Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 186, pp. 1–6.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SMOOTH GLOBAL SOLUTIONS FOR A 1-D
MODIFIED NAVIER-STOKES-FOURIER MODEL
JIANZHU SUN, JISHAN FAN, GEN NAKAMURA
Abstract. We prove the existence of strong global solutions of the 1-D modified compressible Navier-Stokes-Fourier equations proposed by Howard Brenner [1, 2].
1. Introduction
We consider the modified Navier-Stokes-Fourier equations proposed by by Brenner [1, 2]:
∂t ρ + div(ρvm ) = 0,
∂t (ρv) + div(ρv ⊗ vm ) + ∇p = div S,
1
1
∂t (ρ( v 2 + e)) + div(ρ( v 2 + e)vm ) + div(pv) + div q = div(Sv),
2
2
v|∂Ω = 0, vm · n|∂Ω = ∇ρ · n|∂Ω = 0, q · n|∂Ω = ∇θ · n|∂Ω = 0,
(ρ, v, θ)|t=0 = (ρ0 , v0 , θ0 )
in Ω := (0, 1).
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
where ρ is the mass density, v is the fluid-based (Lagrangian) volume velocity, vm
is the mass-based (Eulerian) mass velocity, p = Rρθ is the pressure with positive
constant R > 0, e = CV θ the specific internal energy, θ the temperature, S the
viscous stress tensor, we will adopt the Newton’s rheological law:
2
(1.6)
S := µ ∇v + ∇v T − div vI + η div vI,
3
where µ ≥ 0 and η ≥ 0 stand for the shear and bulk viscosity coefficients, respectively. The relationship between vm and v is a cornerstone of Brenner’s approach.
After a careful study [1, 2], Brenner proposes a universal constitutive equation in
the form:
v − vm = K∇ log ρ,
(1.7)
with K ≥ 0 a purely phenomenological coefficient.
Moreover, we suppose the heat flux obeys Fourier’s law, specifically,
q = −k∇θ,
(1.8)
where k is the heat conductivity coefficient.
2000 Mathematics Subject Classification. 35Q30, 76D03, 76D05.
Key words and phrases. Mass velocity; volume velocity; Navier-Stokes-Fourier equations.
c
2012
Texas State University - San Marcos.
Submitted June 18, 2012. Published October 28, 2012.
1
2
J. SUN, J. FAN, G. NAKAMURA
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We will assume K = 1, Cv = 1, R = 1, µ > 0, η = 0, and
k(θ) := k0 (1 + 4θ3 ),
(1.9)
with a positive constant k0 = 1. (1.9) is physically relevant as radiation heat
conductivity at least for large values of θ (see [8]).
Very recently, Feireisl and Vasseur [4] proved the global-in-time existence of weak
solutions to the problem (1.1)-(1.5). Under the conditions that ρ0 , θ0 , v0 ∈ L∞ (Ω)
and ρ0 ≥ C > 0, θ0 ≥ C > 0 in Ω. Here it should be noted that similar result for
the classical Navier-Stokes-Fourier system ((1.1)-(1.3) with v = vm ) have not yet
been proved. In their proof, they obtained the following global-in-time estimates:
kvkL2 (0,T ;H 1 (Ω)) ≤ C,
kθ
3/2
(1.10)
kL2 (0,T ;H 1 (Ω)) ≤ C,
(1.11)
k∇θkL2 (0,T ;L2 (Ω)) ≤ C,
(1.12)
R
R
where
C is a positive constant depending on Ω ρ0 dx, Ω ρ( 21 v02 + CV θ0 )dx, and
R
ρ s(ρ0 , θ0 )dx, the other norms of ρ0 and v0 , θ0 .
Ω 0
Our aim in this article is to show the existence of a smooth global solution to
the problem (1.1)-(1.5).
Theorem 1.1. Let ρ0 , v0 , θ0 ∈ H 1 (Ω) with inf ρ0 > 0, inf θ0 > 0 in Ω. Then there
exists a unique strong solution (ρ, v, θ) to the problem (1.1)-(1.5) satisfying
(ρ, v, θ) ∈ L∞ (0, T ; H 1 (Ω)) ∩ L2 (0, T ; H 2 (Ω)), (∂t ρ, ∂t v, ∂t θ) ∈ L2 (0, T ; L2 (Ω))
for any given T > 0 and
inf ρ(x, t) > 0,
inf θ(x, t) > 0
in Ω × (0, T ).
