Vanishing di ff usion limit and boundary layers for a nonlinear hyperbolic system with damping and di ff usion

: We consider an initial and boundary value problem for a nonlinear hyperbolic system with damping and di ff usion. This system was derived from the Rayleigh–Benard equation. Based on a new observation of the structure of the system, two identities are found; then, the following results are proved by using the energy method. First, the well-posedness of the global large solution is established; then, the limit with a boundary layer as some di ff usion coe ffi cient tending to zero is justified. In addition, the L 2 convergence rate in terms of the di ff usion coe ffi cient is obtained together with the estimation of the thickness of the boundary layer.


Introduction
We consider the following nonlinear parabolic system with damping and diffusion: with the following initial and boundary conditions: Here σ, α, β and ν are constants with α > 0, β > 0, T > 0, and Q T = (0, 1) × (0, T ).The function ξ(t) is measurable in (0, T ).System (1.1) was originally proposed by Hsieh in [1] as a substitution for the Rayleigh-Benard equation for the purpose of studying chaos, and we refer the reader to [1][2][3] for the physical background.It is worthy to point out that with a truncation similar to the mode truncation from the Rayleigh-Benard equations used by Lorenz in [4], system (1.1) also leads to the Lorenz system.In spite of it being a much simpler system, it is still as rich as the Lorenz system, and some different routes to chaos, including the break of the time-periodic solution and the break of the switching solution, have been discovered via numerical simulations [5].
The mathematical theory of system (1.1) has been extensively investigated in a great number of papers; see [6][7][8][9][10][11][12][13][14][15][16] and the references therein.However, there is currently no global result with large initial data because it is very difficult to treat the nonlinear term of system (1.1) in analysis.Motivated by this fact, we will first study the well-posedness of global large solutions of the problem given by (1.1) and (1.2) for the case in which σν < 0, and then we will discuss the limit problem as α → 0 + , as well as the problem on the estimation of the boundary layer thickness.For the case in which σν > 0, we leave the investigation in the future.
The problem of a vanishing viscosity limit is an interesting and challenging problem in many settings, such as in the boundary layer theory (cf.[17]).Indeed, the presence of a boundary layer that accompanies the vanishing viscosity has been fundamental in fluid dynamics since the seminal work by Prandtl in 1904.In this direction, there have been extensive studies with a large number of references that we will not mention here.As important work on the mathematical basis for the laminar boundary layer theory, which is related to the problem considered in this paper, Frid and Shelukhin [18] studied the boundary layer effect of the compressible or incompressible Navier-Stokes equations with cylindrical symmetry and constant coefficients as the shear viscosity µ goes to zero; they proved the existence of a boundary layer of thickness O(µ q ) with any q ∈ (0, 1/2).It should be pointed out that, for the incompressible case, the equations are reduced to where µ is the shear viscosity coefficient and v and w represent the angular velocity and axial velocity, respectively.Recently, this result was investigated in more general settings; see for instance, [19][20][21] and the references therein.In the present paper, we will prove a similar result to that obtained in [18].Note that every equation in (1.3) is linear.However, equation (1.1) 2 , with a nonconservative term 2ψθ x , is nonlinear.It is the term that leads to new difficulties in analysis, causing all previous results on system (1.1) to be limited to small-sized initial data.
Formally, setting α = 0, we obtain the following system: with the following initial and boundary conditions: Before stating our main result, we first list some notations.Notations: For 1 ≤ p, s ≤ ∞, k ∈ N, and Ω = (0, 1), we denote by L p = L p (Ω) the usual Lebesgue space on Ω with the norm ∥ • ∥ L p , and H k = W k,2 (Ω) and H 1 0 = W 1,2 0 (Ω) as the usual Sobolev spaces on Ω with the norms ∥ • ∥ H k and ∥ • ∥ H 1 , respectively.C k (Ω) is the space consisting of all continuous derivatives up to order k on Ω, C(Q T ) is the set of all continuous functions on Q T , where Q T = (0, 1) × (0, T ) with T > 0, and L p (0, T ; B) is the space of all measurable functions from (0, T ) to B with the norm ∥ • ∥ L p (0,T ;B) , where B = L s or H k .We also use the notations The first result of this paper can be stated as follows.
The second result of this paper is stated as follows.
Theorem 1.3.Under the conditions of Theorem 1.1, any function δ(α) satisfying the condition that δ(α) ↓ 0 and Here, the function θ satisfies the spatially periodic boundary condition that has been considered in some papers, e.g., [3,29,30].One can see from the analysis that Theorems 1.1 and 1.3 are still valid when the boundary condition of θ is the homogeneous Neumann boundary condition or homogeneous Dirichlet boundary condition.
The proofs of the above theorems are based on the uniform estimates given by (1.6).First, based on a key observation of the structure of system (1.1), we find two identities (see Lemma 2.1).In this step, an idea is to transform (1.1) 2 into an equation with a conservative term (see (2.4)).And then, from the two identities, we deduce some basic energy-type estimates (see Lemma 2.2).The condition σν < 0 plays an important role here.With the uniform estimates in hand, we derive the required uniform bound of (α 1/4 ∥ψ x ∥ L ∞ (0,t;L 2 ) + α 3/4 ∥ψ xx ∥ L 2 (Q T ) ) for the study of the boundary layer effect via standard analysis (see [18,19]).See Lemma 2.3 with proof.The uniform bound of ∥(ψ, θ)∥ L ∞ (Q T ) is derived by a more delicate analysis (see Lemma 2.4).Finally, the boundary estimate ∥ω 1/2 ψ x ∥ L ∞ (0,T ;L 2 ) ≤ C is established (see Lemma 2.6), through which we complete the proof for the estimation of the boundary layer thickness.
Before ending the section, let us introduce some of the previous works on system (1.1).It should be noted that most of those works focus on the case when σν > 0. In this direction, Tang and Zhao [6] considered the Cauchy problem for the following system: with the following initial condition: (ψ, θ)(x, 0) = (ψ 0 , θ 0 )(x) → (0, 0) as x → ±∞, where 0 < α < 1 and 0 < ν < 4α(1 − α).They established the global existence, nonlinear stability and optimal decay rate of solutions with small-sized initial data.Their result was extended in [7,8] to the case of the initial data with different end states, i.e., where ψ ± , θ ± are constant states with (ψ + − ψ − , θ + − θ − ) (0, 0).For the initial-boundary value problem on quadrants, Duan et al. [9] obtained the global existence and the L p decay rates of solutions for the problem given by (1.13) and (1.14) with small-sized initial data.For the Dirichlet boundary value problem, Ruan and Zhu [10] proved the global existence of system (1.1) with small-sized initial data and justified the limit as β → 0 + under the following condition: ν = µβ for some constant µ > 0. In addition, they established the existence of a boundary layer of thickness O(β δ ) with any 0 < δ < 1/2.Following [10], some similar results on the Dirichlet-Neumann boundary value problem were obtained in [11].For the case when σν < 0, however, there are few results on system (1.1).Chen and Zhu [12] studied the problem given by (1.13) and (1.14) with 0 < α < 1; they proved global existence with small-sized initial data and justified the limit as α → 0 + .In their argument, the condition σν < 0 plays a key role.Ruan and Yin [13] discussed two cases of system (1.1): α = β and α β, and they obtained some further results that include the C ∞ convergence rate as β → 0 + .We also mention that one can obtain a slightly modified system by replacing the nonlinear term 2ψθ x in (1.1) with (ψθ) x .Jian and Chen [31] first obtained the global existence results for the Cauchy problem.Hsiao and Jian [29] proved the unique solvability of global smooth solutions for the spatially periodic Cauchy problem by applying the Leary-Schauder fixed-point theorem.Wang [32] discussed long time asymptotic behavior of solutions for the Cauchy problem.Some other results on this system is available in [33][34][35] and the references therein.
The rest of the paper is organized as follows.In Section 2, we will derive the uniform a priori estimates given by (1.6).The proofs of Theorem 1.1 and Theorem 1.3 will be given in Section 3 and Section 4, respectively.

