Cauchy problem for isothermal system in a general nozzle with space-dependent friction

1.   Introduction

The following isentropic gas dynamics system in a general nozzle with friction, whose physical phenomena called "choking or choked flow",

is of interest because resonance occurs. This means there is a coincidence of wave speeds from different families of waves (see ^[2,4,6,7,16] and the references cited therein for the details). Here $\rho$ is the density of gas, $u$ the velocity, $P = P (\rho)$ the pressure, $a(x)$ is a slowly variable cross section area at $x$ in the nozzle and $\alpha(x)$ denotes a friction function. For the polytropic gas, $P$ takes the special form $P(\rho) = \frac{1}{\gamma} \rho^\gamma$, where $\gamma > 1$ is the adiabatic exponent and for the isothermal gas, $\gamma = 1$.

The Cauchy problem of system (1.1) with bounded initial data

in the simplest divergent nozzle (with respect to $a'(x) \ge 0$) was first obtained in ^[19] for the usual gases $1 < \gamma \le \frac{5}{3}$, and later, extended in ^[8] to the case of $\gamma > 1$, provided that the initial data are bounded and satisfy the very special condition $z(\rho_0(x), u_0(x)) \le 0$.

When $\gamma = 1$, the global existence of symmetrical weak solutions of the isothermal gas dynamics system (1.1) without a friction ($\alpha = 0$) in the Lagrangian coordinates was well studied in ^{[12,13,20,21]} by using the Glimm scheme method ^[3,15]; and in the Euler coordinates studied in ^[1,9] by using the compensated compactness theory ^[5,14,18]. The global existence of weak solutions of the isothermal gas dynamics system (1.1) with a constant friction was studied in ^[10], where, the maximum principle was used directly to obtain the a-priori dependent-time $L^\infty$ estimate $0 \le \rho \le M (T)$, $|u| \le M (T)$ under the conditions $|A(x)| = \left| \frac{a'(x)}{a(x)} \right| \le M$ and $\alpha \ge 0$.

In this paper, by carefully applying the maximum principle and the viscosity-flux approximation method introduced in ^[11], under the more general conditions $A(x) \in L^1$, $\alpha(x) \in L^1$, we improve the above time-dependent bound $M(T)$ to a constant bound $M$, which ensures that the entropy solutions of the Cauchy problem (1.1) and (1.2) we obtained are stable.

The main result is given in the following

Theorem 1.1. Let $P(\rho) = \rho$, $0 < a_L \le a(x) \le A_L$ for $x$ in any compact set $x \in (-L, L)$, $A(x) = -\frac{a'(x)}{a(x)} \in L^1(\mathbb{R})$ and $\alpha(x) \in L^1(\mathbb{R})$, where $A_L$, $a_L$ are positive constants, but could depend on $L$. Moreover, if

and the bounded initial data satisfy

where $M > 1$ is a constant, then the Cauchy problem (1.1) and (1.2) have a bounded weak solution $(\rho, u)$, which has the following uniform bound

and satisfies system (1.1) in the sense of distributions and the following Lax's entropy condition

where $(\eta, q)$ is a pair of entropy-entropy flux of system (1.1), $\eta$ is convex, and $\phi \in C_0^\infty (\mathbb{R} \times \mathbb{R}^+ - \{ t = 0 \})$ is a nonnegative function.

2.   Proof of Theorem 1.1

Let $v = \rho a(x)$ and rewrite (1.1) as follows

The two eigenvalues of (2.1) are $\lambda_1 = u-1$ and $\lambda_2 = u+1$, with corresponding Riemann invariants

where $m = vu$.

First, we add the viscosity parameter $\varepsilon > 0$ and the flux-approximation parameter $\delta > 0$ to system (2.1) to obtain the following parabolic system

with initial data

where

and $G^\delta$, $G_1^\tau$ are two mollifiers and $\tau > 0$ is the regularity parameter. Then by the conditions given in Theorem 1.1, we have

and

Second, we multiply (2.2) by $(w_v, w_m)$ and $(z_v, z_m)$, respectively, to obtain

and

where $\lambda_1^\delta = u - \frac{v-2\delta}{v}$ and $\lambda_2^\delta = u + \frac{v-2\delta}{v}$.

Let $X(x) = 3(A^\tau(x)+\alpha^\tau(x))$, then $|X(x)|_{L^1(\mathbb{R})} \le \frac{1}{2}$ by the condition (1.3). Making the transformations of $z = z_1 + B(x)$, $w = w_1 + C(x)$, where

for a positive constant $M > 1$, we have from (2.4) and (2.5) that

and

Clearly, we can choose a suitable small positive constant $\varepsilon_1$ and $\varepsilon = o(\varepsilon_1)$, $\tau = o(\varepsilon_1)$ such that the following terms in (2.6) and (2.7) satisfy

Since the initial data $v_0^\delta(x) \ge 2\delta$, we may obtain the a priori estimate $v^{\delta, \varepsilon, \tau}(x) \ge 2\delta$ by applying the maximum principle to the first equation in (2.2) (see the proof of Lemma 2.2 in ^[17]).

