Asymptotic stability of deterministic and stochastic prey-predator models with prey herd immigration

1.   Introduction

In this paper, we are concerned with the motion of the supersonic potential flow in a two-dimensional expanding duct denoted by ${\Omega}$ (See Fig. 1), which is bounded by the lower wall $\Gamma_{\text{low}} = \{(x, y)| y = -f(x), 0\leq x<+\infty\}$, the upper wall $\Gamma_{\text{up}} = \{(x, y)| y = f(x), 0\leq x<+\infty\}$ and the inlet $\Gamma_{\text{in}} = \{(x, y)|x = \varphi(y), y\in[-f(0), f(0)]\}$. Here we assume that $f(x)\in C^{2}([0, +\infty))$ satisfies

and $\varphi(y)\in C^{2}([-f(0), f(0)])$ is an even function which satisfies

At the inlet, the flow velocity is assumed to be along the normal direction of the inlet and its speed is given by $q_{0}(y) \in C^{1}[-f(0), f(0)]$. Moreover, we require that $q_{0}(y)$ satisfies

where $c_1, c_2$ are positive constants, $c_*$ is the critical speed of the flow and $\hat{q}$ is the limit speed of the flow. This means the coming flow is supersonic and does not meet vacuum at the inlet. On the two walls, the flow satisfies the solid wall condition, namely

where $\theta = \arctan\frac{v}{u}$ is the angle of the velocity inclination to the $x-$axis.

The supersonic flow in the duct is described by the 2-D steady isentropic compressible Euler equations:

where $(u, v), p,$ and $\rho$ stand for the velocity, pressure, and density of the flow. For the polytropic gas, the state equation is given by $p = A\rho^{\gamma}$, where A is a positive constant and $\gamma>1$ is the adiabatic exponent. In addition, the gases are assumed as irrotational. Thus the components $(u, v)$ of the velocity satisfy

then for polytropic gas, the following Bernoulli law holds

where $q = \sqrt{u^{2}+v^{2}}$ is the speed of the flow, $c = \sqrt{p'(\rho)}$ is the sound speed, and $\hat{q}$ is the limit speed, which is an identical constant over the whole flow. Therefore the density $\rho$ can be expressed by the function of q and the system (5) can be reduced to a $2\times2$ system with variables $(u, v)$

Such problem (8) with initial data (3) and boundary condition (4) has already been studied by Wang and Xin in [25]. In their paper, by introducing the velocity potential ${\varphi}$ and stream function $\psi$, in terms of hodograph transformation, they proved the global solution with vacuum in the phase space $({\varphi}, \psi)$. Inspired by their paper, in order to understand this problem more intuitively, we establish the global existence of such problem in $(x, y)$-plane, and vacuum will appear in some finite place adjacent to the boundary of duct. In addition, we prove that the vacuum here is not the so-called physical vacuum.

For better stating our results, we firstly give the description of domains ${\Omega}_{non}$ and ${\Omega}_{vac}$, which are the domain before the vacuum appearance and the domain adjoining the vacuum respectively (see Fig. 2 below). Let M and N stand for the first vacuum point on $\Gamma_{\text{up}}\; \text{and}\; \Gamma_{\text{low}}$, where $x_{M} = x_{N}$. Choose a curve whose normal direction coincides with the velocity of the incoming flow at $x = x_{M}$. Denote it as $x = \psi(y)$ with $\psi(\pm f(x_{M})) = x_{M}$. Set $\Omega_{non}$ as the open region bounded by $\Gamma_{\text{in}}, \;\Gamma_{\text{up}}, \;\Gamma_{\text{low}}\; \text{and}\; x = \psi(y)$. Denote $l_{\text{up}}\; \text{and} \; l_{\text{low}}$ as the vacuum boundaries adjacent to $\Gamma_{\text{up}}\; \text{and}\; \Gamma_{\text{low}}$ respectively. Set $\Omega_{vac}$ as the open region bounded by $x = \psi(y), \; l_{\text{up}}\; \text{and} \; l_{\text{low}}$.

Our main results in the paper are:

Theorem 1.1. Assume that $f\in C^{2}([0, +\infty))$ satisfies (1), $\varphi\in C^{2}([-f(0), f(0)])$ satisfies (2) and $q_{0}\in C^{1}[-f(0), f(0)]$ satisfies (3). If $q_{0}$ satisfies

then two alternative cases will happen in the duct, one contains vacuum, and the other does not. More concretely, the two cases are as follows:

(i) When vacuum actually appears in the duct, there exists a global solution $(u, v)\in C(\overline{{\Omega}_{non}\cup{\Omega}_{vac}})$ $\bigcap C^{1}(\overline{{\Omega}_{non}\cup{\Omega}_{vac}}\setminus\{l_{up}, l_{low}\})$ to the problem (8) with (3) and (4). Moreover,

here ${\partial}_{\vec{n}}$ stands for the normal derivative of vacuum boundary. This means the vacuum here is not the physical vacuum.

(ii) If vacuum is absent in the duct, the problem (8) with (3) and (4) has a global solution $(u, v)\in C^1(\bar{{\Omega}})$.

Theorem 1.2. Assume that $f(x)\in C^{2}([0, +\infty))$ satisfies:

If

then the $C^{1}$ solution to the problem (8) with (3) and (4) will blow up at some finite location in the duct.

