GLOBAL EXISTENCE FOR A ONE-DIMENSIONAL MODEL IN GAS DYNAMICS

S. N J A M K E P O (Villetaneuse)

GLOBAL EXISTENCE FOR A ONE-DIMENSIONAL MODEL IN GAS DYNAMICS

Abstract. We prove the existence of a global solution for a one-dimensio- nal Navier–Stokes system for a gas with internal capillarity.

0. Introduction. Several works on Navier–Stokes equations for viscous compressible fluids give global existence and regularity results for small ini- tial data (for an example see Kazhikhov and Shelukin [K-S], Matsumura and Nishida [M-N] and their bibliography). One difficulty lies in the fact that the system is neither parabolic nor hyperbolic, and the regularization (by viscous effects) acts only on the velocity field. Works by Serre [Se1], [Se2]

avoid smallness hypotheses on initial data, for Lagrange’s variables in one space dimension.

In the present paper we consider a system of partial differential equations describing the adiabatic flow of a one-dimensional viscous gas, with the theory of second gradient taking in account the internal capillarity (Germain [G], Gatignol and Seppecher [Ga-S], Seppecher [S], and Serre [Se3]) and a fourth order (positive) viscosity. We denote by ̺, u, f the mass density, the velocity field, and a volume force. γ(∈ ]1, 2[) is the polytropic index of the gas, λ is the (positive) capillarity coefficient, Re, M, ε the Reynolds and Mach numbers, and a fourth order viscosity coefficient (depending on λ).

The system is

̺

t

+ (̺u)

x

= 0, (0.1)

̺(u

t

+ uu

x

) + 1

γM

²

(̺

^γ

)

_x

− λ̺̺

xxx

− 1

Re u

xx

+ εu

xxxx

= ̺f (0.2)

for (x, t) ∈ ]0, 1[ × ]0, T [ (T ∈ R

⁺∗

).

1991 Mathematics Subject Classification: 35Q30, 76N10, 76N15.

Key words and phrases: Navier–Stokes equations, gas dynamics, capillarity.

[203]

We have the following periodic boundary conditions:

̺(0, t) = ̺(1, t), (0.3)

u(0, t) = u(1, t), u

x

(0, t) = u

x

(1, t), u

xx

(0, t) = u

xx

(1, t), (0.4)

u

_xxx

(0, t) = u

_xxx

(1, t), and an initial data.

R e m a r k. Introducing λ and ε does not allow us to define a global solution of the classical Navier–Stokes system by passing to the limit (in λ and ε).

This paper is divided in three parts. In the first part we define the function spaces and a system derived from (0.1)–(0.2) for which we build a solution by an iterative method. In the second one, we show a local existence result. The capillarity coefficient gives regularity to the mass density, and allows us to avoid smallness hypotheses on initial data. In the third part a uniform Gronwall lemma gives the global existence when the volume force is equal to zero.

I. An iterative method. Let I = ]0, 1[, J = ]0, T [, and let f

_x...x

|{z}

k times

be the spatial derivative of order k of f . We consider the function spaces (see for example [A] or [B])

L

^p

(I) = n f :

1\

0

|f (x)|

^p

dx < ∞ o , W

^m,p

(I) = {f : ∀k ∈ [0, m] ∩ N, f

x...x

|{z}

k

∈ L

^p

(I)}, H

^m

(I) = W

^m,2

(I),

H

_per^m

(I) = {f ∈ H

^m

(I) : ∀k ∈ [0, m − 1] ∩ N, f

x...x

|{z}

k

(0) = f

x...x

|{z}

k

(1)},

˙L

²

(I) = n

f ∈ L

²

(I) :

1

\

0

f (x) dx = 0 o , H ˙

_per^m

(I) = H

_per^m

(I) ∩ ˙ L

²

(I).

In the whole work, the scalar products will be L

²

scalar products and C

e

will represent all the Sobolev embedding constants. We shall write L

^p

(resp.

H

^m

) instead of L

^p

(I) (resp. H

^m

(I)) and we shall denote by k k

p

(resp.

k k

H^m

) the norm in L

^p

(resp. H

^m

) and by k k

p,q

the norm in L

^p

(0, t; L

^q

).

Let ̺

0

be a positive constant. We make the following change of unknown

function: ̺(x, t) → ̺(x, t) + ̺

0

. Then (0.1)–(0.2) becomes

(1.1) ̺

t

+ ((̺ + ̺

0

)u)

x

= 0, (1.2) (̺ + ̺

₀

)(u

_t

+ uu

_x

) + 1

γM

²

((̺ + ̺

₀

)

^γ

)

_x

− λ(̺ + ̺

₀

)̺

_xxx

− 1

Re u

xx

+ εu

xxxx

= (̺ + ̺

0

)f.

We add to (1.1)–(1.2) the boundary conditions (0.3)–(0.4), and some initial data.

