Dissipation Limit for the Compressible Navier-Stokes to Euler

Journal of Applied Mathematics and Physics, 2018, 6, 2142-2158 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

Dissipation Limit for the Compressible Navier-Stokes to Euler Equations in One-Dimensional Domains Shangfei Cui College of Science, University of Shanghai for Science and Technology, Shanghai, China

How to cite this paper: Cui, S.F. (2018) Dissipation Limit for the Compressible Navier-Stokes to Euler Equations in OneDimensional Domains. Journal of Applied Mathematics and Physics, 6, 2142-2158. https://doi.org/10.4236/jamp.2018.610180 Received: October 2, 2018 Accepted: October 26, 2018 Published: October 29, 2018

We prove that as the viscosity and heat-conductivity coefficients tend to zero, respectively, the global solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

Keywords

Copyright © 2018 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access

Abstract

Compressible Navier-Stokes Equations, Vanishing Viscosity Limit, Rarefaction Waves, Euler Equations

1. Introduction and the Main Result We study the asymptotic behavior, as the viscosity and heat-conductivity go to zero, respectively, of solutions to the Cauchy problem for the Navier-Stokes equations for a one-dimensional compressible heat-conducting fluid (in Lagrangian coordinates):

0, vt − u x =  u  u + p = ε x  , x  t  v x   2   e + u  + ( up )= κ  θ x  + ε  uu x  , x  2 t  v x  v x

(1.1)

with (discontinuous) initial data = ( u, v, e )( x, 0 )

( u0 , v0 , e0 ) ( x ) ,

x ∈ R,

(1.2)

where v, u , θ , p = p ( e, v ) and e denote the specific volume, the velocity, the DOI: 10.4236/jamp.2018.610180

Oct. 29, 2018

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temperature, the pressure and the internal energy respectively, and ε , κ are the viscosity and heat conductivity coefficients, respectively. At infinity, the initial data u0 , v0 , e0 are assumed to satisfy lim ( u0 , v0 , e0 ) ( x ) = ( u± , v± , e± ) ,

(1.3)

x→±∞

where u± , v± and e± are given constant states. The system (1.1), describing the motion of the fluid, is the conservation laws of mass, momentum and energy. The asymptotic behavior of viscous flows, as the viscosity vanishes, is one of the important topics in the theory of compressible flows. It is expected that a general weak entropy solution to the Euler equations should be (strong) limit of solutions to the corresponding Navier-Stokes equations with same initial data as the viscosity and heat conductivity tend to zero, respectively. For the one-dimensional compressible isentropic Navier-Stokes equations

0, vt − u x =  u   ε x  , ut + p ( v ) x =  v x 

(1.4)

and the corresponding inviscid p-system

0, vt − u x =  0, ut + p ( v ) x =

(1.5)

the vanishing viscosity limit for the Cauchy problem has been studied by several researchers. In [1] Di Perna uses the method of compensated compactness and established almost everywhere convergence of admissible solutions u ε , vε of (1.4) to an admissible solution of (1.5), provided that u ε , vε is uniformly L∞

(

)

(

)

bounded and vε is uniform bounded away from zero. However, this uniform boundedness is difficult to verify in general, and the abstract analysis in [1] gets little information on the qualitative nature of the viscous solutions. In [2] Hoff and Liu investigate the inviscid limit problem for (1.4) in the case that the underlying inviscid flow is a single weak shock wave, and they show that solutions of the compressible Navier-Stokes equations with shock data exist and converge to the inviscid shocks, as viscosity vanishes, uniformly away from the shocks. Based on [2] [3], Xin in [4] shows that the solution to the Cauchy problem for the system (1.4) with weak centered rarefaction wave data exists for all time and converges to the weak centered rarefaction wave solution of the corresponding Euler equations, as the viscosity tends to zero, uniformly away from the initial discontinuity. Moreover, for a given centered rarefaction wave to the Euler equations with finite strength, he constructs a viscous solution to the compressible Navier-Stokes system with initial data depending on the viscosity, such that the viscous solution approaches the centered rarefaction wave as the viscosity goes to zero at the rate ln ε ε 1 4 uniformly for all time away from t = 0 . In the vanishing viscosity limit, the Prandtl boundary layers (characteristic boundaries) are studied for the multidimensional linearized compressible Navier-Stokes equDOI: 10.4236/jamp.2018.610180

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ations by using asymptotic analysis in [5] [6] [7], while the boundary layer stability in the case of non-characteristic boundaries and one spatial dimension is discussed in [8] [9]. We mention that there is an extensive literature on the vanishing artificial viscosity limit for hyperbolic systems of conservation laws, see, for example, [1] [3] [10]-[19], also cf. the monographs [20] [21] [22] and the references therein. We also mention that the convergence of 1-d Broadwell model and the relaxation limit of a rate-type viscoelastic system to the isentropic Euler equations with centered rarefaction wave initial data are studied in [23] [24], respectively. And, in [25], the solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength had been proved exist globally in time, and moreover, as the viscosity and heat-conductivity coefficients tend to zero, the global solution converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

However, in those paper, κ is generally dependent of ε , while in this paper, we will show the dissipation limit in the case that κ and ε are indepen-

dent of each other.

