε.(v.∇v + ∇p) − ρεuε(uε.∇v) = − Z t 0 Z Ω wε,δ.∇vwε,δ+ Z t 0 Z Ω (ρε−√ρε)uε.(v.∇v) + ρεuε.∇p − (√ρεuε− v).∇ v2 2 − Z t 0 Z Ω mε,δ.∇vwε,δ+ (√ρεuε− v).∇v.mε,δ (21)
Finally, we can see that (20) may be rewritten as ||wε,δ(t)||2L2+ ||Φε− ψε,δ(t)||2L2 ≤ ||wε,δ(0)||2L2 + ||Φε− ψε,δ(0)||2L2+ +2Aδ ε+ 2 Z t 0 ||w ε,δ(s)||2L2||∇v(s)||L∞ ds (22) where Aδ ε = − Z Ω (Φε− ϕε)ψε,δ(t) + Z Ω (Φε− ϕε)ψε,δ(0) + Z Ω (√ρε− 1)√ρεuε.(v + mε,δ)(t) − ( p ρ0 ε− 1) p ρ0 εu0ε.(v0+ m0ε,δ) + Z t 0 Z Ω
µǫ∇uε.∇(v + mε,δ) + ξǫdiv(uε)div(mε,δ)
− Z t 0 Z Ω ρεuε(uε.∇mε,δ) − (γ − 1 2 Φ 2 ε)div(∇qε,δ) + Z t 0 Z Ω (ρε−√ρε)uε.(v.∇v) + ρεuε.∇p − (√ρεuε− v).∇ v2 2 − Z t 0 Z Ω mε,δ.∇vwε,δ+ (√ρεuε− v).∇v.mε,δ (23)
For all fixed δ, it is easy to see that Aδ
ε(t) converges to 0 for almost all t,
uniformly in t when ε goes to 0. Then, by Gr¨onwall’s inequality, we deduce that we have for almost all t ∈ (0, T )
||wε,δ(t)||2L2+ ||Φε− ψε,δ(t)||2L2 ≤ ≤h||wε,δ(0)||2L2+ ||Φε− ψε,δ(0)||2L2 + sup 0≤s≤t Aδε(s)ie C Z t 0 ||∇v(s)|| 2 L∞ (24) Then, letting ε go to 0, we obtain
ku − vk2L∞(0,T ;L2(Ω)) ≤ lim ε kwε,δk2L∞(0,T ;L2(Ω))+ ||Φε− ψε,δ||2L∞(0,T ;L2(Ω)) ≤ C0 k˜u0− u0− Q˜u0∗ χδkL2(Ω)+ kϕ0− ϕ0∗ χδkL2(Ω) where C0 = e C Z T 0 ||∇v(s)|| 2 L∞ < +∞.
Then, letting δ go to 0, we deduce that u = v and we obtain also the uniform convergence in t of P (√ρεuε) to v in L2, since
lim ε kP ( √ρ εuε)−vkL∞(0,T ;L2) ≤ C0lim δ h
k˜u0−u0−Q˜u0∗χδkL2(Ω)+kϕ0−ϕ0∗χδkL2(Ω)
i = 0 Moreover, we see that √ρεuε− mε converges uniformly in t, strongly in L2
to v. In fact k√ρεuε− mε− vkL2 ≤ k√ρεuε− mε,δ − vkL2 + kmε− mε,δkL2 + kΦε− ψε,δkL2 and since (Φ0 ε, m0ε = Q p ρ0
εu0ε) converges strongly to (ϕ0, m0) and since L is
an isometry in L2, we deduce that we have
kmε− mε,δkL2 + kΦε− ψε,δkL2 → 0 when δ → 0
uniformly in t and ε.
Finally, we can also deduce the local strong convergence of √ρεuε to v in
Lp(0, T ; L2(B)). Indeed let us denote by B a bounded domain of IRN. Then,
we have for all t
k√ρεuε− vkL2(B) ≤ k√ρεuε− mε,δ− vkL2(B)+ kmε,δkL2(B)
for any q > 2. Then, using the fact that for all δ, kmε,δkLp(0,T ;Lq(B))converges
to 0 as ε goes to 0, we conclude easily by taking the limit in ǫ and then in δ as above.
3
The periodic case
As in the case of the whole space, we can deduce, using the energy bounds, that ρε− 1 converges strongly to 0 in L∞(0, T ; Lγ). Then, using the bound
on √ρεuǫ in L∞(0, T ; L2), we may extract a subsequence which converges
weakly to some u. To pass to the limit in the equation, we need to describe the oscillations in time and show that they will not affect the limit equation.
