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Global regularity of Quantum Navier-Stokes equations
in R 2
Léo Agélas
To cite this version:
Global regularity of Quantum Navier-Stokes equations in R
2
L´
eo Ag´
elas
∗March 1, 2017
Abstract
We prove existence and uniqueness of global strong solutions of Quantum Navier-Stokes equations in R2
for any initial data (n0, u0) with n0 a small perturbation of some positive constant n0 > 0 and such
that n0 is sufficiently large relative to (n0− n0, u0) in some sense. To our knowledge, this result is new
and gives a positive answer to the open problem of existence and smoothness of global solutions of such equations.
1
Introduction
Quantum hydrodynamic models become important and necessary to model and simulate electron trans-port, affected by extremely high electric fields, in ultra-small sub-micron semiconductor devices, such as resonant tunnelling diodes, where quantum effects (like particle tunnelling through potential barriers and built-up in quantum wells [14, 30] take place and dominate the process. They arise in semiclassical me-chanics in the study of semiconductor devices, in which case being derived from the Wigner-Boltzmann equation [13, 34, 14, 18, 26, 20]. Quantum hydrodynamics has engendered substantial activity in the field of theoretical chemical dynamics, where one may refer to Wyatt et. al.([36]) for a comprehensive introductory overview of the numerous recent results emerging from this blossoming field. In quantum chemistry, they arise as solutions to chemical kinetic systems, in which case they are derived from the Schr¨odinger equation by way of Madelung equations [33]. The basic idea emerging from quantum chem-istry is to employ the time-dependent Schr¨odinger equation to solve dynamical properties (probability densities, ”particle” velocities, etc.) of chemical systems. In the same spirit in which the de Broglie-Bohm interpretation (see [4, 5, 6]) of quantum mechanics may be used to recover ”trajectories” of individual fluid elements along the characteristics of motion of the solution (see [36] and [26] for a comprehensive overview).
They are also used to describe superfluids [32], weakly interacting Bose gases [21]. Some other topics of interest in quantum hydrodynamics are quantum turbulence, quantized vortices, second and third sound, and quantum solvents. Later, so-called quantum hydrodynamic equations have been derived by Ferry and Zhou [13] from the Bloch equation for the density matrix and by Gardner [14] from the Wigner equation by a moment method. More recently, dissipative quantum fluid models have been proposed. For instance, the moment method applied to the Wigner-Fokker-Planck equation leads to viscous quantum Euler models [16], and a Chapman-Enskog expansion in the Wigner equation leads under certain assumptions to quantum Navier-Stokes equations [3].
In this paper, we consider the Cauchy problem for Quantum Navier Stokes equations as follows: ∂n ∂t + ∇ · (nu) = 0, ∂nu ∂t + ∇ · (nu ⊗ u) − 2ε 2 n∇ ∆ √ n √ n − 2ε∇ · (nDu) + ∇p(n) = 0 (1)
∗Department of Mathematics, IFP Energies Nouvelles, 1-4, avenue de Bois-Pr´eau, F-92852 Rueil-Malmaison, France
with initial conditions,
n(x, 0) = n0, u(x, 0) = u0, (2)
on O = R2/L
0Z2, L0> 0 a given real and with periodic boundary conditions and we require in addition
that n0and u0 are periodic functions of period L0 where Du = ∇u + ∇u T
2 , p(n) = gn
2 with g > 0 and
ε > 0 is the scaled Planck constant.
From [16, 25], we can assume that the scaled Planck constant ε is of order 10−2.
For the initial data n0, we suppose that it is a small perturbation of some positive constant n0> 0 .
There are only few mathematical results for these viscous quantum hydrodynamic model due to difficulties coming from the third-order derivatives. The existence of classical solutions to the one-dimensional stationary model with ε = 0 and with physical boundary conditions was shown in [24]. The transient equations are considered in [7, 8, 10], and the local-in-time existence and exponential stability of solutions were proved. Global-in-time solutions in one space dimension are obtained if the initial energy is assumed to be sufficiently small. We also mention that in the inviscid case (ε = 0) there is a recent proof of non-global-in-time existence for a quantum hydrodynamic equation in bounded domains with prescribed data corresponding to high boundary and initial energy [15]. Later, Existence of global-in-time weak solutions in one space dimension without smallness conditions is proved in [17]. Concerning the multidimensional case, local-in-time existence theorems have been obtained in [7, 10]. Global-in-time existence of weak solutions in a three-dimensional torus for large data is achieved in [27, 11, 23]. This proof of existence given in [27] relies on the reformulation of the Quantum Navier-Stokes model as a viscous quantum Euler system and vice versa by introducing a new velocity variable ,w, involving gradients of the particle density, w = u + ε∇ log n. Following the results obtained in [28], it is shown provided that the particle density n 6= 0, that the particle density n and the new velocity w solve a viscous Euler system and this new formulation is equivalent to the Quantum Navier-Stokes equations (1)-(2).
In this paper, by using this new viscous Euler system with variables (n, m), m ≡ nw = nu + ε∇n (see (23)), we prove existence and uniqueness of global strong solutions in C([0, +∞[; Hs(O)), s > 1 of the
Quantum Navier-Stokes equations for large initial data for both n0 and u0.
The proof comes from results obtained after introducing the variables ρ (new particle density) and v (new
velocity) defined from n and m as follows ρ(x, t) =
n „ x √ gn0, t gn0 « −n0 n0 and v(x, t) = m „ x √ gn0, t gn0 « n0 √ gn0 .
In Proposition 4.1, under the assumption that kρkL∞ < 1, we show that the blow-up of smooth solutions
(ρ, v) to the viscous Euler equations (30) is controlled by the time integral of the maximum magnitude of the velocity v to the power four. More precisely, in Lemma 4.1, we show the following energy estimates in the homogeneous Sobolev space ˙Hs(Ω), with Ωdef= √gn
0O ( def = R2/(√gn 0L0)Z2 ) and s ≥ 0, k(ρ(t), v(t))kH˙s ≤ k(ρ0, v0)kH˙se CRt 0(kρ(τ )k 2 L∞+kv(τ )k2L∞+kv(τ )k4L∞)dτ, (3)
where (ρ0, v0) are the initial data corresponding to the variables (ρ, v). Furthermore, from a BD entropy
estimates and under the assumption that kρkL∞ ≤3
4, in Lemma 4.2, we show, k(ρ(t), v(t))k2 L2+ ε Z t 0 ∇ v(τ ) 1 + ρ(τ ) 2 L2 + k∇ρ(τ )k2L2 ! dτ . k(ρ0, v0)k2L2. (4)
Since these Inequalities are valid only for kρkL∞ ≤ 3
4, the challenge was then to find bounds for both
kρkL∞ and kvkL∞ allowing to ensure that kρkL∞ ≤ 3
4 and kvkL∞ ≤ 1 as soon as n0 is sufficiently
large relative to (n0− n0, u0) in some sense. It was thus important to obtain inequalities invariant in
respect of any scaling. For this, thanks to Gagliardo-Nirenberg inequalities and under the assumption that kρkL∞ ≤ 3
4 and kvkL∞ ≤ 1, from (3) and (4), we show that for all s > 1,
k(ρ(t), v(t))kL∞ . k(ρ0, v0)k s−1 s L2 k(ρ0, v0)k 1 s ˙ Hse CRt 0(kρ(τ )k 2 L∞+kv(τ )k2L∞)dτ. (5)
It remained then the difficulty to obtain a bound ofRt
0(kρ(τ )k 2
L∞+ kv(τ )k2L∞)dτ depending only on the
σ = −r, for 0 < r < 1, a fractional Gagliardo-Nirenberg inequality (8) and inequality (4), to obtain under the assumption that kρkL∞ ≤3
4 and kvkL∞ ≤ 1,
Z t
0
(kρ(τ )k2L∞+ kv(τ )k2L∞)dτ . k(ρ0, v0)kH˙rk(ρ0, v0)kH˙−re
C1k(ρ0,v0)k2L2.