Remark 1. The condition (9) is equivalent to that the two Riemann invariants are both monotonically increasing along $\Gamma_{\text{in}}$ (see Lemma 2.1). It means the gases are spread at the inlet. In fact, this condition is very important to get the global existence of solution. In order to understand this, we have a glimpse of the Cauchy problem for the Burgers equation

As well known, if the initial data satisfies

then the solution $u(t, x)$ must blow up in finite time, and shock will be formed. On the other hand, by introducing the Riemann invariants $R_{\pm} = {\theta}\pm F(q)$ with $F(q) = \int \frac{\sqrt{q^2-c^2}}{qc}dq$, the equation (8) is actually equivalent to (20) (see Section 2). Thus, we consider the global solution of following Cauchy problem for (20) with initial data

By the results of [16] and [30], there exists a global solution of the Cauchy problem (15) if and only if the two Riemann invariants $R_{\pm}^0(y)$ are both monotonically increasing. Therefore, it seems that posing the condition (9) is reasonable to obtain the global solution. As a contrast, in Theorem 1.2, (12) means that at least one Riemann invariant is strictly monotonic decreasing along some part of the inlet. Thus, the gases have some local squeezed properties at the inlet. Similar to the case in Burgers equation, we prove that the $C^1$ smooth solution to the problem must blow up at some finite location in the straight duct. Actually, for different initial data and structures of boundary, more totally different motions of gases can be found in [7].

Remark 2. By the effect of expanding duct, as far as we know, it is hard to give a necessary and sufficient condition to ensure that vacuum must form at finite location in the duct. In Proposition 4.1, we will give a sufficient condition such that the vacuum will appear in finite place.

Remark 3. For the M-D compressible Euler equations, if the gases are assumed irrotational, by introducing the velocity potential ${\nabla}{\varphi} = \vec{u}$, the Euler system can be changed into a quasi-linear hyperbolic equation (i.e. potential flow equation). It is easy to check that this potential flow equation does not fulfill the "null condition" put forward in [1], [2] and [14]. Thus, in terms of the extensive results of [1], [2] and so on, the classical solution will blow up. On the other hand, if the rotation is involved, in the general case, due to the possible compression of gases, the smooth solutions will blow up and the shock is formed (see [6], [20], [21] and [23]). Meanwhile, if the gases are suitably expanded or expanded into the vacuum, the global solutions can exist(see [4], [5], [10], [11], [22], [24] and [27]-[29]).

Remark 4. If the initial data contains a vacuum, especially for physical vacuum, the local well-posedness results of the compressible Euler equations have been studied in [8], [9], [12], [13], [18] and so on. But in general, such a local classical solution will blow up in finite time as shown in [3], [26] and the references therein.

Remark 5. For the case that Riemann invariants are both constants on $\Gamma_{\text{in}}$ which is a straight segment vertical to the velocity of the incoming flow, the problem has been solved by Chen and Qu in [5]. Here we extend their work to a more general case.

Remark 6. The symmetry of ${\varphi}(y)$ in (2) and the duct with respect to $x-$axis is not essential, by the same analysis as in this paper but more tedious computation, the results can be extended to the non-symmetric case. So for the readers' convenience, we only consider the symmetric case.

The paper is organized as follows. In Section 2, we give some basic structures of the steady plane isentropic flow and discuss the monotonicity of two Riemann invariants $R_{\pm}$ along the inlet. In Section 3, we use the method of the characteristics to divide the duct into several inner and border regions, and the problem is transformed into some Goursat problems in the corresponding inner regions and some boundary value problems in the corresponding border regions. Then the global $C^1$ solution is obtained before vacuum forms if $R_{\pm}$ are monotonically increasing along the inlet. Meanwhile, for the case that at least one of $R_{+}$ or $R_{-}$ is strictly monotonic decreasing along some part of the inlet, we show that the $C^1$ solution to the problem will blow up at some finite location in the straight duct. In Section 4, we solve the problem when vacuum appears and obtain that the vacuum here is not the so-called physical vacuum. Combining the results in Section 3 and Section 4, we finally get Theorem 1.1 and Theorem 1.2. At last, We will give a sufficient condition concerning the geometric shape of the duct to ensure the formation of the vacuum.

2.   Preliminaries

In this section, we start with some basic structures of the steady plane isentropic flow, which can be characterized by the Euler system (5).

Firstly, equation (8) can be written as the matrix form

The characteristic equation of (16) is

So for the supersonic flow, (8) is hyperbolic and we can get two eigenvalues

Correspondingly, the two families of characteristics in the $(x, y)$ plane are defined by

By standard computation, the Riemann invariants $R_{\pm}$ can be defined by

where $F(q) = \int \frac{\sqrt{q^{2}-c^{2}}}{qc}dq$. Since the Jacobian $\frac{\partial(R_{+}, R_{-})}{\partial(u, v)} = \frac{2\sqrt{q^{2}-c^{2}}}{q^{2}c}>0$, then the system (8) without vacuum is equivalent to the following equations

which can also be written as

if we set $\partial_{\pm}$ as the differential operators $\frac{\partial}{\partial x}+\lambda_{\pm}\frac{\partial}{\partial y}$.

Let A be the Mach angle defined by $\sin A = c/q$, then we have that

Under a proper coordinate rotation, the eigenvalues $\lambda_{\pm}$ can always be locally bounded, and the system (21) in the new coordinate is also invariant.

From now on, we will discuss the monotonicity of the Riemann invariants $R_{\pm}$ along the inlet, which will play a key role in the following analysis. Note that $\theta = -\arctan \varphi^{\prime}(y)$ on $\Gamma_{\text{in}}$, then $R_{\pm} = -\arctan \varphi^{\prime}(y)\pm \int^{q_{0}}_{c_{*}} \frac{\sqrt{q^{2}-c^{2}}}{qc}dq$ on $\Gamma_{\text{in}}$. Some properties can be obtained after direct computation.

