The proof of Theorem 2.2 is based on the extension principle. The idea of this principle is explained for example in [24], in the proof of Theorem 2.4, p.233. For the reader’s convenience let us explain it briefly. Theorem 2.1 ensures the existence of a unique solution of our problem on the time domain \(]t_{0},t_{0}+T_{0}[\) with initial functions \(\rho(\cdot,t_{0})\), \(v(\cdot,t_{0})\), \(\omega(\cdot,t_{0})\), and \(\theta(\cdot,t_{0})\). So, we have the existence on the time domain \(]0,T_{1}[\), where \(T_{1}=t_{0}+T_{0}\). After we repeat the procedure for k steps, we will have the existence on the time domain \(]0,T_{k}[\), so we can continue this procedure as long as through a priori estimates we can ensure that \(\rho(\cdot,t_{0})\), \(v(\cdot,t_{0})\), \(\omega (\cdot,t_{0})\), and \(\theta(\cdot,t_{0})\) satisfy the conditions for initial functions for any \(t_{0}\in\,]0,T_{k}[\). We make this principle more formal in the following proposition, as was done for example in [16].
Proposition 3.1
Let
\(T\in\mathbf{R}^{+}\)
and let the function
$$ (x,t)\mapsto(\rho,v,\omega,\theta) (x,t), \quad(x,t)\in Q_{T'} $$
(27)
be the generalized solution of the problem (7)-(12) on the domain
\(Q_{T'}\), for any
\(T'< T\)
with the property
\(\theta>0\)
in
\(\overline{Q}_{T'}\). Then (27) is the generalized solution of the same problem on the domain
\({Q}_{T}\)
with the property
\(\theta>0\)
in
\(\overline{Q}_{T}\).
As it is explained in the book [21], p.40, to be able to use the Proposition 3.1 it is crucial to find a set of global a priori estimates in which the constants are independent of the length of time domain from the local existence theorem. The constants can depend on initial data and the constant T from the Proposition 3.1 only. We shall note these constants by C or \(C_{i}\) where \(i=1,2,\ldots\) and in different places they can take over different values.
3.1 Lower bounds for density and temperature
Following the procedure from the book [21], Chapter 2 we first shall derive some properties of the functions ρ and θ. To be precise we have to show that these two functions are bounded from below which is the hardest part of this paper. We will also show the upper boundedness for the function ρ and derive some important properties for the function θ. Apart from the book [21], in this part of the work we also use the ideas from articles [20] and [22]. In the cases when we will use the results from other papers we will omit the proofs or details of proofs, but we will refer to them appropriately.
In almost all lemmas hereafter we use the lower boundedness of the function r:
Lemma 3.1
(Lemma 3.1 in [19], p.4)
The function
r
defined by (16) satisfies the estimate
$$ r(x,t)\geq a,\quad (x,t)\in Q_{T}, $$
(28)
where
\(a>0\)
is the radius of the smaller boundary sphere of the starting domain.
3.1.1 The ‘energy’ estimate
We first introduce the function
$$ U(x,t)=\frac{v^{2}}{2}+j_{I}\frac{\omega^{2}}{2}+R\psi \biggl(\frac{1}{\rho } \biggr)+c_{v}\psi(\theta), $$
(29)
where
$$ \psi(x)=x-\ln x-1 $$
(30)
is a non-negative and convex function. Let us note that the function U is the generalization of the energy function. The estimate of the energy function is crucial for obtaining the estimates and properties of the functions ρ, v, ω, and θ in the following sections.
Lemma 3.2
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that
$$\begin{aligned} &\int_{0}^{1}U(x,t)\,dx +\int_{0}^{t}\int_{0}^{1} \biggl[\frac{k}{L^{2}}\frac{r^{4}\rho }{\theta^{2}} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2} + \biggl(\lambda+\frac{2}{3}\mu \biggr) \frac{\rho}{\theta} \biggl(\frac {\partial}{\partial x} \bigl(r^{2}v \bigr) \biggr)^{2} \\ &\quad{}\times \biggl(c_{0}+\frac{2}{3}c_{d} \biggr) \frac{\rho}{\theta} \biggl(\frac {\partial}{\partial x} \bigl(r^{2}\omega \bigr) \biggr)^{2} \biggr]\,dx \,d\tau\leq C. \end{aligned}$$
(31)
Proof
Multiplying (7), (8), (9), and (10), respectively, by \(R (-\frac{1}{\rho^{2}}+\frac{1}{\rho} )\), v, \(j_{I}\omega\rho^{-1}\), and \(c_{v} (1-\frac{1}{\theta } )\rho^{-1}\), after addition and integration over \([0,1]\) we obtain
$$\begin{aligned} &\int_{0}^{1} \frac{\partial}{\partial t}U\,dx+\frac{\lambda+2\mu}{L^{2}}\int_{0}^{1} \frac{\rho}{\theta} \biggl[\frac{\partial}{\partial x} \bigl( r^{2}v \bigr) \biggr]^{2}\,dx \\ &\qquad{}+\frac{c_{0}+2c_{d}}{L^{2}}\int_{0}^{1} \frac{\rho}{\theta } \biggl[\frac{\partial}{\partial x} \bigl( r^{2}\omega \bigr) \biggr]^{2}\,dx \\ &\qquad{}+\frac{k}{L^{2}}\int_{0}^{1}\frac{r^{4}\rho}{\theta^{2}} \biggl[\frac{\partial\theta }{\partial x} \biggr]^{2}\,dx+4\mu_{r}\int _{0}^{1}\frac{\omega^{2}}{\rho\theta}\,dx \\ &\quad= \frac{4\mu}{L} \int_{0}^{1}\frac{1}{\theta}\frac{\partial}{\partial x} \bigl(rv^{2} \bigr)\,dx+ \frac{4c_{d}}{L}\int_{0}^{1}\frac{1}{\theta} \frac{\partial }{\partial x} \bigl(r\omega^{2} \bigr)\,dx. \end{aligned}$$
(32)
Integrating (32) over \([0,t]\), using (14) as well as the equality
$$ \frac{\partial}{\partial x} \bigl( rv^{2} \bigr)= \frac{2}{r}v\frac{\partial }{\partial x} \bigl( r^{2}v \bigr)- \frac{3Lv^{2}}{\rho r^{2}}, $$
(33)
which is also valid for the function ω, after some calculations we get
$$\begin{aligned} &\int_{0}^{1}U(x,t)\,dx+ \biggl(\lambda+\frac{2}{3}\mu \biggr)\frac{1}{L^{2}}\int _{0}^{t}\int_{0}^{1} \frac{\rho}{\theta} \biggl[\frac{\partial}{\partial x} \bigl( r^{2}v \bigr) \biggr]^{2}\,dx\,d\tau \\ &\quad{}+\biggl(c_{0}+\frac{2}{3}c_{d} \biggr) \frac {1}{L^{2}}\int_{0}^{t}\int _{0}^{1}\frac{\rho}{\theta} \biggl[\frac{\partial }{\partial x} \bigl( r^{2}\omega \bigr) \biggr]^{2}\,dx\,d\tau \\ &\quad{}+\frac{k}{L^{2}}\int_{0}^{t}\int _{0}^{1}\frac{r^{4}\rho}{\theta^{2}} \biggl[ \frac {\partial\theta}{\partial x} \biggr]^{2}\,dx\,d\tau\leq\int_{0}^{1}U(x,0)\,dx. \end{aligned}$$
(34)
Taking into account (20) we easily conclude the following estimate:
$$ \int_{0}^{1}U(x,0)\,dx\leq C \bigl(1+\bigl\Vert (\rho_{0},v_{0},\omega_{0}, \theta _{0} )\bigr\Vert ^{2}_{\mathrm{L}^{2} (]0,1[)^{4}} \bigr)\leq C, $$
(35)
which together with (34) immediately gives (31). □
Lemma 3.3
Let
\(\alpha_{1}\)
and
\(\alpha_{2}\)
be two positive solutions of the equation
$$ \psi(x)={C} {c^{-1}_{v}}, $$
(36)
where
C
is the same constant as in (31), and
ψ
is the function defined by (30). Then, for any
\(t\in\,]0,T[\)
we have
$$ \alpha_{1}\leq\int_{0}^{1} \theta(x,t)\,dx\leq\alpha_{2} $$
(37)
and there exists a function
\(a:[0,T]\rightarrow[0,1]\)
such that
$$ \alpha_{1}\leq\theta\bigl(a(t),t\bigr)\leq \alpha_{2}. $$
(38)
Proof
From (31) we immediately get
$$ \int_{0}^{1} (\theta-\ln\theta-1 ) (x,t)\,dx\leq{C} {c^{-1}_{v}}. $$
(39)
As the function ψ is convex, we are able to utilize the Jensen inequality and conclude that
$$ \int_{0}^{1}\theta(x,t)\,dx-\ln\int _{0}^{1}\theta(x,t)\,dx-1\leq{C} {c^{-1}_{v}}. $$
(40)
From (40) we easily get (37) and (38). □
3.1.2 Some auxiliary constructions
The aim of this section is to derive a useful representation of the function ρ, which is known in literature as the representation of the Kazhikov type. We will also list all important properties of the functions connected to this representation.
