1. Introduction
In this paper, we study the global existence of cylinder symmetric solutions to the nonlinear compressible Navier-Stokes equations with external forces and heat source in a bounded domain
G = {
r ∈
R+,
0 <
a <
r <
b < +
∞} of
R3, where
r is the radial variable. In the Eulerian coordinates, the system under consideration are expressed as
()
()
()
()
()
where
()
and
ρ is the mass density,
θ is the absolute temperature,
u,
v,
w are the radial velocity, angular velocity, and axial velocity, respectively, and
λ,
μ,
ν,
γ,
CV,
κ,
λ are the constants satisfying
γ,
CV,
κ,
μ > 0, 3
λ + 2
μ ≥ 0 (
ν =
λ + 2
μ).
f1,
f2,
f3, and
g represent external forces and heat source, respectively. For system (
1.1)–(
1.5), we consider the following initial boundary value problem:
()
()
To show the global existence, it is convenient to transform the system (
1.1)–(
1.5) to that in the Lagrangian coordinates. The Eulerian coordinates (
r,
t) are connected to the Lagrangian coordinates (
ξ,
t) by the relation
()
where
and
()
It should be noted that if inf {
ρ0(
s) :
s ∈ (
a,
b)} > 0, then
η is invertible. It follows from (
1.1), (
1.8), and (
1.10) that
()
and
G is transformed into Ω = (0,
L) with
()
Differentiating (
1.11) with respect to
ξ yields
()
In general, for a function
, we easily get
()
()
Without danger of confusion, we denote
still by (
ρ,
u,
v,
w,
θ) and (
ξ,
t) by (
x,
t). We set
τ∶ = 1/
ρ to denote the specific volume. Therefore, by virtue of (
1.13)–(
1.15), system (
1.1)–(
1.8) in the new variables (
x,
t) read
()
()
()
()
()
together with
()
()
By (
1.9) and (
1.13), we have
()
Now let us first recall the related results in the literature. When there were no external forces and heat source, in two or three dimensions, the global existence and large time behavior of smooth solutions to the equations of a viscous polytropic ideal gas have been investigated for general domains only in the case of sufficiently small initial data, see, for example, [1–3]. For any large initial data, the global existence of generalized solutions was shown in [4–7]. Recently, Qin [8] proved the exponential stability in H1 and H2, and Qin and Jiang [9] studied the global existence and exponential stability in H4 with smallness of initial total energy.
When there exist external forces and heat forces, for one-dimensional case, the system is isentropic compressible Navier-Stokes equations. Mucha [10] obtained the exponential stability under various boundary conditions, Yanagi [11] established the existence of classical solutions, and Qin and Zhao [12] proved the global existence and asymptotic behavior of solutions for pressure P = ργ with γ = 1. Later on, Zhang and Fang [13] studied the global existence and uniqueness for γ > 1. For nonisentropic compressible Navier-Stokes equations, Qin and Yu [14] proved the global existence and asymptotic behavior for perfect gas. In two- or three-dimensional case and the external force and heat source f ≠ 0, g ≠ 0, Qin and Wen [15] proved the global existence of spherically symmetric solutions. In this paper, we will prove the global existence of cylinder symmetric solutions with external forces and heat source in a bounded domain in R3.
The notation in this paper will be as follows: , , m ∈ N, H1 = W1,2, denote the usual (Sobolev) spaces on (0, L). In addition, ∥·∥B denotes the norm in the space B; we also put . We denote by Ck(I, B), k ∈ N0, the space of k-times continuously differentiable functions from I⊆R into a Banach space B, and likewise by , the corresponding Lebesgue spaces. Subscripts t and x denote the (partial) derivatives with respect to t and x, respectively. We use C1 to denote the generic positive constant depending on the H1-norm of the initial data and time T.
We suppose that
fi(
r(
x,
t),
t) (
i = 1,2, 3),
g(
r,
t) satisfy, for any
T > 0,
()
()
We are now in a position to state our main theorems.
Theorem 1.1. Assume that (1.24)-(1.25) hold; if , τ0(x) > 0, θ0(x) > 0 on [0, L] and the initial data are compatible with the boundary conditions (1.22), then for problem (1.16)–(1.22) there exists a unique global solution such that, for any T > 0,
()
2. Proof of Theorem
In this section we will complete the proof of Theorem 1.1. To this end, we assume that in this section all assumptions in Theorem 1.1 hold. The proof of Theorem can be divided into the following several lemmas.
Proof. The proof of (2.1) is borrowed from [6, 8]; please refer to (2.1) in [6] or Lemma 2.1 in [8] for detail.
Lemma 2.2. The global solution (τ(t), u(t), v(t), w(t), θ(t)) to problems (1.16)–(1.22) satisfies the following estimates:
()
()
where
()
Proof. Multiplying (1.17)–(1.19) by u, v, and w, respectively, adding up the results, and using (1.16), we have
()
Integrating (
2.5) with respect to
x and
t over
QT = [0,
L] × [0,
t] (
t ∈ [0,
T],
∀
T > 0), using boundary condition (
1.22), we obtain
()
which, by using Gronwall’s inequality and (
1.24)-(
1.25), gives (
2.2).
By (1.16)–(1.20), we can easily obtain
()
Note that constants
ν =
λ + 2
μ and
()
Integrating (
2.7) with respect to
x and
t over
QT, using (
1.22), (
1.24)-(
1.25), and (
2.8), we conclude
()
The proof is complete.
