### Abstract

In this paper, we study a free boundary problem for compressible Navier-Stokes equations
with density-dependent viscosity and a non-autonomous external force. The viscosity
coefficient *μ *is proportional to *ρ*^{θ }with 0 < *θ *< 1, where *ρ *is the density. Under certain assumptions imposed on the initial data and external
force *f*, we obtain the global existence and regularity. Some ideas and more delicate estimates
are introduced to prove these results.

##### Keywords:

Compressible Navier-Stokes equations; Viscosity; Regularity; Vacuum### 1 Introduction

We study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity and a non-autonomous external force, which can be written in Eulerian coordinates as:

with initial data

where *ρ *= *ρ *(*ξ,τ*), *u *= *u*(*ξ*,*τ*), *P *= *P*(*ρ*) and *f *= *f*(*ξ*,*t*) denote the density, velocity, pressure and a given external force, respectively,
*μ *= *μ*(*ρ*) is the viscosity coefficient. *a*(*τ*) and *b*(*τ*) are the free boundaries with the following property:

The investigation in [1] showed that the continuous dependence on the initial data of the solutions to the compressible Navier-Stokes equations with vacuum failed. The main reason for the failure at the vacuum is because of kinematic viscosity coefficient being independent of the density. On the other hand, we know that the Navier-Stokes equations can be derived from the Boltzmann equation through Chapman-Enskog expansion to the second order, and the viscosity coefficient is a function of temperature. For the hard sphere model, it is proportional to the square-root of the temperature. If we consider the isentropic gas flow, this dependence is reduced to the dependence on the density function by using the second law of thermal dynamics.

For simplicity of presentation, we consider only the polytropic gas, i.e. *P*(*ρ*) = *Aρ*^{γ }with *A *> 0 being constants. Our main assumption is that the viscosity coefficient *μ *is assumed to be a functional of the density *ρ*, i.e. *μ *= *cρ*^{θ}, where *c *and *θ *are positive constants. Without loss of generality, we assume *A *= 1 and *c = *1.

Since the boundaries *x *= *a*(*τ*) and *x *= *b*(*τ*) are unknown in Euler coordinates, we will convert them to fixed boundaries by using
Lagrangian coordinates. We introduce the following coordinate transformation

then the free boundaries *ξ *= *a*(*τ*) and *ξ *= *b*(*τ*) become

where

where

and the initial data (1.3) become

Now let us first recall some previous works in this direction. When the external force
*f *≡ 0, there have been many works (see, e.g., [2-9]) on the existence and uniqueness of global weak solutions, based on the assumption
that the gas connects to vacuum with jump discontinuities, and the density of the
gas has compact support. Among them, Liu et al. [4] established the local well-posedness of weak solutions to the Navier-Stokes equations;
Okada et al. [5] obtained the global existence of weak solutions when 0 < *θ *< 1/3 with the same property. This result was later generalized to the case when 0
< *θ *< 1/2 and 0 < *θ *< 1 by Yang et al. [7] and Jiang et al. [3], respectively. Later on, Qin et al. [8,9] proved the regularity of weak solutions and existence of classical solution. Fang
and Zhang [2] proved the global existence of weak solutions to the compressible Navier-Stokes equations
when the initial density is a piece-wise smooth function, having only a finite number
of jump discontinuities.

For the related degenerated density function and viscosity coefficient at free boundaries, see Yang and Zhao [10], Yang and Zhu [11], Vong et al. [12], Fang and Zhang [13,14], Qin et al. [15], authors studied the global existence and uniqueness under some assumptions on initial data.

When *f *≠ 0, Qin and Zhao [16] proved the global existence and asymptotic behavior for *γ *= 1 and *μ *= *const *with boundary conditions *u*(0,*t*) = *u*(1,*t*) = 0; Zhang and Fang [17] established the global behavior of the Equations (1.1)-(1.2) with boundary conditions
*u*(0,*t*) = *ρ*(1,*t*) = 0. In this paper, we obtain the global existence of the weak solutions and regularity
with boundary conditions (1.4)-(1.5). In order to obtain existence and higher regularity
of global solutions, there are many complicated estimates on external force and higher
derivations of solution to be involved, this is our difficulty. To overcome this difficulty,
we should use some proper embedding theorems, the interpolation techniques as well
as many delicate estimates. This is the novelty of the paper.

The notation in this paper will be as follows:

_{B }denotes the norm in the space *B*; we also put

The rest of this paper is organized as follows. In Section 2, we shall prove the global
existence in *H*^{1}. In Section 3, we shall establish the global existence in *H*^{2}. In Section 4, we give the detailed proof of Theorem 4.1.

### 2 Global existence of solutions in *H*^{1}

In this section, we shall establish the global existence of solutions in *H*^{1}.