Therefore, from (5), we obtain,
k(ρ(t), v(t))kL∞. k(ρ0, v0)k s−1 s L2 k(ρ0, v0)k 1 s ˙ Hse C2k(ρ0,v0)kHr˙ k(ρ0,v0)kH−r˙ e C1k(ρ0,v0)k2L2 ,
and then infer the condition on the initial data ensuring that kρkL∞ ≤ 3
4 and kvkL∞ ≤ 1 as soon as n0
is sufficiently large relative to (n0− n0, u0) in some sense, thus yielding to existence and uniqueness of
global strong solutions of (1)-(2).
This paper is organized as follows: In section 2, we introduce some notations. In section 3, we establish some crucial estimates for the proof of Theorem 4.1. In section 4, we give the proof of the Lemmata mentioned previously to obtain the proof of Theorem 4.1. From the latter, we deduce our main Theorem 4.2 with the condition on the initial data (n0− n0, u0) ensuring existence of global strong solutions of the
Quantum Navier Stokes equations (1)-(2).
2
Some notations
We denote A . B, the estimate A ≤ C B where C > 0 is a absolute constant. Given a function f which is periodic with period L, and thus representable as a function on the torus R2
/LZ2, we define the discrete
Fourier transform ˆf : Z27−→ C by the formula,
ˆ f (k) = 1 L2 Z R2/LZ2 e−2πLix·kf (x) dx,
when f is absolutely integrable on R2/LZ2. If f ∈ L2(R2/LZ2), then from Parseval equality, we have,
X k∈Z2 | ˆf (k)|2= 1 L Z L 0 |f (x)|2dx.
Let s ∈ R, we define the Sobolev norm kf kHs(R2/LZ2)of a tempered distribution f : R2/LZ27−→ R by,
kf kHs(R2/LZ2)= X k∈Z2 1 + 2π|k| L 2!s | ˆf (k)|2 !12 ,
and then we denote by Hs(R2/LZ2) the space of tempered distributions with finite Hs(R2/LZ2) norm. On the torus R2
/LZ2, for s > −1, we also define the homogeneous Sobolev norm,
kf kH˙s(R2/LZ2)= X k∈Z2 2π|k| L 2s | ˆf (k)|2 !12 ,
and then we denote by ˙Hs
(R2
/LZ2) the space of tempered distributions with finite ˙Hs
(R2
/LZ2) norm.
We use the Fourier transform to define the fractional Laplacian operator (−∆)α
3
Some estimates
In this section, we give a serie of Lemmata which will be used in the next section. Let Ω = R2\LZ2with
L > 0 a real.
We begin with the following inequality proved in [9] which states that for all f, g ∈ L∞(Ω) ∩ ˙Hs(Ω),
kf gkH˙s. (kf kL∞kgkH˙s+ kf kH˙skgkL∞). (6)
We have also the following inequality proved in [9] for nonlinear composition which states that for s > 0, I an open subset of R with 0 ∈ I and for any real function f ∈ BC[s]+2(I) such that f (0) = 0, we have for all u ∈ Hs(Ω) such that u(x) ∈ I, f (u) ∈ Hs(Ω). More precisely there exists a non decreasing continuous function C depending only on s and max
0≤k≤[s]+2kf (k)k
L∞(I) such that,
kf (u)kHs≤ C(kukL∞)kukHs. (7)
Let us mention also the following Interpolation inequality: for all (α, β) such that 0 ≤ α < 1 < β and for all u ∈ Hβ(Ω), kukL∞(Ω). kukγ˙ Hα(Ω)kuk δ ˙ Hβ(Ω), (8) where γ > 0, δ > 0, γ + δ = 1 and γα + βδ = 1.
Now, we give three crucial Lemmata.
Lemma 3.1 Let σ ∈ R, 0 < |σ| < 1, f, h ∈ ˙Hσ(Ω) ∩ ˙Hσ+1(Ω) and g ∈ ˙H1(Ω). Then, we have,
| hf Dg, hiH˙σ| . (kfkH˙σk∇hkH˙σ+ khkH˙σk∇f kH˙σ)k∇gkL2,
where D is a derivative of first order.
Proof. We set A = −∆. Let δ ≥ 0 such that σ2 + δ > 0, σ2 + δ < 21 and σ2 +12− δ > σ
2 that means −σ 2 < δ < min( 1 2, 1 2− σ 2). Then, we have | hf Dg, hiH˙σ| = |A σ 2(f Dg), Aσ2h | = D Aσ2+δ− 1 2(f Dg), A σ 2+ 1 2−δh E ≤ kAσ2+δ−12(f Dg)kL2kAσ2+12−δhkL2,
where we have used Cauchy-Schwarz inequality. Since 0 < σ2+ δ < 12, we have (see [19], see also (A5) in [29], [12]), kAσ2+δ−12(f Dg)k L2. kA σ 2+δf k L2kA 1 2gk L2. Then, we deduce, | hf Dg, hiH˙σ| . kA σ 2+δf kL2kAσ2+12−δhkL2kA12gkL2. (9)
Thanks to the fractional Gagliardo-Nirenberg inequality given by Corollary 1.5 in [22], we have for any u ∈ H1(Ω) and 0 ≤ θ ≤ 1, kukH˙θ . kuk 1−θ L2 kuk θ ˙ H1. (10)
We use Inequality (10) with u = Aσ2ρ and θ = 2δ, then we get,
kAσ2+δf kL2 = kA σ 2f k˙ H2δ . kA σ 2f k1−2δ L2 kA σ 2f k2δ ˙ H1 = kf k 1−2δ ˙ Hσ k∇f k 2δ ˙ Hσ. (11)
We use Inequality (10) with u = Aσ2h and θ = 1 − 2δ to obtain,
Therefore, from (9), using (11), (12) and Young inequality, we deduce,
| hf Dg, hiH˙σ| . (kfkH˙σk∇hkH˙σ+ khkH˙σk∇f kH˙σ)k∇gkL2,
which concludes the proof.
Lemma 3.2 Let σ ≥ 0, 0 < γ < 1, % ∈ Hσ(Ω) ∩ L∞(Ω) such that k%kL∞(Ω) ≤ γ, then there exists a
constant C > 0 depending only on σ and γ such that, % 1 + % H˙σ ≤ Ck%kH˙σ.