Lemma 3.4
Let
A
be the constant defined by
$$ A=\int_{0}^{1}\frac{1}{\rho_{0}(x)}\,dx. $$
(41)
For the function
ρ
and for any
\(t\in\,]0,T[\)
we have
$$ \int_{0}^{1} \frac{1}{\rho(x,t)}\,dx=A. $$
(42)
Also, there exists a function
g, \(0\leq g(t)\leq1\)
such that
$$ \rho\bigl(g(t),t\bigr)=A^{-1},\quad t\in[0,T]. $$
(43)
Proof
In the same way as in Lemma 2.1 in [21], p.43, from (7) we obtain (42) and (43). □
In the next lemma we use the same procedure as in [21], p.44, as well as some ideas from [22], p.349 in order to make the aforementioned representation of the function ρ.
Lemma 3.5
For the function
ρ
on
\(Q_{T}\)
we have
$$ \rho(x,t)=\frac{\rho_{0}(x)\cdot Y(t)\cdot B(x,t)}{1+ \frac {R}{\lambda+2\mu}\rho_{0}(x)\int_{0}^{t}\theta(x,\tau)\cdot Y(\tau)\cdot B(x,\tau)\,d\tau}, $$
(44)
where
$$ Y(t)=\frac{1}{A\rho_{0}(g(t))} \exp \biggl\{ \frac{R}{\lambda+2\mu}\int _{0}^{t}\rho\bigl(g(t),\tau\bigr)\theta\bigl(g(t), \tau\bigr)\,d\tau \biggr\} $$
(45)
and
$$ B(x,t)=\exp \biggl\{ -\frac{L}{\lambda+2\mu}\int_{g(t)}^{x} \int_{0}^{t}r^{-2}(y,\tau) \frac{\partial v(y,\tau)}{\partial t}\,d\tau d y \biggr\} . $$
(46)
(The constant
A
and the function
g
are from Lemma
3.4.)
Proof
Let us write (7) in the form
$$ \frac{1}{L}\rho\frac{\partial}{\partial x} \bigl(r^{2}v \bigr)=-\frac {\partial }{\partial t}\ln\rho $$
(47)
and insert it in (8). After we integrate the obtained equality over \([0,t]\), \(t\in\,]0,T[\), we get
$$ \begin{aligned}[b] &\frac{\partial }{\partial x} \biggl( \frac{\lambda+2\mu}{L}\ln\rho+\frac {R}{L}\int_{0}^{t} \rho(x,\tau)\theta(x,\tau)\,d\tau \biggr)\\ &\quad= \frac{\lambda+2\mu }{L}\frac{\partial }{\partial x}\ln \rho_{0}(x)-\int_{0}^{t}r^{-2}(x,\tau ) \frac{\partial v(x,\tau)}{\partial t}\,d\tau. \end{aligned} $$
(48)
Now, we integrate (48) over \([g(t),x]\), \(x\in\,]0,1[\) for fixed t and get
$$\begin{aligned} &\frac{\lambda+2\mu}{L}\ln\rho+ \frac{R}{L}\int_{0}^{t}\rho(x,\tau)\theta (x, \tau)\,d\tau \\ &\quad= \frac{\lambda+2\mu}{L}\ln\rho\bigl(g(t),t\bigr)+ \frac{R}{L}\int_{0}^{t}\rho \bigl(g(t),\tau \bigr)\theta\bigl(g(t),\tau\bigr)\,d\tau \\ &\qquad{}+\frac{\lambda+2\mu}{L}\ln\frac{\rho_{0}(x)}{\rho_{0}(g(t))}- \int_{g(t)}^{x}\int_{0}^{t}r^{-2}(y, \tau)\frac{\partial v(y,\tau)}{\partial t}\,d\tau d y. \end{aligned}$$
(49)
Taking into account (43), (45), and (46) we easily get (44). □
Lemma 3.6
There exists
\(C\in\mathbf{R}^{+}\)
such that for
\((x,t)\in Q_{T}\)
we have
$$ \biggl\vert \int_{g(t)}^{x}\int _{0}^{t}r^{-2}(y,\tau)\frac{\partial v(y,\tau )}{\partial t}\,d\tau d y\biggr\vert \leq C, $$
(50)
where the function
g
is defined by (43).
Proof
In the same way as in [22], p.349, (3.35), with the help of (28) and (31) we obtain (50). □
Lemma 3.7
The function
B
defined by (46) has the properties:
$$\begin{aligned}& C^{-1}\leq B(x,t)\leq C, \end{aligned}$$
(51)
$$\begin{aligned}& \frac{\partial B(x,t)}{\partial x}=B(x,t)\varphi(x,t), \end{aligned}$$
(52)
where
$$ \varphi(x,t)=\frac{-L}{\lambda+2\mu}\int_{0}^{t}r^{-2}(y, \tau)\frac {\partial v(y,\tau)}{\partial t}\,d\tau, $$
(53)
for
\((x,t)\in Q_{T}\)
and
\(C\in\mathbf{R}^{+}\).
Proof
Using (50) and (53) from (46) we immediately get (51) and (52). □
Lemma 3.8
There exist constants
\(C_{1},C_{2}\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$ C_{1}\leq Y(t)\leq C_{2}. $$
(54)
Proof
In the same way as in [21], Lemma 2.2, p.45, using (37), (42), (51), and the Gronwall inequality from (44) we get (54). □
Now we introduce the notations analogous to the one in [21], p.46 for the maximal and minimal values of the functions ρ and θ for fixed t:
$$ \begin{aligned} &m_{\rho}(t)=\min _{x\in[0,1]}\rho(x,t),\qquad M_{\rho}(t)=\max_{[0,1]} \rho (x,t), \\ &m_{\theta}(t)=\min_{x\in[0,1]}\theta(x,t),\qquad M_{\theta}(t)=\max_{x\in[0,1]}\theta(x,t). \end{aligned} $$
(55)
The following relationships of the functions from (55) are crucial for deriving the bounds of the function ρ.
Lemma 3.9
There exist positive constants
\(C_{1}\)
and
\(C_{2}\)
such that
$$ M_{\rho}(t)\leq C_{1} \biggl(1+\int _{0}^{t}m_{\theta}(\tau)\,d\tau \biggr)^{-1} $$
(56)
and
$$ m_{\rho}(t)\geq C_{2} \biggl(1+\int _{0}^{t}M_{\theta}(\tau)\,d\tau \biggr)^{-1}. $$
(57)
Proof
In the same way as in Lemma 2.3 in [21], p.46, using (51) and (54) from (44) we immediately get (56) and (57). □
To derive the further properties of the function θ we will need the following result.
Lemma 3.10
(Lemma 2.4 in [21], p.47)
For any
\(\varepsilon>0\)
there exists a constant
\(C_{\varepsilon}>0\), such that for any
\(t\in\,]0,T[\)
we have
$$ M^{2}_{\theta}(t)\leq\varepsilon I_{1}(t)+C_{\varepsilon}\bigl(1+I_{2}(t) \bigr), $$
(58)
where
$$ I_{1}(t)=\int_{0}^{1}r^{4} \rho \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}\,dx,\qquad I_{2}(t)=\int _{0}^{t}I_{1}(\tau)\,d\tau. $$
(59)
Let us mention that we slightly adapted the form of inequality (58) comparing to the one in [21], as well as the form of the function \(I_{1}\), but the proof remains the same.