Let k be a positive integer. We consider the linear system (1.3) (̺

^k

+ ̺

0

)(u

^k+1_t

+ u

^k

u

^k+1_x

) − 1

Re u

^k+1_xx

+ εu

^k+1_xxxx

− (̺

^k

+ ̺

0

)f = −l

_x^k

= −

 1

γM

²

((̺

^k

+ ̺

0

)

^γ

) + λ

2 (̺

^k_x

)

²

− λ(̺

^k

+ ̺

0

)̺

^k_xx



x

, (1.4) ̺

^k+1_t

+ ((̺

^k+1

+ ̺

0

)(u

^k+1

))

x

= 0,

with periodic boundary conditions.

We write (̺

^k

, u

^k

) = (r, v) and (̺

^k+1

, u

^k+1

) = (̺, u), with the initial data (r, v)(x, 0) = (̺, u)(x, 0) = (̺

i

, u

i

) satisfying

(1.5) 0 < ̺

m

≤ ̺

i

(x) + ̺

0

≤ ̺

M

< ∞

for x in I and (̺

_i

, u

_i

) in ˙ H

_per¹

× ˙ L

²

. Let t

_k

∈ R

⁺∗

and J

_k

= ]0, t

_k

[. We make the following hypothesis: r(x, t) + ̺

0

> 0 for (x, t) in I × J

k

, (r + ̺

0

, r

x

) ∈ (L

^∞

(0, t; L

⁴

))

²

for t in J

_k

and f in L

^∞

.

The existence result for (1.3)–(1.4) is given in

Proposition 1.1. Under the previous conditions and for t in J

_k

, the problem (1.3)–(1.4)–(0.3)–(0.4) has a unique solution which satisfies

((r + ̺

0

)

^1/2

u, u

x

) ∈ L

^∞

(0, t; L

²

) × L

²

(0, t; ˙ H

_per¹

), (1.6)

̺(x, t) + ̺

0

≥ 0;

(1.7)

moreover, ̺ + ̺

0

∈ L

^∞

(0, t; H

_per¹

).

P r o o f. The a priori estimates are classical.

(i) We take the scalar product of (1.4) and u

²

/2, and of (1.3) and u. By the Gagliardo–Nirenberg inequalities, we obtain

d

dt (k(r + ̺

0

)

^1/2

uk

²₂

) + 1

2Re ku

x

k

²₂

+ εku

xx

k

²₂

≤ 2Re

(γM

²

)

²

kr + ̺

0

k

^2γ_2γ

+ 9λ

²

Re

2 kr

x

k

⁴₄

+ λ

²

ε kr + ̺

0

k

²₄

kr

x

k

²₄

+ C

_e²

̺

0

Re

2 kf k

²∞

.

Then

(1.8) k((r + ̺

0

)

^1/2

u)(t)k

²₂

+ 1

Re ku

x

k

²_2,2

+ εku

xx

k

²_2,2

≤ k(r

i

+ ̺

0

)

^1/2

u

i

k

²₂

+  λ

²

2ε kr + ̺

₀

k

⁴∞,4

+ λ

²

2ε (1 + 9εRe)kr

_x

k

⁴∞,4

+ C

_e²

̺

₀

Re 2 kf k

²∞

 t

+

 2Re

(γM

²

)

²

k(r + ̺

0

)k

^2γ∞,2γ

 t

and the existence for the velocity field results from the application of the Lions theorem [B].

(ii) For the mass density, we work along the characteristic lines. Let x(t) be a regular solution of dx/dt = u(x(t), t), x(0) = y. Then

(1.9) ̺

0

+ ̺(x(t), t) = (̺

0

+ ̺(y, 0)) exp 

^\t

0

−u

x

(x(τ ), τ ) dτ  . The positivity of the initial data gives the positivity of ̺(x, t) + ̺

0

.

Let p > 1. We take the scalar product of (1.4) and (̺ + ̺

0

)

^p−1

(resp.

of (1.4) and (1/(̺ + ̺

0

))

^p

). Setting g

n

(t) = exp(n

Tt

0

ku

x

k

∞

dτ ) we easily obtain the estimates

k(̺ + ̺

0

)(t)k

p

≤ ̺

M

g

2

(t), (1.10)

1

̺ + ̺

₀

(t)

_p

≤ 1

̺

_m

g

2

(t).

(1.11)

We take the spatial derivative of (1.4), and the scalar product of the result and ̺

x

. By the Gronwall lemma we have

(1.12) k̺

x

(t)k

2

≤ 

k(̺

i

)

x

k

2

+

t

\

0

(k̺ + ̺

0

k

∞

ku

xx

k

2

)(τ ) dτ  g

_5/2

(t) and the proposition follows.

