Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers

In this paper, we study the limiting behavior of solutions to a 1D two-point boundary value problem for viscous conservation laws with genuinely-nonlinear fluxes as \begin{document}$\varepsilon$\end{document} goes to zero. We here discuss different types of non-characteristic boundary layers occurring on both sides. We first construct formally the three-term approximate solutions by using the method of matched asymptotic expansions. Next, by energy method we prove that the boundary layers are nonlinearly stable and thus it is proved the boundary layer effects are just localized near both boundaries. Consequently, the viscous solutions converge to the smooth inviscid solution uniformly away from the boundaries. The rate of convergence in viscosity is optimal.

1. Introduction and main results. In physics and fluid mechanics, it is very important to discuss the asymptotic equivalence between the viscous flows and the associated inviscid flows in the limit of small viscosity [4,7,8,9,12]. The solutions near the boundaries always exhibit very singular behavior as the viscosity is small, which is called boundary layer effects. Boundary layers appear in various physical contexts, such as the theory of rotating fluids, incompressible MHD, the inviscid limit of parabolic systems, or the inviscid limit of Navier-Stokes equations near a boundary. The rigorous mathematical justification of the asymptotic equivalence with boundary layers poses challenging problems in many important cases, see [3,5,6,7] and [9]. In the case of noncharacteristic boundaries, to conquer the difficulty that systems do not admit maximum principles, in [12], the vanishing viscosity limit of one-dimensional quasilinear viscous equations in a half plane is studied by the approach based on the asymptotic analysis and a weighted energy method. Similar as noncharacteristic boundary layers, in [3], the piecewise smooth inviscid solution with a single, entropic, sufficiently weak p-shock is proved to be the limit of a sequence of solutions to the corresponding viscous equation away from the shock. Furthermore, the result holds as well for solutions with finitely many noninteracting entropic shocks. Here, we consider the two-point boundary value problem for the following one dimensional quasilinear equation x u ε , x ∈ (0, 1), t > 0, (1.1) u ε (x = 0, t) = u l (t), t > 0, (1.2) u ε (x = 1, t) = u r (t), t > 0, (1.3) u ε (x, t = 0) = u 0 (x), x ∈ (0, 1), (1.4) where f and u 0 (x) are smooth, and we require that in the region which we are interested in. The aim of this paper is to seek the asymptotic relations between the viscous solution u ε (x, t), to (1.1)- (1.4) and the solution to the associated inviscid hyperbolic problem Case II ∂ u f (u) x=0 < 0 and ∂ u f (u) x=1 < 0. In this case, only values on the left boundary is determined by the solution inside, so a suitable condition should be imposed on {x = 1} for (1.6)-(1.7) ; Case III ∂ u f (u) x=0 > 0 and ∂ u f (u) x=1 > 0. Similar to case ii), a boundary condition is needed on {x = 0} for (1.6)-(1.7) ; Case IV ∂ u f (u) x=0 > 0 and ∂ u f (u) x=1 < 0.
To guarantee the well-posedness of the inviscid problem, boundary conditions should bd imposed on both boundaries for (1.6)-(1.7) .
It is expected that the viscous solution u ε convergence to the inviscid solution u uniformly away from the boundaries. The leading order asymptotic behavior of the viscous solution is governed by where y = x ε and ξ = x−1 ε are the fast variables. We define the strength of the boundary layers in the viscous flow by (1.8) Here, we consider the case that no interactions between the possible boundary layers. We will consider the smooth inviscid flow before shock formation. We first construct a three-term approximate solution to (1.1)-(1.4) by using the method of matched asymptotic analysis and multi-scale expansions. Then using the energy method, we prove that the boundary layers near both boundaries with various types are nonlinearly stable. The analysis depends crucially on the structure of the underlying boundary layers. The stability analysis of the boundary layers is devoted to Case I in Section 2, where the stability analysis of both an expansive boundary layer and a compressive boundary layer is not included due to the opposite sign of the ∂ y ∂ u f and ∂ ξ ∂ u f near the boundaries. Case II-Case IV are discussed in Section 3 and the corresponding inviscid problems are also stated there. The method in this paper can be similarly carried out for systems with suitable modifications, see both weak boundary layers in the two-dimensional case in [10]. Throughout this paper, we use C or O(1) to denote any positive bounded function which is independent of ε.
2. Stability analysis of both boundary layers. According to Case I, we suppose for some suitable ν > 0, the initial data u 0 (x) satisfies Then by continuity, ∃x 1 > 0 and T > 0 such that In this case, the inviscid problem (1.6)-(1.7) is well-defined without any boundary conditions. To be consistency, we assume that In the limit of ε → 0, the discrepancies of boundary conditions lead to the phenomena of boundary layers near both boundaries.