Next we adapt and modify an idea of Qin and Wen [15] for one-dimensional case to give a representation for τ.
Let
()
()
Then, we infer from (
1.16) and (
1.17) that
()
By (
1.16) and (
2.12), we have
()
Integrating (
2.13) with respect to
x and
t over
QT, we obtain
()
where
h0(
x)∶ =
h(
x, 0). It follows from integration of (
1.16) over
QT and use of (
1.22) that
()
If we apply the mean value theorem to (
2.14) and use (
2.15), we conclude there is an
x0(
t)∈[0,
L] such that
()
Therefore, we derive from (
2.11), (
2.14), and (
2.16) that
()
Using (
2.17), we will show the representation of specific volume
τ.
Lemma 2.3. One has the following representation:
()
where
()
Proof. By (1.16) and (1.17), we have
()
Integrating (
2.20) over [
x0(
t),
x]×[0,
t] and using (
2.17), we derive
()
which, when the exponentials are taken, turns into
()
Multiplying (
2.22) by
γθ/
ν and integrating the resulting equation with respect to
t, we arrive at
()
Substituting this into (
2.22), we obtain (
2.18). The proof is complete.
Lemma 2.4. There are positive constants and , such that, for any T > 0,
()
Proof. Recalling the definition D(x, t), we have by (1.24), Cauchy-Schwarz’s inequality, and Lemma 2.1 that
()
which, along with Lemma
2.2, gives
()
By Lemmas
2.1 and
2.2, we easily obtain, for any 0 ≤
s ≤
t,
()
Therefore, we derive from (
2.2), (
2.18), and (
2.26)-(
2.27) that
()
()
which, by using Gronwall’s inequality and (
2.28), gives (
2.24). The proof is complete.
Remark 1. If the initial data or initial energy are small enough, we can obtain the uniform estimate independent of time t about specific volume τ under assumptions of external forces. Moreover, we can prove the large-time behavior of solutions.
Lemma 2.5. Under the assumptions of Theorem 1.1, one has, for any T > 0 and for all t ∈ [0, T],
()
Proof. Multiplying (2.5) by (1/2)(u2 + v2 + w2) + CVθ and then integrating the result over QT, we have
()
where
()
Multiplying (
1.17) by
u3 and then integrating the result over
QT, we get
()
Similarly, multiplying (
1.18) and (
1.19) by
v3 and
w3, respectively, and then integrating over
QT, we have
()
Multiplying (
2.31) and (
2.33) by
and
μ/
ν, respectively, adding up the resulting inequalities, and using (
2.34) to obtain, with the help of (
2.32), the following result:
()
On the other hand, by (
2.3) and (
2.20),
()
In view of (
1.24)-(
1.25) and (
2.36), we apply Gronwall’s inequality to (
2.35) to obtain (
2.30). The proof is complete.
Lemma 2.6. Under the assumptions of Theorem 1.1, one has, for any T > 0,
()
Proof. By means of (1.16), we rewrite (1.17) as
()
Multiplying (
2.38) by (
u/
r) − (
ντx/
τ) in
L2[0,
L] and using Lemmas
2.1–
2.5, we arrive at
()
Integrating it with respect to
t, using Lemmas
2.1-
2.5, (
1.24), and (
2.36), we get
()
We exploit the Gronwall inequality to (
2.40) to obtain (
2.37). The proof is complete.
Lemma 2.7. Under the assumptions of Theorem 1.1, one has, for any T > 0,
()
()
()
Proof. Multiplying (1.17) by ut, integrating the result over QT, using Lemmas 2.1–2.6, and taking into account that , we obtain
()
We derive from (
2.30) that
()
Therefore, using (
1.24), (
2.30), (
2.36), (
2.37), and (
2.44)-(
2.45), we conclude
()
which, by applying the Gronwall inequality, implies
()
By (
1.17), we have
()
Therefore,
()
which, along with (
2.47), gives (
2.41).
Analogously, multiplying (1.18) by vt, integrating the result over QT, and using assumptions (1.24) and Lemmas 2.1–2.6, we deduce
()
In view of (
2.36), we apply Gronwall’s inequality to (
2.50) to obtain
()
By (
1.18) and (
2.51), we easily deduce
()
which, along with (
2.51) and Lemmas
2.1–
2.4, implies (
2.42). The proof of (
2.43) iS similar to that of (
2.41) and (
2.42). The proof is now complete.
Lemma 2.8. Under the assumptions of Theorem 1.1, one has, for any T > 0,
()
Proof. Multiplying (1.20) by θt over QT, we have
()
Using Lemmas
2.1–
2.7, the Cauchy-Schwarz inequality, and the interpolation inequality, we have
()
()
Similarly to (
2.55), by virtue of (
1.25), (
2.30), and (
2.41)–(
2.43), we arrive at
()
Combining (
2.54)–(
2.57), we conclude
()
In view of (
2.36) and (
2.41), we apply Gronwall’s inequality to (
2.58) to obtain
()
Similarly to proof of (
2.41), by Lemmas
2.1–
2.7, (
1.20), (
1.25), and (
2.59), we obtain
()
which, together with (
2.59), implies (
2.53). The proof is complete.
Proof of Theorem 1.1. By Lemmas 2.1–2.8, we complete the proof of Theorem 1.1.
Acknowledgments
The work is in part supported by Doctoral Foundation of North China University of Water Sources and Electric Power (no. 201087) and the Natural Science Foundation of Henan Province of China (no. 112300410040).