3.1.3 Lower bound for the function θ
In the proof of the following lemma we used the adapted approach from [21], Lemma 3.1, p.48, as well as some ideas from Lemma 2.3, p.202 in [20] and Lemma 3.12, p.356 in [22].
Lemma 3.11
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$ m_{\theta}(t)\geq C. $$
(60)
Proof
Multiplying (10) by \(-\theta^{-2}\rho^{-1}\) we get
$$\begin{aligned} \frac{\partial}{\partial t} \biggl( \frac{1}{\theta} \biggr)={}&\frac{k}{c_{v} L^{2}}\frac{\partial}{\partial x} \biggl(r^{4}\rho\frac{\partial}{\partial x} \biggl(\frac{1}{\theta} \biggr) \biggr)-\frac{2k}{c_{v} L^{2}}\frac{\rho r^{4}}{\theta^{3}} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}+\frac {R}{c_{v}L}\frac{\rho}{\theta}\frac{\partial}{\partial x} \bigl( r^{2}v \bigr) \\ &{}-\frac{1}{\theta^{2}} \biggl\{ \frac{\lambda+2\mu}{c_{v}L^{2}}\rho \biggl[\frac {\partial}{\partial x} \bigl( r^{2}v \bigr) \biggr]^{2}-\frac{4\mu}{c_{v}L} \frac {\partial}{\partial x} \bigl(rv^{2} \bigr) \biggr\} \\ &{}-\frac{1}{\theta^{2}} \biggl\{ \frac{c_{2}+2c_{d}}{c_{v}L^{2}}\rho \biggl[\frac {\partial}{\partial x} \bigl( r^{2}\omega \bigr) \biggr]^{2}-\frac {4c_{d}}{c_{v}L} \frac{\partial}{\partial x} \bigl(r\omega^{2} \bigr) \biggr\} -\frac{4\mu_{r}}{c_{v}} \frac{\omega^{2}}{\rho\theta^{2}}, \end{aligned}$$
(61)
which implies the inequality
$$\begin{aligned} \frac{\partial}{\partial t} \biggl( \frac{1}{\theta} \biggr)\leq{}&\frac {k}{c_{v} L^{2}}\frac{\partial}{\partial x} \biggl(r^{4}\rho\frac{\partial }{\partial x} \biggl(\frac{1}{\theta} \biggr) \biggr)+\frac{R^{2}}{4{c_{v}} (\lambda+\frac{2}{3}\mu )}\rho \\ &{}-\frac{\rho}{\theta^{2}} \biggl[\frac {1}{L}\sqrt{\frac{\lambda+\frac{2}{3}\mu}{c_{v}}} \frac{\partial}{\partial x} \bigl( r^{2}v \bigr)-\frac{R}{2\sqrt{c_{v}}\sqrt{\lambda+\frac{2}{3}\mu }}\theta \biggr]^{2}, \end{aligned}$$
(62)
i.e.
$$ \frac{\partial}{\partial t} \biggl( \frac{1}{\theta} \biggr)\leq\frac {k}{c_{v} L^{2}}\frac{\partial}{\partial x} \biggl(r^{4}\rho\frac{\partial }{\partial x} \biggl(\frac{1}{\theta} \biggr) \biggr)+\frac{R^{2}}{4{c_{v}} (\lambda+\frac{2}{3}\mu )}\rho. $$
(63)
After multiplying (63) by \(p\theta ^{-p+1}\), \(p\geq2\), we have
$$ \frac{\partial}{\partial t} \biggl( \frac{1}{\theta^{p}} \biggr)\leq\frac {kp}{c_{v} L^{2}}\frac{\partial}{\partial x} \biggl(r^{4}\rho\frac{\partial }{\partial x} \biggl(\frac{1}{\theta} \biggr) \biggr) \biggl(\frac{1}{\theta } \biggr)^{p-1}+\frac{R^{2}p}{4{c_{v}} (\lambda+\frac{2}{3}\mu )}\rho \biggl(\frac{1}{\theta} \biggr)^{p-1} $$
(64)
which, after integration over \(]0,1[\), gives
$$ \frac{d}{dt}\biggl\Vert \frac{1}{\theta(t)}\biggr\Vert _{\mathrm{L}^{p} (]0,1[ )}^{p}\leq \frac{R^{2}p}{4{c_{v}} (\lambda+\frac{2}{3}\mu )}\int_{0}^{1}\rho \biggl( \frac{1}{\theta} \biggr)^{p-1}\,dx. $$
(65)
After applying the Hölder inequality to the right-hand side of (65) we get
$$ \frac{d}{dt}\biggl\Vert \frac{1}{\theta(t)}\biggr\Vert _{\mathrm{L}^{p} (]0,1[ )}^{p}\leq \frac{R^{2}p}{4{c_{v}} (\lambda+\frac{2}{3}\mu )}\bigl\Vert \rho(t)\bigr\Vert _{\mathrm{L}^{p} (]0,1[ )}\biggl\Vert \frac {1}{\theta(t)}\biggr\Vert _{\mathrm{L}^{p} (]0,1[ )}^{p-1}, $$
(66)
hence we have
$$ \frac{d}{dt}\biggl\Vert \frac{1}{\theta(t)}\biggr\Vert _{\mathrm{L}^{p} (]0,1[ )}\leq\frac{R^{2}}{4{c_{v}} (\lambda+\frac{2}{3}\mu )}\bigl\Vert \rho(t)\bigr\Vert _{\mathrm{L}^{p} (]0,1[ )}. $$
(67)
Now we integrate (67) over \([0,t]\), \(t\in\,]0,T[\) and obtain
$$ \biggl\Vert \frac{1}{\theta(t)}\biggr\Vert _{\mathrm{L}^{p} (]0,1[ )}\leq \biggl\Vert \frac{1}{\theta(0)}\biggr\Vert _{\mathrm{L}^{p} (]0,1[ )}+ \frac{R^{2}}{4{c_{v}} (\lambda+\frac{2}{3}\mu )}\int_{0}^{t} \bigl\Vert \rho(\tau)\bigr\Vert _{\mathrm{L}^{p} (]0,1[ )}\,d\tau, $$
(68)
which implies the assertion of the lemma, analogously to the proof of Lemma 3.1 in [21], p.49. □
With the help of (60) and (56) we immediately get the following property of the function ρ.
Corollary 3.1
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$ M_{\rho}\leq C. $$
(69)
3.1.4 Lower bound for the function ρ
In obtaining the lower bound for the density we were not able to use the method proposed in [21], p.50 which is used in [20], p.203, for the one-dimensional model, so we adapted here the idea from [22], Lemma 3.6, p.348.
Lemma 3.12
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$ m_{\rho}(t)\geq C. $$
(70)
Proof
By using the Cauchy-Schwarz inequality as well as (38), (37), and (28) we get
$$\begin{aligned} \bigl\vert \sqrt{\theta(x,t)}-\sqrt{ \theta\bigl(a(t),t\bigr)}\bigr\vert &\leq C\int_{a(t)}^{x} \frac{1}{\sqrt{\theta}}\biggl\vert \frac{\partial\theta}{\partial x}\biggr\vert \,dy \\ &\leq C \biggl(\int_{0}^{1}\frac{r^{4}\rho}{{\theta^{2}}} \biggl( \frac{\partial\theta }{\partial x} \biggr)^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int_{0}^{1}\frac{\theta}{r^{4}\rho}\,dx \biggr)^{\frac{1}{2}} \\ &\leq C \biggl(\int_{0}^{1} \frac{r^{4}\rho}{{\theta^{2}}} \biggl(\frac{\partial\theta }{\partial x} \biggr)^{2}\,dx \biggr)^{\frac{1}{2}}\frac{1}{\sqrt{m_{\rho}}}. \end{aligned}$$
(71)
Taking into account estimate (38), from (71) we obtain
$$ {\theta(x,t)}\leq C \biggl(1+ \biggl(\int_{0}^{1} \frac{r^{4}\rho}{{\theta^{2}}} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}\,dx \biggr) \frac{1}{m_{\rho}} \biggr), $$
(72)
which we insert into (57) and get
$$ \frac{1}{m_{\rho}(t)}\leq C \biggl( 1+\int_{0}^{t} \int_{0}^{1}\frac{r^{4}\rho}{{\theta ^{2}}} \biggl( \frac{\partial\theta}{\partial x} \biggr)^{2}\,dx\frac{1}{m_{\rho}(\tau)}\,d\tau \biggr). $$
(73)
After we apply the Gronwall inequality to (73) and use estimate (31) we immediately get (70). □
3.2
A priori estimates for derivatives
To be able to derive the estimates of derivatives for functions ρ, v, ω, and θ we will apply the energy method. Therefore, we will make the estimate of the function
$$ \Phi=\frac{1}{2}v^{2}+\frac{1}{2}j_{I} \omega^{2}+c_{v}\theta, $$
(74)
adapting the procedure used in the proof of Lemma 2.4 in [20], p.203.
