Analytical solutions to the 3-D compressible Navier-Stokes equations with free boundaries

In this paper, we study a class of analytical solutions to the 3-D compressible Navier-Stokes equations with density-dependent viscosity coefficients, where the shear viscosity $h(\rho)=\mu\geq0$, and the bulk viscosity $g(\rho)=\rho^{\beta}$ ($\beta>0$). By constructing a class of radial symmetric and self-similar analytical solutions in $\mathbb {R}^{N}$ ($N\geq2$) with both the continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum, we derive that the free boundary expands outward in the radial direction at an algebraic rate in time and we also have shown that such solutions exhibit interesting new information such as the formation of a vacuum at the center of the symmetry as time tends to infinity and explicit regularities, and we have large time decay estimates of the velocity field.

The compressible Navier-Stokes equations with density-dependent viscosity coefficients can be written as

where $\rho(\mathbf{x},t)$, $\mathbf{U}(\mathbf{x},t)$, and $P(\rho )=\rho^{\gamma}$ ($\gamma>1$) stand for the fluid density, velocity, and pressure, respectively. The strain tensor is

and $h(\rho)$ and $g(\rho)$ are the Lamé viscosity coefficients satisfying

During the last several decades, there has been significant progress on the system (1.1) since it was introduced by Liu et al. in [1]. There is a huge literature on the global existence and behaviors of solutions to (1.1) in the case that the viscosity coefficients are both constants. The important progress on the global existence of strong or weak solutions in one spatial dimension or multi-dimension has been made by many authors; refer to [2–8] and references therein.

However, the studies in Hoff and Serre [9], Xin [10], and Liu et al. [1] prove that the compressible Navier-Stokes equations with constant viscosity coefficients behave singularly in the presence of a vacuum. In [1] Liu, Xin and Yang introduced the modified compressible Navier-Stokes equations with density-dependent viscosity coefficients for isentropic fluids. In deriving by Chapman-Enskog expansions from the Boltzmann equation, the viscosity of the compressible Navier-Stokes equations depends on the temperature and thus on the density for isentropic flows. Moreover, it should be emphasized that the viscous Saint-Venant system in the description of the motion for shallow water was also derived recently [11, 12], which is expressed exactly as (1.1) with $N=2$, $h(\rho)=\rho$, $g(\rho)=0$, and $\gamma=2$.

The system (1.1) was first proposed and studied by Vaigant-Kazhikhov in [13]. There is much literature on mathematical studies of the Navier-Stokes equations with density-dependent viscosities. For the one-dimensional case one is referred to [1, 14–24] and references therein. For the periodic problem on the torus $\mathbb {T}^{2}$ and under assumptions that the initial density is uniform away from vacuum and $\beta>3$ where $h(\rho)$ is a positive constant and $g(\rho)=\rho^{\beta}$, Vaigant-Kazhikhov found the well-posedness of the classical solution to (1.1), and the global existence and large time behavior of weak solutions was studied by Perepelitsa in [25]. Later on, Jiu-Wang-Xin [26] improved the result and obtained the global well-posedness of the classical solution to the periodic problem with large initial data permitting vacuum. Then Huang-Li [27] relaxed the index β to $\beta>\frac{4}{3}$ and studied the large time behavior of the solutions. For a 2-D Cauchy problem with vacuum states at far fields, Jiu-Wang-Xin [28] and Huang-Li [29] independently considered the global well-posedness of classical solutions in different weighted spaces. Later on, Jiu-Wang-Xin [30] studied the global well-posedness of the classical solution to the Cauchy problem (1.1) with $h(\rho)$ being a positive constant and $g(\rho)=\rho^{\beta}$ and large data which keeps non-vacuum states $\bar{\rho}>0$ at far fields.

The first successful example is due to Guo et al. in [31] with spherically symmetric initial data and fixed boundary conditions, and later Guo et al. extended it to the free boundary conditions with discontinuously symmetric initial data [32]. Chen and Zhang in [33] showed a local existence result with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state.