Let (̺

i

, u

i

) ∈ ˙ H

_per³

× ˙ H

_per²

and v ∈ L

^∞

(0, t; ˙ H

¹

). We obtain more regu- larity in

Proposition 1.2. The solution of (1.2)–(1.3)–(0.3) satisfies u

x

∈ L

^∞

(0, t; H

¹

), u

xxxx

(r + ̺

0

)

^1/2

∈ L

²

(0, t; L

²

), (1.13)

̺

xx

∈ L

^∞

(0, t; ˙ H

¹

).

(1.14)

P r o o f. We take the scalar product of (1.3) and

_̺+̺¹

0

−1

Re

u

xx

+ εu

xxxx

to obtain

d dt

 1

Re ku

x

k

²₂

+ εku

xx

k

²₂

 + ε

²

4

u

xxxx

(r + ̺

₀

)

^1/2

2

2

≤ 4

ε

⁴

Re

⁴

1 r + ̺

0

2

∞

k(r + ̺

0

)

^1/2

uk

²₂

+ 3kr + ̺

0

k

∞

kvk

²₄

ku

x

k

²₄

+ 3λ

²

kr + ̺

0

k

∞

kr

xxx

k

²₂

+ 3 M

⁴

1

\

0

(r + ̺

0

)

^2γ−3

dx + C

_e²

kf k

²∞

. An integration with respect to time gives

(1.15) 1

Re ku

x

k

²₂

+ εku

xx

k

²₂

+ ε

²

4

u

xxxx

(r + ̺

0

)

^1/2

2

2,2

≤ 1

Re k(u

i

)

x

k

²₂

+ εk(u

i

)

xx

k

²₂

+ 3

M

⁴

k(r + ̺

0

)

^2γ−3

k

1,1

+ 4

ε

⁴

Re

⁴

1 r + ̺

0

2

∞,∞

k(r + ̺

0

)

^1/2

uk

²_2,2

+ 3C

e

k(r + ̺

0

)k

∞,∞

kv

x

k

²∞,2

ku

x

k

²_2,2

+ tC

_e²

kf k

²∞

+ 3λ

²

k(r + ̺

₀

)k

∞,∞

kr

_xxx

k

²_2,2

.

For the mass density, we proceed as for the estimate (1.12) to obtain (1.16) k̺

xx

(t)k

2

≤ k̺

ixx

k

2

g

_5/2

(t)

+ 3C

e



^t\

0

k̺

x

k

2

kr + ̺

0

k

^1/4∞

ku

xx

k

^1/2₂

u

xxxx

(r + ̺

₀

)

^1/2

1/2 2

dτ

 g

5/2

(t)

+ C

_e



t\

0

k̺ + ̺

₀

k

∞

kr + ̺

₀

k

^1/2∞

u

_xxxx

(r + ̺

0

)

^1/2

2

dτ



g

_5/2

(t), (1.17) k̺

xxx

(t)k

2

≤ k(̺

i

)

xxx

k

2

g

_5/2

(t)

+ 6C

e



^t\

0

k̺

xx

k

2

kr + ̺

0

k

^1/4∞

ku

xx

k

^1/2₂

u

xxxx

(r + ̺

₀

)

^1/2

1/2 2

dτ

 g

_5/2

(t)

+



t\

0



4k̺

x

k

∞

ku

xx

k

2

+ k̺ + ̺

0

k

∞

kr + ̺

0

k

^1/2∞

u

_xxxx

(r + ̺

0

)

^1/2

2

 dτ

 g

5/2

(t) and the proposition follows.

The proposition has the following consequence:

Corollary 1.3.

̺ ∈ C([0, t

k

]; ˙ H

_per²

) ∩ L

^∞

(0, t

k

; ˙ H

_per³

), (1.18)

(u, u

x

) ∈ C([0, t

k

]; ˙ H

_per²

) × C([0, t

k

]; ˙ H

_per¹

).

(1.19)

II. Local existence

II.I. Uniform estimates for the sequence (̺

^k

, u

^k

). Let N be an upper bound of

1

Re ku

^k_x

(t)k

²₂

+ εku

^k_xx

(t)k

²₂

+ ε

²

4

u

^k_xxxx

(̺

^k−1

+ ̺

0

)

^1/2

(t)

2

2,2

and

1

Re ku

^k−1_x

(t)k

²₂

+ εku

^k−1_xx

(t)k

²₂

+ ε

²

4

u

^k−1_xxxx

(̺

^k−2

+ ̺

0

)

^1/2

(t)

2

2,2

for t in ]0, min(t

k−1

, t

k

)[. We define b

n

(t) = exp



n  C

_e²

tN Re 2ε



1/2

 . By (1.8), we have the inequalities

t

\

0

ku

^k_x

k

∞

dτ ≤  C

_e²

tN Re 2ε



1/2

,

k(̺

^k

+ ̺

0

)(t)k

2

≤ ̺

M

b

2

(t),

1

̺

^k

+ ̺

0

(t)

