Global regularity of Quantum Navier-Stokes equations in R 2 L´eo Ag´elas

To cite this version: L´eo Ag´elas. Global regularity of Quantum Navier-Stokes equations in R 2. 2017.

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Global regularity of Quantum Navier-Stokes equations in R2 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 transport, 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 mechanics 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 chemistry 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   + ∇ · (nu) = 0, ∂t  √  (1) ∂nu ∆ n   + ∇ · (nu ⊗ u) − 2ε2 n∇ √ − 2ε∇ · (nDu) + ∇p(n) = 0 ∂t n ∗ Department of Mathematics, IFP Energies Nouvelles, 1-4, avenue de Bois-Pr´ eau, F-92852 Rueil-Malmaison, France ([email protected])

1

with initial conditions, n(x, 0) = n0 , u(x, 0) = u0 ,

(2)

on O = R2 /L0 Z2 , L0 > 0 a given real and with periodic boundary conditions and we require in addition T that n and u are periodic functions of period L where Du = ∇u + ∇u , p(n) = gn2 with g > 0 and 0

0

0

2

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 nonglobal-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, +∞[; H s (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 « « „ n

√x

gn

t , gn

0

−n0

√x

m

gn

t , gn

0

0 √0 and v(x, t) = . velocity) defined from n and m as follows ρ(x, t) = n0 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 √ def √ def in the homogeneous Sobolev space H˙ s (Ω), with Ω = gn0 O ( = R2 /( gn0 L0 )Z2 ) and s ≥ 0,

k(ρ(t), v(t))kH˙ s ≤ k(ρ0 , v0 )kH˙ s eC

Rt 0

(kρ(τ )k2L∞ +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∞ ≤ 34 , in Lemma 4.2, we show, !  2 Z t 

v(τ ) 2 2 2

k(ρ(t), v(t))kL2 + ε (4)

∇ 1 + ρ(τ ) 2 + k∇ρ(τ )kL2 dτ . k(ρ0 , v0 )kL2 . 0 L Since these Inequalities are valid only for kρkL∞ ≤ 43 , the challenge was then to find bounds for both kρkL∞ and kvkL∞ allowing to ensure that kρkL∞ ≤ 34 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∞ ≤ 34 and kvkL∞ ≤ 1, from (3) and (4), we show that for all s > 1, s−1

1

C s k(ρ(t), v(t))kL∞ . k(ρ0 , v0 )kL2s k(ρ0 , v0 )kH ˙ se

Rt 0

(kρ(τ )k2L∞ +kv(τ )k2L∞ )dτ

.

(5)

Rt It remained then the difficulty to obtain a bound of 0 (kρ(τ )k2L∞ + kv(τ )k2L∞ )dτ depending only on the initial data and not on the time. We overcome this difficulty thanks to Lemma 4.3 used with σ = r, 2

σ = −r, for 0 < r < 1, a fractional Gagliardo-Nirenberg inequality (8) and inequality (4), to obtain under the assumption that kρkL∞ ≤ 43 and kvkL∞ ≤ 1, Z 0

t

2

(kρ(τ )k2L∞ + kv(τ )k2L∞ )dτ . k(ρ0 , v0 )kH˙ r k(ρ0 , v0 )kH˙ −r eC1 k(ρ0 ,v0 )kL2 .

Therefore, from (5), we obtain, s−1

1

C2 k(ρ0 ,v0 )kH˙ r k(ρ0 ,v0 )kH˙ −r e s k(ρ(t), v(t))kL∞ . k(ρ0 , v0 )kL2s k(ρ0 , v0 )kH ˙ se

C1 k(ρ0 ,v0 )k2 2 L

,

and then infer the condition on the initial data ensuring that kρkL∞ ≤ 34 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 fˆ : Z2 7−→ C by the formula, Z 2π 1 ˆ e− L ix·k f (x) dx, f (k) = 2 L R2 /LZ2 when f is absolutely integrable on R2 /LZ2 . If f ∈ L2 (R2 /LZ2 ), then from Parseval equality, we have, Z X 1 L |fˆ(k)|2 = |f (x)|2 dx. L 0 2 k∈Z

Let s ∈ R, we define the Sobolev norm kf kH s (R2 /LZ2 ) of a tempered distribution f : R2 /LZ2 7−→ R by, kf kH s (R2 /LZ2 ) =



X

1+

k∈Z2

2π|k| L

2 !s

! 21 |fˆ(k)|2

,

and then we denote by H s (R2 /LZ2 ) the space of tempered distributions with finite H s (R2 /LZ2 ) norm. On the torus R2 /LZ2 , for s > −1, we also define the homogeneous Sobolev norm, kf kH˙ s (R2 /LZ2 ) =

! 21 X  2π|k| 2s 2 ˆ |f (k)| , L 2

k∈Z

and then we denote by H˙ s (R2 /LZ2 ) the space of tempered distributions with finite H˙ s (R2 /LZ2 ) norm. We use the Fourier transform to define the fractional Laplacian operator (−∆)α , α > −1 on R2 /LZ2 and we define it as follows, 2α  2π|k| α f (k) = \ (−∆) fˆ(k). L 3

3

Some estimates

In this section, we give a serie of Lemmata which will be used in the next section. Let Ω = R2 \LZ2 with L > 0 a real. We begin with the following inequality proved in [9] which states that for all f, g ∈ L∞ (Ω) ∩ H˙ s (Ω), kf gkH˙ s . (kf kL∞ kgkH˙ s + kf kH˙ s kgkL∞ ).

(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 ∈ H s (Ω) such that u(x) ∈ I, f (u) ∈ H s (Ω). More precisely there exists a non decreasing continuous function C depending only on s and max kf (k) kL∞ (I) such that, 0≤k≤[s]+2

kf (u)kH s ≤ C(kukL∞ )kukH s .