Asymptotic expansions.
To prove the asymptotic equivalence between (1.1)-(1.4) and (1.6)-(1.7), we first need to construct the approximate solution through different scalings and asymptotic expansions in the regions near and away from the boundaries.
In the region away from the boundaries {x = 0} and {x = 1}, the solution of (1.1) may be approximated by the formal series (2.1.1) Substituting this into (1.1) and equating the coefficients of different powers of ε yield with the following initial data for 0 < x < 1 that The problem (2.1.2) and (2.1.5) for the leading order inner function is exactly the inviscid problem (1.6)-(1.7). Therefore, u 0 (x, t) is the unique smooth inviscid solution to (1.6)-(1.7) with u 0 (x, t) ∈ C 2 ([0, T ]; H 6 (0, 1)).
The equation (2.1.3) is a linear hyperbolic equation, then by the method of characteristics, there exists a unique smooth solution u 1 (x, t) such that Similarly, there exists a unique smooth solution u 2 (x, t) such that Since the boundary, {x = 0}, is non-characteristic for the inviscid hyperbolic problem (1.6)-(1.7), we will approximate the viscous solution near the boundary {x = 0} by the following expansion In the matching zone near {x = 0}, both the inner and boundary layer expansions are expected to be valid, then as y → +∞, we obtain The leading boundary function u 0 b (y, t) satisfies the second order nonlinear ODE (2.1.9) and the boundary condition (2.1.12). We impose the boundary condition on y = 0 as In the matching zone near {x = 1}, as ξ → −∞, there hold Proof. First, we consider the problem (2.1.9), (2.1.12), (2.1.15). Set Then we have the following nonlinear ODE system . It has eigenvalues It follows from (H1) that Therefore, for any boundary condition (u, v)(0, t) which is in a neighborhood of 0, the center-stable manifold theorem implies that there exits a unique bounded smooth solution (u, v) to the problem (2.1.30)-(2.1.31) with exponential decay property at infinity. Hence u 0 b (y, t) is bounded and smooth. In fact the exponential decay of ∂ y u 0 b (y, t) can also be seen by the following calculations. We rewrite (2.1.9) as ∂ 2 Integrating over [0, +∞] yields It should be mentioned that the integrations in (2.1.32)-(2.1.33) are near the boundary x = 0, and they are bounded due to (H1)-(H2) and the property of u 0 b . Thus and similarly there exists a bounded smoothū 0 b such that If the boundary layer is assumed to be weak, that means, there exists a small δ 0 > 0 such that To carry out the analysis for strong boundary layers, the structure of the underlying boundary layers should be used. It turns out the monotonicity of the wave speed in the boundary layer plays an important role in our proof. So if δ i (t), i = 1, 2 are not required to be small, we have Definition 2.1. The boundary layers can be expansive in the sense that and compressive boundary layers in the sense that be the leading order boundary layer function, and let t ∈ (0, T ] be fixed. If Proof. Without loss of generality, we only give the proof of the case when u In view of (2.1.34) and (2.1.43), then ∂ y u 0 b ≥ 0. Together with the convexity condition (1.5) we can easily get ) ≥ 0, which implies the boundary layer near {x = 0} is expansive. Now, we come to the higher order boundary layer functions.
Then it follows from the exponential decay property of ∂ y u 0 b that |g(y, t)| ≤ Ce −βy , for some β > 0. (2.1.47) Then the unique solution to (2.1.44)-(2.1.46) is given explicitly by In view of (2.1.47) and the negativity of f near the boundary x = 0, we have u 1 B exists and This finishes the proof of Lemma 2.3.
Using the inner equations (2.1.2) and (2.1.3) and the properties of u i b (y, t), i = 0, 1, we can transform (2.1.11), (2.1.14), (2.1.17) into a linear second order differential equations with a exponential decay source term. Moreover, the new unknown function satisfies the zero condition at infinity, then similar as the proof of the existence of u 1 b (y, t), the unique smooth solution u 2 b (y, t) of (2.1.11), (2.1.14), (2.1.17) can be obtained. It follows similarly that (2.1.21), (2.1.24), (2.1.27) admits a smooth solutionū 2 b (ξ, t). Hence, we have obtained the desired three-term asymptotic inner and boundary layer solutions.

2.2.
A weak boundary layer and a strong expansive boundary layer. We discuss in this subsection the stability of the multiple boundary layers, in which one is a weak boundary layer and the other is a strong expansive boundary layer. Without loss of generality, we assume there is a weak boundary layer near x = 0 and an expansive strong boundary layer near x = 1, which are defined in (1.8) and (2.1.40). We have Theorem 2.1. Assume (H1), (H2) hold and the initial and boundary data in (1.2)-(1.4) are compatible to any order and u 0 (x, t) ∈ C 2 ([0, T ]; H 6 (0, 1)) is the smooth solution to the inviscid problem (1.6) and (1.7). Furthermore, assume that there exists a δ 0 > 0 such that δ 1 (t) ≤ δ 0 , and u r (t) − u 0 (1, t) ≥ 0 for all t ∈ [0, T ]. Then there exists ε 1 > 0, s.t. ∀ 0 < ε < ε 1 , the initial boundary value problem To prove this theorem, we first patch the truncated boundary layer and inner solutions in the previous discussion to get the formal approximate solution to (1. where h(x) is a smooth function and 0 < h(x) < 1. Now, we define the approximate solution to (1.1) as , for some 11 20 < ν < 1. Using the structures of the various orders of boundary layer and inner solutions, u a (x, t) satisfy the following In view of our construction, we have Here, we have used These, together with (2.1), (2.2.6)and lemma 2.1, yield (2.1).
This finishes the construction of the formal approximate solution to (1.1). We now turn to the stability analysis of the multiple boundary layers. We will employ a perturbation analysis to prove Theorem 2.1. The main assumptions are the strength of the boundary layer near {x = 0} is suitably small and the boundary layer near {x = 1} is expansive. We decompose the exact solution u ε (x, t) as (2.2.15) It will be clear by our analysis that the exponent 5/8 can be chosen as any σ ∈ (1/2, 1]. Then ϕ solves

