Blowup for regular solutions and $C^{1}$ solutions of the two-phase model in $\mathbb{R}^{N}$ with a free boundary

1.   Introduction

The compressible inviscid liquid-gas two-phase model in $\mathbb R^{N}$ that will be considered is as follows:

where $\vec{u} = \vec{u}(t, \vec{x})$, $P = P(n, \rho) = n^{\Gamma}+\rho^{\gamma}$ are the velocity and pressure while $n = n(t, \vec{x}) \geq 0$, $\rho = \rho(t, \vec{x}) \geq 0$ are the densities of two phases, where $\gamma$, $\Gamma > 1$.

Euler equations have been used as one of the basic models for studying fluids, plasmas, atmospheric dynamics, and condensed matter in ^{[2,5,6,7,11,30]}, and the blowup analysis for these equations is carried out in ^{[3,4,12,17,21,25,26,28,34,36]}. In 1985, Sideris ^[25] constructed the functional

to prove that the $C^{1}$ solutions of the three-dimensional compressible Euler equations will blow up in a finite time when the initial functional $F(0)$ is sufficiently large. Yuen ^[35] used the energy method to consider the blowup results of the $C^{1}$ solutions and the weakened regular solutions of the Euler equations in $\mathbb{R}^{N}$. In ^[18], the authors rewrote the system in the form of a quasilinear wave equation about the density $\rho$ to study the blowup of solutions to Euler equations. Liu, Wang and Yuen studied the blowup results of solutions to the compressible Euler equations with time-dependent damping with vacuum and $C^1$ solutions of the irrotational compressible Euler equations with time-dependent damping in ^[19]. In ^[24], the author studied the three-dimensional Euler equations with a free boundary subjected to tension.

In the two-phase fluid, because of its wide application in aerospace, micro-technology, chemical engineering and other fields, it has aroused many researchers' interest. Zuber studied the two-phase model firstly in ^[39]. The motion of liquid and gas mixture is studied by the two-phase model in ^[16]. In ^[23], the authors considered a hyperbolic two-phase model. For existence, asymptotic and uniqueness of global weak solutions to the two-phase flow model with vacuum, Yao, Zhang and Zhu used the line method and a priori estimate to obtain relevant results in ^[31,32,33], and there are also studies of these issues in ^{[1,8,13,14,15,27]}. In ^[37], under the assumption of $H^{2}-$norm of the initial perturbation with a constant state is sufficiently small and $L^{1}-$norm is bounded, Zhang and Zhu studied the global existence of Cauchy problem to viscous liquid-gas two-phase flow in three dimensions. Furthermore, for the Cauchy problem of 3D inviscid liquid-gas two-phase flow, Zhang considered the optimal $L^{P}-L^{2}$ $(1\leq p < \frac{6}{5})$ time decay rates of the solutions with the damping on the qualitative behaviors in ^[38]. Wen and Zhu ^[29] considered global existence of weak solutions to two-fluid about the Dirichlet problem in one dimension. In ^[10], Dong et al. considered the energy integration method to prove the singularity of the smooth solutions to the Cauchy problem for the viscous two-phase model in arbitrary dimensions. Furthermore, Dong and Yuen provided the blowup phenomena of self-similar solutions for the inviscid liquis-gas two-phase flow ^[9]. By introducing the definition of regular solutions, Makino and Perthame studied the blowup phenomena of radical symmetric solutions to the Euler-Poisson equations with compact support and with a repulsive force and an attractive force in ^[20,22].

2.   Materials and methods

In this paper, we study the blowup results of regular solutions and $C^{1}$ solutions of the two-phase model in $\mathbb{R}^{N}$ with a free boundary. Our method depends on the energy integration method and a quasi-linear wave equation about $\rho$, the singularity of the two-phase model will form in finite time.

3.   Results

3.1.   Main theorems

In this section, we give the blowup results of regular solutions and $C^{1}$ solutions of the two-phase model in $\mathbb{R}^{N}$ with a free boundary. Firstly, we introduce the definition of regular solutions to the two-phase (1.1).