We next introduce, as in the whole space case, the following group (L(τ), τ ∈ IR) defined by eτ L where L is the operator defined on D′
0 × (D′)N, where D′ 0 = {ϕ ∈ D′, Z ϕ = 0}, by L ϕ v = − divv ∇ϕ . (25)
In the sequel, we will use the following notations
Uε = (ϕε, Q(ρεuε)) and Vε= L(−t/ε)(ϕε, Q(ρεuε))
and for some technical reasons related to the L2 integrability, we will also
use the following approximations ¯ Uε= (Φ ε, Q(√ρεuε)) and V¯ε = L(−t/ε)(Φε, Q(√ρεuε)), which satisfy kUε− ¯Uεk L∞(Lγ+12γ ) → 0 when ε → 0
Let us project the equation (4) on “gradient vector-fields” ∂ ∂tQ(ρǫuǫ) + Q h div(ρǫuǫ⊗ uǫ) i − (µǫ+ ξǫ)∇divuǫ + + a ǫ2∇ ργǫ − γρǫ+ (γ − 1) + 1 ǫ2∇(ρǫ− 1) = 0. (26)
Combining (26) with the conservation of mass, we obtain ǫ ∂ϕǫ ∂t + div Q(ρǫuǫ) = 0, ǫ ∂ ∂tQ(ρǫuǫ) + ∇ϕǫ = ǫFǫ (27) where Fǫ = (µǫ+ξǫ)∇ div uǫ−Q h div (ρǫuǫ⊗uǫ) i −a∇h 1ǫ2(ρ γ ǫ−γρǫ+(γ −1) i . Equation (27) yields that ∂tUε=
1 εLU
ε+ (0, F
ε), which can be rewritten
as ∂tVε = L(−t/ε)(0, Fε). It is easy to check that Fε is bounded in L2(H−s)
(for some s ∈ R), hence Vε is compact in time (the oscillations have been
cancelled). If we had enough compactness in space we could pass to the limit in this equation and recover the following limit system for the oscillating part ∂tV + Q¯ 1(u, ¯V ) + Q2( ¯V , ¯V ) − ν∆ ¯V = 0, (28)
where Q1 and Q2 are respectively a linear and a bilinear forms in ¯V (which
will be defined and computed later on) and the term −ν∆ ¯V explained below. In fact, as in [18] (see also [17]), we consider L(−t/ε)(0, Fε) as an almost
periodic function in τ = t/ε and compute its mean value, which yields (28). Definition 3.1 For all divergence-free vector field u ∈ L2(Ω)N and all V =
(ψ, ∇q) ∈ L2(Ω)N +1, we define the following linear and bilinear symmetric
forms in V Q1(u, V ) = lim τ →∞ 1 τ Z τ 0 L(−s) 0 div u ⊗ L2(s)V + L2(s)V ⊗ u ds (29) and Q2(V, V ) = lim τ →∞ 1 τ Z τ 0 L(−s) 0 div L2(s)V ⊗ L2(s)V + γ−12 ∇(L1(s)V )2 ds. (30) The convergences stated above take place in W−1,1 and can be shown by
using almost-periodic functions (see [17] and the references therein). Indeed the functions inside the integral in (29) and (30) are almost periodic in W−1,1
and Q1(u, V ), Q2(V, V ) are their mean value. We will come back to this issue
in the next section. We also remark that in the equation (28) the viscosity term was divided by 2 and that it applies for both component of the vector
¯
V . This is due to the following proposition, which can be proved easily using almost periodic functions (see also [5] and [3])
Proposition 3.2 Under the same hypothesis on V , we have −ν∆V = limτ →∞τ1 Z τ 0 −L(−s) 0 2ν∆L2(s)V ds (31)
Nevertheless, the fact that the viscosity applies now for both components of V is not sufficient to yield compactness in space for Vε. To recover
com-pactness in space, we will use the regularity of the limit system and extend the method used in [12] to the case of general initial data as was done in [17]. Let V0 be the solution of the following system
∂tV0+ Q1(v, V0) + Q2(V0, V0) − ν∆V0 = 0 V|t=00 = (ϕ0, Q˜u0) (32)
where v is, as in the case of the whole space, the solution of the incompressible Euler equations with initial data u0. The existence of global strong solutions
for the system (32) (and local solutions if the viscosity term is removed) as well as the exact computations of the two forms Q1 and Q2 will be detailed
in the next section. We only need the following two propositions
Proposition 3.3 For all u, V , V1 and V2 (regular enough to define all the
products), we have Z Q1(u, V )V = 0, and Z Q2(V, V )V = 0, (33) Z Q1(u, V1)V2+ Q1(u, V2)V1 = 0 (34) Z Q2(V1, V1)V2+ 2Q2(V1, V2)V1 = 0 (35)
The proof of (33) will be given in the next section, (34) can be shown by applying the first part of (33) to V1+ V2 and to V1− V2. Finally (35) can be
shown by applying the second part of (33) to V1+ X V2 and identifying the
term of degree 1.