(

Our aim in this paper is to study the relation between the solution u κ ,ε , vκ ,ε , eκ ,ε ( x, t ) of the Navier-Stokes equations for a compressible heat-

)

conducting fluid (1.1) and the solution

( u, v, e )( x, t )

of the corresponding in-

viscid Euler equations:

0, vt − u x =  0, ut + px =  2  e + u  + ( up ) = 0,  x  2 t  with the initial data = ( u, v, e )( x, 0 ) satisfying

( u0 , v0 , e0 ) ( x ) ,

x ∈ R,

lim ( u0 , v0 , e0 ) ( x ) = ( u± , v± , e± ) ,

x→±∞

(1.6)

(1.7) (1.8)

( u± , v± , e± ) as in (1.3). It is convenient to work with the equations for the entropy s and the absolute

with the same constant states

temperature θ . The second law of thermodynamics asserts that s de + pdv. θ d= We assume, as is customary in thermodynamics, that given any two of thermodynamics variables ρ , e, θ , s and p, we can obtain the remaining three va-

riables. If we choose ( v, θ ) as independent variables and write

( p, e, s ) = ( p, e, s )( v,θ ) , we deduce that e ( v, θ ) ,= = sv ( v, θ ) pθ ( v, θ ) , = sθ ( v, θ ) θ ev ( v, θ ) θ pθ ( v, θ ) − p ( v, θ ) . θ

Then, a straightforward calculation gives

θ2 u2 θ  st= κ  x  + κ x 2 + ε x , vθ vθ  vθ  x DOI: 10.4236/jamp.2018.610180

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(1.9)

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and

θt +

θ pθ ( v,θ ) u x2 κ  θx  ε ux = + .   eθ ( v, θ ) eθ ( v, θ )  v  x eθ ( v, θ ) v

We may also choose

( v, s )

(1.10)

as independent variables and write

p p= = ( v, s ) , θ θ ( v, s ) .

Thus, instead of (1.1), we shall study the system (1.1)1, (1.1)2 and (1.9), or (1.1)1, (1.1)2 and (1.10). Namely, we shall consider 0, vt − u x =  u  u + p ( v, s ) = ε x  ，  t x  v x   2 2  st= κ  θ x  + κ θ x + ε u x ,    vθ vθ 2  vθ  x

(1.11)

with initial data

u , v, s )( x, 0 ) (= u0 , v0 , s0 ) ( x ) (=

( u− , v− , s− ) , x < 0,  ( u+ , v+ , s+ ) , x > 0,

(1.12)

where u± , v± and s± are the constant states. The corresponding inviscid Euler equations read: 0, vt − u x =  0, ut + p ( v, s ) x =   st = 0.

(1.13)

We assume in this paper that the pressure p is a smooth function of its arguments satisfying

pv ( v, s ) < 0 < pvv ( v, s ) , v > 0.

(1.14)

Notice that the condition (1.14) assures the system (1.13) has characteristic speeds

λ1 =− − pv , λ2 =0, λ3 = − pv , and there are two family of rarefaction waves for the Euler equations (1.13). For illustration, we describe only the 1-rarefaction waves, and thus assume + s= s− ≡ s . The case for the 3-rarefaction waves can be dealt with similarly.

Suppose the end states

( u± , v± , s )

can be connected by 1-rarefaction waves.

The centered 1-rarefaction wave connecting ( u− , v− , s ) to ( u+ , v+ , s ) is the self-similar solution ( u , v, s )( x, t ) = u r , v r , s r ( x t ) of (1.13) defined by (see,

(

)

e.g., [26] [27])

 s r (ξ ) = s ,  v r (ξ ) u r (ξ= u− + ∫v λ1 ( z, s ) dz, )  −  r λ1 v , s ( s, t ) increasing in x,   λ v r , s ( x, t ) =− − pv v r ( x t ) , s ,  1

( (

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) )

(

(1.15)

)

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which is uniquely determined by the system (1.13) and the rarefaction wave initial data

= ( u , v , s ) t =0

( u− , v− , s ) , x < 0,

 u , v , s )( x)  (= ( u , v , s ) , r 0

r 0

r 0

+

+

(1.16)

x > 0.

For the internal energy e ( v, θ ) , we assume eθ ( v, θ ) > 0 for v, θ > 0.

(1.17)

For the sake of convenience, throughout this paper we denote

α = u+ − u− + v+ − v− . In this paper, we prove that the solution of system (1.11) with the centered

rarefaction wave initial data (1.16) of small strength α exists for all time and converges to the centered rarefaction wave of the Euler equation (1.13) as κ , ε

tends to zero respectively, uniformly away from the initial discontinuity. More precisely, the main result of this paper reads:

THEOREM 1.1. Let the constant states ( u± , v± , s ) be connected by a centered u r ( x t ) , v r ( x t ) , s r ( x t ) defined by (1.15). Assume that

(

1-rarefaction wave

)

(1.14) and (1.17) hold. Then, for α small enough, the compressible Navi-

er-Stokes equations (1.11) with the rarefaction wave initial data (1.16) have a global piecewise smooth solution u κ ,ε ( x, t ) , vκ ,ε ( x, t ) , sκ ,ε ( x, t ) , such that

(

)

1) u κ ,ε , θ κ ,ε are continuous for t > 0 , vκ ,ε and u κx ,ε , vκx ,ε , θ xκ ,ε are uniformly Hӧlder continuous in the set x < 0, t ≥ τ and x > 0, t ≥ τ for any

τ > 0 ; utκ ,ε , u κxx,ε , vκxt,ε , θtκ ,ε , θ xxκ ,ε are Hӧlder continuous on compact set ( x, t ) , x ≠ 0, t > 0 . Moreover, the jumps in vκ ,ε at x = 0 satisfy vκ ,ε ( 0, t )  ≤ C1 exp ( − C2t ε ) , and so does the other jumps, where C1 , C2 are positive constants independent of t and ε , and [⋅] denotes jumps in what follows. 2) The solution u κ ,ε , vκ ,ε , sκ ,ε converges to the centered rarefaction wave

(

)

( u , v , s ) as ε → 0, κ → 0 uniformly away from t = 0 , i.e., for any positive h, we have r

r

lim

r

sup

ε →0,κ →0 x∈ ,t >h

( u ( x, t ) , v ( x, t ) , s ( x, t ) ) −  u κ ,ε

κ ,ε

κ ,ε



r

 x  r  x  r  x  0.  ,v  , s   =  t   t   t 

(

)

3) For any fixed ε > 0 and κ > 0 , the solution u κ ,ε , vκ ,ε , sκ ,ε approaches the centered rarefaction wave u r , v r , s r uniformly as time goes to infinity, i.e.,