**Theorem 2.1 ***Let *0 < *θ *< 1, *γ *> 1, *and assume that the initial data *(*ρ*_{0},*u*_{0}) *satisfies *
*and external force f satisfies f*(*r*(*x*,·),·) ∈ *L*^{2n}([0,*T*], *L*^{2n}[0,1]) *for some n *∈ *N satisfying n*(2*n *- 1)/(2*n*^{2 }+ 2*n *- 1) > *θ*, *then there exists a unique global solution *(*ρ *(*x*,*t*),*u*(*x*,*t*)) *to problem (1.8)-(1.11), such that for any T *> 0,

The proof of Theorem 2.1 can be done by a series of lemmas as follows.

**Lemma 2.1 ***Under conditions of Theorem 2.1, the following estimates hold*

*where C*_{1}(*T*) *denotes generic positive constant depending only on *
*time T and *

**Proof **Multiplying (1.8) and (1.9) by *ρ*^{γ-2 }and *u*, respectively, using integration by parts, and considering the boundary conditions
(1.10), we have

Integrating (2.4) with respect to *t *over [0,*t*], using Young's inequality, we have

which, by virtue of Gronwall's inequality and assumption *f*(*r*(*x*,·),·) ∈ *L*^{2n}([0,*T*], *L*^{2n}[0,1]), gives (2.1).

We derive from (1.8) that

Integrating (2.5) with respect to *t *over [0,*t*] yields

Integrating (1.9) with respect to *x*, applying the boundary conditions (1.10), we obtain

Inserting (2.7) into (2.6) gives

Thus, the Hölder inequality and (2.1) imply

and (2.2) follows from (2.8) and (2.9).

Multiplying (1.9) by 2*nu*^{2n-1 }and integrating over *x *and *t*, applying the boundary conditions (1.10), we have

Applying the Young inequality and condition *f*(*r*(*x*, ·), ·) ∈ *L*^{2n}([0,*T*],*L*^{2n}[0,1]) to the last two terms in (2.10) yields

Applying Gronwall's inequality, we conclude

, which, along with (2.11), yields (2.3). The proof of Lemma 2.1 is complete.

**Lemma 2.2 ***Under conditions of Theorem 2.1, the following estimates hold*

**Proof **We derive from (2.5) and (1.9) that

Integrating it with respect to *t *over [0,*t*], we obtain

Multiplying (2.16) by [(*ρ*^{θ}) _{x}]^{2n-1}, and integrating the resultant with respect to *x *to get

here, we use the inequality
*f*, we get from (2.17) that

Hence,

Using the Gronwall inequality to (2.18), we obtain (2.13).

The proof of (2.14) can be found in [3], please refer to Lemma 2.3 in [3] for detail.

**Lemma 2.3 ***Under the assumptions in Theorem 2.1, for any *0 ≤ *t *≤ *T, we have the following estimate*

**Proof **Multiplying (1.9) by *u*_{t}, then integrating over [0,1] × [0,*t*], we obtain

Using integration by parts, (1.8) and the boundary conditions (1.10), we have

Thus,

Using Lemmas 2.1-2.2, we derive

The last term on the right-hand side of (2.21) can be estimated as follows, using (1.8), conditions (1.10) and Lemmas 2.1-2.2,

Inserting the above estimate into (2.21),

which, by virtue of Gronwall's inequality, (2.1) and (2.14), gives (2.19).

**Proof of Theorem 2.1 **By Lemmas 2.1-2.3, we complete the proof of Theorem 2.1.

### 3 Global existence of solutions in *H*^{2}

For external force *f*(*r*, *t*), we suppose

Constant *C*_{2}(*T*) denotes generic positive constant depending only on the *H *^{2}-norm of initial data
*T *and constant *C*_{1}(*T*).

**Remark 3.1 ***By (3.1), we easily know that assumptions (3.1) is equivalent to the following conditions*

*Therefore, the generic constant C*_{2}(*T*) *depending only on the norm of initial data *(*ρ*_{0},*u*_{0}) *in H*^{2}*, the norms of f in the class of functions in (3.2)-(3.3) and time T*.

**Theorem 3.1 ***Let *0 < *θ *< 1, *γ *> 1, *and assume that the initial data satisfies *(*ρ*_{0},*u*_{0}) ∈ *H*^{2 }*and external force f satisfies conditions (3.1), then there exists a unique global
solution *(*ρ *(*x*,*t*),*u*(*x*,*t*)) *to problem (1.8)-(1.11), such that for any T *> 0,

The proof of Theorem 3.1 can be divided into the following several lemmas.