(1.13)
Remark 1.2. The methods for the one-dimensional classical Navier-Stokes-Fourier
equations [6, 7] do not work here. Because their clever method for proving 0 < C1 ≤
ρ ≤ C < ∞ does not work here.
The continuity equation (1.1) can be rewritten as
∂t ρ + div(ρv) = ∆ρ.
(1.14)
The energy equation (1.3) can be rewritten as
∂t (ρθ) + div(ρvm θ) + div q = S : ∇v − p div v.
(1.15)
2. Proof of Theorem 1.1
Since it is easy to prove a local existence result for smooth solution, which is
very similar as that in [3], we omit the details here. We need to prove only the a
priori estimates for smooth solutions and omit the proof of the uniqueness which is
standard.
Since we take x ∈ Ω := (0, 1) and ∂Ω = {0, 1}, it follows that div = ∇ = ∂x ,
∆ = ∂x2 , S := ( 43 µ + η)∂x v and (1.4) becomes
∂ρ = 0,
∂x ∂Ω
First, we note that in 1-D, we have
v|∂Ω = 0,
∇ρ|∂Ω =
kρkL∞ ≤ CkρkH 1 ,
∇θ|∂Ω =
kθkL∞ ≤ CkθkH 1 ,
∂θ = 0.
∂x ∂Ω
kvkL∞ ≤ Ck∇vkL2 .
(2.1)
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EXISTENCE OF SMOOTH GLOBAL SOLUTIONS
3
Lemma 2.1. If (ρ, v, θ) is a strong solution, then
kρkL∞ (0,T ;H 1 ) + kρkL2 (0,T ;H 2 ) ≤ C(T ),
k∂t ρkL2 (0,T ;L2 ) ≤ C(T ),
1
≤ ρ.
C(T )
Proof. Testing (1.14) with ρ, using (1.10) and (2.1), we have
Z
Z
Z
1 d
2
2
ρ dx + |∇ρ| dx = ρv∇ρdx
2 dt
≤ kρkL2 kvkL∞ k∇ρkL2 ≤ Ck∇vkL2 kρkL2 k∇ρkL2
1
≤ k∇ρk2L2 + Ck∇vk2L2 kρk2L2
2
which gives
kρkL∞ (0,T ;L2 ) + kρkL2 (0,T ;H 1 ) ≤ C(T ).
Similarly, testing (1.14) with −∆ρ, using (1.10) and (2.1), we see that
Z
Z
Z
1 d
|∇ρ|2 dx + |∆ρ|2 dx = (ρ div v + v∇ρ)∆ρdx
2 dt
≤ (kρkL∞ k div vkL2 + kvkL∞ k∇ρkL2 )k∆ρkL2
≤ CkρkH 1 k∇vkL2 k∆ρkL2
1
≤ k∆ρk2L2 + Ck∇vk2L2 kρk2H 1
2
∂v
which yields (2.1). Here we have div v = ∇v = ∂x
. Then (2.1) follows easily from
(1.14) and (2.1).
To prove (2.1), we multiply (1.14) by ρ1 to obtain
∂t log ρ − ∆ log ρ = |∇ log ρ|2 − v · ∇ log ρ − div v
1 2 1
= ∇ log ρ − v − v 2 − div v
2
4
1 2
≥ − v − div v.
4
By the classical comparison principle, it is easy to infer that log ρ ≥ w, with w a
solution to the problem
1
∂w ∂t w − ∆w = − v 2 − div v, ∇w|∂Ω =
= 0, w|t=0 = log ρ0 ,
(2.2)
4
∂x ∂Ω
with fixed v satisfying (1.10).
Testing (2.2) with w, using (1.10), we find that
Z
Z
1 d
1
w2 dx + |∇w|2 dx ≤ ( kvkL∞ kvkL2 + k div vkL2 )kwkL2
2 dt
4
≤ C(k∇vkL2 + k∇vk2L2 )kwkL2
which gives
kwkL∞ (0,T ;L2 ) + kwkL2 (0,T ;H 1 ) ≤ C(T ).