Definiton 3.1. (Weakened regular solution) A solution $(n, \rho, \vec{u})$ to the system (1.1) which is in $\mathbb{R}^{N}$ and in the non-vacuum region is regular for $1 < \Gamma < 2$ or $1 < \gamma < 2$, if

with a fixed constant $\eta > 1$.

In the following, we state the blowup results of the two-phase model with a free boundary in $\mathbb{R}^{N}$.

Theorem 3.1. Assume that the fluid enters a bounded open region $\Omega(t)\subsetneq\mathbb{R}^{N}$, with the contacting vacuum boundary$\partial\Omega(t)$. Suppose that the $(n, \rho, \vec{u})$ is a weakened regular solution on $[0, T)\times\Omega(t)$ of the two-phase model (1.1) in $\mathbb{R}^{N}$. If

where the total energy $E = \int_{\Omega(0)}\left(\frac{1}{2}n\left\vert\vec{u}\right\vert ^{2} +\frac{1}{2}\rho\left\vert\vec{u}\right\vert ^{2}+\frac{1}{\Gamma-1}n^{\Gamma}+\frac{1}{\gamma-1}\rho^{\gamma}\right) dV$ and thetotal mass $M = {\int_{\Omega(0)}}(n+\rho) dV > 0$, then the solutions of the two-phase model will blow up in finite time $T$.

Furthermore, we consider the two-phase model (1.1) with the pressure $P = P(n, \rho) = n^{\gamma}+\rho^{\gamma}$. In this paper, we will rewrite the two-phase model (1.1) by forming a quasi-linear wave equation about the density $\rho$ and studying the blowup results for solutions of the two-phase model in $\mathbb{R}^{N}$($N\geq2$) with the initial data

Introducing radial symmetry, the initial data become

where $r = (\sum\limits_{i = 1}^{N}x_{i}^{2})^{\frac{1}{2}}$.

In the following, we give the blowup results of the two-phase model in $\mathbb{R}^{N}(N\geq3)$.

Theorem 3.2. Assume that the initial density $n_{0}\geq 0$ and $\rho_{0}\geq 0$ in (1.2), and $(n_{0}, \rho_{0}, \vec{u_{0}})\in H^{3}(\mathbb R^{N})$. Consider the solutions $(n, \rho, \vec{u})$ of the two-phase model (1.1) in $\mathbb{R}^{N}$($N\geq3$). If the initial conditions

and

are satisfied, where $\frac{\pi ^{\frac{N}{2}}}{\Gamma (1+\frac{N}{2})}$ is the volume ofthe unit ball in $\mathbb{R}^{N}$, then the solutions will blow up on or before the finite time $\frac{2H(0)^{-\frac{\gamma-1}{2}}}{(\gamma-1)C_{0}}$, where $H(0) = \int_{\mathbb R^{N}}\frac{(n_{0}+\rho_{0})}{re^{r}}dx$ and $C_{0} = \sqrt{\frac{1}{2^{\gamma-1}(1+\gamma)}\left[\frac{\Gamma (1+\frac{N}{2})}{(N-2)!\pi ^{\frac{N}{2}}}\right]^{\gamma-1}}$.

Then, under the appropriate assumptions, we will give the blowup results of the two-phase model in the two dimensional cases.

Theorem 3.3. Assume that the initial density $n_{0} \geq 0, \rho_{0}\geq0$ in (1.2), and $(n_{0}, \rho_{0}, \vec{u_{0}})\in H^{3}(\mathbb R^{2})$. Let $K_{0}(r)$ be the modified Bessel function

Consider the solution $(n, \rho, \vec{u})$ of the two-phase model (1.1) in $\mathbb{R}^{2}$. If the initial conditions

and

are satisfied, then the solutions will blow up on or before the finite time $\frac{2G(0)^{-\frac{\gamma-1}{2}}}{(\gamma-1)C_{1}}$, where $G(0) = \int_{\mathbb{R}^{2}}(n_{0}+\rho_{0})K_{0}(r)dx$ and $C_{1} = \sqrt{\frac{1}{2^{\gamma-1}(\gamma+1)}}\left(\int_{\mathbb{R}^{2}}K_{0}(r)dx\right)^{-\frac{\gamma-1}{2}}$.