**Lemma 3.2 ***Under the assumptions in Theorem 3.1, for any *0 ≤ *t *≤ *T, we have the following estimates*

**Proof **Differentiating (1.9) with respect to *t*, multiplying the resulting equation by *u*_{t }in *L*^{2}[0,1], performing an integration by parts, and using Lemma 2.1, we have

Integrating (3.8) with respect to *t*, applying the interpolation inequality, we conclude

On the other hand, by (1.9), we get

We derive from assumption (3.1) and (3.10) that

Inserting (3.11) into (3.9), by virtue of Lemmas 2.1-2.3 and assumption (3.1), we obtain (3.6). We infer from (1.9),

Multiplying (3.12) by *u*_{xx }in *L*^{2}[0,1], we deduce

Using Young's inequality and Sobolev's embedding theorem *W*^{1,1 }↪ *W*^{∞}, Lemma 2.1 and (3.6), we deduce from (3.13) that

whence

Applying embedding theorem, we derive from (3.14) that

which, along with (3.14), gives (3.7). The proof is complete.

**Lemma 3.3 ***Under the assumptions in Theorem 3.1, for any *0 ≤ *t *≤ *T, we have the following estimates*

**Proof **Differentiating (1.9) with respect to *x*, exploiting (1.8), we have

which gives

with

Multiplying (3.18) by *ρ*^{θ-1}*ρ*_{xx}, integrating the resultant over [0,1], using condition (3.1), Young's inequality,
Lemma 3.2 and Theorem 2.1, we deduce

Integrating (3.19) with respect to *t *over [0,t], using Theorem 2.1, Lemma 3.2 and the interpolation inequality, we derive

which, along with Lemma 2.1, gives estimate (3.15).

Differentiating (1.9) with respect to *x*, we can obtain

Integrating (3.21) with respect to *x *and *t *over [0,1] × [0,*t*], applying the embedding theorem, Lemmas 2.1-2.3 and Lemma 3.1, and the estimate
(3.15), we conclude

The proof is complete.

**Proof of Theorem 3.1 **By Lemmas 3.2-3.3, Theorem 2.1 and Sobolev's embedding theorem, we complete the proof
of Theorem 3.1.

### 4 Global existence of solutions in *H*^{4}

For external force *f*(*r*,*t*), besides (3.1), we assume that

**Remark 4.1 ***By (4.1), we easily know that assumptions (4.1) is equivalent to the following conditions*

*Therefore, the generic constant C*_{4}(*T*) *depending only on the norm of initial data *(*ρ*_{0},*u*_{0}) *in H*^{4}*, the norms of f in the class of functions in (4.2)-(4.3) and time T.*

**Theorem 4.1 ***Let *0 < *θ *< 1, *γ *> 1, *and assume that the initial data satisfies *(*ρ*_{0},*u*_{0}) ∈ *H*^{4 }*and external force f satisfies conditions (4.1), then there exists a unique global
solution *(*ρ *(*x*,*t*),*u*(*x*,*t*)) *to problem (1.8)-(1.11), such that for any T *> 0,

The proof of Theorem 4.1 can be divided into the following several lemmas.

**Lemma 4.2 ***Under the assumptions of Theorem 4.1, the following estimates hold for any t *∈ [0,*T*],

**Proof **We easily infer from (1.9) and Theorem 2.1, Theorem 3.1 that

Differentiating (1.9) with respect to *x *and exploiting Lemmas 2.1-2.3, we have

or

Differentiating (1.9) with respect to *x *twice, using Lemmas 2.1-2.3, 3.2-3.3 and the embedding theorem, we have

or

Differentiating (1.9) with respect to *t*, and using Lemmas 2.1-2.3 and (1.8), we deduce that

which together with (4.12) and (4.14) implies

Thus, estimate (4.9) follows from (4.12), (4.14), (4.17) and condition (4.1).

Now differentiating (1.9) with respect to *t *twice, multiplying the resulting equation by *u*_{tt }in *L*^{2}([0,1]), and using integration by parts, (1.8) and the boundary condition (1.10),
we deduce

here, we use
*t*, applying assumption (4.1) and (4.9), we have

which, with Lemmas 2.1-2.3 and Theorem 3.1, implies

If we apply Gronwall's inequality to (4.19), we conclude (4.11). The proof is complete.

**Lemma **4.3 *Under the assumptions of Theorem 4.1, the following estimate holds for any t *∈ [0*,T*],

**Proof **Differentiating (1.9) with respect to *x *and *t*, multiplying the resulting equation by *u*_{tx }in *L*^{2}[0,1], and integrating by parts, we deduce that

where

Employing Theorem 2.1, Theorem 3.1 Lemma 4.2 and the interpolation inequality, we conclude

with

Applying Young's inequality several times, we have that for any *ε *∈ (0,1),

and

Thus we infer from (4.22)-(4.24) that

which, together with Theorem 2.1, Theorem 3.1 and Lemma 4.2, implies

On the other hand, differentiating (1.9) with respect to *x *and *t*, and using Theorem 3.1 and Lemma 4.2, we derive