Proof. If σ = 0, the proof holds. Let us assume that σ > 0. Let λ > 0, %λ(x) = %(λx) and f the
function defined by f (x) = x
1 + x then f ∈ C
∞([−γ, γ]) and k%
λkL∞ ≤ γ therefore thanks to (7), we
deduce that there exists a constant C > 0 depending only on σ, γ such that, %λ 1 + %λ Hσ ≤ Ck%λkHσ. (13)
From (13), we deduce that, %λ 1 + %λ L2 + %λ 1 + %λ H˙σ . k%λkL2+ k%λkH˙σ. (14) Then, we get, 1 λ % 1 + % L2 +λ σ λ % 1 + % H˙σ .λ1k%kL2+ λσ λ k%kH˙σ. (15) We multiply Inequality (15) by λ λσ to obtain, 1 λσ % 1 + % L2 + % 1 + % H˙σ .λ1σk%kL2+ k%kH˙σ. (16)
Then, taking the limit as λ → ∞ in (16), we conclude the proof.
Lemma 3.3 Let σ ∈ R, 0 < |σ| < 1, 0 < γ < 1, f, h ∈ ˙Hσ(Ω) ∩ ˙Hσ+1(Ω), g ∈ ˙H1(Ω) and % ∈ ˙
Hσ(Ω) ∩ Hσ+1(Ω) ∩ L∞(Ω) such that k%kL∞ ≤ γ. Then, we have,
f 1 + %Dg, h ˙ Hσ . (kf k ˙ Hσk∇hkH˙σ+ khkH˙σk∇f kH˙σ)k∇gkL2 + kf kL∞(k%kH˙σk∇hkH˙σ+ khkH˙σk∇%kH˙σ)k∇gkL2,
where D is a derivative of first order.
Then, we deduce, f 1 + %Dg, h ˙ Hσ . Aσ2+δ f 1 + % L2 khk2δ ˙ Hσk∇hk 1−2δ ˙ Hσ k∇gkL2. (17) Since f 1 + % = f − f % 1 + %, then Aσ2+δ f 1 + % L2 = f 1 + % H˙σ+2δ ≤ kf kH˙σ+2δ+ f % 1 + % H˙σ+2δ . Since σ + 2δ > 0, thanks to (6), we have,
f % 1 + % ˙ Hσ+2δ . kf kH˙σ+2δ % 1 + % L∞ + kf kL∞ % 1 + % ˙ Hσ+2δ ≤ kf kH˙σ+2δ k%kL∞ 1 − k%kL∞ + kf kL∞ % 1 + % H˙σ+2δ ≤ γ 1 − γkf kH˙σ+2δ+ kf kL∞ % 1 + % H˙σ+2δ , (18)
where we have used the fact that k%kL∞ ≤ γ. Then, we deduce,
Aσ2+δ f 1 + % L2 . kf kH˙σ+2δ+ kf kL∞ % 1 + % H˙σ+2δ . (19)
Thanks to Lemma 3.2, we have, % 1 + % H˙σ+2δ . k%kH˙σ+2δ. From (19), we infer, Aσ2+δ f 1 + % L2 . kf kH˙σ+2δ+ kf kL∞k%k˙ Hσ+2δ. (20)
Using the same arguments as in (11), we get,
kf kH˙σ+2δ . kf k 1−2δ ˙ Hσ k∇f k 2δ ˙ Hσ k%kH˙σ+2δ . k%k 1−2δ ˙ Hσ k∇%k 2δ ˙ Hσ (21)
From (17), using (20), (21) and Young inequalities, we deduce, f 1 + %Dg, h ˙ Hσ . (kf k ˙ Hσk∇hkH˙σ+ khkH˙σk∇f kH˙σ)k∇gkL2 +kf kL∞(k%k˙ Hσk∇hkH˙σ+ khkH˙σk∇%kH˙σ)k∇gkL2, (22)
which concludes the proof.
4
Global regularity
This section is devoted to the proof of Theorem 4.2. We assume that n0 > 0 and we introduce m0 =
n0u0+ ε∇n0. We begin by noting that the system of equations (1) is equivalent to a system of equations
of type (23).
Indeed, thanks to Theorem 2.1 in [28], under the assumption that n > 0, if (n, u) is solution of the shallow water equations (1) for the initial data (n0, u0) then (n, m) is solution of the system of equations (23) for
the initial (n0, m0), with m = nu + ε∇n.
for the initial data (n0, u0) with u =
m − ε∇n n . Then, we consider the following system of equations,
∂n ∂t + ∇ · m − ε∆n = 0, ∂m ∂t + ∇ · m n ⊗ m − ε∆m + gn∇n = 0. (23) We introduce ˜n defined by ˜n ≡ n − n0 n0 and ˜m ≡ m n0
, from (23), we obtain the following equivalent system of equations, ∂ ˜n ∂t + ∇ · ˜m − ε∆˜n = 0, ∂ ˜m ∂t + ∇ · m˜ 1 + ˜n⊗ ˜m − ε∆ ˜m + gn0n∇˜˜ n + gn0∇˜n = 0. (24)
For any λ ∈ R∗, by using the rescaled solutions ˜nλ(x, t) = ˜n(λx, λ2t) and ˜mλ(x, t) = λ ˜m(λx, λ2t), we
obtain the equivalent system of equations, ∂ ˜nλ ∂t + ∇ · ˜mλ− ε∆˜nλ= 0, ∂ ˜mλ ∂t + ∇ · m˜ λ 1 + ˜nλ ⊗ ˜mλ − ε∆ ˜mλ+ gn0λ2n˜λ∇˜nλ+ gn0λ2∇˜nλ= 0. (25) Setting λ = √1 gn0 in (25), then with ρ(x, t) = n√x gn0, t gn0 − n0 n0 and v(x, t) = 1 n0 √ gn0 m x √ gn0 , t gn0 , provided that ρ > −1, we deduce that the system of equations (23) on O × [0, T0] is equivalent to the
following system of equations on Ω × [0, T ] with Ωdef= √gn0O and T = gn0T0,
∂ρ ∂t + ∇ · v − ε∆ρ = 0, ∂v ∂t + ∇ · v 1 + ρ⊗ v − ε∆v + ρ∇ρ + ∇ρ = 0. (26)
with initial conditions,
ρ(x, 0) = ρ0(x), v(x, 0) = v0(x), (27) where ρ0(x) = n0 x √ gn0 − n0 n0 (28) and v0(x) = 1 n0 √ gn0 m0 x √ gn0 . (29)
Then, in what follows, we study the following system of equations, ∂ρ ∂t + ∇ · v − ε∆ρ = 0, ∂v ∂t + ∇ · v 1 + ρ ⊗ v − ε∆v + (1 + ρ)∇ρ = 0. (30)
with initial conditions,
ρ(x, 0) = ρ0(x), v(x, 0) = v0(x). (31)
Lemma 4.1 Let 0 < δ < 1, ω0 ≡ (ρ0, v0) ∈ Hs(Ω)3, s ≥ 0. If ω ≡ (ρ, v) ∈ C([0, T ]; Hs(Ω)3) ∩
C(]0, T ]; Hs+1(Ω)3) with kρk
L∞(Ω×[0,T ]) ≤ δ is a solution of the system of Equations (30)-(31), then
there exists a constant c1> 0 depending only on δ, s such that for all τ ∈ [0, T ],
kω(τ )k2 ˙ Hs+ ε 2 Z τ 0 kω(σ)k2 ˙ Hs+1dσ ≤ kω0k2H˙se c1 ε Rτ 0 a(σ)dσ, where a(σ) = kω(σ)k2 L∞(1 + kv(σ)k2L∞).