The next proposition is a very simple consequence of the theory of almost-periodic functions (see for instance Lemma 2.3 of [17])
Proposition 3.4 For all u ∈ Lp(0, T ; L2) and V ∈ Lq(0, T ; L2), we have
the following weak convergences (p and q are such that the product are well defined) w − lim ε L(− t ε) 0 div u ⊗ L2(εt)V + L2(tε)V ⊗ u = Q1(u, V ) (36) and w − lim ε L(− t ε) 0 div L2(tε)V ⊗ L2(εt)V +γ−12 ∇(L1(εt)V )2 = Q2(V, V ) (37) Using the symmetry of Q2, we deduce easily the following proposition
Proposition 3.5 Equation (37) of Proposition 3.4 can be extended to the case where we take V1 and V2 using the symmetry of Q2, namely
w − limε L(−εt) 0 div L2(tε)V1⊗ L2(εt)V2+ L2(εt)V2⊗ L2(tε)V1 (38) +L(−εt) 0 γ−1 2 ∇(L1( t ε)V1 L1( t ε)V2) = Q2(V1, V2)
Moreover, the above identity holds for V1 ∈ Lq(0, T ; Hs) and V2 ∈ Lp(0, T ; H−s)
with s ∈ IR and 1/p + 1/q = 1. Its is also possible to extend it to the case where we replace V2 in the left hand side by a sequence V2ε such that V2ε
converges strongly to V2 in Lp(0, T ; H−s).
In order to show the convergence in theorem 1.2, we will try to estimate ||√ρεuε− v − L2( t ε)V 0 ||2L2 + ||Φε− L1( t ε)V 0 ||2L2.
We also introduce the following notations wε = √ρεuε− v − L2(εt)V0 and
βε = Φε − L1(εt)V0. In the sequel, we also note (ψε, mε) = L(εt)V0. The
proof follows the same lines as the proof in the whole space case apart from the fact that the equation satisfied by the gradient part is not trivial and
that we have to use the precise equation satisfied by the oscillating terms. We recall the energy inequality
1 2 Z Ω ρε|uε|2(t) + Φ2ε(t) + Z t 0 Z Ω µε|Duε|2+ ξε(div uε)2 ≤ 1 2 Z Ω ρ0ε|u0ε|2+ Φ2ε(0). (39) as well as the conservation of energy for v
Z Ω 1 2|v| 2(t) = Z Ω 1 2|u0| 2 (40) Using that Z Q1(u, V0)V0 = 0, Z Q2(V0, V0)V0 = 0,
we deduce from (32) the following energy identity Z Ω 1 2|V 0 |2(t) + ν Z t 0 Z Ω|∇V 0 |2(s)ds = Z Ω 1 2|V 0 (t = 0)|2 (41) Next, using the weak formulation of the conservation of mass, we obtain for almost all t
Z Ω ψεϕε(t) + 1 ε Z t 0 Z Ω div(ρǫuǫ)ψε+ div(∇qε)ϕǫ+ − Z t 0 Z ΩL 1( t ε)∂tV 0ϕ ε= Z Ω ϕ0ϕε(0) (42)
while the weak formulation of (4), yields as before the following inequality for almost all t
Z Ω ρεuε.v(t) + Z t 0 Z Ω ρεuε.(v.∇v + ∇p)+ − Z t 0 Z Ω ρεuεuε.∇v + µε Z t 0 Z Ω∇u ε.∇v = Z Ω ρ0εu0ε.u0. (43)
Z Ω ρεuε.∇qε(t) + Z t 0 Z Ω ρεuε.( 1 ε∇ψε) − Z t 0 Z Ω ρεuεuε.∇mε+ − Z t 0 Z ΩL 2( t ε)∂tV 0ρ εuε+ Z t 0 Z Ω
µε∇uε.∇mε+ ξεdiv(uε)div(mε)
− Z t 0 Z Ω (1 εϕε+ γ − 1 2 Φ 2 ε)div(∇qε) = Z Ω ρ0εu0ε.Q˜u0. (44)
Next adding up (39),(40),(41) and subtracting (42),(43),(44), we obtain 1 2 Z Ω|w ε|2(t) + (βε)2(t) + Z t 0 Z Ω µε|Dwε|2+ ξε(div wε)2 ≤ − Z Ω (Φε− ϕε)ψε(t) + Z Ω (Φε− ϕε)ϕ0 + Z Ω (√ρε− 1)√ρεuε.(v + mε)(t) − ( p ρ0 ε− 1) p ρ0 εu0ε.˜u0 +1 2 Z Ω| p ρ0 εu0ε − ˜u0|2+ (Φε− ϕ0)2(0) + Aε+ Bε (45)
where Aε and Bε are given by
Aε = Z t 0 Z Ω−ν|∇V 0 |2+ µε|Dmε|2+ ξε(div mε)2+ µε|Dv|2 + Z t 0 Z Ω−µ
ε∇uε.∇(v + mε) − ξε(divuε)(divmε) − ν∆V0Vε
(46) and Bε = Z t 0 Z Ω ρεuε.(v.∇v + ∇p) − ρεuεuε.∇(v + mε) + Z t 0 Z Ω− γ − 1 2 Φ 2 εdiv(mε) + Q1(v, V0)Vε+ Q2(V0, V0)Vε. (47)
Here, we have used that Z t 0 Z ΩL( t ε)∂tV 0Uε = Z t 0 Z Ω ∂tV0Vε (48)
It is easy to see that to apply a Gr¨onwall’s lemma, one has to estimate Aε and Bε, since one can show (as in the case of the whole space) that the
other terms in the right hand side of (45) converge to 0 uniformly in t. In the following two subsections, we will show that
Bε(t) ≤
||wε(t)||2L2 + ||βε(t)||2L2
||∇(v + mε)||L∞+ rε
and that Aε, rε converge to 0 uniformly in t. Therefore, we conclude as in
the case of the whole space, using that ∇(v + mε) is bounded in L1(0, T ; L∞)
uniformly in ε.