2

≤ 1

̺

m

b

2

(t), from which we deduce the estimates

k̺

^k_xx

(t)k

2

≤ k(̺

i

)

xx

k

2

b

_5/2

(t) + C

1

(tN )

^1/2

ε (1 + (ε

²

k(̺

i

)

x

k

2

)

^2/3

)b

_11/2

(t) (2.1)

+ C

₁

N

^1/2

t

ε

^3/2

(1 + N t

²

)b

_25/2

(t) and

(2.2) k̺

^k_xxx

(t)k

2

≤



k(̺

i

)

xxx

k

2

+ C

₁

(tN )

^1/2

ε (1 + ε(t)

^1/2

k(̺

i

)

xx

k

²₂

)

 b

_7/2

(t)

+ C

1

(tN )

^1/2

ε b

_13/2

(t) + C

1

t

N

^1/2

ε

^7/2

(1 + ε

²

k(̺

i

)

x

k

2

)

^4/3

(1 + (tN )

⁴

)(N t + N ε

²

+ ε)

²

b

30

(t).

Here C

1

> 0 depends only on ̺

m

, ̺

M

and C

e

. We denote by P

12

a polynomial of degree 12 of variable tN , and set

ε

1

= ε

^4(γ−1)

λ

^8γ

M

⁵

Re

²

, ε

2

= 1

M

⁴

max  1

̺

m



2γ−3

, (̺

M

)

^2γ−3

 , ε

3

= max

 ε

2

; M

⁴

Re

²

 λ

²

ε



4γ+1

; λ

²

(1 + Reε) ε

⁴

Re

⁵

; 1

ε

⁷

; ε

⁴

Re

⁵

 . Then we deduce the following estimate from (2.1)–(2.2)–(1.15):

(2.3) 1

Re ku

^k+1_x

(t)k

²₂

+ εku

^k+1_xx

(t)k

²₂

+ ε

²

4

u

^k+1_xxxx

(̺

^k

+ ̺

0

)

^1/2

2 2,2

≤ 1

Re k(u

i

)

x

k

²₂

+ εk(u

i

)

xx

k

²₂

+ C

1

λ

²

k(̺

i

)

xxx

k

²₂

exp



7  C

1²

tN Re 2ε



1/2



+ tε

1

ε

2



1 + kf k

²∞

+ P

12

(tN ) exp



60  C

₁²

tN Re 2ε



1/2



.

For λ small enough, t (depending on N ) small enough (denoted from now on by t

∗

), and for an initial data smaller than N , the right hand side of (2.3) has N as upper bound. By induction on k we obtain the required upper bound.

II.2. Convergence of the sequence (̺

^k

, u

^k

). Let (̺

i

, u

i

) ∈ ˙ H

_per³

× ˙ H

_per²

. We have

Proposition 2.1. The sequence (̺

^k

, u

^k

) is a Cauchy sequence in L

^∞

([0, t

∗

]; ˙ H

_per²

) × C([0, t

∗

]; ˙ H

_per¹

).

P r o o f. We denote by (r

^k+1

, u

^k+1

) the difference (̺

^k+1

− ̺

^k

, u

^k+1

− u

^k

).

Then (r

^k+1

, u

^k+1

) is a solution of the system (2.4) (̺

^k

+ ̺

₀

)(w

^k+1_t

+ u

^k

w

_x^k+1

) − 1

Re w

^k+1_xx

+ εw

^k+1_xxxx

= r

^k

f − (l

^k

− l

^k−1

)

x

− r

^k

u

^k_t

− (r

^k

u

^k

+ ̺

^k−1

w

^k

)u

^k_x

= h

^k+1,k₁

− h

^k+1,k_2,x

− h

^k+1,k₃

− h

^k+1,k₄

, (2.5) r

^k+1_t

+ (̺

^k+1

w

^k+1

)

x

+ (r

^k+1

u

^k

)

x

= 0,

with periodic boundary conditions and a null initial data. We recall that ̺

^k

is a solution of

(2.6) ̺

^k_t

+ ((̺

^k

+ ̺

0

)u

^k

)

x

= 0.

We proceed as in Proposition 1.1 and so the details will be omitted.

(i) We take the scalar product of (2.4) and w

^k+1

, and of (2.7) and

1

2

(w

^k+1

)

²

. Then we obtain a differential inequality with left-hand side 1

2 d

dt (k(̺

^k

+ ̺

0

)

^1/2

w

^k+1

k

²₂

) + 1

Re kw

_x^k+1

k

²₂

+ εkw

^k+1_xx

k

²₂

. (i.a) For the right-hand side, we have first

1

\

0

h

^k+1,1₁

w

^k+1

dx ≤ C

e

kf k

∞

kr

^k

k

2

kw

_x^k+1

k

2

.