(

)

  x   x   x  lim sup u κ ,ε ( x, t ) , vκ ,ε ( x, t ) , sκ ,ε ( x, t ) −  u r   , v r   , s r    = 0. t →∞ 1 x∈   t   t   t 

(

)

To prove Theorem 1.1 and to overcome the difficulties induced by non-isentropy of the flow, we shall adapt and modify the arguments in [25], but we do not use the natural scaling argument, and we do not assume that κ = O ( ε ) . We point out here that in view of Theorem 1.1, an initial jump discontinuity

at x = 0 can be allowed in (1.2). The evolution of this jump discontinuity is an DOI: 10.4236/jamp.2018.610180

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important aspect in our analysis. It has been shown in [28] that the discontinuity evolution follows a curve x = − [u ] [ v ] in x-t plane, and the jump discontinuity in v, u x and θ x decays exponentially in time, while the discontinuity in u and θ are smoothed out at positive time, see [28] for details. We shall exploit this fact in the proof of Theorem 1.1. In Section 2 we reformulate the problem and give the proof of Theorem 1.1, while Section 3 is dedicated to the derivation of a priori estimates used in Section 2. Throughout this paper, we use the following notation:

 − := ⋅

2 ±

( −∞, 0 ) , ≡ ⋅

2

 + :=

( 0, ∞ ) ,

⋅ =

⋅

L2 (  )

,

0

⋅

Lp

=

⋅

Lp (  )

,

∞

2 ( ) + ⋅ L2 ( + ) , ∫± ⋅ dy ≡ ∫−∞ ⋅ dy + ∫0 ⋅ dy.

L2  −

2. Reformulation and the Proof of Theorem 1.1 In this section, we will reduce the proof of Theorem 1.1 to the nonlinear time-asymptotic stability analysis of rarefaction waves for the system (1.11) under non-smooth perturbations. First, we derive some necessary estimates on the rarefaction waves of the Euler equations (1.13) based on the inviscid Burgers equation, in particularly, we construct an explicit smooth 1-rarefaction wave which well approximates a given centered 1-rarefaction wave. We start with the Riemann problem for the Burgers equation:   w2  0,  wt +   =  2 x   r  w ( x, 0 ) = w0 ( x ) ,

(2.1)

where w0r ( x ) is given by

 w , x < 0, w0r ( x ) =  −  w+ , x > 0, If w− < w+ , then the problem (2.1) has the centered rarefaction wave solution w ( x, t ) = wr ( x t ) given by r

 w− , x t ≤ w− ,  w (= x, t )  x t , w− ≤ x t ≤ w+ , w , x t ≥ w . +  + r

To construct a smooth rarefaction wave solution of the Burgers equation which approximates the centered rarefaction wave, we set for δ > 0 ,

= w w= (δ x ) δ ( x)

w+ + w− w+ − w− + tanh (δ x ) 2 2

and for each δ > 0 , we solve the following initial value problem   w2  0,  wt +   =   2 x   w ( x, 0 ) = wδ ( x ) . DOI: 10.4236/jamp.2018.610180

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Next, we state certain properties that will be used later. LEMMA 2.1. (S. JIANG [25]) For each δ > 0 , the problem (2.2) has a unique global smooth solution wδr ( x, t ) , such that 1) w− < wδr ( x, t ) < w+ , ∂ x wδr ( x, t ) > 0 for x ∈ , t ≥ 0, δ > 0 .

2) For any 1 ≤ p ≤ ∞ , there is a constant C ( p ) depending only on p, such that

∂ x wδr ( ⋅ , t )

Lp

∂ 2x wδr ( ⋅ , t )

Lp

∂ 3x wδr ( ⋅ , t )

Lp

{

≤ C ( p ) min ( w+ − w− ) δ 1−1 p , ( w+ − w− )

1 p −1+1 p

t

},

1  ≤ C ( p ) min ( w+ − w− ) δ 2−1 p , δ 1−1 p  , t  1  ≤ C ( p ) min ( w+ − w− ) δ 3−1 p , δ 2−1 p  . t 

3)

lim sup wδr ( x, t ) − wr ( x, t ) = 0.

t →+∞ x∈

Now, set w± = λ1 ( v± , s ) , and we define V ( x, t ) , U ( x, t ) , S ( x, t ) , Θ ( x, t ) , the smooth approximation vr , u r , s r , θ r , by

(

)

V ( x ,t )

wδr ( x, t ) , U ( x, t ) = u± + ∫v λ1 (V ( x, t ) , s ) = S ( x, t ) =s , Θ ( x, t ) = θ (V ( x, t ) , s ) .

±

− pv ( z, s ) dz,

Then, it is not difficult to see that V ( x, t ) , U ( x, t ) , S ( x, t ) , Θ ( x, t ) satisfy

0, Vt − U x =  0, U t + p (V , Θ ) x =   St (V , Θ ) =0,  Θ + ΘpΘ (V , Θ ) U =0, x  t eΘ 

(2.3)

and due to Lemma 2.1, the following lemma holds for V ( x, t ) , U ( x, t ) , S ( x, t ) , Θ ( x, t ) .

LEMMA 2.2. (S. JIANG [25]) The functions V ( x, t ) , U ( x, t ) , S ( x, t ) and Θ ( x, t ) constructed above satisfy:

1) V = U x > 0 for all x ∈ , t ≥ 0 . t 2) For any 1 ≤ p ≤ ∞ , there is a constant C ( p ) depending only on p, such that

(Vx ,U x , Θ x ) ( ⋅ , t ) L

p

{

}

≤ C ( p ) min αδ 1−1 p , α 1 p t −1+1 p ,

(2.4)

1  ≤ C ( p ) min αδ 2−1 p , δ 1−1 p  , t 

(2.5)

(Vxx ,U xx , Θ xx ) ( ⋅ , t ) L

p

(Vxxx ,U xxx , Θ xxx ) ( ⋅ , t ) L

p

1  ≤ C ( p ) min αδ 3−1 p , δ 2−1 p  . t 

(2.6)

3)