3.1
Bounds on
B
εWe recall here that we assume (extracting subsequences if necessary) that √ρ
εuε and uε converges weakly to some u and that Vε = L(−t/ε)Uε as
well as ¯Vε = L(−t/ε) ¯Uε converge (strongly in time) to (0, u) + ¯V . In the
sequel, rε will denote any sequence of functions converging uniformly in t to
0. Rewriting (47), we get Bε = Z t 0 Z Ω−w εwε.∇(v + mε) − γ − 1 2 β 2 εdiv(mε) + Z t 0 Z Ω √ρ εuε.(v.∇v) −√ρεuε.(v + mε).∇(v + mε) + Z t 0 Z Ω−v.( √ρ εuε− v − mε).∇(v + mε) − mε.(√ρεuε− v − mε).∇(v + mε) + Z t 0 Z Ω γ − 1 2 ψ 2 εdiv(mε) − 2γ − 1 2 ψεΦεdiv(mε) + Z t 0 Z ΩQ 1(v, V0)Vε+ Q2(V0, V0)Vε + rε. (49)
Let us compute the limit when ε goes to 0 of the terms appearing in the right hand side. On the one hand, we have
Z t 0 Z Ω mε.mε.∇(v + mε) + γ − 1 2 ψ 2 εdiv(mε) = = − Z t 0 Z Ω h div(mε⊗ mε) + γ − 1 2 ∇(ψ 2 ε) i .(mε+ v) = − Z t 0 Z ΩL(− t ε) 0 div mε⊗ mε +γ−12 ∇(ψε)2 . V0+ 0 v = − Z t 0 Z ΩQ 2(V0, V0). V0+ (0, v) +rε = rε.
On the other hand, using Proposition 3.5, we have − Z t 0 Z Ω (mε⊗√ρεuε+√ρεuε⊗ mε) : ∇(v + mε) + (γ − 1)ψεΦεdiv(mε) = = Z t 0 Z Ω h div(mε⊗√ρεuε+√ρεuε⊗ mε) + (γ − 1)∇(ψεΦε) i .(mε+ v) = Z t 0 Z ΩL(− t ε) 0 div mε⊗√ρεuε+ √ρεuε⊗ mε + (γ − 1)∇(ψεΦε) . V0+ 0 v = Z t 0 Z Ω2Q 2(V0, ¯V ). V0+ (0, v)+Q1(u, V0). V0+ (0, v) +rε = 2 Z t 0 Z ΩQ 2(V0, ¯V ).V0 + rε.
Moreover, it is easy to see that Z t 0 Z Ω h div(v ⊗√ρεuε+√ρεuε⊗ v) i .mε = = Z t 0 Z ΩQ 1(v, ¯V ).V0 + rε. and that Z t 0 Z Ω h div(v ⊗ mε+ mε⊗ v) i .mε = Z t 0 Z ΩQ 1(v, V0).V0 + rε = rε.
Finally, we also have Z t 0 Z ΩQ 1(v, V0)Vε+ Q2(V0, V0)Vε = Z t 0 Z ΩQ 1(v, V0) ¯V + Q2(V0, V0) ¯V + rε.
3.2
Bounds on
A
εWe recall that µε goes to 0 and that ξε goes to 2ν. From the energy bounds
on V0 and on v, we get Z t 0 Z Ω µε|Dmε|2+ µε|Dv|2− µε∇uε.∇(v + mε) → 0
uniformly in t. Besides, integrating by parts and using that mε = ∇qε, we
get Z t 0 Z Ω ξε(div mε)2 = − Z t 0 Z Ω ξε∆mε.mε = − Z t 0 Z ΩL(− t ε) 0 ξε∆mε .V0 = − Z t 0 Z Ω ν∆V0V0 + rε
The same argument yields − Z t 0 Z Ω ξε(div mε)(div uε) = Z t 0 Z ΩL(− t ε) 0 ξε∆uε .V0 = Z t 0 Z Ω ν∆ ¯V V0 + rε Finally, we have Z t 0 Z Ω−ν∆V 0Vε = Z t 0 Z Ω−ν∆V 0V¯ + r ε.