Proof. We take the inner product in ˙Hs(Ω) of the first equation of (30) with ρ, use integrations by parts to obtain, 1 2 d dtkρk 2 ˙ Hs+ εk∇ρk 2 ˙ Hs = hv, ∇ρiH˙s. (32)
Now, we take the inner product in ˙Hs(Ω)2 of the second equation of (30) with v, use integrations by
parts to obtain, 1 2 d dtkvk 2 ˙ Hs+ εk∇vk 2 ˙ Hs = v 1 + ρ⊗ v, ∇v ˙ Hs + ρ 2 2 , ∇ · v ˙ Hs − h∇ρ, viH˙s. (33)
Thanks to Cauchy-Schwarz inequality and Young inequality, we have, v 1 + ρ ⊗ v, ∇v ˙ Hs ≤ v 1 + ρ ⊗ v H˙s k∇vkH˙s ≤ 1 2ε v 1 + ρ⊗ v 2 ˙ Hs +ε 2k∇vk 2 ˙ Hs and ρ2 2 , ∇ · v ˙ Hs ≤ ρ2 H˙s k∇ · vkH˙s 2 ≤ 1 2ε ρ2 2 ˙ Hs+ ε 8k∇ · vk 2 ˙ Hs ≤ 1 2ε ρ2 2 ˙ Hs+ ε 4k∇vk 2 ˙ Hs. Then, we deduce, 1 2 d dtkvk 2 ˙ Hs+ ε 4k∇vk 2 ˙ Hs ≤ 1 2ε v 1 + ρ⊗ v 2 ˙ Hs + 1 2ε ρ2 2 ˙ Hs− h∇ρ, viH˙s. (34)
We sum Equations (32) and (34), to obtain,
Thanks to (6), we have also, vρ 1 + ρ H˙s . kvkL∞ ρ 1 + ρ H˙s + kvkH˙s ρ 1 + ρ L∞ . (39)
Then with (38) and (39), we deduce, v 1 + ρ H˙s . kvkH˙s 1 + ρ 1 + ρ L∞ + kvkL∞ ρ 1 + ρ H˙s . (40)
Therefore, using (40) and (37), we get, v 1 + ρ ⊗ v H˙s . 1 + ρ 1 + ρ L∞ kvkH˙skvkL∞+ kvk2L∞ ρ 1 + ρ H˙s + v 1 + ρ L∞ kvkH˙s ≤ 2 1 − δkvkH˙skvkL∞+ ρ 1 + ρ H˙s kvk2L∞ , (41)
where we have used the fact that kρkL∞ ≤ δ < 1. Then, using (41) and (36), from (35), we deduce,
1 2 d dt(kρk 2 ˙ Hs+ kvk 2 ˙ Hs) + εk∇ρk 2 ˙ Hs+ ε 4k∇vk 2 ˙ Hs . 1 εkvk 2 ˙ Hskvk 2 L∞ +1 ε ρ 1 + ρ 2 ˙ Hs kvk4 L∞+ 1 2εkρk 2 L∞kρk2H˙s. (42)
Thanks to Lemma 3.2, from (42), we deduce that for all t ∈]0, T ], 1 2 d dt(kρ(t)k 2 ˙ Hs+ kv(t)k 2 ˙ Hs) + εk∇ρ(t)k 2 ˙ Hs+ ε 4k∇v(t)k 2 ˙ Hs . 1 εkv(t)k 2 ˙ Hskv(t)k 2 L∞ +1 εkρ(t)k 2 ˙ Hskv(t)k 4 L∞+ 1 2εkρ(t)k 2 L∞kρ(t)k 2 ˙ Hs (43) which implies, 1 2 d dτ(kρ(τ )k 2 ˙ Hs+ kv(τ )k 2 ˙ Hs) +εk∇ρ(τ )k 2 ˙ Hs+ ε 4k∇v(τ )k 2 ˙ Hs . 1ε(kv(τ )k2L∞+ kv(τ )k4L∞+ kρ(τ )kL2∞)(kρ(τ )k2H˙s+ kv(τ )k2H˙s). (44)
Using Gronwall inequality, from (44), we deduce that there exists a constant C > 0 depending only on δ, s such that for all τ ∈ [0, T ],
kρ(τ )k2 ˙ Hs+ kv(τ )k 2 ˙ Hs+ Z τ 0 2εk∇ρ(σ)k2H˙s+ ε 2k∇v(σ)k 2 ˙ Hs dσ ≤ (kρ0k2H˙s+ kv0k 2 ˙ Hs)e CRτ 0 a(σ)dσ, where a(σ) = 1ε(kv(σ)k2
L∞+kv(σ)k4L∞+kρ(σ)k2L∞). Thanks to Plancherel Theorem, we have k∇ρ(τ )kH˙s=
kρ(τ )kH˙s+1 and also k∇v(τ )kH˙s = kv(τ )kH˙s+1 , then we conclude the proof.
In the following Proposition, we prove existence and uniqueness of local strong solutions of (30).
Proposition 4.1 Let ω0≡ (ρ0, v0) ∈ ˙H−r(Ω)3∩ Hs(Ω)3 with 0 ≤ r < 1 < s, kρ0kL∞ < 1. Then there
exists a maximal time of existence T∗> 0 such that there exists a unique strong solution
ω ≡ (ρ, v) ∈ C([0, T∗[; H−r(Ω)3∩ Hs(Ω)3) ∩ C1(]0, T∗[; Hs−1(Ω)3) ∩ C(]0, T∗[; Hs+1(Ω)3) of the system
of Equations (30)-(31). Moreover if T∗< ∞, then either kρkL∞(Ω×[0,T∗])≥ 1 or either
Z T∗
0
kv(τ )k4
Proof.
Let a = 3 + kρ0kL∞ 2(1 + kρ0kL∞)
> 1 and δ = 12(1 + kρ0kL∞) < 1.
We introduce χ(λ) a smooth bump function with values in the interval [0, 1], identically equal to one for −1 ≤ λ ≤ 1 and identically equal to zero for |λ| ≥ a.