(i.b) Then

1

\

0

h

^k+1,k_2,x

w

^k+1

dx ≤  1

M

²

kξ + ̺

0

k

^γ−1∞

kr

^k

k

2

+ λ(C

e

k̺

^k_x

k

∞

+ k̺

^k_xx

k

2

)kr

^k_x

k

2

+ λ(̺

0

+ k̺

^k−1

k

∞

)kr

_xx^k

k

2



kw

_x^k+1

k

2

≤ C

₁^1/2

kr

^k

k

H˙²

kw

_x^k+1

k

2

. (i.c) The term

T1 0

r^k

̺^k−1+̺0

w

^k+1

u

^k_xxxx

dx appears in

T1

0

h

^k+1,k₃

w

^k+1

dx. An integration by parts gives

1

\

0

r

^k

̺

^k−1

+ ̺

₀

w

^k+1

u

^k_xxxx

dx ≤

1

\

0

   

 r

^k

̺

^k−1

+ ̺

₀

w

^k+1



xx

 

  |u

^k_xx

| dx

(| | is the absolute value). Now we use the H¨older inequality and the fact that H

²

is an algebra to obtain an upper bound for k(r

^k

w

^k+1

/(̺

^k−1

+ ̺

0

))

xx

k

2

. Upper bounds for the other terms of

T1

0

h

^k+1,k₃

w

^k+1

dx are classical, and we have

kh

^k+1,k₃

k

2

kw

^k+1

k

2

≤ C

2

kr

^k

k

H¹

kw

_x^k+1

k

2

+ C

3

kr

^k

k

H²

kw

^k+1_xx

k

2

, where C

₂

and C

₃

are positive and depend on N, Re, ̺

_m

, ̺

_M

, λ, ε; more- over, C

2

(resp. C

3

) depends on ku

^k_x

(t)k

2

(resp. ku

^k_xx

(t)k

2

),

Tt

0

C

₂²

(τ ) dτ ≤ C

1

ku

^k_x

k

²_2,2

and

Tt

0

C

₃²

(τ ) dτ ≤ C

1

ku

^k_xx

k

²_2,2

(i.d) For the last term we have

1

\

0

h

^k+1,k₄

w

^k+1

dx ≤ C

e

(kr

^k

k

∞

ku

^k_x

k

²₂

+ ku

^k_xx

k

2

kw

^k_x

k

2

)kw

_x^k+1

k

2

.

For t in [0, t

∗

] an integration with respect to time of the differential inequality

gives

(2.8) k((̺

^k

+ ̺

0

)

^1/2

(u

^k+1

− u

^k

))(t)k

²₂

+ 1

Re k(u

^k+1

− u

^k

)

_x

k

²_2,2

+ εk(u

^k+1

− u

^k

)

_xx

k

²_2,2

≤ C

1

tk(u

^k

− u

^k−1

)

x

k

²∞,2

+ C

1

t

^1/2

(N

^1/2

+ (1 + kf k

∞

)t

^1/2

)k̺

^k

− ̺

^k−1

k

²_L∞(0,t∗; ˙H²)

, and for t

∗

small enough, and a positive constant which we denote again C

₁

, we have the following upper bound for the right-hand side of (2.8):

C

1

t

^1/2

(k̺

^k

− ̺

^k−1

k

²_L∞(0,t∗; ˙H²)

+ k(u

^k

− u

^k−1

)

x

k

²∞,2

).

(ii) We take the scalar product of (2.4) and

_̺k+̺¹ 0

w

_xx^k

. Proceeding as for the estimate (2.8) we obtain

(2.9) k(u

^k+1

− u

^k

)

_x

k

²₂

+ 1 Re

(u

^k+1

− u

^k

)

_xx

(̺

^k

+ ̺

0

)

^1/2

2 2,2

+ ε

(u

^k+1

− u

^k

)

_xxx

(̺

^k

+ ̺

0

)

^1/2

2

2,2

≤ C

1

t

^1/2

(k̺

^k

− ̺

^k−1

k

²_L∞(0,t∗; ˙H²)

+ k(u

^k

− u

^k−1

)

x

k

²∞,2

).

(iii) We take the second spatial derivative of (2.5) and the scalar prod- uct of the result and r

^k+1_xx

. Then we obtain a differential inequality whose integration with respect to time gives

kr

_xx^k+1

k

2

exp 

− C

e t

\

0

ku

^k+1_xx

k

2

dτ 

≤ C

e

(k̺

^k+1_xx

k

H˙¹

kw

_x^k+1

k

2,2

+ k̺

^k+1_x

k

∞,2

kw

^k+1_xx

k

2,2

)t

^1/2

+ C

e



k̺

^k−1

+ ̺

0

k

^1/4∞,∞

×



ku

^k_xx

k

²_2,2

+

u

^k_xxxx

(̺

^k−1

+ ̺

0

)

^1/2

2

2,2



1/2

kr

_x^k+1

k

∞,2

 t

^1/2

+ C

_e



k̺

^k+1

k

∞,∞

k̺

^k

+ ̺

₀

k

^1/2∞,∞

w

^k+1_xxx

(̺

^k

+ ̺

0

)

^1/2

_2,2

 t

^1/2

and the estimate (for t

∗

small enough and a positive constant denoted again by C

1

)

k(̺

^k+1

− ̺

^k

)

xx

k

²₂

≤ C

1

t

^1/2

k̺

^k

− ̺

^k−1

k

²_L∞(0,t∗; ˙H²)

(2.10)

+ C

1

t

^1/2

k(u

^k

− u

^k−1

)

x

k

²∞,2

.