Remark 3.1. For the two-phase model in $\mathbb R^{N}$ with pressure $P = P(n, \rho) = (n+\rho)^{\gamma}$, we can also obtain the same blowup results.

3.2.   Blowup for the two-phase model with a free boundary

In this section, we will show the proof of the blowup result of solutions to the two-phase model in $\mathbb{R}^{N}$. Firstly, we give some lemmas for the conserved energy and the properties of the local second inertia function to the two-phase model.

Lemma 3.1. Assume the density $n$ and $\rho$ have compact support in the region $\Omega(t)$ for non-trivial $C^{1}$ solutions of the two-phase model (1.1) in $\mathbb{R}^{N}$, where $\Omega(t)$ is a moving region, and the region $\Omega(0)$ is bounded by the contacting vacuum boundary, then

and

that is, the total energy energy $E(t)$ is conserved, where $M(t) = \int_{\Omega(t)}(n+\rho) dV$ represents the total mass.

Proof. By $(1.1)_{1}$ and $(1.1)_{2}$, we obtain

Therefore, $M(t) = M(0)$.

Multiplying $\vec{u}$ on both sides of the Eq $(1.1)_{3}$ and integrating over $\Omega(t)$, we obtain

On the one hand, from $(1.1)_{1}$, we have

Due to

we obtain

We note that

Thus, we get

Therefore, we obtain

Similarly, we get

On the other hand,

Integrating over the region $\Omega(t)$ and applying the integration by parts, we obtain

Therefore, we have

Thus, by (3.15), we obtain

Therefore, for non-trivial $C^{1}$ solutions, we obtain

The proof is complete.

Next, we show the second derivative of the local second inertia function for the two-phase model (1.1) in $\mathbb{R}^{N}$ for the solutions on $[0, T)\times\Omega(t)$.

Before stating the following lemma, we first give some physical quantities as follows:

and

where $F_{n}(t)$ and $F_{\rho}(t)$, $H_{n}(t)$ and $H_{\rho}(t)$, $E_{kn}(t)$ and $E_{k\rho}(t)$, $E_{in}(t)$ and $E_{i\rho}(t)$ represent the momentum weight, the local second inertia, the kinetic energy and the internal energy for the two-phase fluid, respectively.

Lemma 3.2. For the two-phase model (1.1) in $\mathbb{R}^{N}$, we have

and

Proof. Differentiating $H(t)$ with respect to $t$, we obtain

Applying the integration by parts to ($3.33$) and using $(1.1)_{1}$, we get

Similarly, we have

Therefore, we obtain

Next, we calculate $H^{''}(t)$ as follows:

By using the integration by parts, we get

Similarly, we have

and

From (3.38) to (3.40) and Gauss formula, we obtain

The proof is complete.

Using the above lemmas, we will give the proof of the blowup results of the solutions to the two-phase model with a free boundary in $\mathbb{R}^{N}$.

Proof of Theorem 3.1. Introducing

for $n(t, x)\neq0$ and $\rho(t, x)\neq0$, we transform the momentum equations (1.1) into

Simplifying the above equation, we have

where $\eta > 1$ is an arbitrary constant, $(\eta-1)\Gamma+1-\eta > 0$ and $(\eta-1)\gamma+1-\eta > 0$.