(7)

Let us mention also the following Interpolation inequality: for all (α, β) such that 0 ≤ α < 1 < β and for all u ∈ H β (Ω), kukL∞ (Ω) . kukγH˙ α (Ω) kukδH˙ β (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 ∈ H˙ σ (Ω) ∩ H˙ σ+1 (Ω) and g ∈ H˙ 1 (Ω). Then, we have, | hf Dg, hiH˙ σ | . (kf kH˙ σ k∇hkH˙ σ + khkH˙ σ k∇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 < δ < min( 12 , 12 − σ2 ). Then, we have

σ  σ | hf Dg, hiH˙ σ | = | A 2 (f Dg), A 2 h | D σ E 1 σ 1 = A 2 +δ− 2 (f Dg), A 2 + 2 −δ h ≤

σ

1

σ

+

1 2

−δ >

σ 2

that means

1

kA 2 +δ− 2 (f Dg)kL2 kA 2 + 2 −δ hkL2 ,

where we have used Cauchy-Schwarz inequality. Since 0 < [29], [12]), σ

σ 2

1

σ 2

+ δ < 12 , we have (see [19], see also (A5) in

σ

1

kA 2 +δ− 2 (f Dg)kL2 . kA 2 +δ f kL2 kA 2 gkL2 . Then, we deduce,

σ

σ

1

1

| hf Dg, hiH˙ σ | . kA 2 +δ f kL2 kA 2 + 2 −δ hkL2 kA 2 gkL2 .

(9)

Thanks to the fractional Gagliardo-Nirenberg inequality given by Corollary 1.5 in [22], we have for any u ∈ H 1 (Ω) and 0 ≤ θ ≤ 1, θ kukH˙ θ . kuk1−θ (10) ˙ 1. L2 kukH σ

We use Inequality (10) with u = A 2 ρ and θ = 2δ, then we get, σ

σ

σ

σ

1−2δ 2δ 2δ 2 kA 2 +δ f kL2 = kA 2 f kH˙ 2δ . kA 2 f k1−2δ ˙ 1 = kf kH ˙ σ. ˙ σ k∇f kH L2 kA f kH

(11)

σ

We use Inequality (10) with u = A 2 h and θ = 1 − 2δ to obtain, σ

1

σ

1−2δ kA 2 + 2 −δ hkL2 = kA 2 hkH˙ 1−2δ . khk2δ ˙ σ k∇hkH ˙σ . H

4

(12)

Therefore, from (9), using (11), (12) and Young inequality, we deduce, | hf Dg, hiH˙ σ | . (kf kH˙ σ k∇hkH˙ σ + khkH˙ σ k∇f kH˙ σ )k∇gkL2 , which concludes the proof.  Lemma 3.2 Let σ ≥ 0, 0 < γ < 1, % ∈ H σ (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 + % ˙ σ ≤ Ck%kH˙ σ . H Proof. If σ = 0, the proof holds. Let us assume that σ > 0. Let λ > 0, %λ (x) = %(λx) and f the x function defined by f (x) = then f ∈ C ∞ ([−γ, γ]) and k%λ kL∞ ≤ γ therefore thanks to (7), we 1+x deduce that there exists a constant C > 0 depending only on σ, γ such that,

%λ

(13)

1 + %λ σ ≤ Ck%λ kH σ . H From (13), we deduce that,

%λ

1 + %λ

L2



%λ

+ . k%λ kL2 + k%λ kH˙ σ . 1 + %λ ˙ σ

(14)

H

Then, we get,

σ

1

% +λ λ 1 + % L2 λ



% 1 λσ

2 + . k%k k%kH˙ σ . L

1 + % ˙ σ λ λ H

λ to obtain, λσ





1 1

% + % . σ k%kL2 + k%kH˙ σ .

σ λ 1 + % L2 1 + % H˙ σ λ

(15)

We multiply Inequality (15) by

(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 ∈ H˙ σ (Ω) ∩ H˙ σ+1 (Ω), g ∈ H˙ 1 (Ω) and % ∈ H˙ σ (Ω) ∩ H σ+1 (Ω) ∩ L∞ (Ω) such that k%kL∞ ≤ γ. Then, we have,   f . (kf k ˙ σ k∇hk ˙ σ + khk ˙ σ k∇f k ˙ σ )k∇gkL2 Dg, h H H H H 1+% ˙σ H + kf kL∞ (k%kH˙ σ k∇hkH˙ σ + khkH˙ σ k∇%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 that means − σ2 < δ < min( 12 , 12 − σ2 ). Similarly as (9), we have,     σ +δ

f f

kA σ2 + 12 −δ hkL2 kA 12 gkL2 . . A 2 Dg, h 1+%

2 1 + % σ ˙ H L From (12), we get, σ

1

1−2δ kA 2 + 2 −δ hkL2 . khk2δ ˙ σ k∇hkH ˙σ . H

5

Then, we deduce,     σ +δ

f f . A 2

Dg, h

1+% 1+% ˙σ

L2

H

1−2δ khk2δ ˙ σ k∇hkH ˙ σ k∇gkL2 . H

(17)



  

σ +δ

f f % % f



Since = ≤ kf kH˙ σ+2δ + f . =f −f , then A 2 1+% 1+% 1 + % L2 1 + % H˙ σ+2δ 1 + % H˙ σ+2δ Since σ + 2δ > 0, thanks to (6), we have,





%

%

f %



∞ . kf k + kf k ˙ σ+2δ L H

1 + % ˙ σ+2δ

1 + % L∞ H

1 + % H˙ σ+2δ

k%kL∞ % ≤ kf kH˙ σ+2δ + kf kL∞ (18)

1 + % ˙ σ+2δ 1 − k%kL∞ H

% γ

≤ kf kH˙ σ+2δ + kf kL∞

1 + % ˙ σ+2δ , 1−γ H

where we have used the fact that k%kL∞ ≤ γ. Then, we deduce,

 



σ +δ f

. kf k ˙ σ+2δ + kf kL∞ %

A 2 H

1 + % ˙ σ+2δ .

1 + % L2 H

(19)

Thanks to Lemma 3.2, we have,

%

1 + % ˙ σ+2δ . k%kH˙ σ+2δ . H

From (19), we infer,

 

σ +δ f

. kf k ˙ σ+2δ + kf kL∞ k%k ˙ σ+2δ .