Lemma 3.13
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$ \int_{0}^{1} \bigl( \Phi^{2}+v^{4}+\omega^{4} \bigr)\,dx+I_{2} \leq C, $$
(75)
where the function
\(I_{2}\)
is defined by (59).
Proof
First we multiply (8), (9), and (10), respectively, by v, \(j_{I}\omega\rho^{-1}\) and \(c_{v}\rho^{-1}\), and integrate them over \(]0,1[\). After addition of the obtained equalities, making use of boundary conditions and (14), we get
$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _{0}^{1}\Phi^{2}(x,t)\,dx={}&{-}\int _{0}^{1} \biggl( \frac{\lambda+2\mu}{L^{2}}\rho r^{4}\frac{\partial\Phi}{\partial x} +\frac{2\lambda}{L}rv^{2}- \frac{R}{L}\rho\theta r^{2}v+\frac{2c_{o}}{L}r\omega ^{2} \\ &{}+ \biggl(\frac{c_{0}+2c_{d}}{L^{2}}-j_{I}\frac{\lambda+2\mu }{L^{2}} \biggr) \rho r^{4}\omega\frac{\partial\omega}{\partial x} \\ &{}+ \biggl(\frac{k}{ L^{2}}-c_{v} \frac{\lambda+2\mu}{L^{2}} \biggr)r^{4}\rho\frac{\partial \theta}{\partial x} \biggr) \frac{\partial\Phi}{\partial x}\,dx, \end{aligned}$$
(76)
which, using the Young inequality, implies
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt}\int _{0}^{1}\Phi^{2}(x,t)\,dx \\ &\qquad{}+\int_{0}^{1} \rho r^{4} \biggl[ \frac{\lambda+2\mu}{L^{2}} \biggl(\frac{\partial\Phi }{\partial x} \biggr)^{2}\,dx+ \biggl( \frac{k}{ L^{2}}-c_{v}\frac{\lambda+2\mu }{L^{2}} \biggr)\frac{\partial\theta}{\partial x} \frac{\partial\Phi }{\partial x}\,dx \biggr] \\ &\quad\leq C_{1}\frac{\varepsilon^{-1}}{2}\int_{0}^{1} \biggl(\bigl(\rho r^{2}\bigr)^{-1}v^{4} +\rho \theta^{2}v^{2}+\bigl(\rho r^{2} \bigr)^{-1}\omega^{4}+\rho r^{4}\omega^{2} \biggl(\frac{\partial\omega}{\partial x} \biggr)^{2} \biggr)\,dx \\ &\qquad{}+2C_{1}\varepsilon\int_{0}^{1}\rho r^{4} \biggl(\frac{\partial\Phi}{\partial x} \biggr)^{2}, \end{aligned}$$
(77)
where \(\varepsilon>0\) is arbitrary.
To simplify (77), using elementary algebraic operations, we derive the following inequality:
$$\begin{aligned} &(A-B) (a+b+c)^{2}+(C-A)c(a+b+c) \\ &\quad\geq(C-3B)c^{2}- \biggl(2B+\frac {(A-2B+C)^{2}}{4B} \biggr)a^{2}-\frac{1}{2B} \biggl[(A-B)^{2}+\frac {(A-2B+C)^{2}}{2} \biggr]b^{2}, \end{aligned}$$
(78)
where \(A, B, C, a, b, c\in\mathbf{R}\), and \(A>B>0\). In (78) we insert \(a=v\frac{\partial v}{\partial x}\), \(b=j_{I}\omega\frac{\partial\omega}{\partial x}\), \(c=c_{v}\frac{\partial\theta}{\partial x}\), \(A=\frac{\lambda+2\mu}{L^{2}}\), \(B=2C_{1}\varepsilon\), \(C=\frac{k}{c_{v}L^{2}}\), and choose ε such that \(A-B>0\) and \(C-3B>0\). For simplicity reasons we denote \(D=C-3B\) and
$$\begin{aligned} C_{2}={}&\max \biggl\{ 2B+\frac{(A-2B+C)^{2}}{4B}, \\ &{} \frac {1}{2B} \biggl[(A-B)^{2}+\frac{(A-2B+C)^{2}}{2} \biggr]+ C_{1}\frac{\varepsilon ^{-2}}{2}\cdot C_{1}\frac{\varepsilon^{-2}}{2} \biggr\} . \end{aligned}$$
(79)
We get
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt}\int _{0}^{1}\Phi^{2}(x,t)\,dx+D\int _{0}^{1} \rho r^{4} \biggl( \frac{\partial\theta}{\partial x} \biggr)^{2}\,dx \\ &\quad\leq C_{2}\int_{0}^{1} \biggl(\bigl(\rho r^{2}\bigr)^{-1}v^{4} +\rho\theta^{2}v^{2}+ \bigl(\rho r^{2}\bigr)^{-1}\omega^{4} \\ &\qquad{}+\rho r^{4}v^{2} \biggl(\frac{\partial v }{\partial x} \biggr)^{2}+ \rho r^{4}\omega^{2} \biggl(\frac{\partial\omega}{\partial x} \biggr)^{2} \biggr)\,dx. \end{aligned}$$
(80)
Using (69), (70), and (28), from (80) we obtain
$$\begin{aligned} & \frac{1}{2}\frac{d}{dt}\int _{0}^{1}\Phi^{2}(x,t)\,dx+D\int _{0}^{1} \rho r^{4} \biggl( \frac{\partial\theta}{\partial x} \biggr)^{2}\,dx \\ &\quad\leq C_{3}\int_{0}^{1} \biggl(v^{4} +\theta^{2}v^{2}+\omega^{4} + r^{4}v^{2} \biggl(\frac{\partial v }{\partial x} \biggr)^{2}+ r^{4} \omega^{2} \biggl(\frac{\partial\omega}{\partial x} \biggr)^{2} \biggr)\,dx. \end{aligned}$$
(81)
To be able to bound the terms \(v^{4}\) and \(\omega^{4}\) on the right-hand side of (81) we multiply (8) and (9), respectively, by \(v^{3}\) and \(\omega^{3}\), integrate over \(]0,1[\) and utilize the Young inequality. After some calculations and making use of (69), (70), and (28), we get
$$\begin{aligned}& \frac{1}{4}\int _{0}^{1}\frac{\partial v^{4}}{\partial t}\,dx +C_{4}\int _{0}^{1} r^{4}v^{2} \biggl( \frac{\partial v }{\partial x} \biggr)^{2}\,dx \leq C_{5}\int _{0}^{1} \bigl(v^{4}+\theta^{2} v^{2} \bigr)\,dx, \end{aligned}$$
(82)
$$\begin{aligned}& \frac{1}{4}\int _{0}^{1}\frac{\partial\omega^{4}}{\partial t}\,dx +C_{6}\int _{0}^{1} r^{4}\omega^{2} \biggl(\frac{\partial\omega}{\partial x} \biggr)^{2}\,dx \leq C_{7}\int _{0}^{1}\omega^{4}\,dx. \end{aligned}$$
(83)
Now, we multiply (82) by \(C_{3}C_{4}^{-1}\) and (83) by \(C_{3}C_{6}^{-1}\). After the addition of the obtained inequalities with (81), we find
$$\begin{aligned} &\frac{d}{dt} \biggl[\int _{0}^{1} \biggl(\Phi^{2}(x,t)+ \frac {C_{3}C_{4}^{-1}}{2}v^{4}+\frac{C_{3}C_{6}^{-1}}{2}\omega^{4} \biggr)\,dx+ 2D\int_{0}^{t}\int_{0}^{1} \rho r^{4} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}\,dx\,d\tau \biggr] \\ &\quad\leq2C_{3}\int_{0}^{1} \bigl( \bigl(1+C_{5}C_{4}^{-1} \bigr) \bigl(v^{4} +\theta ^{2}v^{2} \bigr) + \bigl(1+C_{7}C_{6}^{-1} \bigr) \omega^{4} \bigr)\,dx. \end{aligned}$$