In [32], Guo had discussed the global existence, the Lagrangian structure and large time behavior of weak solutions to the compressible spherically symmetric Navier-Stokes equations (1.1) with the following stress free boundary conditions:

with $h(\rho)=\rho$, $g(\rho)=0$, and $\gamma\in[2,\frac{N}{N-2})$. Notice that the results in [32] can also be generalized to multi-dimensional compressible spherically symmetric Navier-Stokes equation (1.1) with general viscosity coefficients $h(\rho )=\rho^{\theta}$ and $g(\rho)=(\theta-1)\rho^{\theta}$ with $\frac {N-1}{N}<\theta\leq1$ with $N\geq2$ the space dimension or supplying the following free boundary:

On the other hand, there are many references considering the analytical solutions or blowup solutions to the Navier-Stokes equations, Navier-Stokes-Poisson equations, or Euler-Poisson equations (see [34–36]). In [37], Guo has studied the analytical solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients where $h(\rho )=\rho^{\theta}$ and $g(\rho)=(\theta-1)\rho^{\theta}$, $N=3$, and with free boundaries.

In the present paper, we study more precisely the structure and asymptotic behavior of solutions to the free boundary value problem with both the continuous density condition and the stress free condition. Let $h(\rho)=\mu\geq0$ and $g(\rho)=\rho^{\beta}$ ($\beta>0$). At first, we construct a self-similar solution of the form

to the continuity equation with special spherically symmetric initial data and given symmetric sphere separating the initial fluids from vacuum, and we derive that $a(t), f(z)\in C^{1}$ are two functions satisfying some ordinary differential equations. At the same time, the free boundary is shown to expand outward in the radial direction at an algebraic rate in time and the density decays to zero asymptotically in time almost everywhere away from the symmetry center, which leads to the formation of vacuum states as the time goes to infinity. One of our main steps in the analysis is to show that $a(t)$ and $f(z)\in C^{1}$ exist globally, which yields the desired analytical solutions to the free boundary value problem with suitable initial data.

The plan of this paper is as follows. In Section 2 we will give the main results and the self-similar solution to (1.1). Theorems 2.1 and 2.2 will be proved in Sections 3.1 and 3.2, respectively.

Set $h(\rho)=\mu\geq0$ and $g(\rho)=\rho^{\beta}$, $N=3$, in (1.1). Then the isentropic compressible Navier-Stokes system with density-dependent viscosity becomes

for $t\in(0,+\infty)$ and $\mathbf{x}\in\mathbb{R}^{3}$. Here $\rho (\mathbf{x},t)$, $\mathbf{U}(\mathbf{x},t)$, and $P(\rho)=\rho^{\gamma}$ ($\gamma>\frac{4}{3}$). The initial conditions of (2.1) are imposed thus:

In this paper, we are concerned with the following spherically symmetric solutions. To this end, we denote

This leads to the following system of equations:

for $(r,t)\in\Omega$ with

with the initial condition

where $a_{0}>0$ is a constant, and we have the free boundary condition

or, when $\mu=0$,

where

It is easy to get the following usual a priori energy estimate for smooth solutions to (2.4), (2.5), and the boundary conditions (2.7) or (2.8):

Our main result is the following theorem.

Theorem 2.1

Assume that $\gamma=\beta>\frac{4}{3}$. For the radial symmetry compressible Navier-Stokes equations (2.4)-(2.5), with the continuous density condition (2.7), there exists a solution of the form

Here $a(t)\in C^{1}([0,\infty))$ is the free boundary satisfying (2.9) and it exists for all $t\geq0$. Furthermore, $a(t)$ tends to +∞ as $t\rightarrow+\infty$ with the following rates:

Remark 2.1

The solution (2.11) constructed in Theorem 2.1 satisfies the following properties:

where the domain of the fluid is defined by

On the other hand, for the stress free boundary condition (2.8), we have the following.

