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Asymptotic behavior of rarefaction waves for a model system of a radiating gas
© Xiao et al; licensee Springer. 2012
Received: 13 September 2011
Accepted: 10 April 2012
Published: 10 April 2012
This article is concerned with nonlinear stability together with the corresponding convergence rates of rarefaction waves for a hyperbolic-elliptic coupled system arising in the 1D motion of a radiating gas with large initial perturbation. Compared with former results, although we ask the L2(R)-norm of the initial perturbation to be small but the L2(R)-norm of the first- and second-order derivatives of the initial perturbation with respect to the spatial variable x can be large and consequently, the H2(R)-norm of the initial perturbation can be large.
AMS Subject Classifications: 34K25; 35M10.
for some positive constant a and all u ∈ R under consideration and u- < u+ are two given constants.
When , the system (1.1) is a simple model for the 1D motion of a radiating gas, cf. . In fact, u and q in (1.1) represent the velocity and the heat flux of the gas, respectively.
The main purpose of our present article is to study the nonlinear stability of such a w R (t, x). To do so, as in , one needs first to introduce a smooth approximation w(t, x) of the rarefaction wave solution w R (t, x). Since such a w(t, x) tends to w R (t, x) as t → ∞ (cf. Lemma 2.1 below), thus the study of the above problem is then transferred to show that if the initial data u0(x) is a perturbation of w(0, x), the Cauchy problem (1.1), (1.2) admits a unique global solution (u(t, x), q(t, x)) which tends to (w(t, x), -∂ x w(t, x)) as t → ∞. Recall that depending on whether δ = u+ - u-, the strength of the rarefaction waves, and/or certain Sobolev norm, cf. , of the initial perturbation u0(x) - w(0, x) are assumed to be small or not, the corresponding stability results are called local (or global) nonlinear stability of strong (or weak) rarefaction waves, respectively.
hold if one assumes further that .
Since the results obtained in  only implies that weak rarefaction waves of the Cauchy problem (1.1), (1.2) is locally nonlinear stable, thus a question of some interesting is: Can the assumption that is sufficiently small be weakened or even be removed such that rarefaction waves of the Cauchy problem (1.1), (1.2) are global nonlinear stable? Moreover, if the rarefaction waves are nonlinear stable for such a class of large initial perturbation, does the same convergence rate as in  hold for such a class of perturbation?
This article is concentrated on these problems and our main result can be stated as follows.
Here O(1) denotes some positive constant which depends on .
Remark 1.1 Since δ in Theorem 1.1 is assumed to be sufficiently small, the assumption (1.8) does imply that the L2(R)-norm of the initial perturbation is small, but the L2(R)-norm of the first and second derivatives of the initial perturbation with respect to x can be indeed large and consequently, compared with that of , the H2(R)-norm of the initial perturbation can indeed be large. We note, however, that although we obtain some result on the nonlinear stability of rarefaction waves of the Cauchy problem (1.1), (1.2) for a class of initial perturbation which is large in H2(R), the problem on the nonlinear stability of strong rarefaction waves of the Cauchy problem (1.1), (1.2) for any initial perturbation in H2(R) is still not known and will be pursued by the authors in the future.
Remark 1.2 Notice that the decay estimates (1.12) and (1.13) are weaker than that of (1.7), but it is worth to pointing out that when , we can also show that (1.7) holds only under the conditions listed in Theorem 1.1. It would be of some interest to show whether (1.7) holds for general smooth nonlinear flux function f(u).
Now we outline the main ingredients used in deducing our main results. To prove Theorem 1.1, as in , we only need to deduce certain estimates on (ϕ(t, x), ψ(t, x)) = (u(t, x) - w(t, x), q(t, x) + ∂ x w(t, x)) since the estimates on are well-established in .
for some η > 0 and our main trick is to use the smallness of η and δ to control the possible growth of the solution caused by the nonlinearity of the equation. Thus to use the continuation argument to extend the local solution step by step to a global one, a key step is how to design a class of initial perturbation, which is large in the H2-norm, so that the scheme can be continued. It is worth to pointing out in this step that we ask the initial perturbation to satisfy (1.8).
then we can indeed show that (1.12) and (1.13) holds provided that u0(x) - w(0, x) ∈ H1(R) ∩ L1(R). Such a result implies that if one can show that the Cauchy problem (2.12)-(2.14) admits a unique global solution (ϕ(t, x), ψ(t, x)) which satisfies (1.16) and (1.17), then even for any initial perturbation and any δ > 0, one can deduce (1.12) and (1.13) provided that u0(x) - w(0, x) ∈ H1(R) ∩ L1(R).
Before concluding this section, we point out that the study on the nonlinear stability of rarefaction waves to hyperbolic conservation laws with dissipative terms was initiated by Il'in and Oleinik  and since then many excellent results have been obtained, the interested author is referred to [1–17] and the references cited therein.
This article is arranged as follows: reformulation of the problem and some estimates on w R (t, x) together with its smooth approximation will be stated in Section 2, nonlinear stability of the rarefaction waves for a class of large initial data will be discussed in Sections 3 and 4 is devoted to deducing some decay estimate.
Notations: Throughout the rest of this article, C or O(1) will be used to denote a generic positive constants which may vary from line to line. Let's emphasize that unless stated clearly, all the constants are independent of t, x, and δ for sufficiently small δ > 0.