(84)
To finish the proof we need the following inequality which is the direct consequence of (58):
$$ \int_{0}^{1}\theta^{2}v^{2}\,dx \leq C_{8}\epsilon\int_{0}^{1}r^{4} \rho \biggl(\frac {\partial\theta}{\partial x} \biggr)^{2}\,dx+C_{\epsilon}\biggl(1+ \int_{0}^{t}\int_{0}^{1} \rho r^{4} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}\,dx\,d\tau \biggr) $$
(85)
which we insert into (84) and use the suitable ϵ. Hence we have
$$\begin{aligned} &\frac{d}{dt}\int _{0}^{1} \biggl(\Phi^{2}(x,t)+ \frac{C_{3}C_{4}^{-1}}{2}v^{4}+\frac {C_{3}C_{6}^{-1}}{2}\omega^{4}+D\int _{0}^{t} \rho r^{4} \biggl( \frac{\partial\theta}{\partial x} \biggr)^{2}\,d\tau \biggr)\,dx \\ &\quad\leq C_{9}\int_{0}^{1} \biggl( \Phi^{2}(x,t)+\frac{C_{3}C_{4}^{-1}}{2}v^{4} +\frac{C_{3}C_{6}^{-1}}{2} \omega^{4}+D\int_{0}^{t} \rho r^{4} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}\,d\tau \biggr)\,dx+1. \end{aligned}$$
(86)
Using the Gronwall inequality, from (86) we immediately get (75). □
Equations (75) and (58) imply an important property of the function \(M_{\theta}\), which is given in the next corollary.
Corollary 3.2
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that we have
$$ \Vert M_{\theta} \Vert _{\mathrm{L}^{2} (]0,T[ )} \leq C. $$
(87)
Let us notice that (70), (75) and (28) imply
$$ \frac{\partial\theta}{\partial x}\in\mathrm{L}^{2} (Q_{T} ). $$
(88)
We will derive the estimates for the first spatial derivatives of other functions (ρ, v, and ω) in the following two lemmas.
Lemma 3.14
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$ \biggl\Vert \frac{\partial\rho}{\partial x}(t)\biggr\Vert \leq C. $$
(89)
Proof
Taking into account (52) and (53), for the derivative of (44) we get
$$ \frac{\partial\rho}{\partial x}=\rho\varphi-\rho^{2}Y^{-1}B^{-1} \biggl[\frac{d}{dx} \biggl(\frac{1}{\rho_{0}} \biggr)+ \frac {RL}{\lambda+2\mu}\int_{0}^{t}BY \biggl( \frac{\partial\theta}{\partial x}+\theta\varphi \biggr)\,d\tau \biggr]. $$
(90)
With the help of (51), (54), and (69), after integration over \(]0,1[\), (90) implies
$$ \biggl\Vert \frac{\partial\rho}{\partial x}\biggr\Vert ^{2}\leq C \biggl(\|\varphi \|^{2} +\int _{0}^{1}\frac{1}{\rho^{4}_{0}} \bigl( \rho_{0}' \bigr)^{2}\,dx+\int _{0}^{t}\int_{0}^{1} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}\,dx\,d\tau+ \int_{0}^{t}M^{2}_{\theta}\|\varphi\|^{2}\,d\tau \biggr). $$
(91)
Using (28), integration by parts and properties of the initial data, from (53) we obtain
$$ \bigl\| \varphi(t)\bigr\| ^{2}\leq C \biggl(1+\|v\|^{2}+ \int_{0}^{t}\int_{0}^{1}v^{4}\,dx\,d\tau \biggr). $$
(92)
Inserting (92) into (91), using the properties of the initial data as well as (31), (75), (87), and (88), we immediately get (89). □
Lemma 3.15
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$\begin{aligned}& \bigl\| v(t)\bigr\| ^{2}+\int_{0}^{t} \biggl\Vert \frac{\partial v}{\partial x}(\tau)\biggr\Vert ^{2}\leq C, \end{aligned}$$
(93)
$$\begin{aligned}& \bigl\| \omega(t)\bigr\| ^{2}+\int_{0}^{t} \biggl\Vert \frac{\partial\omega}{\partial x}(\tau )\biggr\Vert ^{2}\leq C. \end{aligned}$$
(94)
Proof
After we multiply (8) by v and integrate over \(]0,1[\), we get
$$ \frac{1}{2}\frac{d}{dt}\bigl\Vert v(t)\bigr\Vert ^{2}+\frac{\lambda+2\mu }{L^{2}}\int_{0}^{1}\rho \biggl(\frac{\partial}{\partial x} \bigl( r^{2}v \bigr) \biggr)^{2}\,dx= \frac{R}{L}\int_{0}^{1}\rho\theta \frac{\partial}{\partial x} \bigl( r^{2}v \bigr)\,dx. $$
(95)
Using the Young inequality, (69) and (70) from (95) we obtain
$$ \frac{d}{dt}\bigl\Vert v(t)\bigr\Vert ^{2}+ \int_{0}^{1} \biggl(\frac{\partial }{\partial x} \bigl( r^{2}v \bigr) \biggr)^{2}\,dx\leq CM_{\theta}^{2}. $$
(96)
To simplify the left hand side of (96) we use the inequality
$$ \biggl[\frac{\partial}{\partial x} \bigl(r^{2}v \bigr) \biggr]^{2}\geq\frac {1}{2} \biggl(\frac{\partial v}{\partial x} \biggr)^{2}-Cv^{2}, $$
(97)
which can easily be derived using (14), (28), and (69). After inserting (97) into (96), integrating over \(]0,t[\) and using (31) and (87) we immediately get (93).
Now we prove (94). Let us first notice that (97) is also valid for the function ω. After we multiply (9) by \(\rho^{-1}\omega\) and integrate over \(]0,1[\), we obtain
$$ \frac{1}{2}\frac{d}{dt}\bigl\Vert \omega(t)\bigr\Vert ^{2}+\frac {c_{0}+2c_{d}}{j_{I}L^{2}}\int_{0}^{1} \rho \biggl(\frac{\partial}{\partial x} \bigl( r^{2}\omega \bigr) \biggr)^{2}\,dx=\frac{4\mu_{r}}{j_{I}}\int_{0}^{1} \frac{\omega ^{2}}{\rho}\,dx, $$
(98)
from which, by using (70) and the same procedure as before, we immediately arrive at (94). □
Lemma 3.15 enables us to derive the upper boundedness of the function r, which is crucial for the estimates of the second spatial derivatives.