Now, under the conditions in Theorem 1.1, by using (2.6)–(2.8), we prove the following inequalities

where $b_i(x, t)$, $c_i(x, t)$, $i = 1, 2, 3$, are suitable functions satisfying the necessary conditions $b_3(x, t) \le 0$, $c_3(x, t) \le 0$.

Proof of (2.9). We prove (2.9) in several cases for two different groups of points $(x, t)$, where $\alpha(x) \ge 0$ or $\alpha(x) \le 0$.

We separate $B'(x)B(x) = (1-\varepsilon_1)B'(x)B(x)+\varepsilon_1B'(x)B(x)$ and let the following terms in (2.6)

$\bf{Case \ I.}$ At the points $(x, t)$, where $\alpha(x) \ge 0$, $v(x, t) \le 1$ and $w_1 + 2 \int_{-\infty}^x X(s) ds \le 0$, we have

$\bf{Case \ II.}$ At the points $(x, t)$, where $\alpha(x) \ge 0$, $v(x, t) \le 1$ and $w_1 + 2 \int_{-\infty}^x X(s) ds \ge 0$,

where $e(x, t) = - \frac{1}{4} \alpha^\tau(x) \left(w_1+ 4\int_{-\infty}^x X(s) ds \right) \le 0$, because

$\bf{Case \ III.}$ At the points $(x, t)$, where $\alpha(x) \ge 0$, $v(x, t) > 1$ and $w_1 + 2 \int_{-\infty}^x X(s) ds \le 0$, we have $\frac{v-2\delta}{v} \ge 1-\varepsilon_2 > 0$ for a small $\varepsilon_2 > 0$, and $B'(x)\ln(v) = -X(x) \left(\frac{1}{2} (w_1+z_1) + M \right)$. Then,

because

for small $\varepsilon_1$ and $\varepsilon_2$.

$\bf{Case \ IV.}$ At the points $(x, t)$, where $\alpha(x) \ge 0$, $v(x, t) > 1$ and $w_1 + 2 \int_{-\infty}^x X(s) ds \ge 0$,

because

where $d(x, t)$, $e(x, t)$ are given in (2.10). Thus we obtain the proof of the first inequality in (2.9) at the points $(x, t)$, where $\alpha(x) \ge 0$.

Now we prove the second inequality in (2.9). Let the following terms in (2.7),

At the points $(x, t)$, where $\alpha(x) \ge 0$ and $v(x, t) \le 1$, we have

at the points $(x, t)$, where $\alpha(x) \ge 0$ and $v(x, t) > 1$,

Thus we obtain the proof of (2.9) at the points $(x, t)$, where $\alpha(x) \ge 0$. Similarly, we may prove (2.9) also at the points $(x, t)$, where $\alpha(x) \le 0$.

We now return to the proof of the theorem. Under the conditions given in (1.4), it is clear that $z_1(x, 0) \le 0$, $w_1(x, 0) \le 0$, so, we may apply the maximum principle to (2.9) to obtain the estimates (see ^[9] for the details)

where $M_i$, $i = 1, 2, 3$ are suitable positive constants, independent of $\varepsilon$, $\delta$, $\tau$ and the time $t$.

By applying the general contracting mapping principle to an integral representation of (2.2), with the help of the lower, positive estimate and the $L^\infty$ estimates given in (2.11), we can obtain the existence and uniqueness of smooth solution of the Cauchy problem (2.2) and (2.3). Applying the convergence frame given in ^[5] we have the pointwise convergence

or

Furthermore, in a similar way as given in ^[10], we may prove that the limit $(\rho(x, t), u(x, t))$ satisfies system (1.1) in the sense of distributions and the Lax entropy condition (1.5). So, we complete the proof of Theorem 1.1.

3.   Conclusions

In this paper, we only study the Cauchy problem of the isothermal system, which is corresponding to the adiabatic exponent $\gamma = 1$, in a general nozzle with space-dependent friction $\alpha(x)$. It is more interesting and difficult to study the general adiabatic exponent $\gamma > 1$. We will come back to this topic in a coming article.

Acknowledgments

This paper is partially supported by Zhejiang Province NSFC grant No. LY20A010023 and the NSFC grant No. 12071106 of China. The authors are very grateful to the referees for their many valuable suggestions.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.