(

lim sup (V , U , S , Θ )( x, t ) − v r , u r , s r , θ r t →∞ x∈

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0. ) ( x, t ) =

(2.7)

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S. F. Cui

4)

(Vt ,U t , Θt ) ( x, t ) ≤ C (Vx ,U x , Θ x ) ( x, t ) . Consequently, from Lemmas 2.1 and 2.2, it follows that v r , u r , θ r ( x, t ) as t → ∞ .

verges to

(

)

(2.8)

(V ,U , Θ )( x, t )

con-

The proof of Theorem 1.1 is broken up into several steps. We start with the observation that by making use of the smooth rarefaction wave

(V ,U , Θ )( x, t )

constructed above (e.g. one may take δ = 1 ), one can decompose the solution u κ ,ε , vκ ,ε , θ κ ,ε of (1.1), (1.9) and (1.10) into

(

)

(ϕ ,ψ , φ )( x, t )= ( v − V , u − U ,θ − Θ )( x, t ) , ξ ( x, t )=

s ( x, t ) − s .

Substituting the above decomposition into (1.1), (1.9) and (1.10), we obtain the system for the functions ϕ ,ψ , φ , ξ :

0, ϕt −ψ x =  u  ψ + ( p ( v, θ ) − p (V , Θ ) ) = ε x  , x  t  v x   θ p ( v, θ ) ΘpΘ (V , Θ )  θ pθ ( v,θ )  ψ x +  θ − U x φt + eθ ( v, θ ) eΘ (V , Θ )    eθ ( v, θ )  u x2 κ  θx  ε  , = +  eθ ( v, θ )  v  x eθ ( v, θ ) v   u x2 θ x2  θx  , ξ κ κ ε = + +  t  vθ  vθ vθ 2  x 

(2.9)

with initial data , φ , ξ )( x, 0 ) (ϕ0 ,ψ 0 , φ0 , ξ 0 ) ≡ ( v0 − V0 , u0 − U 0 , θ 0 − Θ0 , s0 − s ) , (ϕ ,ψ=

(ϕ0 ,ψ 0 , φ0 , ξ0 )

(2.10)

and its derivatives are sufficiently smooth away from 2 2 − 2 + x = 0 but up to x = 0 and (ϕ0 ,ψ 0 , φ0 , ξ 0 ) ∈ L (  ) , ϕ0 x ∈ L  ∩ L  . We shall show that the Cauchy problem (2.9), (2.10) possesses a unique global solution (ϕ ,ψ , φ , ξ )( x, t ) in the same function class as for u κ ,ε , vκ ,ε , θ κ ,ε in where

( )

(

(ϕ ,ψ , φ )

Theorem 1.1. Moreover,

( ) )

goes to zero uniformly as t → 0 . This convergence then yields Theorem 1.1 due to Lemmas 2.1 and 2.2.

LEMMA 2.3. (Hoff [28]) Suppose that N ( 0 ) is suitably small so that there exist two positive constants v and v with v ≤ v10 ( x ) ≤ v for all x ∈  . Then, there is a constant T > 0 , such that the Cauchy problem (2.9), (2.10) has

a solution (ϕ ,ψ , φ ) on R × [ 0, T ] in the same function class as for u κ ,ε , vκ ,ε , θ κ ,ε in Theorem 1.1. Moreover, ϕ ,ψ , φ satisfy

(

)

1) There exists a positive constant C, such that

sup t ≥0

( (ϕ,ψ ,φ )(t )

2

+ ϕx (t )

2 ±

)+∫

T

0

(ϕ x ,ψ x , φx ) ( t ) ± dt ≤ C { N 2 ( 0 ) + δ 1 4 } . 2

2) There is a positive constant C, such that

sup

0≤t ≤T

( (ψ ,φ ) (t ) x

x

{

2 ±

+ (ψ xx , φxx ) ( t )

2 ±

)+∫

T

0

2

(ψ xx ,ψ xt , φxx , φxt ) ( t ) ± dt

}

≤ C N 2 ( 0) + δ 1 4 . DOI: 10.4236/jamp.2018.610180

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3) There are constant C1 , C2 > 0 independent of T, such that u   p ( v, e )  = x  ≤ C1 exp {−C2t} v

PROPOSITION 2.4. (A priori estimate)Let the assumptions in Lemma 2.3 be satisfied. Assume that the Cauchy problem (2.9), (2.10) has a solution on R × [ 0, T ] for some t > 0 , in the same function class as in Lemma 2.3. Denote

(ϕ ,ψ , φ )( x, t )

= N 2 ( t0 , t ) : sup

t0 ≤ s ≤t

{ (ϕ ,ψ ,φ )(t )

2

2

2

+ ϕ x ( t ) ± + ψ x ( t ) ± + φx ( t )

2 ±

} , 0 ≤ t ≤ t. 0

Then, there are positive constants η1 and C independent of t1 , such that for each fixed t0 , if

N 2 ( t0 , t1 ) ≤ η1 , then the following estimates hold t

N 2 ( t0 , t1 ) + ∫t

0

(ϕ x ,ψ x , φx ) ( tˆ ) t

≤ CN 2 ( t0 , t0 ) + Cε ∫t t

+ Cκ ∫t

0

( (ϕ , φ ) x

x

2

0

2

2

( (ϕ ,ψ ,ψ ,φ ,φ ) + ψ ) (tˆ) dtˆ + (ϕ , φ ) ) ( tˆ ) dtˆ + Cε + Cκ + Cδ , 2

x

x

x

xx

x

xx

x

14

x

t

2

0

±

sup (ψ x , φx ) ( t ) ± + ∫t (ψ xx , φxx )

0≤t ≤T

dtˆ

( tˆ ) dtˆ

{ (ϕ ,ψ ,φ ) (t ) + (ϕ ,ψ ,φ )(t ) + δ } + Cε ∫ ( (ϕ ,ψ ,ψ , φ , φ ) + ψ ) ( tˆ ) dtˆ + Cκ ∫ ( (ϕ , φ ) + (ψ , φ ) + (φ , φ ) ) ( tˆ ) dtˆ + Cε + Cκ . 2