We notice for all λ δ ≥ a, 1 + λχ λ δ
= 1, and for all λ δ < a, 1 + λχ λδ ≥ 1 − |λ| ≥ 1 − δa = 1
4(1 − kρ0kL∞) > 0. Then, we get for all λ ∈ R,
1 + λχ λ δ
≥1
4(1 − kρ0kL∞) > 0. (46) For the proof, we use some results which deal with existence, uniqueness, regularity of solutions ω = (ρ, v) for nonlinear evolution equations of the form
∂tω = Aω + f (ω), (47)
with initial conditions
ω(0) = ω0. (48)
More precisely, we use Proposition 2.1 in [2] with X = Hs−1(Ω)3, A = ε∆ for our generator of holomorphic
semigroup T (t) = e−tAof bounded linear operators on X and f our locally Lipschitz continuous function on Xα= Hs(Ω)3, α =12 defined by f (ω) = (f1(ω), f2(ω)) with f1(ω) = −∇ · v and f2(ω) = −∇ · v 1 + ρχ(ρδ)⊗ v − ρ∇ρ − ∇ρ. (49)
Indeed, f is locally Lipschitz continuous on Hs(Ω)3, since for all ω
1 = (ρ1, v1) ∈ Hs(Ω)3 and ω2 = (ρ2, v2) ∈ Hs(Ω)3, we have, k∇ · v1− ∇ · v2kHs−1 ≤ kv1− v2kHs k∇ρ1− ∇ρ2kHs−1 ≤ kρ1− ρ2kHs kρ1∇ρ1− ρ2∇ρ2kHs−1 = 1 2k∇((ρ1− ρ2)(ρ1+ ρ2))kHs−1 ≤1 2k(ρ1− ρ2)(ρ1+ ρ2)kHs . kρ1− ρ2kHs(kρ1kHs+ kρ2kHs), (50)
where we have used the fact that for all h1∈ Hs(Ω), h2∈ Hs(Ω) with s > 1 (see [9]),
kh1h2kHs . kh1kHskh2kHs, (51)
We introduce the function g ∈ C∞ defined by g(λ) = 1
1 + λχ(λδ), thanks to (46) we notice that for all k ∈ N, there exists a real Ck > 0 depending only on δ such that
Using (51), we estimate each term on the right hand side of Inequality (53),
k(v1− v2) ⊗ v1g(ρ1)kHs . kv1− v2kHskv1kHs(1 + kg(ρ1) − 1kHs)
kv2⊗ v1(g(ρ1) − g(ρ2))kHs . kv1kHskv2kHskg(ρ1) − g(ρ2)kHs
kg(ρ2)v2⊗ (v1− v2)kHs . kv1− v2kHskv2kHs(1 + kg(ρ2) − 1kHs).
(54)
Thanks to (7) and (52), we deduce that there exists a non-decreasing function C > 0 depending only on s and δ such that,
kg(ρ1) − 1kHs . C(kρ1kL∞)kρ1kHs
kg(ρ2) − 1kHs . C(kρ2kL∞)kρ2kHs.
Furthermore, using Taylor formula at order 1, we get g(ρ2) − g(ρ1) = (ρ2− ρ1)
R1 0 g 0((1 − σ)ρ 1+ σρ2)dσ, then we deduce, kg(ρ1) − g(ρ2)kHs . kρ1− ρ2kHs Z 1 0 kg0((1 − σ)ρ1+ σρ2)kHsdσ . kρ1− ρ2kHs Z 1 0 C1(k(1 − σ)ρ1+ σρ2kL∞)k(1 − σ)ρ1+ σρ2)kHsdσ,
where we have used (7) and C1> 0 is a non-decreasing function depending only on s and δ. Therefore,
we get,
kg(ρ1) − g(ρ2)kHs . kρ1− ρ2kHsC1(kρ1kL∞+ kρ2kL∞)(kρ1kHs+ kσρ2)kHs).
Thanks to (54) and the Sobolev embedding Hs(Ω) ,→ L∞(Ω) since s > 1, from (53), we deduce therefore, k∇ · (g(ρ1)v1⊗ v1) − ∇ · (g(ρ2)v2⊗ v2)kHs
. C2(kρ1kHs, kρ2kHs, kv1kHs, kv2kHs)(kv1− v2kHs+ kρ1− ρ2kHs), (55)
where C2 > 0 is a continuous function on R4. Using (50) and (55), we get that for all ω1 ∈ Hs(Ω)3,
ω2∈ Hs(Ω)3,
kf (ω1) − f (ω2)kHs−1 . C3(kω1kHs, kω2kHs)kω1− ω2kHs, (56)
where C3 > 0 is a continuous function on Ω, which proves that f is well locally Lipschitz continuous
on Hs(Ω)3. Then, we deduce thanks to Proposition 2.1 combined with Theorem 3.1 in [2], that there
exists a maximal time T0> 0 such that there exists an unique solution ω = (ρ, v) ∈ C([0, T0[; Hs(Ω)3) ∩
C1(]0, T0[; Hs−1(Ω)3) of the system of Equations (47)-(48).
Moreover if T0 < ∞ then lim
t→T∗kω(t)kHs = ∞. From (56), we get that for all w ∈ H
s(Ω)3, kf (w)k Hs−1 .
C3(kwkHs, 0)kwkHs, then we deduce from (47) that ω ∈ C(]0, T0[; Hs+1(Ω)).
Then, we write ω under its integral form,
ω(t) = e−εt∆ω0+
Z t
0
e−ε(t−σ)∆f (ω(σ))dσ. (57)
Since s > 1 − r > 0, then by Interpolation inequality, we have Hs(Ω) ,→ ˙H1−r(Ω), then using the same
arguments as for (56), we get that for all w ∈ ˙H−r ∩ Hs, kf (w)k ˙
H−r . C3(kwkHs, 0)kwkHs. Since
ω0∈ H−r(Ω), therefore, from (57), we deduce also that ω ∈ C([0, T0[; H−r(Ω)).
Since kρ0kL∞ < 1 then kρ0kL∞ < δ, moreover ρ ∈ C([0, T0[; L∞(Ω)) due to the Sobolev embedding
Hs(Ω) ,→ L∞(Ω), hence we deduce that there exists a time 0 < T < T0 such that kρk
L∞(Ω×[0,T ]) ≤ δ.
Since kρkL∞(Ω×[0,T ]) ≤ δ then we get χ(ρ
δ) = 1 on [0, T ] and from (49), we deduce in fact that ω is the
unique solution of (30)-(31) on [0, T ]. Then we deduce that there exists a maximal time of existence 0 < T∗ < ∞ such that there exists an unique solution ω0 = (ρ0, v0) ∈ C([0, T∗[; H−r ∩ Hs(Ω)) ∩
∞, then either kρkL∞(Ω×[0,T∗]) ≥ 1 or either lim
t→T∗kω
0(t)k
Hs = ∞ (by using the same arguments as
ω) and in this case thanks to Lemma 4.1 and the fact that for any 0 < T < T∗, RT
0 kv(τ )k 2 L∞dτ ≤ √ TRT 0 kv(τ )k 4 L∞dτ 12 , we get Z T∗ 0 kv(τ )k4
L∞dτ = ∞, then, we conclude the proof.
We derive now a Bresch-Desjardin Entropy type, as in [28].