Adding up the different contributions and integrating by parts, we deduce that Aε = rε converges to 0 uniformly in t.
4
Study of the oscillating part
This section is devoted to the study of the equation satisfied by the gradient part of the momentum. We expect that the following computations will be used in a forthcoming
investigation of a new numerical approach to slightly compressible flows.
4.1
Computation of
Q
2We recall that Ω = TN and that TN is a periodic domain with periods
(2πa1, ..., 2πaN), where for all i, ai > 0. Then, we can decompose L2(Ω) ×
{∇q, q ∈ H1(Ω)} into the following orthogonal basis. (In the sequel L2(Ω)
denotes the space of L2 functions with zero average)
Vk+(X) = p 1 2|TN||k| |k| −sg(k)k eik.X and Vk−(X) = p 1 2|TN||k| |k| sg(k)k eik.X
where k is the vector whose components are defined by ki = k′i/ai for all
1 ≤ i ≤ N with k′
i ∈ ZZ∗and where the notation |k| is defined by |k|2 =
P
ik2i
and |TN| = (2π)NQ
iai. Moreover, sg(k) is a generalization of the function
sign, defined on IRN − {0} by sg(k) = +1 if and only if there exists i,
1 ≤ i ≤ N such that for all j < i, we have kj = 0 and ki > 0. Otherwise
sg(k) = −1. We think that the introduction of this notation yields much simpler formulae. We only point out that we have
L Vk+(X) = isg(k)|k|Vk+(X) and L Vk−(X) = −isg(k)|k|Vk−(X). Decomposing V on this basis, we have
V = X
k′∈ZZN∗
a+kVk++ a−kVk−. (50)
We want to remark here that V+ k = V
+
−k and that a+−k= a+k, since V is a real
function. The same holds with + replaced by −. Then, applying the group L, we obtain
L(s)V = X
k′∈ZZN∗
Next, we can compute easily the term inside the integral in the right hand side of (30) L(−s) X k,l α=±,β=± aαka β l i 0 (k + l).[αβsg(k) sg(l)k ⊗ l] + γ−12 (k + l)|k| |l| × ei(k+l).X 2|TN||k||l|e i(αsg(k)|k|+βsg(l)|l|)s (52) Projection on Vγ
m and on divergence-free vector fields, the function written
above can be rewritten as follows (F = P F + QF ) QF (s) = X k+l=m α,β,γ=± aαka β lχ αβγ klm V γ m exp i(αsg(k)|k| + βsg(l)|l| − γsg(m)|m|)s(53) P F (s) = X k+l=m α,β=± aαkaβlUklmαβ expi(αsg(k)|k| + βsg(l)|l|)s (54)
where χαβγklm is a constant and Uklmαβ = (0, uαβklm is a vector such that m.uαβklm= 0. Moreover, it is easy to see that F is almost-periodic in s with periods of the form αsg(k)|k| + βsg(l)|l| − γsg(k + l)|k + l| and αsg(k)|k| + βsg(l)|l|. Hence, to obtain the limit term in (30), we have to compute the mean value of (52). This is the same as looking at the resonant terms, namely terms which do not depend on s. The resonance condition between ((k, α), (l, β), (m, γ)), namely (Vα k, V β l , Vmγ) is k+ l = m αsg(k)|k| + βsg(l)|l| = γsg(m)|m| (55) Hence, 2k.l = 2αsg(k)βsg(l)|k| |l|, which means that k is parallel to l. Rewriting this product again and using that k is parallel to l, we deduce that k.l = sg(k) sg(l)|k| |l|. This yields that we have α = β and then we can see easily that (55) is equivalent to
k+ l = m, sg(k)|k| + sg(l)|l| = sg(m)|m|
α = β = γ (56)
More precisely, we can only get resonances between the triplet (Vk+, Vl+, V+ m)
and (Vk−, Vl−, V−
the notation sg(k). On the other hand, the possible contribution on the divergence-free part requires the following resonance condition αsg(k)|k| + βsg(l)|l| = 0 and hence |k| = |l|. Next, using the symmetry between k and l, we get
uαβklm+ uβαlkm = Phα(βsg(k)sg(l)((k + l).l k + (k + l).k l)ei(k+l)X + γ − 1
2 (k + l)|k| |l|e
i(k+l)Xi= 0 (57)
The above relationship has already been used by P.-L. Lions and the author in [12] to show the weak convergence of the compressible Navier-Stokes equa-tions towards the incompressible Navier-Stokes equaequa-tions. It means in some sense that the acoustic waves do not perturb the incompressible flow.