A 2 H H

1 + % L2

(20)

Using the same arguments as in (11), we get, kf kH˙ σ+2δ k%kH˙ σ+2δ

1−2δ 2δ . kf kH ˙σ ˙ σ k∇f kH 1−2δ 2δ . k%kH k∇%k ˙ σ ˙ Hσ

From (17), using (20), (21) and Young inequalities, we deduce,   f . (kf k ˙ σ k∇hk ˙ σ + khk ˙ σ k∇f k ˙ σ )k∇gkL2 Dg, h H H H H 1+% ˙σ H +kf kL∞ (k%kH˙ σ k∇hkH˙ σ + khkH˙ σ k∇%kH˙ σ )k∇gkL2 ,

(21)

(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 = n0 u0 + 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. Thanks again to Theorem 2.1 in [28], under the assumption that n > 0, if (n, m) is solution of the system of equations (23) for the initial data (n0 , m0 ) then (n, u) is solution of the shallow water equations (1) 6

m − ε∇n for the initial data (n0 , u0 ) with u = . n Then, we consider the following system of equations,    ∂n + ∇ · m − ε∆n = 0, ∂t m  ∂m   +∇· ⊗ m − ε∆m + gn∇n = 0. ∂t n We introduce n ˜ defined by n ˜≡ of equations,

(23)

n − n0 m and m ˜ ≡ , from (23), we obtain the following equivalent system n0 n0

 ∂n ˜   +∇·m ˜ − ε∆˜ n = 0, ∂t   ∂m ˜ m ˜   ˜ ∇˜ n + gn0 ∇˜ n = 0. +∇· ⊗m ˜ − ε∆m ˜ + gn0 n ∂t 1+n ˜

(24)

For any λ ∈ R∗ , by using the rescaled solutions n ˜ λ (x, t) = n ˜ (λx, λ2 t) and m ˜ λ (x, t) = λm(λx, ˜ λ2 t), we obtain the equivalent system of equations,  ∂n ˜λ   +∇·m ˜ λ − ε∆˜ nλ = 0, ∂t   (25) m ˜λ ∂m ˜λ   +∇· ˜ λ ∇˜ nλ + gn0 λ2 ∇˜ nλ = 0. ⊗m ˜ λ − ε∆m ˜ λ + gn0 λ2 n ∂t 1+n ˜λ   t x   − n0 , n √gn gn 1 1 x t 0 0 Setting λ = √ and v(x, t) = √ in (25), then with ρ(x, t) = m √ , , n0 gn0 n0 gn0 gn0 gn0 provided that ρ > −1, we deduce that the system of equations (23) on O × [0, T0 ] is equivalent to the def √ following system of equations on Ω × [0, T ] with Ω = gn0 O and T = gn0 T0 ,  ∂ρ   + ∇ · v − ε∆ρ = 0, ∂t   (26) ∂v v   +∇· ⊗ v − ε∆v + ρ∇ρ + ∇ρ = 0. ∂t 1+ρ with initial conditions, ρ(x, 0) = ρ0 (x), v(x, 0) = v0 (x), where n0



ρ0 (x) = and

√x gn0



− n0 (28)

n0

1 v0 (x) = √ m0 n0 gn0



(27)

x √ gn0

 .

Then, in what follows, we study the following system of equations,  ∂ρ   + ∇ · v − ε∆ρ = 0, ∂t   v ∂v   +∇· ⊗ v − ε∆v + (1 + ρ)∇ρ = 0. ∂t 1+ρ

(29)

(30)

with initial conditions, ρ(x, 0) = ρ0 (x), v(x, 0) = v0 (x). We thus establish the following energy estimate in H s for any s > 1.

7

(31)

Lemma 4.1 Let 0 < δ < 1, ω0 ≡ (ρ0 , v0 ) ∈ H s (Ω)3 , s ≥ 0. If ω ≡ (ρ, v) ∈ C([0, T ]; H s (Ω)3 ) ∩ C(]0, T ]; H s+1 (Ω)3 ) with kρkL∞ (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 ], Z c1 R τ ε τ kω(σ)k2H˙ s+1 dσ ≤ kω0 k2H˙ s e ε 0 a(σ)dσ , kω(τ )k2H˙ s + 2 0 where a(σ) = kω(σ)k2L∞ (1 + kv(σ)k2L∞ ). Proof. We take the inner product in H˙ s (Ω) of the first equation of (30) with ρ, use integrations by parts to obtain, 1 d (32) kρk2H˙ s + εk∇ρk2H˙ s = hv, ∇ρiH˙ s . 2 dt Now, we take the inner product in H˙ s (Ω)2 of the second equation of (30) with v, use integrations by parts to obtain,    2  v ρ 1 d 2 2 kvkH˙ s + εk∇vkH˙ s = ⊗ v, ∇v + ,∇ · v − h∇ρ, viH˙ s . (33) 2 dt 1+ρ 2 ˙s ˙s H H Thanks to Cauchy-Schwarz inequality and Young   v ≤ ⊗ v, ∇v 1+ρ ˙s H

inequality, we have,

v

k∇vk ˙ s ⊗ v H

1 + ρ

˙s H

2

1

v ⊗ v + ε k∇vk2˙ s ≤

˙s 2

H 2ε 1 + ρ H

and  2  ρ ≤ , ∇ · v 2 ˙s H ≤ ≤

2 k∇ · vkH˙ s

ρ ˙ s H 2

1 2

ρ2 ˙ s + ε k∇ · vk2˙ s H H 2ε 8

ε 1 2

ρ2 ˙ s + k∇vk2˙ s . H H 2ε 4

Then, we deduce, 1 d ε kvk2H˙ s + k∇vk2H˙ s 2 dt 4

≤

2

1

v ⊗ v + 1 ρ2 2˙ s − h∇ρ, vi ˙ s . H

H 2ε 1 + ρ 2ε ˙s H

(34)

We sum Equations (32) and (34), to obtain, 1 d ε (kρk2H˙ s + kvk2H˙ s ) + εk∇ρk2H˙ s + k∇vk2H˙ s 2 dt 4

2

1 v 1

ρ2 2˙ s . ≤ ⊗ v +

H 2ε 1 + ρ 2ε ˙s H

(35)

Thanks to (6), we get, and also

kρ2 kH˙ s . kρkL∞ kρkH˙ s

(36)





v





. v kvkL∞ + v kvk ˙ s . ⊗ v H

1 + ρ

˙ s 1 + ρ ˙ s

1 + ρ ∞ H H L

(37)



v



≤ kvk ˙ s + vρ . H

1 + ρ ˙ s

1 + ρ ˙ s H H

(38)

Furthermore, we have,

8

Thanks to (6), we have also,





vρ



. kvkL∞ ρ + kvk ˙ s ρ . H

1 + ρ ˙ s

1 + ρ ˙ s 1 + ρ L∞ H H Then with (38) and (39), we deduce,







v

ρ



. kvk ˙ s 1 + ρ

∞ H

1 + ρ ˙ s

1 + ρ ∞ + kvkL 1 + ρ ˙ s . H L H Therefore, using (40) and (37), we get,







v

ρ

ρ

v

2





1 + ρ ⊗ v ˙ s . 1 + 1 + ρ ∞ kvkH˙ s kvkL∞ + kvkL∞ 1 + ρ ˙ s + 1 + ρ ∞ kvkH˙ s H L

H L

ρ 2

kvk2L∞ kvkH˙ s kvkL∞ + ≤

1 + ρ ˙ s 1−δ H ,

(39)

(40)

(41)

where we have used the fact that kρkL∞ ≤ δ < 1. Then, using (41) and (36), from (35), we deduce, 1 d ε (kρk2H˙ s + kvk2H˙ s ) + εk∇ρk2H˙ s + k∇vk2H˙ s 2 dt 4

1 kvk2 kvk2 ∞ ε H˙ s 2 L 1 ρ

kvk4L∞ + 1 kρk2L∞ kρk2˙ s . + H ε 1 + ρ H˙ s 2ε .