We take t

∗

small enough to have C

1

t

^1/2∗

< 1/2. Then by (2.8)–(2.10),

(̺

^k

, u

^k

) is a Cauchy sequence in L

^∞

([0, t

∗

]; ˙ H

_per²

) × C([0, t

∗

]; ˙ H

_per¹

), and the

limit (̺, u) is a weak solution of (1.1)–(1.2)–(0.3)–(0.4) in L

^∞

([0, t

∗

]; ˙ H

_per¹

)×

(L

^∞

([0, t

∗

]; L

²

) ∩ L

²

([0, t

∗

]; ˙ H

_per²

)). Moreover,

(2.11) ̺(x, t) + ̺

0

≥ 0

for (x, t) in I × [0, t

∗

].

III. Regularity of the solution. In this section, we show the regularity of the solution for f (6= 0) in L

^∞

, and the global existence for f = 0. We take initial data satisfying

(3.1) (̺

i

, u

i

) ∈ ˙ H

_per³

× ˙ H

_per²

and (̺

i

)

xxx

̺

i

+ ̺

0

∈ L

²

. III.1. The case f 6= 0. We have

Lemma 3.1. The solution satisfies (3.2) k(̺ + ̺

0

)

^1/2

uk

²₂

+ 2

γ(γ − 1)M

²

k̺ + ̺

0

k

^γ_γ

+ λk̺

x

k

²₂

+ 1

Re ku

x

k

²_2,2

+ 2εku

xx

k

²_2,2

≤ k(̺

i

+ ̺

0

)

^1/2

uk

²₂

+ 2

γ(γ − 1)M

²

k̺

i

+ ̺

0

k

^γ_γ

+ λk(̺

i

)

x

k

²₂

+ C

e

̺

0

Rekf k

²∞

t.

P r o o f. We proceed as for Proposition 1.1 to obtain 1

2 d dt



k(̺ + ̺

0

)

^1/2

uk

²₂

+ 2

γ(γ − 1)M

²

k̺ + ̺

0

k

^γ_γ

+ λk̺

x

k

²₂



+ 1

Re ku

x

k

²₂

+ εku

xx

k

²₂

=

1

\

0

(̺ + ̺

0

)f u dx.

By the H¨older and Gagliardo–Nirenberg inequalities, we have

1

\

0

(̺ + ̺

₀

)f u dx ≤ C

_e

k̺ + ̺

₀

k

₁

kf k

∞

ku

_x

k

₂

≤ 1

2Re ku

_x

k

²₂

+ C

_e²

̺

₀

Rekf k

²∞

and the lemma follows from an integration with respect to time.

We have more regularity in Proposition 3.2.

̺

_xxx

(̺ + ̺

0

)

^1/2

, ̺

xxx

, u

xx

∈ L

^∞

(0, t

∗

; L

²

), (3.3)

u

xxx

(̺ + ̺

0

)

^1/2

, u

xxxx

(̺ + ̺

0

)

^1/2

∈ L

²

(0, t

∗

; L

²

).

(3.4)

P r o o f. Set

a

0

= k(̺

i

+ ̺

0

)

^1/2

u

i

k

²₂

+ 2

γ(γ − 1)M

²

k̺

i

+ ̺

0

k

^γ_γ

+ λk(̺

i

)

x

k

²₂

, a

1

= C

e

̺

0

Rekf k

²∞

, a

2

= exp(a

0

),

a

3

= C

_e²

8

 Re ε



1/2

+ C

e

̺

0

Rekf k

²∞

. Interpolating L

^∞

between L

²

and H

²

gives

t

\

0

ku

^k+1_x

k

∞

dτ ≤

t

\

0

 C

_e²

8

 Re ε



1/2

+ 1

Re ku

x

(τ )k

²₂

+ εku

xx

(τ )k

²₂

 dτ and (from Lemma 3.1) we obtain

(3.5) exp 

^t\

0

ku

^k+1_x

k

∞

dτ 

≤ a

2

exp(a

3

t).

We take the third spatial derivative of (1.1) and the scalar product of the result and ̺

xxx

/(̺ + ̺

0

). Then

(3.6) 1 2

d dt



̺

xxx

(̺ + ̺

0

)

^1/2

2 2

 + 3

1

\

0

̺

²_xxx

u

x

̺ + ̺

₀

dx + 6

1

\

0

̺

xx

̺

xxx

u

xx

̺ + ̺

₀

dx +4

1

\

0

̺

x

̺

xxx

u

xxx

̺ + ̺

₀

dx = −

1

\

0

̺

xxx

u

xxxx

dx.