Finally, we deduce that Q2(V, V ) =
X
k+l=m,α=± sg(k)|k|+sg(l)|l|=sg(m)|m|
aαkaαlχαklmVmα (58)
where χ is symmetric in k and l. It only remains to compute χ. By projecting (30) on Vα m, we get χαklm = p(−αsg(m)m) 2|TN|3|k||l||m| · isg(k) sg(l)m.kl +γ − 1 2 i|k||l|m (59) = −iα (γ + 1)sg(m)|m| 4p2|TN| (60)
Inorder to understand more the structure of Q2, we introduce the set P
of prime vectors p, where p ∈ ZZN is such that the N components of p are
prime in their set. This is equivalent to saying that there does not exist any couple (n, q) ∈ N × ZZN such that p = nq and n ≥ 2. Then, we define
Wp(X) =
X
k∈ZZ,k=kp
asg(p)k Vksg(p)(X) (61)
We can associate to the above vector value function the following real value function wp(z) = − γ + 1 4p2|TN| X k∈ZZ,k=kp asg(p)k ei kz (62)
We notice that both Wp and wp are real. Indeed, this is a consequence of
the fact that V is real. Moreover, we remark that sg(p)sg(k)|k| = |p|k and that for all s ∈ IR, we have
2π γ + 1
4p2|TN|kWpkHs(T
N) = |p|skwpkHs(
T). (63)
The following proposition is very easy
Proposition 4.1 For all p, q ∈ P, p 6= q, we have Q2(Wp, Wq) = 0 and
the following differential equations are equivalent
∂tV + Q2(V, V ) − ν∆V = 0 (64)
∀p ∈ P, ∂twp+ |p|∂zwp2 − ν|p|2∂z2wp = 0 (65)
This shows that we have to solve an infinite collection of viscous Burgers equations. However, our initial equation is even more complicated and we will see that this collection of viscous Burgers equations is coupled by a coupling coming from the term Q1.
4.2
Computation of
Q
1We recall that if u is divergence-free it may be written as u =X
k
Ukeik.X (66)
where k.Uk= 0, for all k. Then, using the fact that u is real, we deduce that
Uk = U−k, for all k. Hence, the term inside the integral in the right hand
side of (29) is given by L(−s)X k,l α=± aαkp 1 2|TN||k|i 0 αsg(k)(k + l).[ k ⊗ Ul+ Ul⊗ k] ei(k+l).Xeiαsg(k)|k|s (67) Now, we have to look at the resonant terms as we did in the preceding section for Q2. The resonance condition between ((k, α), l, (m, γ)), namely
(Vα
k, Ul, Vmγ) is
k+ l = m
We just notice that if (Vα
k, Ul, Vmδ) is a resonant triplet than it is also true for
(Vδ
−m, Ul, V−kα ). We will see that this yields the conservation of energy. Of
course, we also get other resonances for instance for (Vα
−k, U−l, V−mδ )... Next, we get Q1(u, V ) = X k+l=m αsg(k)=δsg(m) |k|=|m| aαkσ(k, Ul)Vmδ (69) where σ(k, Ul) = (−δsg(m)m) 2|k||m| · iαsg(k) [m.k Ul+ k.Ul k] (70) = −i m.k |k||m| Ul.k (71)
Moreover, using the fact that l.Ul = 0, we get that σ(−m, Ul) = −σ(k, Ul). It
must be noticed that while Q2(V, V ) is formed by resonances between modes
oscillating in the same direction, Q1(u, V ) is formed by resonances between
modes oscillating with the same frequency. This induces some coupling in the limit equation (32) which can be seen as a an infinite collection of coupled viscous Burgers equations. However, we will see that for suitable choices of the periods of the domain this coupling is low. Next, we introduce the following set, that we call the set of trivial resonances
Ap = {q ∈ P, ∀ i, qi = ±pi} (72)
Proposition 4.2 For almost all choices of (a1, ..., aN) ∈ IR+N, Q1(u, V )
reduces to trivial resonances, namely Q1(u, V ) = X k∈Am,αsg(k)=δsg(m) aαkσα(k, Um−k)Vmδ (73) More precisely if 1 a2 1, ..., 1 a2
N are Q independent then the above conclusion holds.
The proof of this proposition is very easy. In fact, if a12 1, ...,
1 a2
N are Q
inde-pendent, then the equation
N X i=1 k′2 i a2 i = N X i=1 m′2 i a2 i (74)
has only trivial solutions, namely k′2 i = m
′2
i , which means that k ∈ Am.
Moreover in this case the resonance also holds for the prime represent of (k, α) and (m, δ), namely q and p such that sg(q) = α, k = kq and sg(p) = δ, m = mp, since k = m in this case. However, in the general case, the resonances couple all the different modes.