(42)

Thanks to Lemma 3.2, from (42), we deduce that for all t ∈]0, T ], 1 d ε (kρ(t)k2H˙ s + kv(t)k2H˙ s ) + εk∇ρ(t)k2H˙ s + k∇v(t)k2H˙ s 2 dt 4

1 . kv(t)k2H˙ s kv(t)k2L∞ ε 1 1 2 + kρ(t)k2H˙ s kv(t)k4L∞ + kρ(t)k2L∞ kρ(t)kH˙ s ε 2ε (43)

which implies, 1 d (kρ(τ )k2H˙ s + kv(τ )k2H˙ s ) 2 dτ

ε +εk∇ρ(τ )k2H˙ s + k∇v(τ )k2H˙ s 4 1 . (kv(τ )k2L∞ + kv(τ )k4L∞ + kρ(τ )k2L∞ )(kρ(τ )k2H˙ s + kv(τ )k2H˙ s ). ε

(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 ], Z τ  R ε 2 2 2 C 0τ a(σ)dσ 2 k∇v(σ)k , kρ(τ )k2H˙ s + kv(τ )k2H˙ s + 2εk∇ρ(σ)kH + ˙ s dσ ≤ (kρ0 kH ˙ s + kv0 kH ˙ s )e ˙s H 2 0 where a(σ) = 1ε (kv(σ)k2L∞ +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 ∩ H s (Ω)3 with 0 ≤ r < 1 < s, kρ0 kL∞ < 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 ∩ H s (Ω)3 ) ∩ C 1 (]0, T ∗ [; H s−1 (Ω)3 ) ∩ C(]0, T ∗ [; H s+1 (Ω)3 ) of the system of Equations (30)-(31). Moreover if T ∗ < ∞, then either kρkL∞ (Ω×[0,T ∗ ]) ≥ 1 or either Z

T∗

kv(τ )k4L∞ dτ = ∞.

0

9

(45)

Proof.

3 + kρ0 kL∞ > 1 and δ = 12 (1 + kρ0 kL∞ ) < 1. 2(1 + kρ0 kL∞ ) 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. λ λ

We notice for all ≥ a, 1 + λχ λδ = 1, and for all < a, 1 + λχ λδ ≥ 1 − |λ| ≥ 1 − δa = δ δ 1 ∞ ) > 0. Then, we get for all λ ∈ R, (1 − kρ k 0 L 4   1 λ ≥ (1 − kρ0 kL∞ ) > 0. 1 + λχ (46) δ 4 Let a =

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 ∂t ω = Aω + f (ω),

(47)

with initial conditions ω(0) = ω0 .

(48)

s−1

3

More precisely, we use Proposition 2.1 in [2] with X = H (Ω) , A = ε∆ for our generator of holomorphic semigroup T (t) = e−tA of bounded linear operators on X and f our locally Lipschitz continuous function on Xα = H s (Ω)3 , α = 21 defined by f (ω) = (f1 (ω), f2 (ω)) with   v f1 (ω) = −∇ · v and f2 (ω) = −∇ · ⊗ v − ρ∇ρ − ∇ρ. (49) 1 + ρχ( ρδ ) Indeed, f is locally Lipschitz continuous on H s (Ω)3 , since for all ω1 = (ρ1 , v1 ) ∈ H s (Ω)3 and ω2 = (ρ2 , v2 ) ∈ H s (Ω)3 , we have, k∇ · v1 − ∇ · v2 kH s−1 k∇ρ1 − ∇ρ2 kH s−1 kρ1 ∇ρ1 − ρ2 ∇ρ2 kH s−1

≤ kv1 − v2 kH s ≤ kρ1 − ρ2 kH s 1 = k∇((ρ1 − ρ2 )(ρ1 + ρ2 ))kH s−1 2 1 ≤ k(ρ1 − ρ2 )(ρ1 + ρ2 )kH s 2 . kρ1 − ρ2 kH s (kρ1 kH s + kρ2 kH s ),

(50)

where we have used the fact that for all h1 ∈ H s (Ω), h2 ∈ H s (Ω) with s > 1 (see [9]), kh1 h2 kH s . kh1 kH s kh2 kH s ,

(51)

1 , thanks to (46) we notice that for all 1 + λχ( λδ ) k ∈ N, there exists a real Ck > 0 depending only on δ such that We introduce the function g ∈ C ∞ defined by g(λ) =

kg (k) kL∞ ≤ Ck .

(52)

Then, we get, k∇ · (g(ρ1 )v1 ⊗ v1 ) − ∇ · (g(ρ2 )v2 ⊗ v2 )kH s−1

10

≤ kg(ρ1 )v1 ⊗ v1 − g(ρ2 )v2 ⊗ v2 kH s ≤ k(g(ρ1 )v1 − g(ρ2 )v2 ) ⊗ v1 kH s + kg(ρ2 )v2 ⊗ (v1 − v2 )kH s ≤ k(v1 − v2 ) ⊗ v1 g(ρ1 )kH s +kv2 ⊗ v1 (g(ρ1 ) − g(ρ2 ))kH s +kg(ρ2 )v2 ⊗ (v1 − v2 )kH s .

(53)

Using (51), we estimate each term on the right hand side of Inequality (53), k(v1 − v2 ) ⊗ v1 g(ρ1 )kH s kv2 ⊗ v1 (g(ρ1 ) − g(ρ2 ))kH s kg(ρ2 )v2 ⊗ (v1 − v2 )kH s

. kv1 − v2 kH s kv1 kH s (1 + kg(ρ1 ) − 1kH s ) . kv1 kH s kv2 kH s kg(ρ1 ) − g(ρ2 )kH s . kv1 − v2 kH s kv2 kH s (1 + kg(ρ2 ) − 1kH s ).