We study the solutions near the contacting vacuum boundary point $\vec{x}_{0}(t)$ in the region $\Omega(t)$. Firstly, we consider the local second inertial function

By Lemma 3.2, we obtain

with the conserved total energy $E = \int_{\Omega(0)}\left(\frac{1}{2}n\left\vert \vec{u}\right\vert ^{2} +\frac{1}{2}\rho\left\vert \vec{u}\right\vert ^{2}+\frac{1}{\Gamma-1}n^{\Gamma}+\frac{1}{\gamma-1}\rho^{\gamma}\right) dV$ by Lemma 3.1, where $C_{1} = \min\{2, N(\Gamma-1), N(\gamma-1)\}$.

Therefore, we obtain

For an arbitrary point $\vec{x}_{0} (t)\in\partial\Omega(t)$ with the contacting vacuum, by applying Lemma 3.1 and $\rho(0, \vec{x}_{0}) = 0$, we obtain

We consider the governing differential dynamic system at the contacting vacuum point $\vec{x}_{0}(t)$:

where $\frac{dx_{0, i}(t)}{dt} = u_{i}(t, \vec{x}_{0}(t))$.

The solutions of the above system are as follows:

Then, we get

By applying the Euclidean norm, we obtain

Therefore, we have

Furthermore, we obtain

Similarly, we obtain

Therefore,

that is

with the conserved total mass $M = { \int_{\Omega(0)}} (n+\rho) dV > 0$ in Lemma 3.1.

By (3.50), we have

When $t\rightarrow +\infty$, we obtain

Therefore, we have

that is

which contradicts with inequality (3.2) in Theorem 3.1. Therefore, we conclude that the solutions for the two-phase model (1.1) in $\mathbb{R}^{N}$ will blow up in finite time.

This completes the proof.

3.3.   Blowup for the two-phase model with vacuum

For proofing the blowup results of solutions to the two-phase model in $\mathbb{R}^{N}(N\geq2)$, we first introduce the properties of the two-phase model.

Lemma 3.3. (Theorem 2.1 in ^[18])Assume that $n_{0}\geq0$, $\rho_{0}\geq0$ and $(n_{0}, \rho_{0}, \vec{u_{0}})\in H^{3}(\mathbb R^{N})$, then there exists a unique solution $(n, \rho, \vec{u})$ to the two-phase model on some time interval $[0, T)$, which satisfies

and

Lemma 3.4. (Lemma 4 in ^[35])From the mass equations $(1.1)_{1}$ and $(1.1)_{2}$, we obtain

and

Next, we give the lemma to show the property of the modified Bessel function $K_{0}(r)$.

Lemma 3.5. (Lemma 3.1 in ^[18])The modified Bessel function $K_{0}(r) = \int_{0}^{\infty}e^{-r cosh t}dt$ satisfies

for some constants $C_{k}$ depending only on $k > 1$.

Then, we give the proof of the blowup results of solutions to the two-phase model in $\mathbb{R}^{N}(N\geq2)$.

Proof of Theorem 3.2. We consider the solution $(n, \rho, \vec{u})$ that satisfies the conditions in Theorem 3.2. From the two-phase system $(1.1)$, we obtain

Then, multiplying Eq ($3.67$) by $\frac{1}{re^{r}}$ and integrating over $\mathbb{R}^{N}$ to obtain

By the divergence theorem, we obtain

By applying the assumption $(3.5)$ and Lemma 3.4, we obtain $n(t, \vec{0})\equiv 0, \rho(t, \vec{0})\equiv 0$. Using the continuity of $P$, we get that for any sufficiently small $\delta > 0$, there is $0 < \varepsilon < 1$, such that $P(t, \vec{x}) < \delta$ while $r = \varepsilon$, then,

where $\alpha (N) = N(N-2)\frac{\pi ^{\frac{ N}{2}}}{\Gamma (1+\frac{N}{2})}$ and $\frac{\pi ^{\frac{N}{2}}}{\Gamma (1+\frac{N}{2})}$ is the volume of the unit ball in $\mathbb{R}^{N}$.