The scalar product of (1.2) and u

xxxx

/(̺ + ̺

0

) gives (3.7) 1

2 d

dt (ku

xx

k

²₂

) + 1 Re

u

xxx

(̺ + ̺

0

)

^1/2

2 2

+ ε

u

xxxx

(̺ + ̺

0

)

^1/2

2 2

=

1\

0



f − uu

x

− ̺

x

M

²

(̺ + ̺

0

)

^2−γ



u

xxxx

dx − 1 Re

1

\

0

̺

x

u

xx

u

xxx

(̺ + ̺

0

)

²

dx

+ λ

1

\

0

̺

xxx

u

xxxx

dx.

From these two equations, we deduce a differential inequality whose left- hand side is

1 2

d dt



ku

_xx

k

²₂

+ λ

̺

_xxx

(̺ + ̺

0

)

^1/2

2 2

 + 1

Re

u

_xxx

(̺ + ̺

0

)

^1/2

2 2

+ ε

u

_xxxx

(̺ + ̺

0

)

^1/2

2 2

and (by the Gagliardo–Nirenberg inequalities) the right-hand side has the

following upper bound:

3ε

₁

Re

u

_xxx

(̺ + ̺

0

)

^1/2

2

2

+ 4ε

2

ε

u

_xxxx

(̺ + ̺

0

)

^1/2

2

2

+ λ(ε

₃

+ 3ku

_x

k

∞

+ 10C

_e

ku

_xx

k

₂

)

̺

_xxx

(̺ + ̺

0

)

^1/2

2 2

+ ̺

²₀

4ε

2

ε kf k

²∞

+ C

_e⁴

4ε

2

ε k(̺ + ̺

0

)

^1/2

uk

²₂

ku

x

k

2

ku

xx

k

2

+ 1

4ε

2

εM

⁴

k̺ + ̺

0

k

^2γ−1∞

k̺

x

k

²₂

+ C

_e⁴

64ε

³₁

Re k̺ + ̺

0

k

∞

1

̺ + ̺

0

6

∞

k̺

x

k

⁴₂

ku

xx

k

²₂

+ λ

³

C

_e¹²

64ε

²₂

ε

3

ε

²

k̺ + ̺

0

k

²∞

k̺

x

k

⁶₂

. Let us take ε

2

= 1/6 and ε

2

= 1/8.

We denote by a

4

and a

5

real numbers depending on C

e

, a

2

, Re, M , λ, ε, γ, ̺

M

, ̺

m

, and set a

6

=

^2̺_ε²⁰

kf k

²∞

. Finally, we define

h

1

(t) = C

e

ku

xx

k

2

+ ε

3

+ 1

ε (a

0

+ a

1

t) + a

4

(a

³₀

+ a

³₁

t

³

) exp(7a

3

t), h

₂

(t) = a

₆

+ a

₅

(a

³₀

+ a

³₁

t

³

) exp(3a

₃

t).

There exist positive functions g

1

and g

2

depending on time so that

t

\

0

h

1

(τ ) dτ ≤ t

^1/2

2 g

1

(t),

t

\

0

h

2

(τ ) dτ ≤ t 2 g

2

(t).

The differential inequality is (3.8) d

dt



ku

xx

k

²₂

+ λ

̺

xxx

(̺ + ̺

0

)

^1/2

2 2

 + 2

Re

u

xxx

(̺ + ̺

0

)

^1/2

2

2

+ 2ε

u

xxxx

(̺ + ̺

0

)

^1/2

2 2

≤ 2



ku

xx

k

²₂

+ λ

̺

xxx

(̺ + ̺

0

)

^1/2

2 2



h

1

(t) + 2h

2

(t) and (by the Gronwall lemma) we obtain

(3.9) ku

xx

(t)k

²₂

+ λ

̺

xxx

(̺ + ̺

0

)

^1/2

(t)

2

2

≤ k(u

i

)

xx

k

²₂

exp(t

^1/2

g

1

(t)) +

 λ

(̺

i

)

xxx

(̺

i

+ ̺

0

)

^1/2

2 2

+ tg

2

(t)



exp(t

^1/2

g

1

(t)) for t ∈ [0, t

∗

]. A direct integration of (3.9) with respect to time gives (3.4).

Now we take the third spatial derivative of (1.1) and the scalar product

of the result and ̺

xxx

. Then the estimate (3.4) and an integration with

respect to time show that ̺

xxx

belongs to L

^∞

(0, t

∗

; L

²

). This completes the

proof of the proposition.