Lemma 2.1. For $R_{\pm}$ defined on $\Gamma_{in}$ above, we have that $R_{\pm}$ are both monotonically increasing along $\Gamma_{in}$ if and only if the initial speed $q_{0}$ satisfies

Proof. Since $R_{\pm} = -\arctan \varphi^{\prime}(y)\pm \int^{q_{0}}_{c_{*}} \frac{\sqrt{q^{2}-c^{2}}}{qc}dq$ on $\Gamma_{\text{in}}$, then direct computation yields

Thus $R_{\pm}$ are monotonically increasing along $\Gamma_{\text{in}}$ if and only if (24) are nonnegative, which is equivalent to (23).

Due to Lemma 2.1, the following analysis will be focused on these two cases.

Case I. $R_{\pm}$ are both monotonically increasing on $\Gamma_{\text{in}}$;

Case II. At least one of $R_{+}$, $R_{-}$ is strictly monotonic decreasing along some part of $\Gamma_{\text{in}}$.

The next lemma shows the partial derivatives $\frac{\partial \lambda_{+}}{\partial R_{-}}$ and $\frac{\partial \lambda_{-}}{\partial R_{+}}$ are positive, which will be often used in the following analysis.

Lemma 2.2. As the $q, c, \theta, and\; A$ defined above, for the supersonic flow, we have

Proof. By using (22) and the chain rule, it follows from direct computation that

thus (25) holds true for the supersonic flow.

3.   The $C^{1}$ solution before vacuum formation

In this section, the existence of the $C^{1}$ solution to the problem before vacuum formation will be proved for case Ⅰ. We shall use characteristics to divide the whole region into several inner and border regions, where solutions are constructed successively by the method of characteristics. As illustrated in Fig. 3, we assume that the $C_{-}$ characteristic issuing from point $A_{1}$ intersects the lower wall at point $B_{2}$ and the $C_{+}$ characteristic issuing from point $B_{1}$ intersects the upper wall at point $A_{2}$. These two characteristics intersect at point $C_{1}$. Denote the inner region surrounded by $A_{1}B_{1}, A_{1}C_{1}, B_{1}C_{1}$ as region $D_{1}$, which is the domain of determination of $A_{1}B_{1}$.

Next we solve the solution in the border region $D_{2}$($D_{2}^{\prime}$, resp.)bounded by $A_{1}C_{1}, A_{1}A_{2}, A_{2}C_{1}$($B_{1}C_{1}, B_{1}B_{2}, B_{2}C_{1}$, resp.). Note that $R_{-}$ ($R_{+}$, resp.) has already been defined in region $D_{2}$($D_{2}^{\prime}$, resp.). By the fixed wall condition $R_{+}+R_{-} = 2\arctan(\pm f^{\prime}(x))$, we can know the value of $R_{+}$ ($R_{-}$, resp.) in region $D_{2}$($D_{2}^{\prime}$, resp.).

Let the $C_{-}$ characteristic issuing from $A_{2}$ and the $C_{+}$ characteristic issuing from $B_{2}$ intersect at $C_{2}$. Define the inner region $D_{3}$ with boundaries $A_{2}C_{1}, B_{2}C_{1}, A_{2}C_{2} \ \text{and}\; B_{2}C_{2}$. If the $C_{-}$ characteristic issuing from $A_{2}$ approaches the lower wall at $B_{3}$ and the $C_{+}$ characteristic issuing from $B_{2}$ approaches the upper wall at $A_{3}$, we can get two border regions $D_{4}$($D_{4}^{\prime}$, resp.) with boundaries $A_{2}A_{3}, A_{2}C_{2}, A_{3}C_{2}$ ($B_{2}B_{3}, B_{2}C_{2}, B_{3}C_{2}$, resp.). Repeat this process until the vacuum forms. The inner regions $D_{1}$, $D_{3}$ and the border regions $D_{2}$($D_{2}^{\prime}$, resp.), $D_{4}$($D_{4}^{\prime}$, resp.) are mainly considered in this section.

Lemma 3.1. If (9) holds true, then for the $C^{1}$ solution to the problem we have $\partial_{+}R_{+}\geq 0, \partial_{-}R_{-}\leq 0$ in the duct before vacuum forms.

Proof. We will prove the conclusion between these inner and border regions separately.

Step 1. Inner region $D_{1}$.

For any fixed two points $E, F$ on $\Gamma_{\text{in}}$ and any $C_{-}$ characteristic l in region $D_{1}$, suppose the $C_{+}$ characteristics issuing from $E, F$ intersect l at $E_{1}, F_{1}$ and intersect $A_{1}A_{2}$ at $E_{2}, F_{2}$ respectively. Since $R_{-}$ is constant along the $C_{+}$ characteristic and $R_{-}$ is monotonically increasing along $\Gamma_{\text{in}}$, then the value of $R_{-}$ at $E_{1}$ is no less than the value of $R_{-}$ at $F_{1}$, which implies

along l in region $D_{1}$. Similarly, $\partial_{+}R_{+}$ is nonnegative in region $D_{1}$.

Step 2. Border region $D_{2}$.

Next we consider the solution in border region $D_{2}$ bounded by $A_{1}C_{1}, A_{1}A_{2}, A_{2}C_{1}$. From step 1, we know that $R_{-}$ is monotonically decreasing along $A_{1}C_{1}$. By the same argument in region $D_{1}$, we can get that $\partial_{-}R_{-}\leq 0$ in region $D_{2}$. Since the value of $R_{-}$ at $E_{2}$ is no less than the value of $R_{-}$ at $F_{2}$, which means that $R_{-}$ is monotonically decreasing along $A_{1}A_{2}$. Due to the solid wall condition

on $A_{1}A_{2}$, $R_{+}$ is monotonically increasing along $A_{1}A_{2}$. Therefore we can get $\partial_{+}R_{+}\geq 0$ in region $D_{2}$ by the same argument in region $D_{1}$. By analogous method, we also have $\partial_{+}R_{+}\geq 0, \partial_{-}R_{-}\leq 0$ in region $D'_2$.