H l (R) (l ≥ 0) and L p (R) stand for the usual Sobolev space with the norm and the usual Lebesgue space with the norm , respectively. While , ||·|| l will be used to denote the usual L p (R)-norm, H l (R)-norm and for j ∈ N. For simplicity, and will usually be denoted by and ||f(t)|| l , respectively. Especially, .
2 Reformulation of the problem
Now we collect some properties of w R (t, x) and w(t, x) given in .
For the reformulated problem (2.12)-(2.14), we can deduce the following result.
Here D i (i = 1, 2, 3) are positive constants independent of the positive constant δ and the constants β, γ satisfying (1.9).
For the corresponding decay estimates, we have the following result.
Here , and D1(E, T), Cl+1(ϕ0, δ), Dl+1(E, T), (l = 1,..., S - 1) are positive constants which depend only on E and T.
Here , , and (l = 0, 1, ...) are positive constants independent of t, x, and δ for sufficiently small δ > 0.
Here D i (i = 4, 5, 6) are some positive constants which are independent of x, t, and δ for δ > 0 sufficiently small.
The above analysis implies that if the assumptions listed in Theorem 1.1 are satisfied, then the conditions listed in Theorems 2.1 and 2.2 are also satisfied. Consequently, we have from Theorems 2.1 and 2.2 that there exists a positive constant δ0 > 0 such that if 0 < δ ≤ δ0, the Cauchy problem (1.1), (1.2) admits a unique global solution (u(t, x), q(t, x)) = (w(t, x) + ϕ(t, x), ψ(t, x) - ∂ x w(t, x)) and (ϕ(t, x), ψ(t, x)) satisfy (2.20)-(2.22).
This proves (1.12). (1.13) can be proved similarly. This completes the proof of Theorem 1.1.
3 The proof of Theorem 2.1
This section is devoted to proving Theorem 2.1. For this purpose, we first show the local solvability of the Cauchy problem (2.12)-(2.14)
Lemma 3.1 (Local existence) Let ϕ0(x) ∈ H2(R), then there exists t1 > 0 which depends only on ||ϕ0(x)||2 such that the Cauchy problem (2.12)-(2.14) admits a unique solution satisfying
Lemma 3.1 can be proved by employing the standard method and we omit the details for brevity.
where η is a positive constant.
Based on (3.3) and (3.4), we have
hold for any t ∈ [0, T]. Here are positive constants independent of x, T, and δ for sufficiently small δ > 0.
Substituting (3.11)-(3.13) into (3.10) and (3.6) follows immediately from Lemma 2.1.
Here and in the rest of this article, (···) x denotes some terms which vanish after integration with respect to x over x ∈ R.
and (3.7) follows immediately from (3.29) and the above inequality.
we can deduce that (3.8) follows immediately. Thus Lemma 3.2 is proved.
for 0 ≤ t ≤ t1.
hold for any t ∈ [0, t1]. Here D > 0 is the constant appearing in the Gagliardo-Nirenberg inequality and
holds for any t ∈ [0, t1].
hold for t ∈ [0, t1].
for t ∈ [t1, t1 + t2].
hold for any t ∈ [0, t1 + t2]. Here
holds for any t ∈ [0, t1 + t2].
which means that the conditions listed in Lemma 3.2 hold for T = t1 + t2 and and thus we have from Lemma 3.2 that (3.50) holds for any t ∈ [0, t1 + t2].
Now take ϕ(t1 + t2, x) as initial data, we can employ Lemma 3.1 once more to deduce that (ϕ(t, x), ψ(t, x)) can be extended to the time step t = t1 + 2t2 and (3.51) holds for t ∈ [t1 + t2, t1 + 2t2]. Since the constant C in (3.50) is independent of t, we can thus repeat the above process to extend (ϕ(t, x), ψ(t, x)) to a global one.
Having obtained (3.54), the proof of (2.19) is standard, we thus omit the details for brevity.
4 The proof of Theorem 2.2
The main purpose of this section is devoted to deducing the corresponding convergence rates of the global solution (ϕ(t, x), ψ(t, x)) of the Cauchy problem (2.12)-(2.14) to the rarefaction waves under the assumptions that the Cauchy problem (2.12)-(2.14) admits a unique global solution (ϕ(t, x), ψ(t, x)) which satisfies (1.16), (1.17).
Before stating our main result, we first deduce certain estimates on (ϕ(t, x), ψ(t, x)) based on the assumption (1.16)
Lemma 4.1 can be proved by employing the standard energy estimates and thus we omit the details for brevity.
Remark 4.1 If the constant E in (1.16) is assumed to be sufficiently small, then the constant C(E, T) can be chosen independent of T > 0. This fact follows easily from the proof of Lemma 3.2.
The following lemma is concerned with the L1 estimate on ϕ(t, x).
where C0 is independent of t, x, and δ.
Here D l , D l (l = 1, ..., S) are positive constants defined in (2.21) and (2.22), respectively.
and consequently we can show that the decay estimates (2.20)-(2.22) can be improved such that the term ln(2 + t) in the right-hand side can be replaced by 1.
On the other hand, the assumptions in Theorem 4.1 tell us that ϕ(t, x) satisfies (1.17).
where C∞ is the constant in (2.8) for p = ∞.