To be able to use the notations Wp to analyze the limit system, we define
Cp(u, Wq) which is the contribution of Wq on Wp
Cp(u, Wq) = X k+l=m k=kq,m=mp m|p|=k|q| asg(q)k σ(k, Ul)Vmsg(p) (75)
Indeed, using that k = kq, m = mp, the second equation in (68) yields sg(q)sg(k)|k| = sg(p)sg(m)|m| (76) from which we deduce the condition m|p| = k|q|. We also introduce the corresponding notation for wp
cp(u, wq) =
X
k+l=m,k=kq,m=mp,m|p|=k|q|
asg(q)k σ(k, Ul)eim z (77)
With these notations, we can see that
Proposition 4.3 Solving the system (32) for V = Pp∈PWp is equivalent
to solving the system (ICV B) of infinite coupled viscous Burgers equations (ICV B) ∀p ∈ P, ∂twp+ |p|∂zwp2 − ν|p| 2∂2 zwp+ X q∈P cp(u, wq) = 0 (78)
We also notice that, when ν = 0, the system reduces to an infinite coupled Burgers equations
(ICB) ∀p ∈ P, ∂twp+ |p|∂zw2p+
X
q∈P
cp(u, wq) = 0 (79)
The proof of the existence of solutions will rely on the energy estimates. Therefore, we begin by proving Proposition 3.3, which is the essence of the conservation of the energy.
4.3
Proof of Proposition 3.3
Let us start by proving the first part of equation (33). It is easy to see that if (Vα
k, Ul, Vmδ) is resonant then it is also true for (V−mδ , Ul, V−kα ). Moreover
using that V is real, we deduce that a+−k = a+k and hence the following computations are obvious
Z Q1(u, V )V = X k+l=m,αsg(k)=δsg(m) |k|=|m| aαkσ(k, Ul)aαm = 1 2 X k+l=m,αsg(k)=δsg(m) |k|=|m| −i m.k |k||m| Ul.kaαka α −m− Ul.maαka α −m = 0
For the second part of equation (33), we just remark that if (Vα
k, Vlα, Vmα)
is a resonant triplet, it is also true for (Vα
k, V−mα , V−lα) and (V−mα , Vlα, V−kα ). Hence Z Q2(V, V )V = X k+l=m,α=± aαkaαlχαklmaα−m = 1 3 X k+l=m,α=± −iα (γ + 1) 4p2|TN| sg(m)|m|aαka α la α −m+ +sg(−l)|l|aαka α −maαl + sg(−k)|k|a α la α −maαk = 0.
4.4
Existence of global solutions for the coupled
sys-tem
(ICV B)
In this subsection, we give a sketch of proof of existence and uniqueness of solutions to (ICV B) and hence for (32). Let u ∈ L∞(Hs), with s ≥ N
2 + 1
be a given function (the regularity of u can be weakened but u ∈ L∞(Hs) is
the regularity we get from the fact that u is a solution of the Euler system) and V0 = PpWp0 ∈ Hs−1, we also define wp0 as above. Then, the following
Theorem 4.4 There exists a unique global strong solution for (ICV B), with ∀p ∈ P, wp ∈ L∞(Hs−1(T)) ∩ L2(Hs(T)) (80)
And hence, we obtain a unique strong solution for (32), with V =PpWp ∈
L∞(Hs−1(TN)) ∩ L2(Hs(TN))
We are just going to give the a priori bounds we can derive for this system. The existence result is then easily deduced by solving some approximated systems. For instance, we can solve (ICV BM) for all M ∈ N, defined by
PM = {p ∈ P, |p| ≤ M} and (ICV BM) ∀p ∈ PM, ∂twp+ |p|∂zw2p− ν|p|2∂z2wp+ X q∈PM cp(u, wq) = 0 (81) Of course all the a priori estimates we are going to show for (ICV B) can be easily extended to (ICV BM). Then, we have just to take the limit M → ∞
and use a compactness method to pass to the limit. To solve (ICV BM) for
fixed M, we can use any classical type of regularization. For instance, one can use a Galerkin approximation method.
Now, we turn to the proof of the a priori bounds. If Proposition 3.3 is sufficient to get L2 energy estimates, we need the following Proposition to
get higher order estimates.