Repeating this process in inner region $D_{2i+1}(i\geq 1)$ and border region $D_{2i+2}$ ($D_{2i+2}^{\prime}$, resp)($i\geq 1$) until the vacuum forms, thus we prove Lemma 3.1.

We can ensure that the hyperbolic direction is always in the x-direction by a proper rotation of coordinates if necessary. Without loss of generality, we may assume that $u^{2}>c^{2}$, which implies that $\lambda_{+}>\lambda_{-}$.

Lemma 3.2. If (9) holds true, then we have $\partial_{y}R_{\pm}\geq 0$ in the duct before vacuum forms.

Proof. In terms of the definition of $\partial_{\pm}$, one has

Thus we have

Combining this with Lemma 3.1 yields the result.

Corollary 1. Under the assumption that $u^2>c^{2}$, by (21) and (29), we have that

Lemma 3.3. Let ${\partial}_{l} = u{\partial}_{x}+v{\partial}_{y}$ stand for the derivative along the streamline, under the assumption that $u^2>c^{2}$, then the speed of the flow is monotonically increasing along the streamline, that is to say ${\partial}_{l}q\geq 0$.

Proof. It follows from (29) that ${\partial}_{l} = u{\partial}_{x}+v{\partial}_{y}$ can be expressed as

which implies that

Since $u^{2}>c^{2}$, we know that $\theta\pm A$ is away from $\pm \frac{\pi}{2}$. Thus the coefficients of the right hand part of (33) are positive. So it remains to determine the symbol of $\partial_{\pm}q$. Due to (20) and Lemma 3.1, one has that $\partial_{+}R_{+} = \partial_{+}(R_{+}-R_{-}) = 2F^{\prime}(q)\partial_{+}q\geq 0,$ which immediately indicates that $\partial_{+}q\geq 0$. Similarly, one can get that $\partial_{-}q\geq 0$. Then the proof of Lemma 3.3 is completed.

Corollary 2. Combining Lemma 3.3 and the Bernoulli law yields that the sound speed is monotonically decreasing along the streamline, that is to say ${\partial}_{l}c\leq 0$.

We now focus on the solution in the inner regions $D_{1}$ and $D_{2i-1}(i\geq 2)$ bounded by $A_{i}C_{i-1}, B_{i}C_{i-1}, A_{i}C_{i} \;\text{and}\;B_{i}C_{i}$. We only need to discuss the region $D_{2i-1}(i\geq 2)$, since same method can be used in region $D_{1}$.

Set $R_{+}(x, y) = W_{+}(x, y)$ on $A_{i}C_{i-1}$ and $R_{-}(x, y) = W_{-}(x, y)$ on $B_{i}C_{i-1}$. Thus the solution in the region $D_{2i-1}$ should be determined by solving a Goursat problem, which is given by

By standard iteration method, the local existence of the problem (34) can be achieved, one can also see [4] and [15] for more details. In order to get the global solution to the problem (34) in the region $D_{2i-1}(i\geq 2)$, some prior estimates about $R_{\pm}\;\text{and}\; \nabla R_{\pm}$ are needed.

Lemma 3.4. If (9) holds true, then $|\partial_{+}R_{+}|$ is monotonically decreasing along $C_{-}$ characteristics and $|\partial_{-}R_{-}|$ is monotonically decreasing along $C_{+}$ characteristics before vacuum formation.

Proof. It comes from the definition of $\partial_{\pm}$ that

Due to

and $\partial_{y}R_{+}\geq 0$, then $\partial_{-}\partial_{+}R_{+}\leq 0$, which implies $\partial_{+}R_{+}$ is monotonically decreasing along $C_{-}$ characteristics. By the same argument, we can get that $\partial_{-}R_{-}$ is monotonically increasing along $C_{+}$ characteristics.

In terms of Lemma 3.1, one has ${\partial}_+R_+\ge0$ and ${\partial}_-R_-\le0$. This means that $|\partial_{+}R_{+}|$ is monotonically decreasing along $C_{-}$ characteristics and $|\partial_{-}R_{-}|$ is monotonically decreasing along $C_{+}$ characteristics before vacuum formation. Thus we complete the proof.

Lemma 3.5. Suppose that $R_{\pm}(x, y)$ are the classical solutions to the problem (34), then $R_{\pm}\; \text{and}\; {\partial}_{\pm} R_{\pm}$ are uniformly bounded in the region $D_{2i-1}(i\geq 2)$ if (9) holds true.

Proof. Firstly, we give the estimates of $|R_{\pm}|$.

For any point $(x, y)$ in $D_{2i-1}(i\geq 2)$, by the method of characteristics, we have that

in the region $D_{2i-1}(i\geq 2)$. To be more specific, we shall point out that $\|W_{+}(x, y)\|_{C({A_{i}C_{i-1}})}$ and $\| W_{-}(x, y)\|_{C({B_{i}C_{i-1}})}$ listed above only depend on $\|f\|_{C^{1}}$ and $\|R_{\pm}\|_{C(\Gamma_{\text{in}})}$.

Fix $(x_{0}, y_{0})\in A_{i}C_{i-1}$, let $y = y^{(1)}_{-}(x)$ be the $C_{-}$ characteristic from $(x_{0}, y_{0})$, which approaches either $\Gamma_{\text{up}}$ or $\Gamma_{\text{in}}$ at a point $(x_{1}, y_{1})$. If $y_{1} = f(x_{1})$, then there exists a $C_{+}$ characteristic $y = y^{(2)}_{+}(x)$ from $(x_{1}, y_{1})$, which approaches either $\Gamma_{\text{low}}$ or $\Gamma_{\text{in}}$ at a point $(x_{2}, y_{2})$. If $y_{2} = -f(x_{2})$, then there exists a $C_{-}$ characteristic $y = y^{(3)}_{-}(x)$ from $(x_{2}, y_{2})$, which approaches either $\Gamma_{\text{up}}$ or $\Gamma_{\text{in}}$ at a point $(x_{3}, y_{3})$. Since the two eigenvalues $\lambda_{\pm}$ have the uniform upper or lower bound, then there exists a positive N such that

If $y(x_{N-1}) = f(x_{N-1})$ (see Fig. 5), then we get

Similarly, if $y(x_{N-1}) = -f(x_{N-1})$ (see Fig. 6), then we get

Combining (39) and (40) yields that

Analogously, the estimate of $\|W_{-}\|_{C({B_{i}C_{i-1}})}$ can also be obtained as (41).

