Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data

Theorem 2.1

For any given ( v − , u − , θ − ) , suppose that ( v + , u + , θ + ) satisfies (2.3), μ , κ satisfy (1.3), let ( V , U , Θ ) be defined in (2.12), assume that ∣ θ + m − θ − m ∣ ≤ C ( γ − 1 ) α , ∣ θ ± − θ ± m ∣ ≤ C ( γ − 1 ) β for some constants α > 5 4 , β > 0 , and there exists ( γ − 1 ) -independent positive constants v ̲ , Θ ̲ , Θ ¯ , M 0 such that

Then, there exists a small positive constant δ 0 such that if

then the Cauchy problem (1.1)–(1.3) admits a unique global solution ( v , u , θ ) satisfying ( ϕ , ψ , ζ ) ∈ X ( [ 0 , + ∞ ) ) and

Remark 1

The difference ∣ θ + m − θ − m ∣ , ∣ θ ± − θ ± m ∣ are naturally bounded by C ( γ − 1 ) from the physical point of view (see [10,21]). If μ , κ are constants, we only need that α > 1 2 . And we can check from (2.5) and (2.16) that

Moreover, without loss of generality, we assume that θ ± ≤ θ ± m since θ ± ≥ θ ± m can be discussed in the same way. We deduce from the mean value theorem that

where R θ ± A < η < R θ ± m A . Thus, from (2.11), we have δ ± ≤ C ( γ − 1 ) β ≤ C δ 0 β ≪ 1 .

Remark 2

Theorem 2.1 means that if γ − 1 is smaller, the initial perturbation can be larger, it is a generalization of the results in [17, 18,38]. Particularly, all the results obtained in [18, 38] postulate that the initial perturbation ( ϕ 0 , ψ 0 , ζ 0 ) ∈ H 3 ( R ) , but we require only that ( ϕ 0 , ψ 0 , ζ 0 ) ∈ H 2 ( R ) .

Remark 3

We can easily observe that (1.4) satisfies (1.3), for each b ∈ R . Therefore, our result in Theorem 2.1 holds for (1.4) with any b ∈ R , as in [29], can cover the case when the transport coefficients are derived from the Boltzmann equation.

Remark 4

In 2021, Sun et al. [36] studied an initial and boundary value problem of (1.1) for μ = θ α , κ = θ β , α , β ≥ 0 . And the nonlinearly exponential stability of the solution is obtained for the large data provided α ≪ 1 . In fact, we can also obtain the similar result in Theorem 2.1 for α ≪ 1 , β ≥ 0 with large perturbation. Note that our derivation of the uniform bounds on v , θ relies heavily on the assumption that γ − 1 or α sufficiently small. Thus, it is an interesting and difficult problem to extend the result to the case with transport coefficients μ , κ satisfying μ = θ α , κ = θ β for α > 0 . One would conjecture that the result in Theorem 2.1 should be improved to 0 < α < c for some fixed constant c . However, this will be left for future.

Remark 5

Experimental results for gases at very high temperature show that both the viscosity μ and the heat-conductivity κ may depend on density and temperature (see [26,40]). Even for this case, similar result in Theorem 2.1 still holds if μ ( v , θ ) and κ ( v , θ ) satisfy

where μ ˜ ( v ) = min Θ ̲ ≤ θ ≤ Θ ¯ μ ( v , θ ) , ℬ ( v ) = ∫ 1 v s − ln s − 1 s μ ˜ ( s ) d s . Particularly, (2.18)–(2.20) involve the following cases:

Particularly, the transport coefficients μ , κ satisfying (2.18)–(2.20) can include μ = c 1 h ( v ) θ b , κ = c 2 h ( v ) θ b in [11] as a special example. In [38], they consider μ = c 1 h ( v ) θ b , κ = c 2 h ( v ) θ b , h ( v ) ≥ c ( v l 1 + v − l 2 ) , ∣ b ∣ ≪ 1 , but their result cannot cover the model satisfying (1.4) since the parameters l 1 and l 2 are assumed to be positive. Note that our result in Theorem 2.1 gives a positive answer of the problem in [38] for l 1 = l 2 = 0 and any b ∈ R if γ − 1 ≪ 1 .

Lemma 3.1

Assume that δ c = ∣ θ + m − θ − m ∣ ≤ δ ¯ for a small positive constant δ ¯ . Then the viscous contact wave ( V c , U c , Θ c ) has the following properties:

where ξ = x ( γ − 1 ) ( 1 + t ) and c 1 , c 2 are positive constants depending only on θ ± m , but independently of γ − 1 .

Lemma 3.2

For any given ( v − , u − , θ − ) , suppose that ( v + , u + , θ + ) satisfies (2.3). Then the viscous contact wave constructed in (2.6) and (2.8) and the smooth rarefaction waves constructed in (2.10) satisfying the following properties:

1. ( U ± r ) x ≥ 0 .

2. For any p ∈ [ 1 , + ∞ ] , there exists a positive constant C = C ( p , v − , u − , θ − ) such that for small δ ,

   ‖ ( ( V ± r ) x , ( U ± r ) x , ( Θ ± r ) x ) ‖ L p ≤ C min δ , δ 1 p t − 1 + 1 p , ‖ ( ( V ± r ) x x , ( U ± r ) x x , ( Θ ± r ) x x ) ‖ L p ≤ C min { δ , t − 1 } , ‖ ( ( V ± r ) x x x , ( U ± r ) x x x , ( Θ ± r ) x x x ) ‖ L p ≤ C min { δ , δ − 1 + 1 p t − 2 + 1 p } .

3. There exists a positive constant C = C ( v − , u − , θ − ) such that for

   c 0 = 1 10 min { ∣ λ − ( v − m , s − ) ∣ , λ + ( v + m , s + ) , c 2 λ − 2 ( v − m , s − ) , c 2 λ + 2 ( v + m , s + ) } ,

   ∀ k ≥ 0 , we have

   ∣ ∂ x k ( V ± r − v ± m ) ∣ + ∣ ∂ x k ( U ± r ) ∣ + ∣ ∂ x k ( Θ ± r − θ ± m ) ∣ ≤ C δ e − c 0 ( ∣ x ∣ + t ) , in Ω c , ∣ ∂ x k ( V c − v ± m ) ∣ + ∣ ∂ x k + 1 U c ∣ + ∣ ∂ x k ( Θ c − θ ± m ) ∣ ≤ C δ c ( γ − 1 ) − k 2 e − c 0 ( γ − 1 ) − 1 ( ∣ x ∣ + t ) ∣ ( V ∓ r ) x ∣ + ( U ∓ r ) x + ∣ ( Θ ∓ r ) x ∣ + ∣ V ∓ r − v ∓ m ∣ + ∣ Θ ∓ r − θ ∓ m ∣ ≤ C δ e − c 0 ( ∣ x ∣ + t ) , in Ω ± .

4. For the rarefaction wave ( v ± r , u ± r , θ ± r ) x t determined by (2.1) and (2.9), it holds that

   lim t → + ∞ sup x ∈ R ( V ± r , U ± r , Θ ± r ) ( x , t ) − ( v ± r , u ± r , θ ± r ) x t = 0 .