Similarly, testing (2.2) with −∆w, using (1.10), we infer that
Z
Z
Z
Z
1 2
1 d
2
2
|∇w| dx + |∆w| dx ≤
∇v · ∇wdx + | div v · ∆wdx|
2 dt
4
4
J. SUN, J. FAN, G. NAKAMURA
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1
kvkL∞ k div vkL2 k∇wkL2 + k div vkL2 k∆wkL2
2
1
≤ k∆wk2L2 + Ck div vk2L2 + Ck∇vk2L2 k∇wkL2
2
≤
which yields
kwkL∞ (0,T ;H 1 ) ≤ C(T ).
This yields
log ρ ≥ w ≥ −C(T ) > −∞
and thus (2.1) holds. The proof is complete.
Using (1.1), (1.2), (2.1), (2.1), p := Rρθ, (1.11), (1.12) and the method in [4], it
is easy to verify the following lemma.
Lemma 2.2 ([4]). If (ρ, v, θ) is a weak solution, then
kvkL∞ (0,T ;Lm (Ω)) ≤ C(T )
for some m > 2.
(2.3)
It follows from (1.11) and (2.1) that
kθkL3 (0,T ;L∞ (Ω)) ≤ C(T ).
(2.4)
Lemma 2.3. If (ρ, v, θ) is a strong solution, then
kvkL∞ (0,T ;H 1 ) + kvt kL2 (0,T ;L2 ) ≤ C(T ),
(2.5)
kvkL2 (0,T ;H 2 ) ≤ C(T ).
(2.6)
Proof. We start rewriting the momentum equation (1.2) in the form
1
(2.7)
ρ(∂t v + vm · ∇v) + R∇(ρθ) = µ∆v + µ∇ div v.
3
Testing (2.7) with vt , using (2.1), (2.1), (1.12), (2.3) and (2.4), we deduce that
Z
Z
1
1 d
2
2
µ|∇v| + µ(div v) dx + ρvt2 dx
2 dt
3
Z
Z
= − ρvm · ∇v · vt dx − R ∇(ρθ) · vt dx
Z
Z
Z
= − ρv · ∇v · vt dx + ∇ρ · ∇v · vt dx − R (ρ∇θ + θ∇ρ)vt dx
(2.8)
≤ kρkL∞ kvkL∞ k∇vkL2 kvt kL2 + k∇ρkL∞ k∇vkL2 kvt kL2
+ R(kρkL∞ k∇θkL2 + kθkL∞ k∇ρkL2 )kvt kL2
≤ Ck∇vk2L2 kvt kL2 + Ck∆ρkL2 k∇vkL2 kvt kL2
+ C(k∇θkL2 + kθkL∞ )kvt kL2 .
On the other hand, using (2.7) and the H 2 -theory of second order elliptic equations,
we have
kvkH 2 ≤ Ckρ∂t v + ρvm · ∇v + R∇(ρθ)kL2
≤ C(kvt kL2 + kv · ∇vkL2 + k∇ρkL∞ k∇vkL2 + k∇θkL2 + kθkL∞ )
≤ C(kvt kL2 +
k∇vk2L2
+ k∆ρkL2 k∇vkL2 + k∇θkL2 + kθk
L∞
(2.9)
).
Now using (2.3), Young’s inequality and the Gagliardo-Nirenberg inequality [5],
1
2(1−α)
2(1−α)
kvkH 2 + C,
k∇vk2L2 ≤ Ckvk2α
≤ CkvkH 2
≤
Lm kvkH 2
2C
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EXISTENCE OF SMOOTH GLOBAL SOLUTIONS
with 1 − α =
m+2
3m+2
<
1
2
5
, we obtain
kvkH 2 ≤ C(kvt kL2 + k∆ρkL2 k∇vkL2 + k∇θkL2 + kθkL∞ + C).
(2.10)
Combining (2.8), (2.9) and (2.10) and using Gronwall’s inequality, we obtain (2.5)
and (2.6). This completes the proof.
Lemma 2.4. Let K(θ) := θ + θ4 . If (ρ, v, θ) is a strong solution, then
kK(θ)kL∞ (0,T ;L2 ) + kK(θ)kL2 (0,T ;H 1 ) ≤ C(T ).
(2.11)
Proof. We start by rewriting the energy equation (1.15) in the form:
ρ∂t K(θ) + ρvm · ∇K(θ) − ∆K(θ) = (S : ∇v − p div v)K 0 (θ).