≤C

0

2

x

±

x

x

t

t0

14

±

2

x

x

x

x

xx

x

xx

x ±

±

t

t0

0

2

±

x

x

2

x

±

xx

±

Proof of Theorem 1.1. By the systems (2.3) and (2.9), Lemma 2.2, Cauchy-Schwarz’s and Sobolev’s inequalities, we easily find that ∞

2

∫0 (ϕt ,ψ t , φt ) ( t ) L

∞

dt < ∞,

which together with Lemma 2.3 yields lim sup (ϕ ( x, t ) ,ψ ( x, t ) , φ ( x, t ) ) → 0 . t →∞

x

Hence, In view of Lemma 2.2, we have proved Theorem 1.1. □

3. Uniform a Priori Estimates In this section we derive the key a priori estimates given in Proposition 2.4. First, we introduce the normalized entropy: 

= η ( v, u , s , V , U , S ) :  e ( v, θ ) +

u2   U2   −  e (V , Θ ) +  2   2 

 − {− p (V , Θ )( v − V ) + U ( u − U ) + Θ ( s − S )} ,

where we have used the fact that ev ( v, s ) = 0. − p ( v, θ ) , es ( v, s ) = An easy computation implies that η satisfies the equation: DOI: 10.4236/jamp.2018.610180

2150

Journal of Applied Mathematics and Physics

S. F. Cui

 Θψ x2 Θφx2  +κ  x vθ 2   vθ Θp (V , Θ ) Θp (V , Θ )   − U x  pV (V , Θ ) ϕ − pΘ (V , Θ ) Θ ϕ+ Θ ξ e eΘ Θ   φφx   ψ U xx φΘ xx   ψψ x = ε +κ + ε +κ v vθ  x  v vθ   ψψ xVx 2φψ xU x φϕ Θ φφ Θ   ψϕ xU x +  −ε −ε +ε − κ x2 x + κ x 2 x  2 2 v θ v v vθ vθ    φU x2 ψ U xVx φΘ2x φΘ xVx φΘ2x  + ε −ε + 2 − κ 2 − κ 2 . v2 vθ vθ vθ   vθ

ηt + {( p ( v,θ ) − p (V , Θ ) )ψ } +  ε

(3.1)

Employing (3.1), one has LEMMA3.1. Suppose that the assumptions of Proposition 2.4 hold. Then,

(ϕ ,ψ , φ )( t ) ≤C

2

(

t

+ ∫t

0

( (ϕ ,ψ ,φ ) (t )

2

0

2

Vt (ϕ , φ ) + (ψ x , φx ) t

+ N ( t0 , t ) ∫t φx 0

2 ±

2 ±

) (tˆ) dtˆ

( tˆ ) dtˆ ) + Cδ

14

∫t (ϕ x ,ψ x , φx ) ± ( tˆ ) dtˆ 2 23 t + Cκ N ( t0 , t ) ∫ (ϕ x , φx ) ± ( tˆ ) dtˆ. t + C ε N ( t0 , t )

t

23

2

(3.2)

0

0

PROOF. Integrating (3.1) with respect to t and x, we get

(ϕ ,ψ , φ )( t )

2

t

+ ∫t

0

(

2

Vt (ϕ , φ ) + (ψ x , φx )

2 ±

) (tˆ) dtˆ

j =4   2 ≤ C  (ϕ ,ψ , φ ) ( t0 ) + ∑ R j  , j =1  

(3.3)

where

R1 = = R2 = R3 = R4

t



0



φφ    ψψ x + κ x   ( x, tˆ ) dxdtˆ, v vθ  x  ˆ ˆ + κ φΘ xx ) ( x, t ) dxdt ,

∫t ∫±  ( − ( p ( v,θ ) − p (V , Θ ) )ψ ) x +  ε

∫t ∫± (ε ψ U xx t ∫t ∫± (ε ( U xψϕ x + U xψ xφ + Vxψψ x ) + κ ( Θ xϕ xφ + Θ xφφx ) ) ( x, tˆ ) dxdtˆ, t 2 2 2 ∫t ∫± ( φΘ x + ε ( U x φ + U xVxψ ) + κ ( Vx Θ xφ + Θ xφ ) ) ( x, tˆ ) dxdtˆ. t

0

0

0

Recalling the definition of N ( t0 , t ) and applying Lemma 2.2, for given α , R j can be estimated as follows. = R 1

t



∫t   − p ( v,θ ) + p (V , Θ ) + ε

ψx 

 φφ   ψ + κ  x   ( tˆ ) dxd= tˆ 0,  v   vθ  

0

(

t

R2 ≤ C ∫t ε ψ 0

t

(

12

+ C ∫t κ φ t

0

(

ψx

12

12

12

φx

t

2

(

+ C ∫t κ N ( t0 , t ) φx

(

0

t

≤ Cε N ( t0 , t ) ∫t ψ x

(

2

0

t

+ Cκ N ( t0 , t ) ∫t φx DOI: 10.4236/jamp.2018.610180

2151

0

L1

xx L1

≤ C ∫t ε N ( t0 , t ) ψ x 0

) (tˆ ) dtˆ Θ ) (tˆ ) dtˆ + U ) (tˆ ) dtˆ + Θ ) (tˆ ) dtˆ

U xx

43 xx L1

2

43 xx L1

( tˆ ) dtˆ + δ ) ( tˆ ) dtˆ + δ ) , 14

2

14

Journal of Applied Mathematics and Physics

S. F. Cui t

R4 ≤ C ∫t

0

(φ

12

φx

12

( (ψ ,φ ) + Cκ ∫ ( (ϕ , φ ) t

+ Cε ∫t

(

12

(ϕ x , φ x )

t0

t

) (tˆ ) dtˆ

(ψ x , φx )