Theorem 2.2

Assume that $\gamma=2\beta-\frac{1}{3}$, $\gamma>\frac{5}{3}$. Then the free boundary value problem for the radial symmetry compressible Navier-Stokes system, (2.4)-(2.5), with the stress free condition (2.8), has a unique solution with the free boundary $r=a(t)$ given by

and

where $f(z)>0$, $f(z)\in C([0,1])\cap C^{1}((0,1])$ satisfies

Remark 2.2

It can be verified easily that the solution (2.15) in Theorem 2.2, satisfies the following properties:

Moreover, for $0\leq r\leq a(t)$, we have

First, we will give a self-similar solution to the continuity equation (2.4).

Lemma 3.1

For any two $C^{1}$ functions $f(z)\geq0$ and $a(t)>0$, define

Then $(\rho,u)(r,t)$ solves the continuity equation (2.4), i.e.,

Here, we can choose $a(t)$ as a free boundary which satisfies the condition (2.7) or (2.8). So in the following we will determine the form of the function $f(x)$ and then prove the global existence of the free boundary $a(t)$. To this end, denoting $z=\frac{r}{a(t)}$, one can obtain from (2.5) that

Next, we will solve (3.3) according to the free boundary conditions (2.7) or (2.8), respectively.

The continuous boundary condition

Assume that $\gamma=\beta>\frac{4}{3}$. We require

Then it follows from (3.3)-(3.4) that

Using the boundary condition (2.7) yields $f(z)=[\frac {\gamma-1}{2\gamma}(1-z^{2})]^{\frac{1}{\gamma-1}}$, and then

Clearly, if $a(t)>0$ is a free boundary satisfying the condition (2.7), then it is straightforward to check that $(\rho,u)$ defined by (3.7) is a solution of (2.4)-(2.5), where $a(t)$ can be determined by

with $a_{0}>0$ and $a_{1}$ being the initial location and slope of the free boundary.

Thus, it remains to solve the boundary value problem (3.8). We start with estimates on solutions of (3.8).

Lemma 3.2

Let $a(t)\in C^{1}[0,T]$ be a solution to (3.8) for $T>0$. Then there exist two uniform positive constants $C_{1}$ and $C_{2}>0$, independent of T, such that

Proof

We first verify the fact that $a(t)\geq1$ for all $t\in[0,T]$. Note that (3.7) implies

We introduce

Due to $a'(t)=u(a(t),t)$ and (3.10), a direct computation gives

Equations (2.4) and (3.10) yield

One has

Thus, substituting the above estimates into (3.12) and using the Cauchy-Schwarz inequality, one deduces

where one has used

Note also that the conservation of total mass implies that

In the case $\gamma\geq\frac{5}{3}$, (3.13) yields

and so

and consequently

Thus, as a consequence of (3.16) and the conservation of mass, we have, for any $t>0$,

which implies

For $1<\gamma<\frac{5}{3}$, (3.13) yields

which yields

Here

As in (3.15)-(3.17), one can show that

Next, we derive an upper bound for $a(t)$. It follows from (3.8), (3.20) that

Then

This yields (3.9) and completes the proof of Lemma 3.2. □

We are now ready to give the existence and uniqueness of the solution to the boundary value problem (3.8).

Lemma 3.3

There exists a sufficiently small T such that (3.8) has a solution $a(t)$, which is unique in $C^{1}[0,T]$ and satisfied with $0<\frac{1}{2}a_{0}<a(t)<2a_{0}$.