Corollary 3.3
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\((x,t)\in\overline{Q}_{T}\)
we have
Proof
From (16) and (15), using the Gagliardo-Ladyzhenskaya inequality as well as the Young inequality together with (31), we get
$$ r(x,t)\leq C \biggl(1+\int_{0}^{t} \biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{2} \biggr)\,dx $$
(100)
from which, using (93) we obtain the assertion of the lemma. □
Lemma 3.16
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$\begin{aligned}& \biggl\Vert \frac{\partial v}{\partial x}(t)\biggr\Vert ^{2}+ \int_{0}^{t}\biggl\Vert \frac {\partial^{2} v}{\partial x^{2}}(\tau) \biggr\Vert ^{2}\,d\tau\leq C, \end{aligned}$$
(101)
$$\begin{aligned}& \biggl\Vert \frac{\partial\omega}{\partial x}(t)\biggr\Vert ^{2}+ \int_{0}^{t}\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}}(\tau) \biggr\Vert ^{2}\,d\tau\leq C, \end{aligned}$$
(102)
$$\begin{aligned}& \biggl\Vert \frac{\partial\theta}{\partial x}(t)\biggr\Vert ^{2}+ \int_{0}^{t}\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}(\tau) \biggr\Vert ^{2}\,d\tau\leq C. \end{aligned}$$
(103)
Proof
Multiplying (8) by \(\frac{\partial^{2} v}{\partial x^{2}}\) and integrating over \(]0,1[\), we get
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \biggl\Vert \frac{\partial v}{\partial x}(t)\biggr\Vert ^{2}+\frac{\lambda+2\mu}{L^{2}} \int_{0}^{1}r^{4}\rho \biggl( \frac{\partial^{2} v}{\partial x^{2}} \biggr)^{2}\,dx \\ &\quad= \frac{R}{L}\int_{0}^{1}r^{2} \theta\frac {\partial^{2} v}{\partial x^{2}}\frac{\partial\rho}{\partial x}\,dx+ \frac{R}{L}\int _{0}^{1}r^{2}\rho\frac{\partial^{2} v}{\partial x^{2}} \frac {\partial\theta}{\partial x}\,dx \\ &\qquad{}+2(\lambda+2\mu)\int_{0}^{1}\frac {v}{r^{2}\rho} \frac{\partial^{2} v}{\partial x^{2}}\,dx-\frac{\lambda+2\mu }{L^{2}}\int_{0}^{1}r^{4} \frac{\partial\rho}{\partial x}\frac{\partial v}{\partial x}\frac{\partial^{2} v}{\partial x^{2}}\,dx \\ &\qquad{}-\frac{4(\lambda+2\mu )}{L}\int_{0}^{1}r \frac{\partial v}{\partial x}\frac{\partial^{2} v}{\partial x^{2}}\,dx. \end{aligned}$$
(104)
Using the Hölder, Gagliardo-Ladyzhenskaya, and Young inequalities as well as (89) and (99), we obtain the estimates of the integrals on the right-hand side of (104) as follows:
$$\begin{aligned}& \biggl\vert \frac{R}{L}\int_{0}^{1}r^{2} \theta\frac{\partial^{2} v}{\partial x^{2}}\frac{\partial\rho}{\partial x}\,dx\biggr\vert \leq CM_{\theta}\biggl\Vert \frac {\partial^{2} v}{\partial x^{2}}\biggr\Vert \biggl\Vert \frac{\partial\rho }{\partial x} \biggr\Vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{2}+ CM^{2}_{\theta}, \end{aligned}$$
(105)
$$\begin{aligned}& \biggl\vert \frac{R}{L}\int_{0}^{1}r^{2} \rho\frac{\partial^{2} v}{\partial x^{2}}\frac{\partial\theta}{\partial x}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{2}+ C\biggl\Vert \frac{\partial \theta}{\partial x}\biggr\Vert ^{2}, \end{aligned}$$
(106)
$$\begin{aligned}& \biggl\vert 2(\lambda+2\mu)\int_{0}^{1} \frac{v}{r^{2}\rho}\frac{\partial^{2} v}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{2}+ C\Vert v\Vert ^{2}, \end{aligned}$$
(107)
$$\begin{aligned}& \begin{aligned}[b] \biggl\vert -\frac{\lambda+2\mu}{L^{2}}\int _{0}^{1}r^{4}\frac{\partial\rho}{\partial x} \frac{\partial v}{\partial x}\frac{\partial^{2} v}{\partial x^{2}}\,dx\biggr\vert &\leq C\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{\frac {1}{2}}\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{\frac{1}{2}}\int_{0}^{1} \biggl\vert \frac{\partial\rho}{\partial x}\frac{\partial^{2} v}{\partial x^{2}}\biggr\vert \,dx\\ &\leq C\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{\frac{1}{2}}\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{\frac{3}{2}}\biggl\Vert \frac{\partial\rho}{\partial x}\biggr\Vert \\ &\leq \varepsilon\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}} \biggr\Vert ^{2}+ C\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{2}, \end{aligned} \end{aligned}$$
(108)
$$\begin{aligned}& \biggl\vert -\frac{4(\lambda+2\mu)}{L}\int_{0}^{1}r \frac{\partial v}{\partial x}\frac{\partial^{2} v}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{2}+ C\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{2}. \end{aligned}$$
(109)
Inserting (105)-(109) into (104), and by using the small enough \(\varepsilon>0\) we get
$$ \frac{1}{2}\frac{d}{dt}\biggl\Vert \frac{\partial v}{\partial x}(t)\biggr\Vert ^{2}+C_{1}\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{2}\leq C_{2} \biggl(1+M_{\theta}^{2}+\Vert v\Vert ^{2}+\biggl\Vert \frac{\partial\theta }{\partial x}\biggr\Vert ^{2}+\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{2} \biggr), $$
(110)
from which by integration over \(]0,t[\) and by using the properties of the initial data, (88) and (93) we get the assertion (101).
Now we prove (102). By multiplying (8) by \(\rho ^{-1}\frac{\partial^{2} \omega}{\partial x^{2}}\) and integrating over \(]0,1[\), we obtain
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \biggl\Vert \frac{\partial\omega}{\partial x}(t)\biggr\Vert ^{2}+ \frac{c_{0}+2c_{d}}{j_{I}L^{2}}\int_{0}^{1}r^{4}\rho \biggl(\frac {\partial^{2} \omega}{\partial x^{2}} \biggr)^{2}\,dx \\ &\quad=\frac{4\mu_{r}}{j_{I}}\int_{0}^{1}\frac{\omega}{\rho} \frac{\partial^{2} \omega}{\partial x^{2}}\,dx+2\frac{c_{0}+2c_{d}}{j_{I}}\int_{0}^{1} \frac{\omega}{r^{2}\rho}\frac {\partial^{2} \omega}{\partial x^{2}}\,dx \\ &\qquad{}-\frac{c_{0}+2c_{d}}{j_{I}L^{2}}\int_{0}^{1}r^{4} \frac{\partial\rho}{\partial x}\frac{\partial\omega }{\partial x}\frac{\partial^{2} \omega}{\partial x^{2}}\,dx-\frac {4(c_{0}+2c_{d})}{j_{I}L} \int_{0}^{1}r\frac{\partial\omega}{\partial x} \frac {\partial^{2} \omega}{\partial x^{2}}\,dx. \end{aligned}$$
(111)
In the same way as before, we get the estimates
$$\begin{aligned}& \biggl\vert \frac{4\mu_{r}}{j_{I}}\int_{0}^{1} \frac{\omega}{\rho}\frac{\partial^{2} \omega}{\partial x^{2}}\,dx\biggr\vert \leq C\biggl\Vert \frac{\partial^{2} \omega }{\partial x^{2}}\biggr\Vert \Vert \omega \Vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}}\biggr\Vert ^{2} + C\Vert \omega \Vert ^{2}, \end{aligned}$$
(112)
$$\begin{aligned}& \biggl\vert 2\frac{c_{0}+2c_{d}}{j_{I}}\int_{0}^{1} \frac{\omega}{r^{2}\rho}\frac {\partial^{2} \omega}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac {\partial^{2} \omega}{\partial x^{2}}\biggr\Vert ^{2}+ C\Vert \omega \Vert ^{2}, \end{aligned}$$
(113)
$$\begin{aligned}& \biggl\vert -\frac{c_{0}+2c_{d}}{j_{I}L^{2}}\int_{0}^{1}r^{4} \frac{\partial\rho }{\partial x}\frac{\partial\omega}{\partial x}\frac{\partial^{2} \omega }{\partial x^{2}}\,dx\biggr\vert \leq \varepsilon\biggl\Vert \frac{\partial^{2} \omega }{\partial x^{2}}\biggr\Vert ^{2}+ C\biggl\Vert \frac{\partial\omega}{\partial x}\biggr\Vert ^{2}\biggl\Vert \frac {\partial\rho}{\partial x}\biggr\Vert ^{4}, \end{aligned}$$
(114)
$$\begin{aligned}& \biggl\vert -\frac{4(c_{0}+2c_{d})}{j_{I}L}\int_{0}^{1}r \frac{\partial\omega }{\partial x}\frac{\partial^{2} \omega}{\partial x^{2}}\,dx\biggr\vert \leq \varepsilon\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}}\biggr\Vert ^{2}+ C\Vert \omega \Vert ^{2}. \end{aligned}$$
(115)
By inserting (112)-(115) into (111), for small enough \(\varepsilon>0\), we get
$$ \frac{1}{2}\frac{d}{dt}\biggl\Vert \frac{\partial\omega}{\partial x}(t)\biggr\Vert ^{2}+C_{1}\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}}\biggr\Vert ^{2}\leq C_{2} \biggl(1+ \Vert \omega \Vert ^{2}+\biggl\Vert \frac{\partial \omega}{\partial x}\biggr\Vert ^{2} \biggr), $$
(116)
from which, by integration over \(]0,t[\), using the properties of the initial data and (94), we obtain (102).
