Asymptotic analysis of an advection-diffusion equation and application to boundary controllability

1. Introduction – Problem statement

Let T > 0 and Q T : = ( 0 , 1 ) × ( 0 , T ) . This work is concerned with the scalar advection-diffusion equation (1) y t ε − ε y x x ε + M y x ε = 0 , ( x , t ) ∈ Q T , y ε ( 0 , t ) = v ε ( t ) , y ε ( 1 , t ) = 0 , t ∈ ( 0 , T ) , y ε ( x , 0 ) = y 0 ε ( x ) , x ∈ ( 0 , 1 ) , where ε > 0 is the diffusion coefficient, M ∈ R ⋆ is the transport coefficient, v ε = v ε ( t ) is the control function, y 0 ε is the initial data, and y ε = y ε ( x , t ) is the associated state.

For any y 0 ε ∈ H − 1 ( 0 , 1 ) and v ε ∈ L 2 ( 0 , T ) , there exists a unique solution y ε to (1), with the regularity y ε ∈ L 2 ( Q T ) ∩ C ( [ 0 , T ] ; H − 1 ( 0 , 1 ) ) . As ε → 0 + , system (1) “degenerates” into a transport equation in the following sense: if v ε ⇀ v in L 2 ( 0 , T ) and if the initial data y 0 ε is independent of ε, then the solution y ε of (1) weakly converges in L 2 ( Q T ) towards y solution of the equation (2) y t + M y x = 0 , ( x , t ) ∈ Q T , y ( 0 , t ) = v ( t ) , t ∈ ( 0 , T ) if M > 0 , y ( 1 , t ) = 0 , t ∈ ( 0 , T ) if M < 0 , y ( x , 0 ) = y 0 ( x ) , x ∈ ( 0 , 1 ) . We refer to [2], Proposition 1.

We are interested in this work with a precise asymptotic description of the solution y ε when ε is small. As a first motivation, we can mention that system (1) can be seen as a simple example of complex models where the diffusion coefficient is very small compared to the others. We have notably in mind the Stokes system where ε stands for the viscosity coefficient. A second motivation comes from the numerical approximation of (1) that may be not straightforward for small values of ε (we refer to [3]). A third motivation comes from the asymptotic controllability property of (1) recently studied in [2,5,12,13] and which exhibits some apparently surprising behaviors. We also mention [6] for the multi-dimensional case. More precisely, for any final time T > 0 , ε > 0 and y 0 ε ∈ H − 1 ( 0 , 1 ) , there exist control functions v ε ∈ L 2 ( 0 , T ) such that the corresponding solution to (1) satisfies y ε ( · , T ) = 0 in H − 1 ( 0 , 1 ) (see [4,9]). This raises the question of the asymptotic behavior as ε → 0 of the cost of control defined by (3) K ( ε , T , M ) : = sup ‖ y 0 ‖ L 2 ( 0 , 1 ) = 1 { min u ∈ C ( y 0 , T , ε , M ) ‖ u ‖ L 2 ( 0 , T ) } , where C denotes the (non-empty) set of null controls C ( y 0 , T , ε , M ) : = { v ∈ L 2 ( 0 , T ) ; y = y ( v ) solves ( 1 ) and satisfies y ε ( · , T ) = 0 in H − 1 ( 0 , 1 ) } . The minimal time T M for which this cost is uniformly bounded with respect to ε is unknown. It is proved in [2] that T M ∈ [ 1 , 4.3 ] / M for M > 0 and T M ∈ [ 2 , 57.2 ] / | M | for M < 0 . Precisely, if T < 1 / M (resp. T < 2 / | M | ) for M > 0 (resp. M < 0 ), then the cost K ( ε , T , M ) blows up exponentially as ε → 0 + : such behavior is achieved with the following initial condition (4) y 0 ε ( x ) = K ε e M x 2 ε sin ( π x ) , K ε = ( 2 π 2 ε 3 ( e M ε − 1 ) M ( M 2 + 4 π 2 ε 2 ) ) − 1 / 2 = O ( ε − 3 / 2 e − M 2 ε ) , so that ‖ y 0 ε ‖ L 2 ( 0 , 1 ) = 1 . This data get concentrated at x = 1 (resp. x = 0 ) for M > 0 (resp. M < 0 ). The bounds for T M have then been improved in [5] and in [13] successively. These bounds for M < 0 are apparently not expected since, first the cost of control K ( 0 , T , M ) for (2) is zero as soon as T ⩾ 1 / | M | and second, because we can check that the L 2 -norm of y ε , solution of (1) with v ε ≡ 0 satisfies the inequality (5) ‖ y ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ ‖ y ε ( · , 0 ) ‖ L 2 ( 0 , 1 ) e − c ε t , ∀ t > 1 | M | , for some constant c > 0 independent of ε (see Proposition 3.1). In other words, the null function v ε ≡ 0 is an approximate null control for (1) for T > 1 / | M | as ε goes to zero. Precisely, (5) implies that for any η > 0 , there exists an ε 0 > 0 , such that for all ε < ε 0 , ‖ y ε ( · , t ) ‖ L 2 ( 0 , 1 ) < η for all t > 1 / | M | . One may then conclude that the controllability property for (1) and the limit as ε → 0 + do not commute. However, it should be noted that the initial condition (4) does not fall in the framework of the weak convergence result stated above as it depends on ε! Nevertheless, the time T M and more generally the behavior of the control of minimal L 2 -norm (which appears in (3)) remains unclear for ε small: there is a kind of balance between the term − ε y x x ε which favors the diffusion (and so the null controllability) for ε large and the term M y x ε which enhances the complete transport of the solution out of the domain ( 0 , 1 ) for ε small.