(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 ) − 1kH s

.

C(kρ1 kL∞ )kρ1 kH s

kg(ρ2 ) − 1kH s

.

C(kρ2 kL∞ )kρ2 kH s .

Furthermore, using Taylor formula at order 1, we get g(ρ2 ) − g(ρ1 ) = (ρ2 − ρ1 ) then we deduce, Z 1 kg 0 ((1 − σ)ρ1 + σρ2 )kH s dσ kg(ρ1 ) − g(ρ2 )kH s . kρ1 − ρ2 kH s

R1 0

g 0 ((1 − σ)ρ1 + σρ2 )dσ,

0

Z . kρ1 − ρ2 kH s

1

C1 (k(1 − σ)ρ1 + σρ2 kL∞ )k(1 − σ)ρ1 + σρ2 )kH s dσ, 0

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 )kH s

. kρ1 − ρ2 kH s C1 (kρ1 kL∞ + kρ2 kL∞ )(kρ1 kH s + kσρ2 )kH s ).

Thanks to (54) and the Sobolev embedding H s (Ω) ,→ L∞ (Ω) since s > 1, from (53), we deduce therefore, k∇ · (g(ρ1 )v1 ⊗ v1 ) − ∇ · (g(ρ2 )v2 ⊗ v2 )kH s . C2 (kρ1 kH s , kρ2 kH s , kv1 kH s , kv2 kH s )(kv1 − v2 kH s + kρ1 − ρ2 kH s ),

(55)

where C2 > 0 is a continuous function on R4 . Using (50) and (55), we get that for all ω1 ∈ H s (Ω)3 , ω2 ∈ H s (Ω)3 , kf (ω1 ) − f (ω2 )kH s−1 . C3 (kω1 kH s , kω2 kH s )kω1 − ω2 kH s , (56) where C3 > 0 is a continuous function on Ω, which proves that f is well locally Lipschitz continuous on H s (Ω)3 . Then, we deduce thanks to Proposition 2.1 combined with Theorem 3.1 in [2], that there exists a maximal time T 0 > 0 such that there exists an unique solution ω = (ρ, v) ∈ C([0, T 0 [; H s (Ω)3 ) ∩ C 1 (]0, T 0 [; H s−1 (Ω)3 ) of the system of Equations (47)-(48). Moreover if T 0 < ∞ then lim∗ kω(t)kH s = ∞. From (56), we get that for all w ∈ H s (Ω)3 , kf (w)kH s−1 . t→T

C3 (kwkH s , 0)kwkH s , then we deduce from (47) that ω ∈ C(]0, T 0 [; H s+1 (Ω)). Then, we write ω under its integral form, Z t −εt∆ ω(t) = e ω0 + e−ε(t−σ)∆ f (ω(σ))dσ.

(57)

0

Since s > 1 − r > 0, then by Interpolation inequality, we have H s (Ω) ,→ H˙ 1−r (Ω), then using the same arguments as for (56), we get that for all w ∈ H˙ −r ∩ H s , kf (w)kH˙ −r . C3 (kwkH s , 0)kwkH s . Since ω0 ∈ H −r (Ω), therefore, from (57), we deduce also that ω ∈ C([0, T 0 [; H −r (Ω)). Since kρ0 kL∞ < 1 then kρ0 kL∞ < δ, moreover ρ ∈ C([0, T 0 [; L∞ (Ω)) due to the Sobolev embedding H s (Ω) ,→ L∞ (Ω), hence we deduce that there exists a time 0 < T < T 0 such that kρkL∞ (Ω×[0,T ]) ≤ δ. Since kρkL∞ (Ω×[0,T ]) ≤ δ then we get χ( ρδ ) = 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 ω 0 = (ρ0 , v 0 ) ∈ C([0, T ∗ [; H −r ∩ H s (Ω)) ∩ C 1 (]0, T ∗ [; H s−1 (Ω)) ∩ C(]0, T ∗ [; H s+1 (Ω)) of the system of equations (30)-(31). We get that if T ∗ < 11

∞, then either kρkL∞ (Ω×[0,T ∗ ]) ≥ 1 or either lim∗ kω 0 (t)kH s = ∞ (by using the same arguments as t→T RT ω) and in this case thanks to Lemma 4.1 and the fact that for any 0 < T < T ∗ , 0 kv(τ )k2L∞ dτ ≤ Z T∗  21 √ R T T 0 kv(τ )k4L∞ dτ , we get kv(τ )k4L∞ dτ = ∞, then, we conclude the proof. 0

 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 kρ0 kL∞ ≤ 34 . If ω ≡ (ρ, v) ∈ C([0, T ]; L2 (Ω))3 ∩ C 1 (]0, T ]; L2 (Ω)3 ) with ρ ∈ L∞ (Ω × [0, T ]) such that kρkL∞ (Ω×[0,T ]) ≤ 43 is a solution of the system of Equations (30)-(31), then we have for all t ∈ [0, T ], !

  2 Z t

v(τ ) 1 1 2 2

∇

+ k∇ρ(τ )k 2 dτ ≤ 2kω0 k2 2 . kω(t)kL2 + ε L L

2 4 4 1 + ρ(τ ) 0 L Proof. We can write the system of equations (30) as follows,    ∂η + ∇ · (ηw) − ε∆η = 0, ∂t ∂(ηw)   + ∇ · (ηw ⊗ w) − ε∆(ηw) + ∇p(η) = 0, ∂t

(58)

η2 v , η = 1 + ρ and p(η) = . By using the enthalpy h(η) = η − 1, we notice ηh0 (η) = p0 (η), 1+ρ 2 then from Equation (15) of [28], we obtain, where w =

d dt where H(η) =

Rη 1

d dt

Z  Ω

 Z 1 2 η|w| + H(η) + (εη|∇w|2 + |∇η|2 ) = 0, 2 Ω

h(τ )dτ , which yields to, Z  Ω

|v|2 ρ2 + 2(1 + ρ) 2



Z +ε Ω

!   2 v + |∇ρ|2 = 0. (1 + ρ) ∇ 1+ρ

We integrate Equation just above on [0, t] to obtain, for all t ∈ [0, T ],

2 Z t

1

v(t) 1

p

+ kρ(t)k2L2 + ε 2 1 + ρ(t) 2 2 0 L

=

2

1

√ v0

2 1 + ρ0 L2

+

1 kρ0 k2L2 . 2

L

and kρ0 kL∞ ≤ 43 , then we deduce that for all t ∈ [0, T ], !