To prove (103) we multiply (10) by \(\rho ^{-1}\frac{\partial^{2} \theta}{\partial x^{2}}\). After integrating over \(]0,1[\), we obtain
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \biggl\Vert \frac{\partial\theta}{\partial x}(t)\biggr\Vert ^{2}+\frac{k}{c_{v} L^{2}} \int_{0}^{1}r^{4}\rho \biggl( \frac{\partial^{2} \theta}{\partial x^{2}} \biggr)^{2}\,dx \\ &\quad=-\frac{4k}{c_{v} L}\int _{0}^{1}r\frac {\partial\theta}{\partial x}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx \\ &\qquad{}-\frac{k}{c_{v} L^{2}}\int_{0}^{1}r^{4} \frac{\partial\rho}{\partial x}\frac {\partial\theta}{\partial x}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx \frac{2R}{c_{v}}\int _{0}^{1}\frac{\theta v}{r}\frac{\partial^{2} \theta }{\partial x^{2}}\,dx+ \frac{R}{c_{v}L}\int_{0}^{1}r^{2}\rho \theta\frac{\partial v}{\partial x}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx \\ &\qquad{}-\frac{4(\lambda +\mu)}{c_{v}}\int_{0}^{1}\frac{v^{2}}{r^{2}\rho} \frac{\partial^{2} \theta}{\partial x^{2}}\,dx-\frac{4\lambda}{c_{v}L}\int_{0}^{1}rv \frac{\partial v}{\partial x}\frac {\partial^{2} \theta}{\partial x^{2}}\,dx \\ &\qquad{}-\frac{\lambda+2\mu}{c_{v}L^{2}}\int_{0}^{1}r^{4} \rho \biggl(\frac{\partial v}{\partial x} \biggr)^{2}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx- \frac {4(c_{0}+c_{d})}{c_{v}}\int_{0}^{1}\frac{\omega^{2}}{r^{2}\rho} \frac{\partial^{2} \theta }{\partial x^{2}}\,dx \\ &\qquad{}-\frac{4c_{0}}{c_{v}L}\int_{0}^{1}r\omega \frac{\partial \omega}{\partial x}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx -\frac{c_{0}+2c_{d}}{c_{v}L^{2}}\int _{0}^{1}r^{4}\rho \biggl( \frac{\partial\omega }{\partial x} \biggr)^{2}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx \\ &\qquad{}-\frac {4\mu_{r}}{c_{v}} \int_{0}^{1}\frac{\omega^{2}}{\rho}\frac{\partial^{2} \theta }{\partial x^{2}}. \end{aligned}$$
(117)
Analogously to before we conclude the following:
$$\begin{aligned}& \biggl\vert -\frac{4k}{c_{v} L}\int_{0}^{1}r \frac{\partial\theta}{\partial x}\frac {\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq C\biggl\Vert \frac{\partial \theta}{\partial x}\biggr\Vert \biggl\Vert \frac{\partial^{2}\theta}{\partial x^{2}}\biggr\Vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C \biggl\Vert \frac{\partial\theta}{\partial x}\biggr\Vert ^{2}, \end{aligned}$$
(118)
$$\begin{aligned}& \biggl\vert -\frac{k}{c_{v} L^{2}}\int _{0}^{1}r^{4}\frac{\partial\rho}{\partial x} \frac {\partial\theta}{\partial x}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq C\biggl\Vert \frac{\partial\theta}{\partial x}\biggr\Vert ^{\frac{1}{2}}\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{\frac {3}{2}}\biggl\Vert \frac{\partial\rho}{\partial x}\biggr\Vert \leq \varepsilon \biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C \biggl\Vert \frac{\partial\theta}{\partial x}\biggr\Vert ^{2}, \end{aligned}$$
(119)
$$\begin{aligned}& \biggl\vert \frac{2R}{c_{v}}\int_{0}^{1} \frac{\theta v}{r}\frac{\partial^{2} \theta }{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} \theta }{\partial x^{2}}\biggr\Vert ^{2}+ CM^{2}_{\theta} \Vert v\Vert ^{2}\leq \varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}} \biggr\Vert ^{2}+ CM^{2}_{\theta}, \end{aligned}$$
(120)
$$\begin{aligned}& \biggl\vert \frac{R}{c_{v}L}\int_{0}^{1}r^{2} \rho\theta\frac{\partial v}{\partial x}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon \biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ CM^{2}_{\theta}\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{2}\leq\varepsilon\biggl\Vert \frac {\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ CM^{2}_{\theta}, \end{aligned}$$
(121)
$$\begin{aligned}& \biggl\vert -\frac{4(\lambda+\mu)}{c_{v}}\int_{0}^{1} \frac{v^{2}}{r^{2}\rho}\frac {\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac {\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C\bigl\Vert v^{2} \bigr\Vert ^{2}\leq \varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C, \end{aligned}$$
(122)
$$\begin{aligned}& \biggl\vert -\frac{4\lambda}{c_{v}L}\int_{0}^{1}rv \frac{\partial v}{\partial x}\frac {\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac {\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C \biggl(\Vert v\Vert ^{2}+\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{6} \biggr)\leq\varepsilon \biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C, \end{aligned}$$
(123)
$$\begin{aligned}& \begin{aligned}[b] \biggl\vert -\frac{\lambda+2\mu}{c_{v}L^{2}}\int _{0}^{1}r^{4}\rho \biggl( \frac{\partial v}{\partial x} \biggr)^{2}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert &\leq \varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C \biggl(\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{2}+\biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{6} \biggr)\\ &\leq\varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C \biggl(1+\biggl\Vert \frac {\partial^{2} v}{\partial x^{2}}\biggr\Vert ^{2} \biggr), \end{aligned} \end{aligned}$$
(124)
$$\begin{aligned}& \biggl\vert -\frac{4(c_{0}+c_{d})}{c_{v}}\int_{0}^{1} \frac{\omega^{2}}{r^{2}\rho}\frac {\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac {\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C, \end{aligned}$$
(125)
$$\begin{aligned}& \biggl\vert -\frac{4c_{0}}{c_{v}L}\int_{0}^{1}r \omega\frac{\partial\omega}{\partial x}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C, \end{aligned}$$
(126)
$$\begin{aligned}& \biggl\vert -\frac{c_{0}+2c_{d}}{c_{v}L^{2}}\int_{0}^{1}r^{4} \rho \biggl(\frac{\partial\omega }{\partial x} \biggr)^{2}\frac{\partial^{2} \theta}{\partial x^{2}}\,dx\biggr\vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C \biggl(1+\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}}\biggr\Vert ^{2} \biggr), \end{aligned}$$
(127)
$$\begin{aligned}& \biggl\vert -\frac{4\mu_{r}}{c_{v}}\int_{0}^{1} \frac{\omega^{2}}{\rho}\frac{\partial^{2} \theta}{\partial x^{2}}\biggr\vert \leq\varepsilon\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ C. \end{aligned}$$
(128)
By inserting (118)-(128) into (117) and taking the sufficiently small \(\varepsilon>0\), we get
$$ \frac{1}{2}\frac{d}{dt} \biggl\Vert \frac{\partial\theta}{\partial x}(t)\biggr\Vert ^{2}+C_{1} \biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}\leq C_{2} \biggl(1+M^{2}_{\theta}+ \biggl\Vert \frac{\partial\theta }{\partial x}\biggr\Vert ^{2}+\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}} \biggr\Vert ^{2}+\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}}\biggr\Vert ^{2} \biggr), $$
(129)
which, after integration over \(]0,t[\), and making use of (87), (88), (101), and (102) as well as the properties of initial data, gives the assertion (103). □
In the next lemma we estimate the time derivatives of the functions ρ, v, ω, and θ.