One may tackle this problem and the determination of the minimal uniform controllability time T M by numerical methods: this consists in approximating the cost K ( ε , T , M ) for various values of ε and T > 0 , the ratio 1 / M being fixed. This has been done in [15] for ε in the range [ 10 − 3 , 10 − 1 ] and suggests that for M > 0 , T M is equal to 1 / M achieved with initial conditions concentrating at x = 0 closed to the function x → K ˜ ε sin ( π x ) e − M x 2 ε for some constant K ˜ ε = O ( ε − 3 / 2 ) . The case M < 0 for which the transport term acts “against” the control is more involved, the underlying approximated problem being highly ill-conditioned. Smaller values of ε are difficult to consider numerically: in view of (5), the norm ‖ y ε ( · , t ) ‖ L 2 ( 0 , 1 ) decreases very fast under the zero “numeric”, which is of the order O ( 10 − 16 ) when the double digit precision is used.

An alternative theoretical approach is to analyze, through an asymptotic analysis with respect to the parameter ε, the structure of the (unique) control of minimal L 2 -norm, the initial condition y 0 ε being fixed. In this respect, we may use the fact that such control is characterized by the following optimality system (6) y t ε − ε y x x ε + M y x ε = 0 , − φ t ε − ε φ x x ε − M φ x ε = 0 , ( x , t ) ∈ Q T , y ε ( x , 0 ) = y 0 ε ( x ) , y ε ( x , T ) = 0 , x ∈ ( 0 , 1 ) , v ε ( t ) = y ε ( 0 , t ) = ε φ x ε ( 0 , t ) , t ∈ ( 0 , T ) , y ε ( 1 , t ) = φ ε ( 0 , t ) = φ ε ( 1 , t ) = 0 , t ∈ ( 0 , T ) , φ ε being the adjoint solution. We are then faced to the asymptotic analysis of a partial differential system with respect to a small parameter, in the spirit for instance of the book [10] in the closed context of optimal control theory. In view of (5), such asymptotic analysis should be as precise as possible in order to fill the gap between the approximate null controllability achieved with v ε ≡ 0 and the null controllability leading to exponentially large controls. However, in spite of the apparent simplicity of the system (1), such analysis is not straightforward because, as ε goes to zero, the direct and adjoint solutions exhibit boundary layers in the transition parabolic-hyperbolic. For example, for M > 0 , in agreement with the structure of the weak limit (2), the solution y ε (resp. φ ε ) exhibits a first boundary layer of size O ( ε ) at x = 1 (resp. x = 0 ). Moreover, the solution y ε (resp. φ ε ) exhibits a second boundary layer of size O ( ε ) along the characteristic { ( x , t ) ∈ Q T , x − M t = 0 } (resp. { ( x , t ) ∈ Q T , x − M ( t − T ) − 1 = 0 } ). A third singular behavior due to the initial condition y 0 ε occurs for y ε in the neighborhood of the points ( x 0 , t 0 ) = ( 0 , 0 ) and ( x 1 , t 1 ) = ( 1 , 0 ) .

Remark however that the boundary layer for y ε on the characteristic does not occur if and only if the function v ε and the initial condition y 0 ε satisfy some compatibility conditions at the point ( x 0 , t 0 ) . Remark also that the optimal control v ε , supported on { 0 } × ( 0 , T ) , lives in the first boundary layer for  φ ε .

