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):
with (discontinuous) initial data
where and e denote the specific volume, the velocity, the temperature, the pressure and the internal energy respectively, and are the viscosity and heat conductivity coefficients, respectively. At infinity, the initial data are assumed to satisfy
where and 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
and the corresponding inviscid p-system
the vanishing viscosity limit for the Cauchy problem has been studied by several researchers. In  Di Perna uses the method of compensated compactness and established almost everywhere convergence of admissible solutions of (1.4) to an admissible solution of (1.5), provided that is uniformly bounded and is uniform bounded away from zero. However, this uniform boundedness is difficult to verify in general, and the abstract analysis in  gets little information on the qualitative nature of the viscous solutions. In  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   , Xin in  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 uniformly for all time away from . In the vanishing viscosity limit, the Prandtl boundary layers (characteristic boundaries) are studied for the multidimensional linearized compressible Navier-Stokes equations by using asymptotic analysis in    , while the boundary layer stability in the case of non-characteristic boundaries and one spatial dimension is discussed in   . We mention that there is an extensive literature on the vanishing artificial viscosity limit for hyperbolic systems of conservation laws, see, for example,    -  , also cf. the monographs    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   , respectively. And, in  , 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, is generally dependent of , while in this paper, we will show the dissipation limit in the case that and are independent of each other.
Our aim in this paper is to study the relation between the solution of the Navier-Stokes equations for a compressible heat-conducting fluid (1.1) and the solution of the corresponding inviscid Euler equations:
with the initial data
with the same constant states as in (1.3).
It is convenient to work with the equations for the entropy s and the absolute temperature . The second law of thermodynamics asserts that
We assume, as is customary in thermodynamics, that given any two of thermodynamics variables and p, we can obtain the remaining three variables. If we choose as independent variables and write , we deduce that
Then, a straightforward calculation gives
We may also choose as independent variables and write
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
with initial data
where and are the constant states. The corresponding inviscid Euler equations read:
We assume in this paper that the pressure p is a smooth function of its arguments satisfying
Notice that the condition (1.14) assures the system (1.13) has characteristic speeds
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 . The case for the 3-rarefaction waves can be dealt with similarly.
Suppose the end states can be connected by 1-rarefaction waves. The centered 1-rarefaction wave connecting to is the self-similar solution of (1.13) defined by (see, e.g.,   )
which is uniquely determined by the system (1.13) and the rarefaction wave initial data
For the internal energy , we assume
For the sake of convenience, throughout this paper we denote
In this paper, we prove that the solution of system (1.11) with the centered rarefaction wave initial data (1.16) of small strength exists for all time and converges to the centered rarefaction wave of the Euler equation (1.13) as 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 be connected by a centered 1-rarefaction wave defined by (1.15). Assume that (1.14) and (1.17) hold. Then, for small enough, the compressible Navier-Stokes equations (1.11) with the rarefaction wave initial data (1.16) have a global piecewise smooth solution , such that
1) are continuous for , and are uniformly Hӧlder continuous in the set and for any ; are Hӧlder continuous on compact set . Moreover, the jumps in at satisfy
and so does the other jumps, where are positive constants independent of t and , and denotes jumps in what follows.
2) The solution converges to the centered rarefaction wave as uniformly away from , i.e., for any positive h, we have
3) For any fixed and , the solution approaches the centered rarefaction wave uniformly as time goes to infinity, i.e.,
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  , but we do not use the natural scaling argument, and we do not assume that .
We point out here that in view of Theorem 1.1, an initial jump discontinuity at can be allowed in (1.2). The evolution of this jump discontinuity is an important aspect in our analysis. It has been shown in  that the discontinuity evolution follows a curve in x-t plane, and the jump discontinuity in and decays exponentially in time, while the discontinuity in u and are smoothed out at positive time, see  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. 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:
where is given by
If , then the problem (2.1) has the centered rarefaction wave solution given by
To construct a smooth rarefaction wave solution of the Burgers equation which approximates the centered rarefaction wave, we set for ,
and for each , we solve the following initial value problem
Next, we state certain properties that will be used later.
LEMMA 2.1. (S. JIANG  ) For each , the problem (2.2) has a unique global smooth solution , such that
1) , for .
2) For any , there is a constant depending only on p, such that
Now, set , and we define , the smooth approximation , by
Then, it is not difficult to see that satisfy
and due to Lemma 2.1, the following lemma holds for .
LEMMA 2.2. (S. JIANG  ) The functions and constructed above satisfy:
1) for all .
2) For any , there is a constant depending only on p, such that
Consequently, from Lemmas 2.1 and 2.2, it follows that converges to as .
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 constructed above (e.g. one may take ), one can decompose the solution of (1.1), (1.9) and (1.10) into
Substituting the above decomposition into (1.1), (1.9) and (1.10), we obtain the system for the functions :
with initial data
where and its derivatives are sufficiently smooth away from but up to and , .
We shall show that the Cauchy problem (2.9), (2.10) possesses a unique global solution in the same function class as for in Theorem 1.1. Moreover, goes to zero uniformly as . This convergence then yields Theorem 1.1 due to Lemmas 2.1 and 2.2.
LEMMA 2.3. (Hoff  ) Suppose that is suitably small so that there exist two positive constants and with for all . Then, there is a constant , such that the Cauchy problem (2.9), (2.10) has a solution on in the same function class as for in Theorem 1.1. Moreover, satisfy
1) There exists a positive constant C, such that
2) There is a positive constant C, such that
3) There are constant independent of T, such that
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 for some , in the same function class as in Lemma 2.3. Denote
Then, there are positive constants and C independent of , such that for each fixed , if
then the following estimates hold
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
which together with Lemma 2.3 yields . 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:
where we have used the fact that .
An easy computation implies that satisfies the equation:
Employing (3.1), one has
LEMMA3.1. Suppose that the assumptions of Proposition 2.4 hold. Then,
PROOF. Integrating (3.1) with respect to t and x, we get
Recalling the definition of and applying Lemma 2.2, for given can be estimated as follows.
where we have used Sobolev’s inequality:
and the following inequality:
Substituting the above estimates for into (3.3), we obtain (3.2). This completes the proof.
LEMMA3.2. Suppose that the assumptions of Proposition 2.4 hold. Then
PROOF. Multiplying the second equation of (2.9) by , one obtains
Integrating (3.5) with respect to over , we infer
where can be bounded as follows, using Sobolev’s imbedding theorem and Lemma2.3 (iii) (see  for detail).
Inserting the estimates for into (3.6), we arrive at
Finally, combining Lemma 3.1 with Lemma 3.2, we conclude
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 in the space .
LEMMA 3.3.Suppose that the assumptions of Proposition 2.4 hold. Then,
PROOF. Multiplying the second equation of (2.9) by , one obtains
which, by integrating with respect to x and t, leads to
The terms on the right hand side of (3.11) can be bounded as follows (see  for detail),
Substituting the above estimates into (3.11), we obtain (3.9).
Similarly, we can bound the derivatives of as follows.
LEMMA 3.4Assume that the assumptions of Proposition 2.4 hold. Then,
PROOF. Multiplying the third equation of (2.9) by , then integrating with respect to x and t, utilizing (3.7) and (3.8), we deduce that
where the right hand side can be estimated as follows (see  for detail),
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.
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