Lemma 3.17
There exists a constant
\(C\in\mathbf{R}^{+}\)
such that for any
\(t\in\,]0,T[\)
we have
$$\begin{aligned}& \int_{0}^{t}\biggl\Vert \frac{\partial\rho}{\partial t}(\tau)\biggr\Vert ^{2}\,d\tau\leq C, \end{aligned}$$
(130)
$$\begin{aligned}& \int_{0}^{t}\biggl\Vert \frac{\partial v}{\partial t}(\tau)\biggr\Vert ^{2}\,d\tau\leq C, \end{aligned}$$
(131)
$$\begin{aligned}& \int_{0}^{t}\biggl\Vert \frac{\partial\omega}{\partial t}(\tau)\biggr\Vert ^{2}\,d\tau\leq C, \end{aligned}$$
(132)
$$\begin{aligned}& \int_{0}^{t}\biggl\Vert \frac{\partial\theta}{\partial t}(\tau)\biggr\Vert ^{2}\,d\tau\leq C. \end{aligned}$$
(133)
Proof
From (7), by integrating over \(]0,1[\), using (28), (99), and (69) we get
$$ \biggl\Vert \frac{\partial\rho}{\partial t} \biggr\Vert ^{2} = \frac{1}{L^{2}}\int_{0}^{1}\rho^{4} \biggl[ \frac{\partial}{\partial x} \bigl(r^{2}v \bigr) \biggr]^{2}\,dx \leq C \biggl(\|v\|^{2}+ \biggl\Vert \frac{\partial v}{\partial x}\biggr\Vert ^{2} \biggr). $$
(134)
Using (93) and (101) from (134) we can easily conclude (130).
In a similar way, from (8), using estimates (28), (99), (69), (70),(69), (70), and (89), we get the inequality
$$\begin{aligned} \biggl\Vert \frac{\partial v}{\partial t} \biggr\Vert ^{2} \leq& C\int_{0}^{1} \biggl(r^{4}\rho^{2} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}+r^{4}\theta^{2} \biggl( \frac{\partial\rho}{\partial x} \biggr)^{2}+\frac{v^{2}}{r^{4}\rho^{2}}+r^{8} \biggl(\frac{\partial\rho}{\partial x} \biggr)^{2} \biggl(\frac{\partial v}{\partial x} \biggr)^{2} \\ &{} +r^{2} \biggl(\frac{\partial v}{\partial x} \biggr)^{2}+ r^{8}\rho^{2} \biggl(\frac{\partial^{2} v}{\partial x^{2}} \biggr)^{2} \biggr)\,dx \\ \leq &C \biggl(1+M_{\theta}^{2}+ \biggl\Vert \frac{\partial\theta}{\partial x} \biggr\Vert ^{2}+\|v\|^{2}+\biggl\Vert \frac{\partial v}{\partial x} \biggr\Vert ^{2} +\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}} \biggr\Vert ^{2} \biggr). \end{aligned}$$
(135)
Using (87), (88), (93), and (101) from (135) we can easily conclude (131).
To prove (132), we multiply (9) by \(\rho^{-1}\) and using the same procedure as in (135), we get
$$ \biggl\Vert \frac{\partial\omega}{\partial t} \biggr\Vert ^{2} \leq C \biggl(1+\Vert \omega \Vert ^{2}+\biggl\Vert \frac{\partial \omega}{\partial x} \biggr\Vert ^{2} +\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}} \biggr\Vert ^{2} \biggr) $$
(136)
which, using (94) and (102) leads to (132).
Analogously, from (10) we derive
$$\begin{aligned} \biggl\Vert \frac{\partial\theta}{\partial t} \biggr\Vert ^{2} \leq& C \biggl(1+\int_{0}^{1} \biggl( \biggl(\frac{\partial^{2} \theta}{\partial x^{2}} \biggr)^{2}+ \biggl( \frac{\partial\theta}{\partial x} \biggr)^{2} + \biggl(\frac{\partial\rho}{\partial x} \biggr)^{2} \biggl(\frac{\partial\theta}{\partial x} \biggr)^{2}+{ \theta^{2} v^{2}} \\ &{}+ \theta^{2} \biggl(\frac{\partial v}{\partial x} \biggr)^{2}+ v^{2} \biggl(\frac{\partial v}{\partial x} \biggr)^{2}+ \biggl(\frac{\partial v}{\partial x} \biggr)^{4}+ \omega^{2} \biggl(\frac{\partial\omega}{\partial x} \biggr)^{2}+ \biggl( \frac{\partial\omega}{\partial x} \biggr)^{4} \biggr)\,dx \biggr). \end{aligned}$$
(137)
Using (89), (93), (101), (102), and (103) as well as the Gagliardo-Ladyzhenskaya and the Young inequalities, from (137) we get
$$\begin{aligned} \biggl\Vert \frac{\partial\theta}{\partial t} \biggr\Vert ^{2} \leq& C \biggl(1+M_{\theta}^{2}+\biggl\Vert \frac{\partial\theta}{\partial x} \biggr\Vert ^{2}+\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}}\biggr\Vert ^{2}+ \biggl\Vert \frac{\partial\theta}{\partial x} \biggr\Vert \biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}} \biggr\Vert \biggl\Vert \frac{\partial\rho}{\partial x} \biggr\Vert ^{2} \\ &{}+ \biggl\Vert \frac{\partial v}{\partial x} \biggr\Vert ^{3}\Vert v \Vert +\biggl\Vert \frac{\partial \omega}{\partial x} \biggr\Vert ^{3}\Vert \omega \Vert +\biggl\Vert \frac{\partial v}{\partial x} \biggr\Vert ^{3} \biggl\Vert \frac{\partial^{2} v}{\partial x^{2}} \biggr\Vert +\biggl\Vert \frac{\partial\omega}{\partial x} \biggr\Vert ^{3}\biggl\Vert \frac{\partial^{2} \omega}{\partial x^{2}} \biggr\Vert \biggr) \\ \leq& C \biggl(1+M_{\theta}^{2}+\biggl\Vert \frac{\partial^{2} \theta}{\partial x^{2}} \biggr\Vert ^{2} +\biggl\Vert \frac{\partial^{2} v}{\partial x^{2}} \biggr\Vert + \biggl\Vert \frac {\partial^{2} \omega}{\partial x^{2}} \biggr\Vert \biggr), \end{aligned}$$
(138)
from which, using (87), (101), (102), and (103), we obtain the assertion (133). □
3.3 Final proof of Theorem 2.2
Corollary 3.1 and Lemma 3.14 gives the assertion
$$ \rho\in\mathrm{L}^{\infty}\bigl(0,T;\mathrm{H}^{1} \bigl(]0,1[ \bigr) \bigr). $$
(139)
From Lemmas 3.13, 3.15, and 3.16 we have
$$ v,\omega,\theta\in\mathrm{L}^{2} \bigl(0,T; \mathrm{H}^{2} \bigl(]0,1[ \bigr) \bigr)\cap\mathrm{L}^{\infty}\bigl(0,T;\mathrm{H}^{1} \bigl(]0,1[ \bigr) \bigr). $$
(140)
Using inclusion (139) and (140) as well as the results from Lemma 3.17 we get
$$ \rho, v, \omega, \theta\in\mathrm{H}^{1} (Q_{T} ). $$
(141)
Now, using Lemmas 3.11 and 3.12 as well as inclusions (139), (140), and (141) in accordance with the Proposition 3.1, we have the statement of Theorem 2.2.