Proof

The lemma can be proved by a fixed point argument. In fact, set

Then (3.8) can be rewritten as

Let $T_{1}$ be a positive small constant to be determined. Define

Then, for any $a_{1}(t)$ and $a_{2}(t)\in X$, since $\beta>1$, we have

where $L=(\frac{1}{2}a_{0})^{1-3\beta}3^{3(\beta-1)}(\frac{3}{2-3\beta }+T_{1})$ is a constant. We now define a mapping on X by

Then $\mathbb{T}a(t)\in C^{1}[0,T_{1}]$, and, for any $t< T_{1}$, one can deduce that

if

and

if

Thus, if $T_{1}\leq\min\{T_{2},T_{3}\}$, then $\mathbb{T}a(t)\in X$. Furthermore, since

$\mathbb{T}$ will be a contraction mapping if $(\frac{3}{2-3\beta }+T_{1})T_{1}<(\frac{1}{2}a_{0})^{3\beta-1}3^{3(1-\beta)}$, i.e.,

The above argument shows that $\mathbb{T}: X\rightarrow X$ is a contraction with the sup-norm for any $T_{1}=\min\{T_{2},T_{3},T_{4}\}$. By the contraction mapping theorem, there exists a unique $a(t)\in C^{1}[0,T_{1}]$ such that $\mathbb{T}a(t)=a(t)$ and then $a'(t)=g(a(t))$, which yields (3.8). This completes the proof of the lemma. □

Now, Theorem 2.1 follows from Lemma 3.3, the a priori estimates, Lemma 3.2, and the standard continuity argument. This completes the proof of Theorem 2.1.

The stress free boundary condition

First, it follows from the free boundary (2.8) and from (2.4) that

Using the ansatz in (3.1) shows that

Then (3.1) becomes

Set $\alpha=f(1)$, and then (3.22) tells us that $\alpha >0$. Recall the assumption that $\beta=\frac{1}{2}(\gamma+\frac {1}{3})$, $\gamma>\frac{5}{3}$. Then (3.3) becomes

Denoting $g(z)=(\frac{f(z)}{\alpha})^{\frac{3\gamma-5}{6}}$ for any $z\in[0,1]$, then the above equality becomes

Next, we will prove that (3.25) can be solved on $[0,1]$. To this end, we start with a priori estimates and the uniqueness.

Lemma 3.4

For any $\gamma>\frac{5}{3}$, let $g(z)$ be a solution to the system (3.25) in $C([0,1])\cap C^{1}((0,1])$. Then

for all $z\in(0,1]$. Furthermore, such a solution is unique.

Proof

If $g^{\frac{3\gamma-1}{3\gamma-5}}(z)-\frac{3\gamma+1}{6\gamma}=0$, then (3.25) implies that z must be zero, i.e. $g^{\frac{3\gamma-1}{3\gamma-5}}(0)=\frac{3\gamma+1}{6\gamma}$. Namely, if $z\neq0$, then one has $g^{\frac{3\gamma-1}{3\gamma-5}}(z)\neq\frac {3\gamma+1}{6\gamma}$.

If for any $z\in(0,1]$, $g^{\frac{3\gamma-1}{3\gamma-5}}(z)\in[0,\frac {3\gamma+1}{6\gamma})$, then (3.25) implies that $g'(z)\leq 0$ and thus

which is a contradiction. Thus we can deduce that $g^{\frac{3\gamma -1}{3\gamma-5}}(z)>\frac{3\gamma+1}{6\gamma}$ for all $z\in(0,1]$ and together with (3.25) to get $g'(z)\geq0$ and consequently

It remains to prove the uniqueness.

To this end, let $\bar{g}(z)\in C([0,1])\cap C^{1}((0,1])$ be another solution to (3.25) with $\bar{g}(1)=1$ and $(\frac{3\gamma+1}{6\gamma})^{\frac{3\gamma-5}{3\gamma-1}}< g(z)\leq1$ for all $z\in(0,1]$.

Define $w(z)=g(z)-\bar{g}(z)$. Then $w(z)$ solves the following problem:

Set

Here $I\neq\emptyset$ because of $1\in I$. Define $z_{0}=\inf I$ and then $z_{0}\in[0,1]$. Obviously, the uniqueness of solutions to the system (3.25) will be shown by proving that $z_{0}\equiv0$ and using a continuity argument.