  2 Z t

1 1 v(τ ) 2 2

∇

+ k∇ρ(τ )k 2 dτ ≤ 2 kv0 k2 2 + 1 kρ0 k2 2 , + kρ(t)kL2 + ε L L L

2 4 1 + ρ(τ ) L2 2 0

Since kρkL∞ (Ω×[0,T ]) ≤ 1 2 kv(t)kL2 4

!

  2

p v(τ ) 2

+ k∇ρ(τ )k 2 dτ

1 + ρ(τ )∇ L

1 + ρ(τ ) 2

3 4

which concludes the proof. 

  2 Rt Rt

v(τ ) The following Lemma will help us to express 0 k(ρ, v)(σ)k2L∞ dσ in terms of 0 (k∇ρ(τ )k2L2 + ∇ 1+ρ(τ ) )dτ . L2

Lemma 4.3 Let ω0 ≡ (ρ0 , v0 ) ∈ H˙ σ (Ω)3 ∩ L2 (Ω)3 , σ ∈ R, 0 < |σ| < 1. If ω ≡ (ρ, v) ∈ C([0, T ]; H˙ σ (Ω)3 ) ∩ C(]0, T ]; H 1+σ (Ω)3 ) ∩ L∞ (Ω × [0, T ])3 is a solution of the system of 12

RT Equations (30)-(31) with kωkL∞ (Ω×[0,T ])3 ≤ 43 and such that 0 a(τ )dτ < ∞, then there exists a constant C > 0 such that for all s ∈ [0, T ], Z s Rt kω(s)k2H˙ σ + ε kω(τ )k2H˙ σ+1 dτ ≤ kω0 k2H˙ σ eC 0 a(τ )dτ , 0

where a(τ ) =

k∇ρ(τ )k2L2

  2

v(τ ) + ∇ 1+ρ(τ ) 2 L

ε

.

Proof. We take the inner product in H˙ σ (Ω) of the first equation of (30) with ρ, use integrations by parts to obtain, 1 d (59) kρk2H˙ σ + εk∇ρk2H˙ σ = hv, ∇ρiH˙ σ . 2 dt Now, we take the inner product in H˙ σ (Ω) of the second equation of (30) with v, to obtain, 1 d kvk2H˙ σ + εk∇vk2H˙ σ 2 dt

    v =− ∇· − hρ∇ρ, viH˙ σ − h∇ρ, viH˙ σ . ⊗ v ,v 1+ρ ˙σ H

(60)

We sum Equations (59) and (60), to obtain, 1 d (kρk2H˙ σ + kvk2H˙ σ ) + εk∇ρk2H˙ σ + εk∇vk2H˙ σ 2 dt

 =− ∇·



  v − hρ∇ρ, viH˙ σ . ⊗ v ,v 1+ρ ˙σ H

(61)

Thanks to Lemma 3.1, we get, − hρ∇ρ, viH˙ σ . (kρkH˙ σ k∇vkH˙ σ + kvkH˙ σ k∇ρkH˙ σ )k∇ρkL2 . We estimate now the first term at the right hand side of Equation (61). We notice ∇ ·     v v · ∇ v + ∇ · 1+ρ 1+ρ v, then we deduce,  − ∇·



  v ⊗ v ,v 1+ρ ˙σ H

 = −



v 1+ρ

      v v − ∇· . · ∇ v, v v, v 1+ρ 1+ρ ˙σ ˙σ H H

Thanks to Lemma 3.1, we get,

     

v v

. − ∇· . kvkH˙ σ k∇vkH˙ σ ∇ v, v

2 1+ρ 1 + ρ σ ˙ H L Thanks to Lemma 3.3, we have,    v − · ∇ v, v 1+ρ ˙σ H

(62)  ⊗v =

. kvkH˙ σ k∇vkH˙ σ k∇vkL2

(63)

(64)

+kvkL∞ (kρkH˙ σ k∇vkH˙ σ + kvkH˙ σ k∇ρkH˙ σ )k∇vkL2 .

Using inequalities (62)-(64) and Young inequalities, from (61), we deduce that for all t ∈ [0, T ], 1 d (kρ(t)k2H˙ σ + kv(t)k2H˙ σ ) 2 dt

ε ε + k∇ρ(t)k2H˙ σ + k∇v(t)k2H˙ σ 2 2 1 1 . kρ(t)k2H˙ σ k∇ρ(t)k2L2 + kv(t)k2H˙ σ k∇ρ(t)k2L2 ε ε   2 !

1 v(t)

+ kv(t)k2H˙ σ k∇v(t)k2L2 + ∇

ε 1 + ρ(t) L2 1 kv(t)k2L∞ + kv(t)k2L∞ kρ(t)k2H˙ σ k∇v(t)k2L2 + kv(t)k2H˙ σ k∇v(t)k2L2 . ε ε 13

(65)

Using the fact that kvkL∞ (Ω×[0,T ])2 ≤ 34 , from (65), we infer that there exists a constant C > 0 depending only on σ such that, 1 d (kρ(s)k2H˙ σ + kv(s)k2H˙ σ ) 2 ds

ε ε + k∇ρ(s)k2H˙ σ + k∇v(s)k2H˙ σ 2 2   2

v(s) k∇ρ(s)k2L2 + ∇ 1+ρ(s)

2 + k∇v(s)k2L2 L ≤C (kρ(s)k2H˙ σ + kv(s)k2H˙ σ ), ε (66)

which implies, 1 d (kρ(s)k2H˙ σ + kv(s)k2H˙ σ ) ≤ C 2 ds

  2

v(s) k∇ρ(s)k2L2 + ∇ 1+ρ(s)

+ k∇v(s)k2L2 L2

ε

(kρ(s)k2H˙ σ + kv(s)k2H˙ σ ). (67)