Therefore, for $N\geq3$, we obtain

Similarly, we get

Thus, we obtain

Using H$\ddot{o}$lder inequality, we get

and

Therefore, (3.73) becomes

Applying the integration by parts, we obtain

Therefore, we obtain

where $H(t) = \int_{\mathbb R^{N}}\frac{(n+\rho)(t, r)}{re^{r}}dx$. Using integration by parts and (3.7), we obtain

Therefore,

Multiplying the both sides of (3.78) by $2H^{'}(t)$, we obtain

Integrate over $[0, t]$, we obtain

Applying the assumption (3.7), we obtain

where $C_{0} = \sqrt{\frac{1}{2^{\gamma-1}(1+\gamma)}\left[\frac{\Gamma (1+\frac{N}{2})}{(N-2)!\pi ^{\frac{ N}{2}}}\right]^{\gamma-1}}$.

Therefore,

By integrating over $[0, t]$, we obtain

Therefore,

which means that

From (3.6), we have

Applying the mass conservation, we obtain

where $\mathbb B_{r_{0}}$ is the N-dimensional ball centered at the origin with any given radius $r_{0}$.

When $t\to \frac{2H(0)^{-\frac{\gamma-1}{2}}}{(\gamma-1)C_{0}}$, $H(t)$ cannot be bounded. Therefore, we obtain that $\int_{r\leq r_{0}}(n+\rho)(t, r)rdr$ cannot be bounded as $t\to \frac{2H(0)^{-\frac{\gamma-1}{2}}}{(\gamma-1)C_{0}}$, hence the solutions will blow up on or before the finite time $t\to \frac{2H(0)^{-\frac{\gamma-1}{2}}}{(\gamma-1)C_{0}}$.

The proof is finished.

In the following, we will prove Theorem 3.3.

Proof of Theorem 3.3. We consider the solution $(n, \rho, \vec{u})$ satisfies the conditions in Theorem 3.3. From (3.67), we obtain

Then, multiplying Eq ($3.90$) by $K_{0}(r)$ and taking the integration over $\mathbb{R}^{2}$ to obtain

Using the same method as that we obtain from (3.69) to (3.73), we obtain from Lemma 3.5 that

Using the integration by parts, we obtain

Therefore,

where $G(t) = \int_{\mathbb R^{2}}(n+\rho)(t, r)K_{0}(r)dx$.

Using integration by parts, $(1.1)_{1}$, $(1.1)_{2}$ and (3.11), we obtain

Similarly, we obtain $G^{'}(t)\geq0$.

Multiplying both sides of (3.94) by $2G^{'}(t)$, we obtain

Integrate over $[0, t]$ and using the assumption (3.11), we obtain

where $C_{1} = \sqrt{\frac{1}{2^{\gamma-1}(1+\gamma)}}\left(\int_{\mathbb R^{2}}K_{0}(r)dx\right)^{-\frac{\gamma-1}{2}}$.

Therefore,

By integrating over $[0, t]$, we obtain

Therefore,

which means that

From (3.10), we have

Applying the mass conservation, we obtain

where $\mathbb B_{r_{0}}$ is the 2-dimensional ball centered at the origin with any given radius $r_{0}$.

When $t\to \frac{2G(0)^{-\frac{\gamma-1}{2}}}{(\gamma-1)C_{1}}$, $G(t)$ cannot be bounded. Thus, the solutions will blow up on or before the finite time $t\to \frac{2G(0)^{-\frac{\gamma-1}{2}}}{(\gamma-1)C_{1}}$. Therefore, we conclude that the solutions of the two-phase model in $\mathbb{R}^{2}$ will blow up.

4.   Conclusions

In this paper, we study the blowup results of solutions to the two-phase model in $\mathbb{R}^N$. Our method depends on the energy integration method and a quasi-linear wave equation about $\rho$, the singularity of the two-phase model will form in finite time.

Acknowledgments

The first author is very grateful to the second and corresponding authors for their support of Beijing Natural Science Foundation under Grant No. 1202006 and the Small Grant for Academic Staff (MIT/SGA08/20-21), Department of Mathematics and Information Technology, the Education University of Hong Kong.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.