III.2. The case f = 0. We have to show the existence of uniform (with respect to time) and strictly positive bounds of ̺(x, t) + ̺

0

. We define

̺

_min

= min

_x∈I

(̺

₀

+ ̺

_i

(x)) with 0 < ̺

_min

< ̺

₀

, and f (̺) =

 2λ

γ(γ − 1)(γ + 2)

²

M

²



1/2

(̺

^(γ+2)/2₀

− (̺ + ̺

0

)

^(γ+2)/2

).

Let us make the following hypothesis:

(H) We suppose the existence of µ

1

, µ

2

, µ

3

, µ

4

(∈ ]0, 1[) so that X

4

i=1

µ

i

≤ 1, 1

2 k(̺

i

+ ̺

0

)

^1/2

u

i

k

²₂

≤ µ

1

f (̺

min

− ̺

0

), 2

^γ

γ(γ − 1)M

²

k̺

i

k

^γ_γ

≤ µ

2

f (̺

min

− ̺

0

), 2

^γ

̺

^γ₀

γ(γ − 1)M

²

≤ µ

3

f (̺

min

− ̺

0

), λ

2 k(̺

i

)

x

k

²₂

≤ µ

4

f (̺

min

− ̺

0

).

We have Lemma 3.3.

̺(t) + ̺

0

∈ C

⁰

(I), (3.10)

̺(x, t) + ̺

₀

≥ ̺

_min

> 0 (3.11)

uniformly for t in R

⁺

.

P r o o f. (i) For (3.10) we proceed as in Lemma 3.1 to obtain

(3.12) 1

γ(γ − 1)M

²

k(̺ + ̺

0

)(t)k

^γ_γ

+ λ

2 k̺

x

(t)k

²₂

≤ 1

2 k(̺

_i

+ ̺

₀

)

^1/2

u

_i

k

²₂

+ 1

γ(γ − 1)M

²

k̺

_i

+ ̺

₀

k

^γ_γ

+ λk(̺

_i

)

_x

k

²₂

from which we deduce the estimates (uniform with respect to time)

(3.13) ̺(t) + ̺

0

∈ L

^γ

, ̺

x

(t) ∈ L

²

.

The first one results from the embedding W

^1,1

(I) → C

⁰

(I), and we shall write ̺

max

= k̺ + ̺

0

k

∞

.

(ii) For (3.11) we follow a proof by Eden–Milani–Nicolaenko [E-M-N] for which we need the hypothesis (H). We define ̺

∗

(t) = min

x∈I

(̺

0

+ ̺(x, t)).

Then ̺

∗

(t) ≤ ̺

0

(else, we should have ̺

0

=

T1

0

(̺(x, t) + ̺

0

) dx = k̺(t) +

̺

0

k

1

> ̺

∗

(t) > ̺

0

, which is impossible). Since ̺(·, t) is continuous on

I, there exist a

m

(t) and x

m

(t) in I so that ̺(a

m

(t), t) = 0, ̺(x

m

(t), t) =

̺

∗

(t) − ̺

0

, and f (̺(a

m

(t), t)) = 0. We can write f (̺(x

m

(t), t)) − f (̺(a

m

(t), t)) =

̺(xm(t),t)

\

̺(am(t),t)

∂f

∂̺ d̺.

Then

f (̺(x

m

(t), t)) = −

 λ

2γ(γ − 1)(γ + 2)

²

M

²



1/2 x^m(t)

\

am(t)

(̺ + ̺

0

)

^γ/2

̺

z

dz.

Now we have

(3.14) 0 ≤ f (̺(x

m

(t), t))

≤

 λ

2γ(γ − 1)(γ + 2)

²

M

²



k̺(t) + ̺

₀

k

^γ/2_γ

k̺

_x

(t)k

₂

≤ f (̺

_min

− ̺

₀

).

Therefore f (̺) is nonnegative, nonincreasing and has a strictly positive up- per bound. Then we obtain the estimate

(3.15) ̺

min

≤ ̺(·, t) + ̺

0

and the proof of the lemma follows.

Theorem 3.4. In the case f = 0 the problem (1.1)–(1.2)–(0.3)–(0.4) with initial data satisfying (3.1) has a unique, global and regular solution.

P r o o f.

(i) Regularity of the solution. The scalar product of (1.4) and −̺

xxx

gives (3.16)

1\

0

((̺ + ̺

₀

)u

_t

)

_x

̺

_xx

dx + 1 M

²

1

\

0

(̺ + ̺

₀

)

^γ−1

̺

_xx

dx

+ λ

1

\

0

(̺ + ̺

0

)̺

²_xxx

dx

= − γ − 1 M

²

1

\

0

(̺ + ̺

₀

)

^γ−2

̺

²_x

̺

_xx

dx

+

1\

0

(̺ + ̺

0

)uu

x

̺

xxx

dx − 1 Re

1

\

0

u

xx

̺

xxx

dx

+ ε

1

\

0

̺

_xxx

u

_xxxx

dx.