Next, we give the estimates of $|{\partial}_{\pm}R_{\pm}|$.

It follows from Lemma 3.4 that

As the process shown in the estimates of $|R_{\pm}|$, we shall point out that $\|\partial_{+}W_{+}\|_{C({A_{i}C_{i-1}})}\; \text{and} \; \|\partial_{-}W_{-}\|_{C({B_{i}C_{i-1}})}$ listed above only depend on $\|f\|_{C^{2}}\; \text{and} \|R_{\pm}\|_{C^{1}(\Gamma_{\text{in}})}$. To this end, we need to derive the boundary condition of $\partial_{\pm} R_{\pm}$. It follows from (32) and the solid wall condition that

By the process of the reflection shown in the estimates of $|R_{\pm}|$ and Lemma 3.4, if $y(x_{N-1}) = f(x_{N-1})$, we can get

Since $u^{2}>c^{2}$, then the coefficients

are uniformly bounded. Denote the maximum one of these bounds as the constant G, thus (44) can be expressed as

Similarly, if $y(x_{N-1}) = -f(x_{N-1})$, then we also have

Combining (45), (46) and (31) yields that

Analogously, the same estimate of $\|\partial_{-}W_{-}\|_{C({B_{i}C_{i-1}})}$ holds true.

Thus, we complete the proof of Lemma 3.5.

Corollary 3. Based on Lemma 3.5, we have that $|\nabla R_\pm|\lesssim M$, where $M$ depends on $\|f\|_{C^{2}}, \; \|R_{\pm}\|_{C^{1}(\Gamma_{in})}$ and the lower bound of the sound speed c in $D_{2i-1}(i\geq2)$.

Proof. Substituting (31) into (47) gives that

Combining the local existence and the Corollary 3 induces the global $C^{1}$ solution to the problem (34) in the inner region $D_{2i-1}(i\geq 2)$ before vacuum forms.

We now turn to consider the border region $D_{2i}(i\geq 1)$, which is bounded by a part of the upper wall $A_{i}A_{i+1}$, the $C_{-}$ characteristic $A_{i}C_{i}$, and the $C_{+}$ characteristic $A_{i+1}C_{i}$ (see Fig. 7). Then the problem in region $D_{2i}(i\geq 1)$ can be generalized as the following problem

where $Z_{-}(x, y)$ is a known function. The local existence of the problem (49) can be achieved by [15]. In order to get the global solution to the problem (49) in the region $D_{2i}(i\geq 1)$, some prior estimates about $|R_{\pm}| \text{and} |\nabla R_{\pm}|$ are needed.

Lemma 3.6. Suppose that $R_{\pm}(x, y)$ are the classical solution to the problem (49), then $R_{\pm}\; and\; {\partial}_{\pm} R_{\pm}$ are uniformly bounded in the region $D_{2i}(i\geq 1)$.

Proof. For any point $(x, y)$ in $D_{2i}(i\geq 1)$, by the method of characteristics and the solid wall condition, we have that

in the region $D_{2i}(i\geq 1)$. It follows from Lemma 3.4 that

Moreover, by (43), we have that

By the same process shown in Lemma 3.5, we know that $\|\partial_{-}Z_{-}\|_{C({A_{i}C_{i}})}$ can be controlled by $\|R_{\pm}\|_{C^{1}(\Gamma_{\text{in}})}$. And thus the proof is finished.

Corollary 4. Based on Lemma 3.6, we have that $|\nabla R_\pm|\lesssim M$, where M depends on $\|f\|_{C^{2}}, \; \|R_{\pm}\|_{C^{1}(\Gamma_{in})}$ and the lower bound of the sound speed c in $D_{2i}(i\geq1)$.

Combining the local existence and the Corollary 4 induces the global $C^{1}$ solution to the problem (49) in the border region $D_{2i}(i\geq 1)$ before vacuum forms. The problem in the border region $D_{2i}^{\prime}(i\geq 1)$ can be solved by the same way exactly. After solving the problem in these inner and border regions successively, we have that

Theorem 3.7. For case I, there exists a global $C^{1}$ solution to the problem in $(\overline{\Omega}_{non}\setminus\{M, N\})$ and

Moreover, there exists a constant $C$ which only depends on the initial data and $\|f\|_{C^2}$, such that

So far we have established the global $C^{1}$ solution before vacuum formation for case Ⅰ, where the initial speed $q_{0}$ satisfies (9). The existence of solution after vacuum formation will be established in Section 4.

Now we begin to consider the problem in case Ⅱ. We will prove Theorem 1.2.

Proof of Theorem 1.2. Firstly, we consider the special case $f''(x) = 0$, i.e. the duct is straight.

Suppose that $R_{+}$ is strictly monotonic decreasing along arc MN on $\Gamma_{\text{in}}$(see Fig. 8). Assume that the $C_{-}$ characteristics issuing from M and N intersect $\Gamma_{\text{low}}$ at $M_{1}\; \text{and}\; N_{1}$ respectively. Then it follows from the same argument in Lemma 3.1 that $\partial_{+}R_{+}<0$, which implies that $\partial_{y}R_{+}< 0$ in the region $MNM_{1}N_{1}$.