The main purpose of this work, devoted to the case M > 0 , is to perform an asymptotic analysis of the direct problem (1), assuming v ε fixed and satisfying appropriate compatibility conditions at the initial time t = 0 with the initial condition y 0 ε as x = 0 . We therefore focus on the boundary layers appearing at x = 1 , employing the matched asymptotic expansion method described in the book [18].

This work is organized as follows. In Section 2, for a fixed function v ε , we perform the asymptotic analysis of the direct problem (1). Assuming that the initial condition is independent of ε and that the control function v ε is given in the form v ε = ∑ k = 0 m ε k v k , we construct an asymptotic approximation w m ε of the solution y ε . The matched asymptotic expansion method is used in Section 2.1 to define an outer solution (out of the boundary layer) and an inner solution. Upon regularity assumptions on the functions v k , k = 0 , … , m and y 0 ε , plus compatibility conditions between the functions v k and y 0 ε at ( x 0 , t 0 ) , we prove that w m ε is a regular and strong convergent approximation of y ε , as ε → 0 + . The error estimate involves the initial boundary layer function, exponentially small with respect to ε (see Lemma 2.8). The analysis is done in the case m = 2 in Section 2.2 (see Theorem 2.1) and in the general case in Section 2.3 leading to the following estimate (see Theorem 2.2): (7) ‖ y ε ( · , t ) − w m ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ c m ε 2 m + 1 2 γ + c m ( ε 1 2 + ε ( 2 m + 3 ) γ 2 ) e − M 2 2 ε γ t , ∀ t ∈ [ 0 , T ] , for some constant c m independent of ε and γ ∈ ( 0 , 1 / 2 ] . Note also that the asymptotic approximation w m ε is very useful, for instance, from a numerical viewpoint: the inner and outer solutions are constructed by using explicit formulae, and w m ε is obtained by combining the inner and outer solutions through a process of matching. In Theorem 2.3, we provide sufficient conditions on the control functions v k and on y 0 allowing to pass to the limit, as m → ∞ , with ε small enough but fixed. This leads to the following decomposition y ε ( x , t ) = w ε ( x , t ) + θ ˜ ε ( x , t ) , ( x , t ) ∈ Q T where w ε is an infinite sum of explicit solutions of transport equations, and θ ˜ ε is the initial layer corrector, defined as the solution of a non-homogeneous advection-diffusion equation (see (76)) and satisfying ‖ θ ˜ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ c ε 1 2 e − M 2 2 ε γ t , for all t ∈ [ 0 , T ] , for some constant c independent of ε. A similar analysis is conducted for the adjoint solution φ ε in Section 2.5. In Section 3, we then use (7) to construct explicitly approximate controls leading to polynomially small (with respect to ε) state at any time T > 1 / M . In Section 4, we discuss the case of initial conditions which depend on ε, in particular the one defined by y 0 ε ( x ) = e − M x 2 ε sin ( π x ) exhibited in [15]. We conclude in the final Section 5 with some remarks on the use of such analysis in order to investigate the coupled system (6).

As far as we know, this type of asymptotic analysis at any order m (including the case m → ∞ ) has not been done in the literature, most of the time, only the first terms ( m ⩽ 2 ) are considered. Note also that there are few works in the literature dealing both with asymptotic analysis and controllability. The chapter 3 of [11] entitled “Exact controllability and singular perturbation” studies the controllability property of the equation y ″ + ε Δ 2 y − Δ y = 0 as ε → 0 + and identifies the limit control problem. We mention the recent work [14] where the controllability of a Burgers equation y t − y x x + y y x = 0 in small time is discussed, leading after a change of variable to a small parameter in front of the linear second order term. We also mention [16] where a vanishing viscosity method is employed to study the sensitivity of an optimal control problem.

In the sequel, we shall use the following notations: L ε ( y ) : = y t − ε y x x + M y x , L ε ⋆ ( φ ) : = − φ t − ε φ x x − M φ x .

2. Matched asymptotic expansions and approximate solutions

In this section we consider the solution of problem (1). We apply the method of matched asymptotic expansions to construct approximate solutions. We refer to [7,17,18] for a general presentation of the method. Then we apply the same procedure to construct asymptotic approximate solutions of the adjoint solution φ ε , see problem (6).

