Abstract
In this paper, we study a free boundary problem for compressible NavierStokes equations with densitydependent viscosity and a nonautonomous 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 NavierStokes equations; Viscosity; Regularity; Vacuum1 Introduction
We study a free boundary problem for compressible NavierStokes equations with densitydependent viscosity and a nonautonomous 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 NavierStokes 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 NavierStokes equations can be derived from the Boltzmann equation through ChapmanEnskog expansion to the second order, and the viscosity coefficient is a function of temperature. For the hard sphere model, it is proportional to the squareroot 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 is the total initial mass, and without loss of generality, we can normalize it to 1. So in terms of Lagrangian coordinates, the free boundaries become fixed. Under the coordinate transformation, Eqs. (1.1)(1.2) are now transformed into
where . The boundary conditions (1.4)(1.5) become
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., [29]) 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 wellposedness of weak solutions to the NavierStokes 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 NavierStokes equations when the initial density is a piecewise 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:
denote the usual (Sobolev) spaces on [0,1]. In addition,  · _{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(2n  1)/(2n^{2 }+ 2n  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^{γ2 }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 2nu^{2n1 }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}]^{2n1}, and integrating the resultant with respect to x to get
here, we use the inequality . Using Young's inequality and assumptions of external of 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 2.12.2, we derive
The last term on the righthand side of (2.21) can be estimated as follows, using (1.8), conditions (1.10) and Lemmas 2.12.2,
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 2.12.3, 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 , time 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 2.12.3 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^{θ1}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 2.12.3 and Lemma 3.1, and the estimate (3.15), we conclude
The proof is complete.
Proof of Theorem 3.1 By Lemmas 3.23.3, 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 2.12.3, we have
or
Differentiating (1.9) with respect to x twice, using Lemmas 2.12.3, 3.23.3 and the embedding theorem, we have
or
Differentiating (1.9) with respect to t, and using Lemmas 2.12.3 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 . Integrating (4.18) with respect to t, applying assumption (4.1) and (4.9), we have
which, with Lemmas 2.12.3 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
Inserting (4.27) into (4.26), employing Theorem 2.1, Theorem 3.1 and Lemma 4.2, we conclude
Similarly, by Theorem 2.1, Theorem 3.1, Lemma 4.2 and the embedding theorem, we get that for any ε ∈ (0,1),
By virtue of assumption (4.1), Theorem 2.1 and Theorem 3.1, we derive that
which, combined with (4.21) and (4.27)(4.29), gives
Integrating (4.30) with respect to t, picking ε small enough, using Theorem 2.1 and Theorem 3.1, Lemma 4.2 and assumption (4.1), we complete the proof of estimate (4.20).
Lemma 4.4 Under the assumptions of Theorem 4.1, the following estimates hold for any t ∈ [0,T],
Proof Differentiating (3.18) with respect to x, we have
where
An easy calculation with the interpolation inequality, Theorem 2.1 and Theorem 3.1, gives
and
By virtue of Theorem 2.1 and Theorem 3.1, we infer from (4.36)(4.37), (4.20) and assumption (4.1) that
Now multiplying (4.34) by ρ^{θ1}ρ_{xxx }in L^{2}[0,1], we obtain
Integrating (4.39) with respect to t, using Theorem 2.1 and Theorem 3.1, assumption (4.1) and (4.38), we can get
By virtue of Theorem 2.1 and Theorem 3.1, we infer from (4.10), (4.15) and (4.40) that
Differentiating (1.9) with respect to t, using Theorem 2.1 and Theorem 3.1 and Lemmas 4.24.3, we infer that for any t ∈ [0, T],
which, combined with (4.15), (4.40) and (4.42), gives
Differentiating (4.34) with respect to x, we see that
with
and
Using the embedding theorem, (1.8), Theorem 2.1, Theorem 3.1 and Lemmas 4.14.2, we can deduce that
Inserting (4.46) into (4.47), we have
By virtue of Theorems 2.1, 3.1, Lemmas 4.24.3, we derive from (4.40)(4.43) and assumption (4.1) that
Multiplying (4.44) by ρ^{θ1}ρ_{xxxx }in L^{2}[0,1], we get
Integrating (4.50) with respect to t, using condition (4.1) and (4.49), we conclude
Differentiating (1.9) with respect to x three times, using Theorems 2.1, 3.1, Lemmas 4.24.3 and the interpolation inequality, we infer
Thus we conclude from (1.8), (4.27), (4.41), (4.43), (4.51) and assumption (4.1) that
Thus (4.31) follows from (4.40) and (4.51), we can derive estimate (4.32)(4.33) from Theorem 2.1, Theorem 3.1, Lemmas 4.24.3, (4.41), (4,43) and (4.53). The proof is complete.
Proof of Theorem 4.1 Using (1.8),Theorem 2.1, 3.1 and Lemmas 4.24.4 and the proper interpolation inequality, we readily get estimate (4.4)(4.8) and complete the proof from Theorem 4.1.
Corollary 4.5 Under assumptions of Theorem 4.1 and some suitable compatibility conditions, the global solution (ρ (x,t),u(x,t)) to problem (1.8)(1.11) is the classical solution verifying
Proof By the embedding theorem, we easily prove the corollary from Theorem 4.1.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed to each part of this work equally.
Acknowledgements
The work is in part supported by Doctoral Foundation of North China University of Water Sources and Electric Power (No. 201087), the Natural Science Foundation of Henan Province of China (No. 112300410040) and the NNSF of China (No. 11101145).
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