A priori estimates
In the section, we will derive the uniform a priori estimates by (1.6), and we suppose that the solution (ψ, θ) is smooth enough on Q T .From now on, we denote by C a positive generic constant that is independent of α ∈ (0, α 0 ] for α 0 > 0. First of all, we observe that ( ψ, θ) := (2ψ, 2θ/ν) solves the following system: with the following initial and boundary conditions: From the system, we obtain the following crucial identities for our analysis.
Proof.Let W = ψ x .Differentiating (1.1) 1 with respect to x, we obtain Multiplying it by h η (W) and integrating over Q t , we get (2.20) By applying |h η (s)| ≤ 1, we have By using h ′ η (s) ≥ 0 for s ∈ R, we have
By applying Lemmas 2.3 and 2.4, we obtain Similarly, we have Therefore, From the embedding theorem, W 1,1 → L ∞ , and the Hölder inequality, we have ; therefore, by the Hölder inequality and Young's inequality, we obtain the following for any ε ∈ (0, 1): , and taking ε sufficiently small, we obtain (2.30) by using the Gronwall inequality.The proof of the lemma is completed.□ As a consequence of Lemma 2.5 and (2.32), we have that α 1/2 Let the assumptions of Theorem 1.1 hold.Then, for any t ∈ [0, T ], where ω is the same as that in (1.7).
Proof.Multiplying (1.1) 1 by ψ xx ω(x) and integrating over Q t , we have where

Existence and uniqueness
For any fixed α > 0, the global existence of strong solutions for the problem given by (1.1) and (1.2) can be shown by a routine argument (see [29,36]).First, by a smooth approximation of the initial data satisfying the conditions of Theorem 1.1, we obtain a sequence of global smooth approximate solutions by combining the a priori estimates given by (1.6) with the Leray-Schauder fixed-point theorem (see [36,Theorem 3.1]).See [29] for details.And then, a global strong solution satisfying (1.6) is constructed by means of a standard compactness argument.

Vanishing diffusion limit
The aim of this part is to prove (1.10) and (1.11).Given the uniform estimates given by (1.6) and Aubin-Lions lemma (see [37]), there exists a sequence {α n } ∞ n=1 , tending to zero, and a pair of functions (ψ, θ) satisfying (1.8) such that, as α n → 0 + , the unique global solution of the problem given by (1.1) and (1.2) with α = α n , still denoted by (ψ, θ), converges to (ψ, θ) in the following sense: where M(Q T ) is the set of Radon measures on Q T .From (3.4), one can directly check that (ψ, θ) is a global solution for the problem given by (1.4) and (1.5) in the sense of (1.8) and (1.9).
Then, the Gronwall inequality gives .
In summary, we have completed the proof of Theorem 1.1.

Use of AI tools declaration
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.