≤ C ∫t N ( t0 , t ) φx 0

2

12

0

t

t

Θx

2

(

2

12

83

+ Θx

Θx

) (tˆ) dtˆ

2

) (tˆ ) dtˆ

2

+ Cε ∫t N ( t0 , t ) (ψ x , φx ) + (U x , Vx ) 0

(

+ Cκ ∫t N ( t0 , t ) (ϕ x , φx ) + Θ x ( tˆ ) t

0

(

t

2

0

( + Cκ ( N ( t , t ) ∫

( tˆ ) dtˆ + δ

t

0

and

)

14

( tˆ ) dtˆ + δ

2

+ Cε N ( t0 , t ) ∫t (ψ x , φx ) 0

83

83

2

≤ C N ( t0 , t ) ∫t φx

14

(ϕ x , φx ) ( tˆ ) dtˆ + δ 1 4

t

2

t0

)

(U x ,Vx ) ( tˆ ) dtˆ

12

) (tˆ) dtˆ

) (tˆ) dtˆ ) ),

( (ψ ,φ ,ϕ ) (ψ ,φ ,ϕ ) (U ,V ) ) (tˆ) dtˆ + Cκ ∫ ( (ϕ , φ ) (ϕ , φ ) Θ ) ( tˆ ) dtˆ ≤ Cε ∫ { N ( t , t ) (ψ , φ , ϕ ) + (U ,V ) } ( tˆ ) dtˆ + Cκ ∫ { N ( t , t ) (ϕ , φ ) + Θ } ( tˆ ) dtˆ ≤ Cε { N ( t , t ) ∫ (ψ , φ , ϕ ) ( tˆ ) dtˆ + δ } + Cκ { N ( t , t ) ∫ (ϕ , φ ) ( tˆ ) dtˆ + δ } . t

R3 ≤ Cε ∫t

32

12

x

0

t

x

x

x

t

x

x

2

23

x

0

t

x

x

2

x

x

2

x

t0

23

0

x

14

x

t

2

x

t0

x

4

x

t

23

0

4

x

23

0

t0

x

32

12

t0

t0

x

14

x

where we have used Sobolev’s inequality: u

np ( n − p )

L

n 1 ( n ) ≤ C Du Lp ( n ) , 1 ≤ p < n, for all u ∈ Cc (  )

Hӧlder’s inequality: uv

L1(U )

≤ u

Lp (U )

for 1 ≤ p, q ≤ ∞,

v

Lq (U )

,

1 1 + = 1, u ∈ Lp (U ) , v ∈ Lq (U ) . p q

Young’s inequality:

1 q v q , q L (U ) 1 1 for 1 ≤ p, q ≤ ∞, + = 1, u ∈ Lp (U ) , v ∈ Lq (U ) . p q uv

L1(U )

≤

1 u p

p

Lp (U )

+

and the following inequality: t

∫t

0

∂ ixU ( tˆ )

a +b Lp

dtˆ ≤ sup ∂ ixU ( tˆ ) tˆ∈[t0 ,t ]

≤ Cδ ( DOI: 10.4236/jamp.2018.610180

2152

i −1 p )a

t

∫t

0

a Lp

t

∫t

0

∂ ixU ( tˆ )

∂ ixU ( tˆ )

b Lp

b Lp

dtˆ

dtˆ.

Journal of Applied Mathematics and Physics

S. F. Cui

Substituting the above estimates for R j , ( j = 1, , 4 ) into (3.3), we obtain (3.2). This completes the proof. □ LEMMA3.2. Suppose that the assumptions of Proposition 2.4 hold. Then t

2

ϕ x ( t ) ± + ∫t ϕ x

2

0

≤C

{ (ϕ ,ψ )(t ) x

t

+ Cε ∫t

0

±

( tˆ ) dtˆ

2

0

±

t

2

+ ψ ( t ) ± + ∫t

0

(φx ,ψ x ,ψ xx ) ± ( tˆ ) dtˆ + δ 2

2

 1  ϕ x 2 ϕ x   ψψ x  =    − ψ  +  v   v  x 2 v  t ϕ2 ψ2  ϕ φ +  x pv ( v, θ ) + x  + x x pθ v  v  v

ϕx , one obtains v

 ψ xxϕ x ψ xϕ x2   U xϕ x2   2 − 3 − 3  v   v   v  ψϕ U ψψ V  ( v,θ ) +  x2 x − 2x x  v  v 

{

}

Vx ( pv ( v,θ ) − pV (V , Θ ) ) + Θ x ( pθ ( v,θ ) − pΘ (V , Θ ) ) v  ϕψ ψ ϕ2   ϕ U ϕ U V   ψ ϕ V ϕ 2U  + ε  − x 2 xx + x 3 x  + ε  − x 2 xx + x 3x x  + ε  x 3x x + x 3 x  . v v  v v v     v 

+

Integrating (3.5) with respect to x, t over 2

t

ϕ x ( t ) ± + ∫t ϕ x 0

2 ±

(3.4)

(ϕ x ,ψ x ,ψ xx ) ± ( tˆ ) dtˆ.

PROOF. Multiplying the second equation of (2.9) by −

ϕx

}

( tˆ ) dtˆ

 t 2 2 ≤ C  (ϕ x ,ψ )( t0 ) ± + ψ ( t ) ± + ∫t 0 

(

(3.5)

( t0 , t ) ×  , we infer

2

(ψ x , φx ) ± +

Vt (ϕ , φ )

2 ±

)

j =11  (3.6) ˆ ˆ d t t + R ( ) ∑ j , j =5 

with t

R5 = − ∫t

0

 ψψ x  ( x, tˆ ) dxdtˆ, v  x

∫± 

∫t ∫± ( ψ U xϕ x

+ Vxψψ x

∫t ∫± ( ψ xxϕ x

+ ψ xϕ x2

t

= R6

0

t

= R7

0

= R8

) ( x, tˆ ) dxdtˆ,

) ( x, tˆ ) dxdtˆ,

U xϕ x ( x, tˆ ) dxdtˆ, R9 ∫= t ∫± t

2

0

∫± ( U xxϕ x

t

= R10 ε ∫t

0

t

R11 ε ∫t =

0

+ U xVxϕ x

∫± ( Vxψ xϕ x

+ U xϕ x2

ε R7 ,

) ( x, tˆ ) dxdtˆ,

) ( x, tˆ ) dxdtˆ,

where R j , ( j = 5, ,11) can be bounded as follows, using Sobolev’s imbedding theorem and Lemma2.3 (iii) (see [25] for detail). t