Let $y_{-}^{(0)}(x)$ be the $C_{-}$ characteristic from any point $(x_{0}, y_{0})$ on $MN$, i.e.

Then $R_{+}$ is constant along the curve $y = y_- ^{(0)}(x)$, which means that $R_+(x, y_{-}^{(0)}(x)) = R_+(x_0, {\varphi}^{-1}(x_0))$. Differentiating the first equation in (20) with respect to y yields that

Use the second equation in (20), we can get $\frac{{\partial} R_-}{{\partial} x}+{\lambda}_-\frac{{\partial} R_-}{{\partial} y} = ({\lambda}_–{\lambda}_+)\frac{\partial R_{-}}{\partial y}$, i.e.

Substitute it into the above identity to obtain that

Define $H_{0}(x): = \exp{\{\int^{x}_{x_{0}}\frac{\partial_{x}R_{-}+\lambda_{-}\partial_{y}R_{-}}{\lambda_{-}-\lambda_{+}}\frac{\partial \lambda_{-}}{\partial R_{-}}(s, y_{-}^{(0)}(s))ds\}}$, then (56) can be transformed into

where $\frac{d}{d x}$ stands for the derivative along the curve $y = y_{-}^{(0)}(x)$.

If $y = y_{-}^{(0)}(x)$ intersects $\Gamma_{\text{low}}$ at point $(x_{1}, y_{1})$, then by integrating (57) from $x_{0}$ to $x_{1}$, we get that

Since $R_{+}$ is constant along $y = y_{-}^{(0)}(x)$, thus the term $\frac{1}{\lambda_{-}-\lambda_{+}}\frac{\partial \lambda_{-}}{\partial R_{-}}$ is only the function of $R_{-}$. If we set $h_{0}(R_{-}) = \int^{R_{-}}_{0}\frac{1}{\lambda_{-}-\lambda_{+}}\frac{\partial \lambda_{-}}{\partial R_{-}}d R_{-}$, then $H_{0}(x)$ can be expressed by

and $H_{0}(x)$ is bounded.

Note that $\partial_{y} R_{+}(x_{0}, y_{0})<0$ and $\frac{\partial \lambda_{-}}{\partial R^{+}}>0$. If we denote the positive lower bound of $\frac{\partial \lambda_{-}}{\partial R^{+}}H_{0}^{-1}$ by $K_{0}$, then we have that

By the boundary condition (43), one has that

The coefficient $\frac{\cos(\theta+A)}{\cos(\theta-A)}(x_{1}, y_{1})$ is greater than 1, which implies that

Analogously, let $y_{+}^{(1)}(x)$ be the $C_{+}$ characteristic from $(x_{1}, y_{1})$ on $N_{1}M_{1}$, i.e.

one can obtain the similar equation of $\partial_{y}R_{-}$ as (57) by

where $H_{1}(x) = \exp{\{\int^{x}_{x_{1}}\frac{\partial_{x}R_{+}+\lambda_{+}\partial_{y}R_{+}}{\lambda_{+}-\lambda_{-}}\frac{\partial \lambda_{+}}{\partial R_{+}}(s, y_{+}^{(1)}(s))ds\}}$ and $\frac{d}{d x}$ stands for the derivative along the curve $y = y_{+}^{(1)}(x)$. In addition, $H_{1}(x)$ can be written as $H_{1}(x) = \exp\{h_{1}(R_{+}(x, y^{(1)}_{+}(x))-h_{1}(R_{+}(x_{1}, y_{1})\}$ where $h_{1}(R_{+}) = \int^{R_{+}}_{0}\frac{1}{\lambda_{+}-\lambda_{-}}\frac{\partial \lambda_{+}}{\partial R_{+}}d R_{+}$. If $y = y_{+}^{(1)}(x)$ intersects $\Gamma_{\text{up}}$ at point $(x_{2}, y_{2})$, integrating (64) from $x_{1}$ to $x_{2}$ yields that

Note that $\partial_{y} R_{-}(x_{1}, y_{1})<0$ and $\frac{\partial \lambda_{+}}{\partial R^{-}}>0$. If we denote the positive lower bound of $\frac{\partial \lambda_{+}}{\partial R^{-}}H_{1}^{-1}$ by $K_{1}$ and let $K_{1}^{*} = \min\{K_{0}, K_{1}\}$, then we have that

By the same process and after series of reflection if necessary, the denominator of (66) will tend to $-\infty$ at some finite location in the duct. Thus the $C^{1}$ solution to the problem will blow up at some finite location in the straight duct.

Next, we consider the case $f''(x)\le0$.

The proof is very similar to the case $f''(x) = 0$, except for using following boundary condition to replace (61) by

Thus we have

This is same as (62). So we omit the details of proof for the case $f''(x)\le0$.

Therefore, we complete the proof of Theorem 1.2.

4.   Formation of vacuum

Due to the conservation of mass, vacuum can not appear in the inner regions of the duct. So if vacuum exists, it must locate on the two walls. On the other hand, since the two walls are the streamlines, Corollary 2 shows that the sound speed is monotonically decreasing along the two walls. When the sound speed decreases to zero, it corresponds to the formation of vacuum.

First, we list some properties of the vacuum boundary, one can also see [4] and [25] for more details.

Lemma 4.1. When the vacuum area actually appears, the first vacuum point M and N must form at $\Gamma_{\text{up}}\; \text{and}\; \Gamma_{\text{low}}$ respectively. Furthermore, the boundary of the vacuum $l_{\text{up}}\; \text{or}\; l_{\text{low}}$ is a straight stream line which is tangential to $\Gamma_{\text{up}}\; \text{or}\; \Gamma_{\text{low}}$ at M or N, where $f^{\prime\prime}(x_{M}) = f^{\prime\prime}(x_{N})>0$.