If not, then $z_{0}\in(0,1]$, and $w(z_{0})=0$. For any $z\in(0,z_{0})$, (3.26) tells us that

Integrating (3.28) over $[z,z_{0}]$ shows

Since for any $\gamma>\frac{5}{3}$ and $\frac{6(\gamma-1)}{3\gamma -5}>2$, a Taylor expansion gives

for sufficiently small $w(z)$. Putting (3.31) into (3.30) and using the fact $w(z_{0})=0$, one has

for z close to $z_{0}$. Notice that

by virtue of (3.29). Then one can easily deduce that $w(z)\equiv0$, $\forall z\in(z_{0}-\delta,z_{0})$ for some $\delta>0$. This contradicts to $z_{0}=\inf I$. Thus $z_{0}\equiv0$ and the proof of Lemma 3.4 is completed. □

Now we are ready to give an existence result to system (3.25).

Lemma 3.5

For any $\gamma>\frac{5}{3}$, there is a positive function $y=g(z)$ in $C([0,1])\cap C^{1}((0,1])$ satisfying (3.25).

Proof

We can rewrite (3.25) as follows:

We look for a solution to (3.33) such that

Set $\mathbb{R}=\{(z,g(z))|0\leq1-z\leq a, 0\leq1-g(z)\leq b\}$ for small $a\in(0,1)$, $b\in(0,1-(\frac{3\gamma+1}{6\gamma})^{\frac {3\gamma-5}{3\gamma-1}})$. Then we can easily deduce that

where M is a positive constant only depending on γ, a, b.

Since $G(g(z),z)$ is continuous in $\mathbb{R}$, by choosing $h=\min\{ a,\frac{b}{M}\}$, one can show that the solution to the initial value problem (3.32) exists in the neighborhood $0\leq1-z\leq h$.

Similarly, we can extend this solution from the left of the neighborhood $0\leq1-z\leq h$ step by step. Let the maximum interval of the existence of solutions be $(\alpha,1]$ for some $\alpha\geq0$; with $g(z)\in C([\alpha,1])\cap C^{1}((\alpha,1])$, we will show that $\alpha \equiv0$.

If not, then $\alpha\in(0,1)$. By Lemma 3.4, one has $g(\alpha)>(\frac{3\gamma+1}{6\gamma})^{\frac{3\gamma-5}{3\gamma-1}}$ and (3.33)₁ is well defined in the small neighborhood of $z=\alpha$ for $C^{1}$ function $g(z)$. Thus similar arguments to those above show that the solution of this initial value problem (3.32) in the neighborhood $\vert \alpha-z\vert \leq h_{0}$ for small $h_{0}>0$ exists. That is to say, the solution $g(z)$ can be extended to the interval $[\alpha-h_{0},1]$, which contradicts the fact that $(\alpha ,1]$ is the maximum interval of the existence of solutions. Then we have obtained a $C([0,1])\cap C^{1}((0,1])$ solution $g(z)$ to the system (3.25). □

Finally, by virtue of (3.23) and Lemmas 3.4-3.5, we have obtained the global existence of $y=f(z)\in C([0,1])\cap C^{1}((0,1])$ to (3.24) and the proof of Theorem 2.2 is complete. According to Lemma 3.4, $f(0)>(\frac{3\gamma+1}{6\gamma})^{\frac{6}{3\gamma-1}}f(1)$, and $f(z)=\rho(r,t)a^{3}(t)$, we can deduce from (3.21)-(3.22) the following.

Corollary 3.1

The density at the origin of the center has the following estimate:

and then

The authors would like to thank the anonymous referees for their valuable suggestions which helped to improve the presentation of the paper. This research is partially supported by National Natural Science Foundation of China 11331005, 11201371, and also supported by Research Fund for the Doctoral Program of Higher Education of China 20136101110015.

Affiliations

1. School of Mathematics, Northwest University, Xi’an, 710127, China

   • Mei Wang
   •  & Wei Li

2. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan Province, 454000, P.R. China

   • Zilai Li

Corresponding author

Correspondence to Mei Wang.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Reprints and Permissions