Then, using Gronwall Inequality, from (67), we deduce for all s ∈ [0, T ], Rt

(68) kρ(s)k2H˙ σ + kv(s)k2H˙ σ ≤ (kρ0 k2H˙ σ + kv0 k2H˙ σ )e2C 0 b(τ )dτ ,

  2

v(τ ) 2 k∇ρ(τ )k2L2 + ∇ 1+ρ(τ ) 2 + k∇v(τ )kL2 L where b(τ ) = . After integrating (66) and using (68), we ε deduce that for all s ∈ [0, T ], Z s Rt
2 2 kρ(s)kH˙ σ + kv(s)kH˙ σ + εk∇ρ(τ )k2H˙ σ + εk∇v(τ )k2H˙ σ dτ ≤ (kρ0 k2H˙ σ + kv0 k2H˙ σ )e2C 0 b(τ )dτ . (69) 0

   v(τ ) v(τ ) v(τ ) (1 + ρ(τ )) = ∇ (1 + ρ(τ )) + ⊗ ∇ρ(τ ), then We notice that ∇v(τ ) = ∇ 1 + ρ(τ ) 1 + ρ(τ ) 1 + ρ(τ )   v(τ ) (1 + |ρ(τ )|) + |v(τ )| |∇ρ(τ )|. we get, |∇v(τ )| ≤ ∇ 1 + ρ(τ ) 1 − |ρ(τ )| Since kρkL∞ (Ω×[0,T ]) ≤ 43 and kvkL∞ (Ω×[0,T ])2 ≤ 43 , then we deduce for a.e τ ∈ [0, T ],   2 v(τ ) 2 7 v(τ ) + 3|∇ρ(τ )|. Therefore, we deduce that b(τ ) . k∇ρ(τ )kL2 +k∇( 1+ρ(τ ) )kL2 and |∇v(τ )| ≤ ∇ ε 4 1 + ρ(τ ) 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 ∩ H s (Ω)3 with 0 < r < 1 < s and kω0 kL∞ ≤ 21 . Then there exists a real α1 > 0 depending only on r, s, v0 , ρ0 such if α1 ≤ 21 , then there exists a unique global strong solution ω ≡ (ρ, v) ∈ C([0, ∞[; H˙ −r (Ω)3 ∩ H s (Ω)3 ) ∩ C 1 (]0, ∞[; H s−1 (Ω)3 ) ∩ C(]0, ∞[; H s+1 (Ω)3 ) of the system of Equations (30)-(31) with kωkL∞ (Ω×[0,+∞[)3 ≤ 21 . Furthermore, ω ∈ C ∞ (Ω×]0, ∞[)3 and the real α1 is given by, s−1

1

s α1 = C1 kω0 kL2s kω0 kH ˙ se

c1 ε

α0

,

where, α0

=

C

2

kω0 kH˙ r kω0 kH˙ −r e ε2 kω0 kL2 ,

with c1 > 0, C > 0 and C1 > 0 constants depending only on r, s. Moreoever, for all t ≥ 0, Z t kω(τ )k2L∞ (Ω) dτ ≤ c1 α0 . 0

14

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 ∩ H s (Ω)3 ) ∩ C 1 (]0, T ∗ [; H s−1 (Ω)3 ) ∩ C(]0, T ∗ [; H s+1 (Ω)3 ) of the system of Equations (30)-(31). Moreover if T ∗ < ∞, then either kρkL∞ (Ω×[0,T ∗ ]) ≥ 1 or either T∗

Z

kv(τ )k4L∞ = ∞.

(70)

0

Let us assume that T ∗ < ∞. Since s > 1, we get the Sobolev embedding H s (Ω) ,→ L∞ (Ω) and then ω ∈ C([0, T ∗ [; L∞ (Ω)3 ). Furthermore, since kω0 kL∞ < 43 , then we deduce that there exists a maximal time 0 < T ≤ T ∗ such that for all t ∈ [0, T [, kω(t)kL∞ < 34 . 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 ], Z c1 R t 2 ε t kω(t)k2H˙ s + kω(τ )k2H˙ s+1 dτ ≤ kω0 k2H˙ s e ε 0 kω(τ )kL∞ dτ , (72) 2 0 Thanks to (8), we have for a.e t ∈ [0, T ], 1

1

2 2 . kω(t)kH ˙ 1+r kω(t)kH ˙ 1−r .

kω(t)kL∞

Thanks to Cauchy-Schwarz inequality, we infer, Z

t

kω(τ )k2L∞ dτ

Z .

0

0

t

kω(τ )k2H˙ 1+r dτ

 12 Z

t

0

kω(τ )k2H˙ 1−r (Ω) dτ

Thanks to Lemma 4.3 used first with σ = r and after with σ = −r, we deduce, Z t Rt kω(τ )k2L∞ dτ . kω0 kH˙ r kω0 kH˙ −r eC 0 b(τ )dτ ,

 12 .

(73)

0

  2

v(τ ) k∇ρ(τ )k2L2 + ∇ 1+ρ(τ ) 2

L . Thanks to (73) and Lemma 4.2, ε we get that there exists a constant c2 > 0 depending only on r, s such that for all t ∈ [0, T ], Z t kω(τ )k2L∞ dτ ≤ c2 α0 ,

where C > 0 is a constant and b(τ ) =

0

where, α0

2

4C

= kω0 kH˙ r kω0 kH˙ −r e ε2 kω0 kL2 .

Then, from (72), we deduce that for all t ∈ [0, T ], kω(t)kH˙ s ≤ kω0 kH˙ s e

c1 ε

c2 α0

.

(74)

Thanks again to Lemma 4.2, we have for all t ∈ [0, T ], kω(t)kL2

. kω0 kL2 . 15

(75)

Thanks to (8), we have for all t ∈ [0, T ], s−1

kω(t)kL∞

1

(76)

s . kω(t)kL2s kω(t)kH ˙ s.

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 ], s−1

kω(t)kL∞

1

s ≤ C1 kω0 kL2s kω0 kH ˙ se

C2 ε

α0

.

(77)

We assume in what follows that, s−1

1

s C1 kω0 kL2s kω0 kH ˙ se

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 ∗ [, 1 (79) kω(t)kL∞ ≤ , 2 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 ∩ H s (Ω)3 ) ∩ C 1 (]0, ∞[; H s−1 (Ω)3 ) ∩ C(]0, ∞[; H s+1 (Ω)3 ) of the system of Equations (30) for the initial data ω0 ≡ (ρ0 , v0 ) ∈ H˙ −r (Ω)3 ∩ H s (Ω)3 , moreover kωkL∞ (Ω×[0,+∞[) ≤ 21 . 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 ∩ H s+k (Ω)3 ) ∩ C 1 (]k , ∞[; H s+k−1 (Ω)3 ) ∩ C(]k , ∞[; H s+k+1 (Ω)3 ) for any k ∈ N. Since for all k ∈ N, k < , then we get ω ∈ C([, ∞[; H˙ −r (Ω)3 ∩ H s+k (Ω)3 ) ∩ C 1 ([, ∞[; H s+k−1 (Ω)3 ) ∩ C([, ∞[; H s+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 α1 given in Theorem 4.1 with ω0 ≡ (ρ0 , v0 ) given by (28) and (29). n0 − n0 m0 Let n ˜0 = and m ˜0 = . We recall that m0 = n0 u0 + ε∇n0 . Then, we get, m ˜ 0 = (1 + n ˜ 0 )u0 + n0 n0 def √ ε∇˜ n0 . We recall that Ω = gn0 O. Then, we notice, for all x ∈ Ω,     x 1 x ρ0 (x) = n ˜0 √ m ˜0 √ and v0 (x) = √ . gn0 gn0 gn0 By introducing, def

ω ˜0 =



n ˜0

 p gn0 , m ˜0 ,

(80)

we deduce that for all x ∈ Ω, ω0 (x) = √

1 ω ˜0 gn0

 √

x gn0

 .