Based on Theorem 3.7, we have obtained the $C^{1}$ solution to the problem for $(x, y)\in (\overline{\Omega}_{non}\setminus\{M, N\})$ and $\partial_{+}R_{+}\geq 0, \partial_{-}R_{-}\leq 0 \;\text{in}\; (\overline{\Omega}_{non}\setminus\{M, N\})$. We now begin to solve the problem when vacuum forms. Set $R_{+}(x, y) = \tilde{R}_{+}(y)$ and $R_{-}(x, y) = \tilde{R}_{-}(y)$ on $x = \psi(y)$. Then $\tilde{R}_{\pm}(y)\in C^{1}(-f(x_{M}), f(x_{M}))$ and $\tilde{R}_{\pm}(y)$ are monotonically increasing along $x = \psi(y)$. Since the problem is singular at M and N, we choose $Q^{n}_{+} = (x_{n}, y_{n}), \; Q^{n}_{-} = (x_{n}, -y_{n})$ on $x = \psi(y)$ and $\tilde{R}^{n}_{\pm}(y)\in C^{1}[-f(x_{M}), f(x_{M})]$ such that

Consider the problem in the region ${\Omega}_{vac}$ bounded by $x = \psi(y), l_{\text{up}}\; \text{and}\; l_{\text{low}}$ with initial value $\tilde{R}^{n}_{\pm}(y)$. The local existence can be obtained by the method in [15]. For any point $(x, y)\in {\Omega}_{vac}$, the back$-C_{+}$ characteristic from it must intersect $\Gamma_{\text{low}}\; \text{or}\; \Gamma_{\text{in}}$ and the back$-C_{-}$ characteristic from it must intersect $\Gamma_{\text{up}}\; \text{or}\; \Gamma_{\text{in}}$. Then by Theorem 3.7, one has that ${\partial}_{\pm} R^{n}_{\pm}$ are uniformly bounded in ${\Omega}_{vac}$. Thus we can get the global $C^{1}$ solution $R^{n}_{\pm}$ to the problem in ${\Omega}_{vac}$ with initial value $\tilde{R}^{n}_{\pm}(y)$.

Let $y = y_{+}^{n}$ stand for the $C_{+}$ characteristic from $(x_{n}, y_{n})$ and $y = y_{-}^{n}$ stand for the $C_{-}$ characteristic from $(x_{n}, -y_{n})$. Then $y_{+}^{n}$($y_{-}^{n}$, resp.) tends to $l_{\text{up}}$($l_{\text{low}}$, resp.) as $n\rightarrow \infty$ and we can set

To sum up, we get that

Theorem 4.2. Assume that $\tilde{R}_{\pm}(y)\in C^{1}(-f(x_{M}), f(x_{M}))$ and $\tilde{R}_{\pm}(y)$ are monotonically increasing along $x = \psi(y)$. Then $R_{\pm}(x, y)$ defined by (70) is the unique solution to the problem in ${\Omega}_{vac}$. Furthermore, $R_{\pm}(x, y)\in C^{1}(\overline{{\Omega}_{vac}}\backslash\{l_{up}\cup l_{low}\})$ and

Next, we consider the regularity near vacuum boundary. In fact, there are many results about physical vacuum singularity. A vacuum boundary (or singularity) is called physical if

in a small neighborhood of the boundary, where $\vec{n}$ is the normal direction of the vacuum boundary. This definition of physical vacuum was motivated by the case of Euler equations with damping studied in [17] and [19]. But the following Theorem shows that the vacuum here is not the physical vacuum.

Theorem 4.3. Let $\partial_{\vec{n}}$ stand for the normal derivative on the vacuum boundary, then for any point $(x, y)$ near the vacuum line, we have $\partial_{\vec{n}}c^{2}(x, y)$ tends to zero as $(x, y)$ approaches to the vacuum line along the normal direction except the vacuum point M and N.

Proof. For convenience, we only consider the case near the vacuum line $l_{\text{low}}$, which is tangential to $\Gamma_{\text{low}}$ at N. First, we derive the expression of $\partial_{\vec{n}}c^{2}$, where $\vec{n} = (n_{1}, n_{2})$. Acting $\partial_{\vec{n}}$ on the Bernoulli law (7) yields that

By the relation $R_{+}-R_{-} = 2F(q)$, one has that

Substituting it into (73) gives that

Combining this with (30) yields that

Construct a line $NN_{h}$ vertical to $l_{\text{low}}$ at the first vacuum point N, where h represents the distance between N and $N_{h}$. Let $C^{h}_{-}$ be the $C_{-}$ characteristic through $N_{h}$ and intersect $\Gamma_{\text{up}}$ at $P_{h}$. Fix a point S on $l_{\text{low}}$. Make a line from S vertical to $l_{\text{low}}$ and it intersects $C^{h}_{-}$ at $S_{h}$. Then we have that $N_{h}\rightarrow N, S_{h}\rightarrow S$ as $h\rightarrow 0$. It follows from (35) that

Integrating (77) from $P_{h}$ to $S_{h}$ yields that

This together with (26) yields that

where

Since $u^{2}>c^{2}$, then $T_{1}\; \text{and} \; T_{2}$ have the uniform positive lower bound denoted as $g_{0}$

We claim that $\overline{\lim\limits_{h\rightarrow 0}}\partial_{+}R_{+}(S_{h}) = 0 \;\text{and}\; \varliminf\limits_{h\rightarrow 0}\partial_{-}R_{-}(S_{h}) = 0$, which immediately implies that $\lim\limits_{h\rightarrow 0}\partial_{\vec{n}}c^{2}(S_{h}) = 0$. For convenience, we only prove $\overline{\lim\limits_{h\rightarrow 0}}\partial_{+}R_{+}(S_{h}) = 0$, for the case $\varliminf\limits_{h\rightarrow 0}\partial_{-}R_{-}(S_{h}) = 0$ can be obtained by the same argument.