Lemma 4.2 Let ω0 ≡ (ρ0, v0) ∈ L2(Ω)3 with ρ0 ∈ L∞(Ω) such that kρ0kL∞ ≤ 3
4. If ω ≡ (ρ, v) ∈
C([0, T ]; L2(Ω))3∩ C1(]0, T ]; L2(Ω)3) with ρ ∈ L∞(Ω × [0, T ]) such that kρk
L∞(Ω×[0,T ])≤34 is a solution
of the system of Equations (30)-(31), then we have for all t ∈ [0, T ],
1 4kω(t)k 2 L2+ ε Z t 0 1 4 ∇ v(τ ) 1 + ρ(τ ) 2 L2 + k∇ρ(τ )k2L2 ! dτ ≤ 2kω0k2L2.
Proof. We can write the system of equations (30) as follows, ∂η ∂t + ∇ · (ηw) − ε∆η = 0, ∂(ηw) ∂t + ∇ · (ηw ⊗ w) − ε∆(ηw) + ∇p(η) = 0, (58) where w = v 1 + ρ, η = 1 + ρ and p(η) = η2
2 . By using the enthalpy h(η) = η − 1, we notice ηh
0(η) = p0(η),
then from Equation (15) of [28], we obtain,
d dt Z Ω 1 2η|w| 2+ H(η) + Z Ω (εη|∇w|2+ |∇η|2) = 0, where H(η) =Rη
1 h(τ )dτ , which yields to,
d dt Z Ω |v|2 2(1 + ρ)+ ρ2 2 + ε Z Ω (1 + ρ) ∇ v 1 + ρ 2 + |∇ρ|2 ! = 0.
We integrate Equation just above on [0, t] to obtain, for all t ∈ [0, T ],
1 2 v(t) p1 + ρ(t) 2 L2 +1 2kρ(t)k 2 L2+ ε Z t 0 p 1 + ρ(τ )∇ v(τ ) 1 + ρ(τ ) 2 L2 + k∇ρ(τ )k2L2 ! dτ = 1 2 v0 √ 1 + ρ0 2 L2 + 1 2kρ0k 2 L2. Since kρkL∞(Ω×[0,T ])≤3 4 and kρ0kL∞ ≤ 3
4, then we deduce that for all t ∈ [0, T ],
1 4kv(t)k 2 L2+ 1 2kρ(t)k 2 L2+ ε Z t 0 1 4 ∇ v(τ ) 1 + ρ(τ ) 2 L2 + k∇ρ(τ )k2L2 ! dτ ≤ 2 kv0k 2 L2+ 1 2kρ0k 2 L2,
which concludes the proof.
The following Lemma will help us to expressRt
0k(ρ, v)(σ)k 2 L∞dσ in terms of Rt 0(k∇ρ(τ )k 2 L2+ ∇ v(τ ) 1+ρ(τ ) 2 L2)dτ . Lemma 4.3 Let ω0≡ (ρ0, v0) ∈ ˙Hσ(Ω)3∩ L2(Ω)3, σ ∈ R, 0 < |σ| < 1.
Equations (30)-(31) with kωkL∞(Ω×[0,T ])3 ≤34 and such thatR
T
0 a(τ )dτ < ∞, then there exists a constant
C > 0 such that for all s ∈ [0, T ],
kω(s)k2 ˙ Hσ+ ε Z s 0 kω(τ )k2 ˙ Hσ+1dτ ≤ kω0k2H˙σe CRt 0a(τ )dτ, where a(τ ) = k∇ρ(τ )k2 L2+ ∇ v(τ ) 1+ρ(τ ) 2 L2 ε . Proof.
We take the inner product in ˙Hσ(Ω) of the first equation of (30) with ρ, use integrations by parts to
obtain, 1 2 d dtkρk 2 ˙ Hσ+ εk∇ρk 2 ˙ Hσ = hv, ∇ρiH˙σ. (59)
Now, we take the inner product in ˙Hσ(Ω) of the second equation of (30) with v, to obtain,
1 2 d dtkvk 2 ˙ Hσ+ εk∇vk 2 ˙ Hσ = − ∇ · v 1 + ρ⊗ v , v ˙ Hσ − hρ∇ρ, viH˙σ − h∇ρ, viH˙σ. (60)
We sum Equations (59) and (60), to obtain, 1 2 d dt(kρk 2 ˙ Hσ+ kvk 2 ˙ Hσ) + εk∇ρk 2 ˙ Hσ+ εk∇vk 2 ˙ Hσ = − ∇ · v 1 + ρ⊗ v , v ˙ Hσ − hρ∇ρ, viH˙σ. (61)
Thanks to Lemma 3.1, we get,
− hρ∇ρ, viH˙σ . (kρkH˙σk∇vkH˙σ + kvkH˙σk∇ρkH˙σ)k∇ρkL2. (62)
We estimate now the first term at the right hand side of Equation (61). We notice ∇ · v 1+ρ ⊗ v = v 1+ρ· ∇ v + ∇ ·1+ρv v, then we deduce, − ∇ · v 1 + ρ ⊗ v , v ˙ Hσ = − v 1 + ρ· ∇ v, v ˙ Hσ − ∇ · v 1 + ρ v, v ˙ Hσ .
Thanks to Lemma 3.1, we get,
− ∇ · v 1 + ρ v, v ˙ Hσ . kvkH˙σk∇vkH˙σ ∇ v 1 + ρ L2 . (63)
Thanks to Lemma 3.3, we have,
− v 1 + ρ · ∇ v, v ˙ Hσ . kvkH˙σk∇vkH˙σk∇vkL2 +kvkL∞(kρkH˙σk∇vkH˙σ+ kvkH˙σk∇ρkH˙σ)k∇vkL2. (64)
Using the fact that kvkL∞(Ω×[0,T ])2 ≤ 3
4, from (65), we infer that there exists a constant C > 0 depending
only on σ such that, 1 2 d ds(kρ(s)k 2 ˙ Hσ+ kv(s)k 2 ˙ Hσ) + ε 2k∇ρ(s)k 2 ˙ Hσ+ ε 2k∇v(s)k 2 ˙ Hσ ≤ C k∇ρ(s)k2 L2+ ∇ v(s) 1+ρ(s) 2 L2+ k∇v(s)k 2 L2 ε (kρ(s)k 2 ˙ Hσ + kv(s)k 2 ˙ Hσ), (66) which implies, 1 2 d ds(kρ(s)k 2 ˙ Hσ+ kv(s)k 2 ˙ Hσ) ≤ C k∇ρ(s)k2 L2+ ∇ v(s) 1+ρ(s) 2 L2+ k∇v(s)k 2 L2 ε (kρ(s)k 2 ˙ Hσ + kv(s)k 2 ˙ Hσ). (67) Then, using Gronwall Inequality, from (67), we deduce for all s ∈ [0, T ],
kρ(s)k2 ˙ Hσ+ kv(s)k 2 ˙ Hσ ≤ (kρ0k2H˙σ+ kv0k2H˙σ)e 2CRt 0b(τ )dτ, (68) where b(τ ) = k∇ρ(τ )k2 L2+ ∇ v(τ ) 1+ρ(τ ) 2 L2+ k∇v(τ )k 2 L2
ε . After integrating (66) and using (68), we deduce that for all s ∈ [0, T ],
kρ(s)k2 ˙ Hσ + kv(s)k 2 ˙ Hσ+ Z s 0 εk∇ρ(τ )k2H˙σ+ εk∇v(τ )k 2 ˙ Hσ dτ ≤ (kρ0k 2 ˙ Hσ + kv0k 2 ˙ Hσ)e 2CRt 0b(τ )dτ. (69) We notice that ∇v(τ ) = ∇ v(τ ) 1 + ρ(τ )(1 + ρ(τ )) = ∇ v(τ ) 1 + ρ(τ ) (1 + ρ(τ )) + v(τ ) 1 + ρ(τ )⊗ ∇ρ(τ ), then we get, |∇v(τ )| ≤ ∇ v(τ ) 1 + ρ(τ ) (1 + |ρ(τ )|) + |v(τ )| 1 − |ρ(τ )||∇ρ(τ )|.