Proposition 4.5 For all p, q, ∈ P and all j ∈ N, we have Z T ∂zjcp(u, wq)|p|2j∂jzwp+ Z T ∂zjcq(u, wp)|q|2j∂zjwq= 0. (82)
The proof of this proposition relies upon the following observations. Since for all resonant triplet (Vα
k, Ul, Vmδ) (with sg(q) = α, k = kq and sg(p) = δ,
m = mp), we know that (Vδ
−m, Ul, V−kα ) is also resonant and σ(−m, Ul) =
−σ(k, Ul), we obtain
aδ−mσ(−m, Ul)|k|2jaαk+ a α
kσ(k, Ul)|m|2jaδ−m = 0 (83)
Here, we have also used that |m|2j = |m|2j|p|2j = |k|2j = |k|2j|q|2j. Then,
Using Proposition 3.3, we get the following L2 energy estimate for all t 1 2 X p∈P kwpk2L2(t) + ν Z t 0 X p∈P |p|2k∂zwpk2L2 ≤ 1 2 X p∈P kwpk2L2(0) = CkV0k2L2 (84) Next, for all j ∈ N and all p ∈ P, we have
1 2∂tk∂ jw pk2L2+ν|p|2k∂j+1wpk2L2+|p| Z T ∂j+1(w2p)∂jwp+ Z T X q ∂jcp(u, wq)∂jwp = 0 (85) To estimate R T ∂j+1(w2
p)∂jwp, we need the following Proposition
Proposition 4.6 Using interpolation inequality, we get Z
T
∂j+1(wp2)∂jwp ≤ CkwpkL∞k∂jwpkL2k∂j+1wpkL2. (86)
Indeed, by Hlder inequality, we find Z T ∂j+1(wp2)∂ j wp ≤ X s+r=j+1 j≥r,s≥1 j + 1 r k∂rwpkL4k∂swpkL4k∂jwpkL2 + 2kwpkL∞k∂j+1wpkL2k∂jwpkL2. (87)
Then, using Gagliardo-Nirenberg’s inequality, we deduce
k∂rwpkL4 ≤ CkwpkL1−θ∞k∂j+1wpkθL2 (88)
where 14 − s = θ(12 − (j + 1)). And, since r + s = j + 1, we also have
k∂swpkL4 ≤ CkwpkLθ∞k∂j+1wpk1−θL2 (89)
and the Proposition is proved.
Then, multiplying (85) by |p|2j and summing up in p, we get
1 2∂tkV k 2 Hj+ νkV k2Hj+1 ≤ X p kwpkL∞|p|2j+1k∂jwpkL2k∂j+1wpkL2 ≤ Cν X p kwpkL∞|p|2jk∂jwpk2L2 + ν 2 X p |p|2j+2k∂jwpk2L2
where we have used Cauchy Schwarz inequality. This yields for all t ∂tkV k2Hj + νkV k2Hj+1 ≤ C ν(supp kwpk 2 L∞)kV k2Hj ≤ Cν(X p kwpk2L∞)kV k2Hj. (90)
Then, by Gagliardo-Nirenberg’s inequality, we have for all t, X p kwpk2L∞ ≤ X p kwpkL2k∂zwpkL2 ≤ X p kwpk2L2 1/2X p k∂zwpk2L2 1/2
Integrating in time and using again Cauchy-Schwarz’ inequality, we obtain Z T 0 X p |wp|2L∞ ≤ C√T √ν (91)
Finally, by Grnwall inequality, we find for all t
kV k2Hj(t) + ν Z t 0 kV k 2 Hj+1 ≤ CkV0k2Hjexp( C√t ν3/2) (92)
We want to remark, that it is possible to get an estimate independent of t, by noticing that since wp has a zero average then kwpkL∞ ≤ k∂zwpkL2. This
yields the following estimate kV k2Hj(t) + ν Z t 0 kV k 2 Hj+1 ≤ CkV0k2Hjexp( C ν2) (93)
The argument, we presented here uses the fact that j ∈ N. However, it can be easily adapted to the case where s ∈ R, hence we get that
kV k2 Hs(t) + ν Z t 0 kV k 2 Hs+1 ≤ CkV0k2Hsexp( C√t ν3/2) (94) and we conclude.
4.5
Existence of local solutions for the coupled system
(ICB)
In this case the proof is the same, apart from the fact that we can no longer use the effect of the viscosity. Hence, inequality (86) should be replaced by
Z T ∂j+1(wp2)∂jwp ≤ k∂zwpkL∞k∂jwpk2L2. (95) Indeed, computing ∂j+1(w2 p), we have ∂j+1(w2p) = 2wp∂j+1wp+ 2(j + 1)∂wp∂jwp+ X r+s=j+1 j−1≥r,s≥2 ∂rwp∂swp (96)
The first term in the left hand side can be treated as follows Z T wp∂j+1wp∂jwp = Z T wp∂(∂jwp)2 = − Z T ∂wp(∂jwp)2 (97)
For the others, we conclude as in the proof of Proposition 4.6. Besides, inequality (90) must be replaced by
∂tkV k2Hj ≤ CkV k3Hj (98)
In fact, since j > 3
2, we deduce that k∂zwpkL∞ ≤ k∂ jw
pkL2. Finally, it is
easy to see that (98) yields the local existence for (ICB).
Acknowledgment
The author wishes to thank P.L.Lions for many discussions about this work.
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