α ˜1 Then, we get α1 = √ , with, gn0 def

s−1

1

s α ˜ 1 = C1 k˜ ω0 kL2s k˜ ω0 kH ˙ se

c1 ε

α ˜0

,

(81)

where, 4C

2

α ˜ 0 = k˜ ω0 kH˙ r k˜ ω0 kH˙ −r e ε2 k˜ω0 kL2 . Then, an immediate consequence of Theorem 4.1 with initial data ω0 ∈ H˙ −r (Ω)3 ∩ H s (Ω)3 with 0 < r < 1 < s is the result which follows. 16

√ Corollary 4.1 Let (n0√ −n0 , m0 ) ∈ H˙ −r (O)3 ∩H s (O)3 with 0 < r < 1 < s. If k˜ ω0 kL∞ ≤ 12 gn0 (˜ ω0 given 1 α1 given by (81)), then there exists a unique global strong solution (n, m) by (80)) and if α ˜ 1 ≤ 2 gn0 (˜ of the system of Equations (23) for the initial data (n0 , m0 ) such that (n − n0 , m) ∈ C([0, ∞[; H˙ −r (O)3 ∩ n0 H s (O)3 ) ∩ C 1 (]0, ∞[; H s−1 (O)3 ) ∩ C(]0, ∞[; H s+1 (O)3 ) and kn − n0 kL∞ (O×[0,+∞[) ≤ . Furthermore, 2 ∞ 3 (n − n0 , m) ∈ C (O×]0, ∞[) . √ Let us give a sufficient condition ensuring that α ˜ 1 ≤ 12 gn0 . 2 2 For this, we assume n0 − n0 ∈ H s+1 (O) ∩ W 1, 1+r (O), u0 ∈ H s (O)2 ∩ L 1+r (O)2 with 0 < r < 1 < s and we assume also that there exists a constant C > 0, √ 2 kn0 − n0 kH s+1 + kn0 − n0 k 1, 1+r ≤ C min(n0 , n0 ). (82) W

We recall that α ˜ 1 is given by, s−1

α ˜1 ω ˜0

1

2 4C kω 2 ˜ 0 kL2

c1

ω0 kH˙ r k˜ ω0 kH˙ −r e ε s ε k˜ ω0 kH = C1 k˜ ω0 kL2s k˜ ˙ se   n0 − n0 √ n0 u0 + ε∇n0 √ , g, = n0 n0

, (83)

where C1 > 0 and c1 > 0 are constant. n0 u0 n0 − n0 Since = u0 + u0 and s > 1, we get, n0 n0



 

n0 u0



. ku0 kH s 1 + n0 − n0

n0 s

n0 s . ku0 kH s . H H Then, we have, k˜ ω0 kH s



n0 u0

n0 − n0 √

+ ε k∇n0 kH s

g+ ≤ √

n0 H s n0 √ n0 H s . C( g + ε) + ku0 kH s .

For any f ∈ H˙ −r (O), we can write kf kH˙ −r (O) as follows kf kH˙ −r (O) =

(84) sup

|hf, ϕi|.

˙ r (O),kϕk ˙ r ϕ∈C0∞ (O)∩H H (O) =1

2 2 Since for all ϕ ∈ L 1−r (O) ∩ H˙ r (O), we have kϕk 1−r . kϕkH˙ r (see [1]), then we deduce that for all L 2 −r ˙ 2 f ∈ L 1+r (O), f ∈ H (O) and kf kH˙ −r (O) . kf k 1+r .

L

(O)

Then, we get, k˜ ω0 kH˙ −r

. k˜ ω k 2

0 L 1+r

n0 − n0

n0 u0 ε √



2 ≤ √ g+ + k∇n0 k 1+r (85)

2 2 L n0 L 1+r n0 √ n0 L 1+r 2 , . C( g + ε) + (1 + C)ku0 k 1+r L





n0

n0 − n0

n0 − n0



where we have used the fact that ≤ 1 + . 1 +

n0 ∞

n0 ∞

n0 s ≤ 1 + C. L L H Under the assumption (82), thanks to (84) and (85), from (83) we deduce that there exists a real C2 > 0 2 depending only on r, s, g, ε, ku0 kH s , ku0 k 1+r such that α ˜ 1 ≤ C2 . √ L We get also k˜ ω0 kL∞ . k˜ ω0 kH s ≤ C( g + ε) + ku0 kH s . By using Corollary 4.1 and the first equation of (23) to obtain again more regularity on h, we deduce our main Theorem, 2

2

Theorem 4.2 Let n0 − n0 ∈ W 1, 1+r (O) ∩ H s+1 (O),√u0 ∈ L 1+r (O)2 ∩ H s (O)2 with 0 < r < 1 < s. 2 If kn0 − n0 kH s+1 + kn0 − n0 k 1, 1+r ≤ C0 min(n0 , n0 ) with C0 > 0 a constant, then there exists a W 2 real C > 0 depending only on C0 , r, s, g, ε, ku0 kH s , ku0 k 1+r such that if n0 ≥ C then there exists a L

17

unique global strong solution (n, u) of the system of Equations (1) for the initial data (n0 , u0 ) such that n0 (n − n0 , u) ∈ C([0, ∞[; H˙ −r (O)3 ∩ H s (O)3 ) ∩ C 1 (]0, ∞[; H s−1 (O)3 ) and kn − n0 kL∞ (O×[0,+∞[) ≤ . 2 ∞ 3 Moreover, (n − n0 , u) ∈ C (O×]0, ∞[) .

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