R5 ≤ C ∫t ψ x 0

( tˆ ) dtˆ + C ,

(ϕ x ,ψ x ,ψ xx ) ( tˆ ) dtˆ + C ,

t

R7 ≤ C ∫t

2

2

0

(ϕ x ,ψ x ,ψ xx ) ( tˆ ) dtˆ + Cε ,

t

R9 ≤ Cε ∫t

2

0

t

R10 ≤ Cε ∫t (ψ x , ϕ x ) 0

DOI: 10.4236/jamp.2018.610180

2153

2

t

R6 ≤ C ∫t

0

(ϕ x ,ψ x ) ( tˆ ) dtˆ + C , 2

t

R8 ≤ C ∫t ϕ x

2

0

t

R10 ≤ Cε ∫t ϕ x 0

( tˆ ) dtˆ + C , 2

( tˆ ) dtˆ + Cε ,

( tˆ ) dtˆ + Cε . Journal of Applied Mathematics and Physics

S. F. Cui

Inserting the estimates for R j , ( j = 5, ,11) into (3.6), we arrive at t

2

ϕ x ( t ) ± + ∫t ϕ x

2

0

≤C

{ (ϕ ,ψ )(t ) x

( tˆ ) dtˆ

2 ±

(φx ,ψ x ,ψ xx ) ± ( tˆ ) dtˆ + δ

t

2

+ ψ ( t ) ± + ∫t

2

0

}

(3.7)

(ϕ x ,ψ x ,ψ xx ) ± ( tˆ ) dtˆ.

t

+ Cε ∫t

0

±

2

0

□ Finally, combining Lemma 3.1 with Lemma 3.2, we conclude

≤C

t

2

(ϕ ,ψ , φ , ϕ x ) ( t )

+ ∫t

{ (ϕ ,ψ ,φ ) (t )

2

0

2

Vt (ϕ , φ ) + (ϕ x ,ψ x , φx ) 2

t

+ ϕ x ( t ) ± + ∫t ψ xx

2 ±

0

2 ±

) (tˆ) dtˆ

( tˆ ) dtˆ + δ }

(3.8)

(ϕ x , φx ,ψ x ,ψ xx ) ± ( tˆ ) dtˆ + Cκ ∫t (ϕ x , φx ) ± ( tˆ ) dtˆ + Cε .

t

+ Cε ∫t

0

(

t

2

0

2

0

Comparing with the standard energy estimate for the compressible Navier-Stokes equations, we refer (3.8) to the basic energy estimate. Next, we proceed to estimate higher order derivatives of ψ , φ in the space

(

( )) .

L t0 , t ; L2  ± ∞

LEMMA 3.3.Suppose that the assumptions of Proposition 2.4 hold. Then, t

2

ψ x ( t ) ± + ∫t ψ xx

2

0

≤C

{ (ϕ ,ψ ,φ ) (t )

±

2

0

t

+ Cκ ∫t

0

( tˆ ) dtˆ + ψ x ( t0 )

±

2 ±

} + Cε ∫

t

(ϕ x , φx ,ψ x ,ψ xx ) ± ( tˆ ) dtˆ 2

t0

(3.9)

(ϕ x , φx ) ± ( tˆ ) dtˆ + Cε . 2

PROOF. Multiplying the second equation of (2.9) by −ψ xx , one obtains

 ψ x2  ψ xx2 − + ψ ψ ε ( )   t x x v  2 t =

( pv ( v,θ ) ϕ x + pθ ( v,θ ) φx )ψ xx + ε

ϕ xψ xψ xx 2

+ε

Vxψ xψ xx v2

v U xϕ xψ xx +ε + Vx { pv ( v, θ ) − pV (V , Θ )}ψ xx v2 U ψ VUψ + Θ x { pθ ( v, θ ) − pΘ (V , Θ )}ψ xx − ε xx xx + ε x x2 xx , v v

(3.10)

which, by integrating with respect to x and t, leads to t

2

ψ x ( t ) ± + ε ∫t ψ xx

2 ±

0

t

2

≤ ψ x ( t0 ) ± + ∫t t

+ C ∫t

0

0

∫± (ψ tψ x ) x ( x, tˆ ) dxdtˆ

( (ϕ ,ψ ,φ ) x

t

+ Cε ∫t

0

t

+ Cε ∫t

0

( tˆ ) dtˆ

x

x

2 ±

+

Vt (ϕ , φ )

2

) (tˆ) dtˆ

∫± ( ϕ xψ xψ xx

+ Vxψ xψ xx + U xϕ xψ xx

∫± ψ xx ( U xx

+ U xVx

(3.11)

) ( x, tˆ ) dxdtˆ

) ( x, tˆ ) dxdtˆ,

The terms on the right hand side of (3.11) can be bounded as follows (see [25] DOI: 10.4236/jamp.2018.610180

2154

Journal of Applied Mathematics and Physics

S. F. Cui

for detail),

∫t ∫± (ψ tψ x ) x ( x, tˆ ) dxdtˆ ≤ C ∫t ( ψ xx t

t

0

0

t

ε ∫t

0

t

ε ∫t

0

t

ε ∫t

0

2

+ ϕ +ψ

∫± (ϕ xψ xψ xx ) ( x, tˆ ) dxdtˆ ≤ Cε ∫t ( ψ xx t

∫± ψ xx ( Vxψ x

±

∫± ψ xx ( U xx

) ( x, tˆ ) dxdtˆ ≤ Cε ∫t ( ψ xx t

+ U xϕ x

+ U xVx

2

+ ψx

+ N ( t0 , t ) ψ x

2

0

2

0

) ( x, tˆ ) dxdtˆ ≤ Cε ∫t

t

ψ xx

0

2 ±

2 ±

) (tˆ ) dtˆ, ) (tˆ ) dtˆ,

+ δ (ϕ x ,ψ x )

) (tˆ ) dtˆ,

( tˆ ) dtˆ + Cεδ .