Since kρkL∞(Ω×[0,T ])≤34 and kvkL∞(Ω×[0,T ])2 ≤34, then we deduce for a.e τ ∈ [0, T ],
|∇v(τ )| ≤ 7 4 ∇ v(τ ) 1 + ρ(τ )
+ 3|∇ρ(τ )|. Therefore, we deduce that b(τ ) .
k∇ρ(τ )k2 L2+k∇( v(τ ) 1+ρ(τ ))k 2 L2 ε and
thanks to Plancherel Theorem, we have k∇ρ(τ )kH˙σ = kρ(τ )kH˙σ+1, k∇v(τ )kH˙σ = kv(τ )kH˙σ+1 , then from
(69), we conclude the proof.
Then, gathering the previous results, we prove our Theorem 4.1 which deals with existence and uniqueness of global strong solutions of (30).
Theorem 4.1 Let ω0 ≡ (ρ0, v0) ∈ ˙H−r(Ω)3∩ Hs(Ω)3 with 0 < r < 1 < s and kω0kL∞ ≤ 1
2. Then there
exists a real α1> 0 depending only on r, s, v0, ρ0 such if α1≤ 12, then there exists a unique global strong
solution ω ≡ (ρ, v) ∈ C([0, ∞[; ˙H−r(Ω)3∩ Hs(Ω)3) ∩ C1(]0, ∞[; Hs−1(Ω)3) ∩ C(]0, ∞[; Hs+1(Ω)3) of the
system of Equations (30)-(31) with kωkL∞(Ω×[0,+∞[)3 ≤ 1
2. Furthermore, ω ∈ C
∞(Ω×]0, ∞[)3 and the
real α1 is given by,
α1= C1kω0k s−1 s L2 kω0k 1 s ˙ Hse c1 εα0, where, α0 = kω0kH˙rkω0kH˙−re C ε2kω0k 2 L2,
with c1> 0, C > 0 and C1> 0 constants depending only on r, s. Moreoever, for all t ≥ 0,
Z t
0
kω(τ )k2
Proof.
Thanks to Proposition 4.1, there exists a maximal time of existence T∗> 0 such that there exists a unique strong solution ω ≡ (ρ, v) ∈ C([0, T∗[; ˙H−r(Ω)3∩ Hs(Ω)3) ∩ C1(]0, T∗[; Hs−1(Ω)3) ∩ C(]0, T∗[; Hs+1(Ω)3)
of the system of Equations (30)-(31). Moreover if T∗< ∞, then either kρkL∞(Ω×[0,T∗])≥ 1 or either
Z T∗
0
kv(τ )k4
L∞ = ∞. (70)
Let us assume that T∗< ∞.
Since s > 1, we get the Sobolev embedding Hs(Ω) ,→ L∞(Ω) and then ω ∈ C([0, T∗[; L∞(Ω)3).
Further-more, since kω0kL∞ < 3
4, then we deduce that there exists a maximal time 0 < T ≤ T
∗ such that for all
t ∈ [0, T [, kω(t)kL∞ <3
4.
Let us show that T = T∗. Indeed if T < T∗, then we get,
kω(T )kL∞ =
3
4. (71)
Thanks to Lemma 4.1 and the fact that kvkL∞(Ω×[0,T ]) ≤ 1 , there exists a constant c1 > 0 depending
only on s such that for all t ∈ [0, T ],
kω(t)k2 ˙ Hs+ ε 2 Z t 0 kω(τ )k2 ˙ Hs+1dτ ≤ kω0k 2 ˙ Hse c1 ε Rt 0kω(τ )k 2 L∞dτ, (72)
Thanks to (8), we have for a.e t ∈ [0, T ],
kω(t)kL∞ . kω(t)k 1 2 ˙ H1+rkω(t)k 1 2 ˙ H1−r.
Thanks to Cauchy-Schwarz inequality, we infer,
Z t 0 kω(τ )k2 L∞dτ . Z t 0 kω(τ )k2 ˙ H1+rdτ 12Z t 0 kω(τ )k2 ˙ H1−r(Ω)dτ 12 .
Thanks to Lemma 4.3 used first with σ = r and after with σ = −r, we deduce, Z t 0 kω(τ )k2 L∞dτ . kω0kH˙rkω0kH˙−reC Rt 0b(τ )dτ, (73)
where C > 0 is a constant and b(τ ) =
k∇ρ(τ )k2 L2+ ∇ v(τ ) 1+ρ(τ ) 2 L2
ε . Thanks to (73) and Lemma 4.2, we get that there exists a constant c2> 0 depending only on r, s such that for all t ∈ [0, T ],
Z t 0 kω(τ )k2 L∞dτ ≤ c2α0, where, α0 = kω0kH˙rkω0kH˙−re 4C ε2kω0k 2 L2.
Then, from (72), we deduce that for all t ∈ [0, T ],
kω(t)kH˙s≤ kω0kH˙se c1
εc2α0. (74)
Thanks again to Lemma 4.2, we have for all t ∈ [0, T ],
Thanks to (8), we have for all t ∈ [0, T ], kω(t)kL∞ . kω(t)k s−1 s L2 kω(t)k 1 s ˙ Hs. (76)
Then, using (75) and (74), from (76), we deduce that there exists two constant C1 > 0 and C2 > 0
depending only on r, s such that for all t ∈ [0, T ],
kω(t)kL∞ ≤ C1kω0k s−1 s L2 kω0k 1 s ˙ Hse C2 εα0. (77)
We assume in what follows that,
C1kω0k s−1 s L2 kω0k 1 s ˙ Hse C2 ε α0≤ 1 2. (78)
Then thanks to (77) and (78), we obtain a contradiction with (71), therefore T = T∗. Owing to (78), this means for all t ∈ [0, T∗[,
kω(t)kL∞ ≤
1
2, (79)
which leads to a contradiction with (70) and then T∗ = ∞. Therefore, under the assumption (78), we deduce that there exists a unique global strong solution ω ≡ (ρ, v) ∈ C([0, ∞[; ˙H−r(Ω)3∩ Hs(Ω)3) ∩
C1(]0, ∞[; Hs−1(Ω)3) ∩ C(]0, ∞[; Hs+1(Ω)3) of the system of Equations (30) for the initial data ω 0 ≡
(ρ0, v0) ∈ ˙H−r(Ω)3∩ Hs(Ω)3, moreover kωkL∞(Ω×[0,+∞[)≤ 1
2.