2.25)
where we have used the boundary conditions (2.2.17). In view of the structure of the approximate solution u a (x, t) , we have where we have used that the exponential decay property of the leading boundary layer profile (2.1.38). Using (2.2.11) and (2.1) yields Similarly, we get Using the expansive property ∂ ξ ∂ u f ≥ 0 due to Lemma 2.2, we get

RUNNING HEADING WITH FORTY CHARACTERS OR LESS 899
In view of (2.2.19), we have By Young's inequality and (2.2.13), we have Collecting all the estimates, we have Thus v(x, t) solves by v(x, t) and integrate the resulting equation over (0, 1) to get after integration by parts that The former five integrals can be treated similarly as in Lemma 2.5 and the last two integrals can be treated as follows.
Remark 2.3. The method in this subsection can be applied to both weak boundary layers case and both expansive strong boundary layers case. In such two cases, we have the same estimate of the error term as stated in Proposition 2.1. If the both boundary layers are strong and compressive, the analysis here can not be applied, and this case will be discussed in the following part.

Both strong compressive boundary layers.
In this subsection, we will study the stability of both strong compressive boundary layers. We have and sup To prove the asymptotic limit, we need to find a new approximate solution to (1.1). Now, we define the approximate solution to (1.1) as  d(x, t) is a higher order correction term to be determined. Using the structures of the various orders of boundary layer and inner solutions, u a (x, t) satisfies the following problem Here q i , i = 1, 2, 3, 4 are the same as in the section 2.2, and with d(x, t) being the solution of the diffusion problem By the standard parabolic theory, we have the following estimates for the linear diffusion wave d(x, t).
Lemma 2.8. Let d(x, t) be the solution of (2.3.6), the following estimates hold for all t ∈ [0, T ] that It follows from our construction thatū a has the following property.
We decompose the solution as u ε (x, t) =ū a (x, t) + ε 5/8 v(x, t), x ∈ (0, 1), t ∈ [0, T ]. (2.3.11) Hence v solves 14) It turns out that it is more convenient to integrate this error problem once with respect to x. Set then we obtain from (2.3.12)-(2.3.14) the following integrated error equation and We thus have to find a small solution ϕ(x, t) to the problem (2.3.15)-(2.3.17). This will be given by the following result.

3.20)
provided that sup 0≤t≤T ∂ x ϕ L ∞ (0,1) ≤ C. The third integral on the left hand side of the above equation can be written as where we have used We now estimate J i , i = 1, 2, 3, 4, 5 as follows.
where we have used that the compressibility assumption on the boundary layers that ∂ y ∂ u f (u 0 b ) ≤ 0. In view of (2.2.11) in the matching zone, we obtain By the structure of the approximate solution, we have The compressibility condition on the boundary layer near {x = 1} gives Next, we have for some η > 0. Collecting all the estimates, we have By choosing suitably small η > 0 and using Gronwall's inequality, we have Proof. We first prove (2.3.33). Set ψ = ∂ t ϕ, then ψ solves the following problem Multiply (2.3.35) by ψ and integrate the resulting equation over (0, 1) to obtain Similar estimates as in Lemma 2.10 leads to By choosing suitably small η > 0 and using Gronwall's inequality, we have This proves (2.3.33). Together with (2.1) we get Finally, we sketch the L 2 estimate on the second order derivatives of ϕ.
Remark 2.4. The analysis in the previous two subsections can be applied to the case that a weak boundary layers and a strong compressive boundary layer. Without loss of generality, we may assume the weak boundary layer is near the boundary {x = 0} and the strong compressive boundary layer is near the boundary {x = 1}. Using the same approximate solution (2.3.5) , then by noticing that the term J 1 in Lemma 2.10 can be treated as where the first term can be absorbed by the viscosity term with δ 0 being suitably small, we can finally obtain the same estimate of the error term as stated in Proposition 2.2.
In the process of stability analysis, noticing the term like J 5 in Lemma 2.5 vanishes, we have the same estimates of the error terms as in Proposition 2.1 and Proposition 2.2.
In the process of stability analysis, since u 0 b (y, t) = u 0 (0, t), then the term like J 1 in Lemma 2.5 or Lemma 2.10 vanishes, we still have the same estimates of the error term as in Proposition 2.1 and Proposition 2.2.