We will use the proof by contradiction to obtain above assertions. Assume that $\overline{\lim\limits_{h\rightarrow 0}}\partial_{+}R_{+}(S_{h}) = \bar{g}>0$, where $\bar{g}$ is a positive constant. Then there exists a sequence $S_{\bar{g}_{n}}$ on $S_{h}S$ such that $\partial_{+}R_{+}(S_{\bar{g}_{n}})\geq \frac{\bar{g}}{2}>0 \;\text{and}\; |NN_{\bar{g}_{n}}|\leq\frac{h}{n}$, where $N_{\bar{g}_{n}}$ is the intersection of the $C_{-}$ characteristic $C_{-}^{\bar{g}_{n}}$ through $S_{\bar{g}_{n}}$ and the segment $NN_{h}$. Let the $C_{-}$ characteristic $C_{-}^{\bar{g}_{n}}$ intersect $\Gamma_{\text{up}}$ at $P_{\bar{g}_{n}}$. Then it follows from Lemma 3.4 that

By the fact that $\partial_{-}q\geq 0$ and the Bernoulli law, we can get $\partial_{-}c\leq 0$, which implies that the value of c on $N_{\bar{g}_{n}}S_{\bar{g}_{n}}$ is no greater than the value of c at $N_{\bar{g}_{n}}$ denoted by $c(N_{\bar{g}_{n}})$. Thus we have $\lim\limits_{n\rightarrow \infty}c(N_{\bar{g}_{n}}) = 0$ and

where $s_{0} = |NS|$. Combining it with (79) yields that

Note that $P_{\bar{g}_{n}}$ is away from the vacuum as $n\rightarrow\infty$. If $\lim\limits_{n\rightarrow\infty}P_{\bar{g}_{n}} = P$, then $\lim\limits_{n\rightarrow\infty}\partial_{+}R_{+}(P_{\bar{g}_{n}}) = \partial_{+}R_{+}(P)$ and $\partial_{+}R_{+}(P)$ is bounded. Then let $n\rightarrow\infty$ in (82), we can get that

which is a contradiction. Thus, it follows

This together with (71) gives that

Combining it with (76) yields that

Note that $c^2\in C(\bar{{\Omega}}_{vac})$, by the derivative limit theorem, we have that

Similarly, one has that

Thus, we prove the Theorem 4.2.

So far we have established the solution to the problem when vacuum actually appears. When it comes to the case that there is no vacuum in the duct, by the same prior estimates in Lemma 3.5 and Lemma 3.6, one only needs to check whether the duct can be covered by characteristics in finite steps.

Theorem 4.4. Assume that $f\in C^{2}([0, +\infty))$ satisfies (1), $\varphi\in C^{2}([-f(0), f(0)])$ satisfies (2) and $q_{0}\in C^{1}[-f(0), f(0)]$ satisfies (3), (9). If there is no vacuum in the duct, then there exists a global $C^{1}$ solution in the duct.

Proof. It remains to check whether the duct can be covered by characteristics in finite steps. Since the duct is convex, then there exists $i_{0}$ such that we can find an arc $EF$ on $A_{i_{0}}A_{i_{0}+1}$ and $f^{\prime\prime}>0$ on $EF$. Let the $C_-$ characteristics $C^E_-, C^F_-$ intersect $C_{i_{0}}A_{i_{0}+1}$ at $E_1, F_1$, intersect $B_{i_{0}+1}C_{i_{0}+1}$ at $E_2, F_2$ and intersect $B_{i_{0}+1}B_{i_{0}+2}$ at $E_3, F_3$. By Lemma 3.1 and (28), we know that $\partial_{+}R_+>0$ on $E_1F_1$ and $E_2F_2$. Combining it with $\partial_+R_- = 0$ yields that $\tilde{E_1}\tilde{F_1}, \tilde{E_2}\tilde{F_2}$ which are the images of $E_1F_1, E_2F_2$ in $(u, v)$ plane have positive length. That is to say

Similarly, we can get that

By the induction method, we have that

Suppose that the image of the $C_+$ characteristic through $C_{i_{0}}$ in $(u, v)$ plane intersects the critical circle at $C^{0}_{i_{0}}$ and intersects the limiting circle at $C^{1}_{i_{0}}$. Then

where $\mu = \sqrt{\frac{\gamma-1}{\gamma+1}}$. By the same method in [5], we have that

where $\lfloor.\rfloor$ stands for the maximal integer no more than the number inside the bracket.

Proof of Theorem 1.1. Under the assumptions of Theorem 1.1, Theorem 1.1 comes from Theorem 3.1 and Theorem 4.1-4.3.

Finally, we give a sufficient condition to ensure that vacuum must form at finite location in the duct.

Proposition 1. If $f^{\prime}_{\infty}>f^{\prime}(0)+\frac{2}{(\gamma-1)M_{B_{1}}\sqrt{1+(f^{\prime}(0))^{2}}}$, where $M_{B_{1}}$ represents the Mach number at $B_{1}$, then the vacuum will form at the finite locations on $\Gamma_{low}$.

Proof. We will use the proof by contradiction. Assume that there is no vacuum on $\Gamma_{\text{low}}$. Let $\frac{d}{ds}$ stand for the derivative along ${\Gamma}_{\text{low}}$. By the relation $R_{+}-R_{-} = 2F(q)$, one has that

This together with (29) and (43) yields that

By (95) and the Bernoulli law (7), we get that

Since

then

which is a contradiction. Thus, we complete the proof.

Acknowledgments

The authors would like to thank the referee for his (or her) very helpful suggestions and comments that lead to an improvement of the presentation.