It remains to prove that ω ∈ C∞(Ω×]0, ∞[)3.
Let > 0, k ∈ N and k = (1 − 2−k), notice
k+1 > k. By considering for each k ∈ N, the system of equations (30) for the initial data ω(k), then
using a recurrence argument combined with Proposition 4.1 and uniqueness of the solution ω, we infer ω ∈ C([k, ∞[; ˙H−r(Ω)3∩ Hs+k(Ω)3) ∩ C1(]k, ∞[; Hs+k−1(Ω)3) ∩ C(]k, ∞[; Hs+k+1(Ω)3) for any k ∈ N.
Since for all k ∈ N, k < , then we get ω ∈ C([, ∞[; ˙H−r(Ω)3∩ Hs+k(Ω)3) ∩ C1([, ∞[; Hs+k−1(Ω)3) ∩
C([, ∞[; Hs+k+1(Ω)3
) for all k ∈ N. Then, thanks to Sobolev embedding and the system of equations (30), we deduce that ω ∈ C∞(Ω×]0, ∞[)3, which allows us to conclude the proof.
Let us compute the real α1given in Theorem 4.1 with ω0≡ (ρ0, v0) given by (28) and (29).
Let ˜n0= n0− n0 n0 and ˜m0= m0 n0
. We recall that m0= n0u0+ ε∇n0. Then, we get, ˜m0 = (1 + ˜n0)u0+
ε∇˜n0. We recall that Ω def
= √gn0O. Then, we notice, for all x ∈ Ω,
ρ0(x) = ˜n0 x √ gn0 and v0(x) = 1 √ gn0 ˜ m0 x √ gn0 . By introducing, ˜ ω0 def = n˜0pgn0, ˜m0 , (80)
we deduce that for all x ∈ Ω,
ω0(x) = 1 √ gn0 ˜ ω0 x √ gn0 . Then, we get α1= ˜ α1 √ gn0 , with, ˜ α1 def = C1k˜ω0k s−1 s L2 k˜ω0k 1 s ˙ Hse c1 εα˜0, (81) where, ˜ α0= k˜ω0kH˙rk˜ω0kH˙−re 4C ε2k˜ω0k 2 L2.
Then, an immediate consequence of Theorem 4.1 with initial data ω0∈ ˙H−r(Ω)3∩ Hs(Ω)3with 0 < r <
Corollary 4.1 Let (n0−n0, m0) ∈ ˙H−r(O)3∩Hs(O)3with 0 < r < 1 < s. If k˜ω0kL∞ ≤1 2 √ gn0(˜ω0given by (80)) and if ˜α1≤ 12 √
gn0 ( ˜α1 given by (81)), then there exists a unique global strong solution (n, m)
of the system of Equations (23) for the initial data (n0, m0) such that (n − n0, m) ∈ C([0, ∞[; ˙H−r(O)3∩
Hs(O)3) ∩ C1(]0, ∞[; Hs−1(O)3) ∩ C(]0, ∞[; Hs+1(O)3) and kn − n
0kL∞(O×[0,+∞[)≤
n0
2 . Furthermore, (n − n0, m) ∈ C∞(O×]0, ∞[)3.
Let us give a sufficient condition ensuring that ˜α1≤12
√ gn0.
For this, we assume n0− n0∈ Hs+1(O) ∩ W1,
2
1+r(O), u0∈ Hs(O)2∩ L 2
1+r(O)2 with 0 < r < 1 < s and
we assume also that there exists a constant C > 0, kn0− n0kHs+1+ kn0− n0k
W1,1+r2 ≤ C min(n0,
√
n0). (82)
We recall that ˜α1 is given by,
˜ α1 = C1k˜ω0k s−1 s L2 k˜ω0k 1 s ˙ Hse c1 εk˜ω0kHr˙ k˜ω0kH−r˙ e 4C ε2k ˜ω0k 2 L2 , ˜ ω0 = n0− n0 √ n0 √ g,n0u0+ ε∇n0 n0 , (83)
where C1> 0 and c1> 0 are constant.
Since n0u0 n0 = u0+ n0− n0 n0 u0and s > 1, we get, n0u0 n0 Hs . ku0kHs 1 + n0− n0 n0 Hs . ku0kHs. Then, we have, k˜ω0kHs ≤ n0− n0 √ n0 Hs √ g + n0u0 n0 Hs + ε n0 k∇n0kHs . C(√g + ε) + ku0kHs. (84)
For any f ∈ ˙H−r(O), we can write kf kH˙−r(O) as follows kf kH˙−r(O)= sup
ϕ∈C∞
0 (O)∩ ˙Hr(O),kϕkHr (O)˙ =1
|hf, ϕi|.
Since for all ϕ ∈ L1−r2 (O) ∩ ˙Hr(O), we have kϕk
L1−r2 . kϕkH˙r (see [1]), then we deduce that for all
f ∈ L1+r2 (O), f ∈ ˙H−r(O) and kf k˙
H−r(O). kf k L 2 1+r(O). Then, we get, k˜ω0kH˙−r . k˜ω0k L1+r2 ≤ n0− n0 √ n0 L1+r2 √ g + n0u0 n0 L1+r2 + ε n0 k∇n0k L1+r2 . C(√g + ε) + (1 + C)ku0k L1+r2 , (85)
where we have used the fact that n0 n0 L∞ ≤ 1 + n0− n0 n0 L∞. 1 + n0− n0 n0 Hs ≤ 1 + C.
Under the assumption (82), thanks to (84) and (85), from (83) we deduce that there exists a real C2> 0
depending only on r, s, g, ε, ku0kHs, ku0k
L1+r2 such that ˜α1≤ C2.
We get also k˜ω0kL∞ . k˜ω0kHs≤ C(√g + ε) + ku0kHs.
By using Corollary 4.1 and the first equation of (23) to obtain again more regularity on h, we deduce our main Theorem,
Theorem 4.2 Let n0− n0∈ W1,
2
1+r(O) ∩ Hs+1(O), u0∈ L 2
1+r(O)2∩ Hs(O)2 with 0 < r < 1 < s.
If kn0− n0kHs+1+ kn0− n0k
W1,1+r2 ≤ C0min(n0,
√
n0) with C0 > 0 a constant, then there exists a
real C > 0 depending only on C0, r, s, g, ε, ku0kHs, ku0k
unique global strong solution (n, u) of the system of Equations (1) for the initial data (n0, u0) such that
(n − n0, u) ∈ C([0, ∞[; ˙H−r(O)3∩ Hs(O)3) ∩ C1(]0, ∞[; Hs−1(O)3) and kn − n0kL∞(O×[0,+∞[) ≤
n0
2 . Moreover, (n − n0, u) ∈ C∞(O×]0, ∞[)3.
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