Substituting the above estimates into (3.11), we obtain (3.9). □ Similarly, we can bound the derivatives of φ as follows. LEMMA 3.4Assume that the assumptions of Proposition 2.4 hold. Then, t

2

φx ( t ) ± + ∫t φxx

2 ±

0

( tˆ ) dtˆ

( (ϕ ,ψ ,φ ) (t ) + φ (t ) + δ ) + Cε ∫ ( (φ ,ψ ,ψ ) + ψ ) ( tˆ ) dtˆ + Cκ ∫ ( (ϕ , φ ) + (φ , φ ) ) ( tˆ ) dtˆ + Cε + Cκ . 2

≤C

x

0

t

0

±

(3.12)

2

xx

t0

x

xx

x ±

±

t

2

x

t0

x

x

±

xx

±

PROOF. Multiplying the third equation of (2.9) by −φxx , then integrating with respect to x and t, utilizing (3.7) and (3.8), we deduce that t

2

φx ( t ) ± + ∫t φxx 0

2 ±

( tˆ ) dtˆ

t

2

≤ φx ( t0 ) ± + Cε ∫t ψ x 0

t

+ Cκ ∫t

∫± (φtφx ) x ( x, tˆ ) dxdtˆ

t

(tˆ)dtˆ + C ∫t

0

ϕ xφx ( x, tˆ ) dxdtˆ + Cε ∫t

t

∫± φxx

0

2 ±

0

∫± φxx

ψ x2 ( x, tˆ ) dxdtˆ

∫± U xφxx ( ϕ + φ ) ( x, tˆ ) dxdtˆ t + Cκ ∫t ∫± φxx ( Vxφx + Θ xϕ x ) ( x, tˆ ) dxdtˆ t

+ C ∫t

0

0

t

+ Cε ∫t

0

t

+ Cκ ∫t

0

∫± φxx

ψ xU x ( x, tˆ ) dxdtˆ + Cε ∫t

t

∫± φxx ( Θ xVx

+ Θ xx

0

∫± φxx

U x2 ( x, tˆ ) dxdtˆ

(3.13)

) ( x, tˆ ) dxdtˆ,

where the right hand side can be estimated as follows (see [25] for detail),

1

t

t

∫t ∫± (φtφx ) x dxdtˆ ≤ 16 ∫t 0

t

κ ∫t

0

0

φxx

κ

( tˆ ) dtˆ + C ∫

t

≤

0

ε

t

t0

∫± ϕ xφxφxx ( x, tˆ ) dxdtˆ ≤ 16 ∫t ε ∫t

t 0

φxx

2 ±

(ψ

2

2

+ φ + φx

( tˆ ) dtˆ + Cκ N ( t , t ) ∫ 0

t

t0

2 ±

) (tˆ ) dtˆ,

φx

2 ±

( tˆ ) dtˆ,

2 ∫± ψ x φxx ( x, tˆ ) dxdtˆ t

16 ∫t

φxx

0

2 ±

( tˆ ) dtˆ + Cε N ( t , t ) ∫ 0

∫t ∫± U xφxx ( ϕ t

0

≤

DOI: 10.4236/jamp.2018.610180

2 ±

1 t φxx 16 ∫t0

2155

2 ±

+φ

t

t0

(ψ

2 xx ±

+ ψx

2 ±

) (tˆ ) dtˆ,

) ( x, tˆ ) dxdtˆ

( tˆ ) dtˆ + Cδ ∫

t

t0

Vt (ϕ , φ ) ( tˆ ) dtˆ,

Journal of Applied Mathematics and Physics

S. F. Cui t

κ ∫t ≤ t

ε ∫t

∫± φxx

0

t

ε ∫t

0

0

∫± φxx ( Vxφx

κ

t

16 ∫t

0

φxx

2 ±

+ Θ xϕ x

( tˆ ) dtˆ + Cκδ ∫ (ϕ , φ ) ( tˆ ) dtˆ, t

x

t0

U xψ x ( x, tˆ ) dxdtˆ ≤

∫± φxx

) ( x, tˆ ) dxdtˆ

ε

t

16 ∫t

0

φxx

2 ±

x

( tˆ ) dtˆ + Cεδ ∫

ε t U x2 ( x, tˆ ) dxdtˆ ≤ ∫t φxx 16 0

t

t0

2 ±

ψ x ( tˆ ) dtˆ,

( tˆ ) dtˆ + Cεδ

14

,

and t

κ ∫t

0

∫± φxx ( Θ xVx

+ Θ xx

κ

) ( x, tˆ ) dxdtˆ ≤ 16 ∫t

t 0

φxx

2 ±

( tˆ ) dtˆ + Cκδ

14

.

Substitution of the above estimates into (3.13) gives Lemma 3.4 immediately. □ Now, combining Lemma 3.1 - 3.4, we obtain Proposition 2.4.

Conflicts of Interest The authors declare no conflicts of interest regarding the publication of this paper.

References

DOI: 10.4236/jamp.2018.610180

[1]

Di Perna, R.J. (1983) Convergence of Approximate Solutions to Conservation Laws. Archive for Rational Mechanics and Analysis, 83, 27-70. https://doi.org/10.1007/BF00251724

[2]

Hoff, D. and Liu, T.P. (1989) The Inviscid Limit for Navier-STOKES equations of Compressible Isentropic Flow with Shock Data. Indiana University Mathematics Journal, 38, 861-915. https://doi.org/10.1512/iumj.1989.38.38041

[3]

Goodman, J. and Xin, Z. (1992) Viscous Limits for Piecewise Smooth Solutions to Systems of Conservation Laws. Archive for Rational Mechanics and Analysis, 121, 235-265. https://doi.org/10.1007/BF00410614

[4]

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