(2.12)
Testing (2.12) with K(θ), using (1.1), (2.5), (2.1) and (2.1), we find that
Z
Z
1 d
ρK 2 (θ)dx + |∇K(θ)|2 dx
2 dt
Z
= (S : ∇v − p div v)K 0 (θ)K(θ)dx
≤ kSkL2 k∇vkL2 kK 0 (θ)K(θ)kL∞ + CkρkL∞ k div vkL2 kK(θ)k2L4
7/4
≤ CkK(θ)kL∞ + CkK(θ)k2L4
1
7/8
7/8
≤ CkK(θ)kL2 kK(θ)kH 1 + k∇K(θ)k2L2 + CkK(θ)k2L2
8
1
≤ k∇K(θ)k2L2 + CkK(θ)k2L2 + C
4
which yields (2.11). Here we have used the Gagliardo-Nirenberg inequalities:
1/2
1/2
3/4
1/4
kK(θ)kL∞ ≤ CkK(θ)kL2 kK(θ)kH 1 ,
kK(θ)kL4 ≤ CkK(θ)kL2 kK(θ)kH 1 .
This completes the proof.
Lemma 2.5. If (ρ, v, θ) is a strong solution, then
kθkL∞ (0,T ;H 1 ) + kθkL2 (0,T ;H 2 ) ≤ C(T ),
(2.13)
kθt kL2 (0,T ;L2 ) ≤ C(T ).
(2.14)
Proof. We start by rewriting the energy equation (2.12) in the form:
S : ∇v − p div v 0
1
K (θ).
∂t K(θ) + vm · ∇K(θ) − ∆K(θ) =
ρ
ρ
Testing the above equation with −∆K(θ), using (2.5), (2.6), (2.1), (2.1) and (2.11),
we deduce that
Z
Z
1 d
1
2
|∇K(θ)| dx +
|∆K(θ)|2 dx
2 dt
ρ
Z
S : ∇v − p div v 0 =
(v − ∇ log ρ)∇K(θ) −
K (θ) ∆K(θ)dx
ρ
1
≤ kvkL∞ k∇K(θ)kL2 + ρ ∞ k∇ρkL∞ k∇K(θ)kL2
L
1
+ k kL∞ kS : ∇vkL2 kK 0 (θ)kL∞ + Ck div vkL2 kK(θ)kL∞ k∆K(θ)kL2
ρ
6
J. SUN, J. FAN, G. NAKAMURA
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3/4
≤ C(kK(θ)kH 1 + kρkH 2 kK(θ)kH 1 + k∇vk2L4 kK(θ)kL∞ )k∆K(θ)kL2
1/2
3/8
≤ C(kK(θ)kH 1 + kρkH 2 kK(θ)kH 1 + kvkH 2 kK(θ)kH 1 )k∆K(θ)kL2
1
3/4
≤ k∆K(θ)k2L2 + CkK(θ)k2H 1 + Ckρk2H 2 kK(θ)k2H 1 + CkvkH 2 kK(θ)kH 1
2
which yields (2.13). Here we have used the Gagliardo-Nirenberg inequalities:
3/2
1/2
1/2
1/2
k∇vk2L4 ≤ Ck∇vkL2 kvkH 2 , kK(θ)kL∞ ≤ CkK(θ)kL2 kK(θ)kH 1 ,
kθkL∞ (0,T ;L∞ ) ≤ CkθkL∞ (0,T ;H 1 ) ,
k∇θkL∞ (0,T ;L2 ) ≤ Ck∇K(θ)kL∞ (0,T ;L2 ) ,
k∆θkL2 (0,T ;L2 ) ≤ Ck∆K(θ)kL2 (0,T ;L2 ) .
Equation (2.14) follows easily from (2.12), (2.13), (2.5), (2.6), and (2.1). This
completes the proof.
Acknowledgments. This work is partially supported by grant 11171154 from the
NSFC .
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Jianzhu Sun
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037,
China
E-mail address: jzsun@njfu.com.cn
Jishan Fan
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037,
China
E-mail address: fanjishan@njfu.com.cn
Gen Nakamura
Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan
E-mail address: gnaka@math.sci.hokudai.ac.jp