Let us consider the problem (8) y t ε − ε y x x ε + M y x ε = 0 , ( x , t ) ∈ Q T , y ε ( 0 , t ) = v ε ( t ) , t ∈ ( 0 , T ) , y ε ( 1 , t ) = 0 , t ∈ ( 0 , T ) , y ε ( x , 0 ) = y 0 ( x ) , x ∈ ( 0 , 1 ) , where y 0 and v ε are given functions. We assume that M > 0 and v ε is in the form v ε = ∑ k = 0 m ε k v k , the functions v 0 , v 1 , … , v m being known. We construct an asymptotic approximation of the solution y ε of (8) by using the method of matched asymptotic expansions. We assume here that the initial condition y ε ( x , 0 ) is independent of ε but the procedure is very similar for y ε ( · , 0 ) of the form y ε ( · , 0 ) = ∑ k = 0 m ε k y 0 k . The case M < 0 can be treated similarly.

In the sequel, c, c 1 , c 2 , … , will stand for generic constants that do not depend on ε. When the constants c, c 1 , c 2 , … , depend in addition on some other parameter p we will write c p , c 1 ( p ) , c 2 ( p ) , … .

2.1. Formal asymptotic expansions

Let us consider two formal asymptotic expansions of y ε :

the outer expansion ∑ k = 0 m ε k y k ( x , t ) , ( x , t ) ∈ Q T ,

the inner expansion ∑ k = 0 m ε k Y k ( z , t ) , z = 1 − x ε ∈ ( 0 , ε − 1 ) , t ∈ ( 0 , T ) .

We will construct outer and inner expansions which will be valid in the so-called outer and inner regions, respectively. Here the boundary layer (inner region) occurs near x = 1 , it is of O ( ε ) size, and the outer region is the subset of ( 0 , 1 ) consisting of the points far from the boundary layer, it is of O ( 1 ) size. There is an intermediate region between them, with size O ( ε γ ) , γ ∈ ( 0 , 1 ) . To construct an approximate solution we require that inner and outer expansions coincide in the intermediate region, then some conditions must be satisfied in that region by the inner and outer expansions. These conditions are the so-called matching asymptotic conditions.

Putting ∑ k = 0 m ε k y k ( x , t ) into equation (8)₁, the identification of the powers of ε yields ε 0 : y t 0 + M y x 0 = 0 , ε k : y t k + M y x k = y x x k − 1 , for any 1 ⩽ k ⩽ m . Taking the initial and boundary conditions into account we define y 0 and y k ( 1 ⩽ k ⩽ m ) as functions satisfying the transport equations, respectively, (9) y t 0 + M y x 0 = 0 , ( x , t ) ∈ Q T , y 0 ( 0 , t ) = v 0 ( t ) , t ∈ ( 0 , T ) , y 0 ( x , 0 ) = y 0 ( x ) , x ∈ ( 0 , 1 ) , and (10) y t k + M y x k = y x x k − 1 , ( x , t ) ∈ Q T , y k ( 0 , t ) = v k ( t ) , t ∈ ( 0 , T ) , y k ( x , 0 ) = 0 , x ∈ ( 0 , 1 ) . The solution of (9) is given by (11) y 0 ( x , t ) = y 0 ( x − M t ) , x > M t , v 0 ( t − x M ) , x < M t . Using the method of characteristics we find that, for any 1 ⩽ k ⩽ m , (12) y k ( x , t ) = ∫ 0 t y x x k − 1 ( x + ( s − t ) M , s ) d s , x > M t , v k ( t − x M ) + ∫ 0 x / M y x x k − 1 ( s M , t − x M + s ) d s , x < M t .

Remark 1.

We actually verify that we have explicitly (13) y 1 ( x , t ) = t y 0 ( 2 ) ( x − M t ) , x > M t , v 1 ( t − x M ) + x M 3 ( v 0 ) ( 2 ) ( t − x M ) , x < M t , and (14) y 2 ( x , t ) = t 2 2 y 0 ( 4 ) ( x − M t ) , x > M t , v 2 ( t − x M ) + x M 3 ( v 1 ) ( 2 ) ( t − x M ) − 2 x M 5 ( v 0 ) ( 3 ) ( t − x M ) + x 2 2 M 6 ( v 0 ) ( 4 ) ( t − x M ) , x < M t . Here and in the sequel, f ( i ) denotes the derivative of order i of the real function f.

Now we turn back to the construction of the inner expansion. Putting ∑ k = 0 m ε k Y k ( z , t ) into equation (8)₁, the identification of the powers of ε yields ε − 1 : Y z z 0 ( z , t ) + M Y z 0 ( z , t ) = 0 , ε k − 1 : Y z z k ( z , t ) + M Y z k ( z , t ) = Y t k − 1 ( z , t ) , for any 1 ⩽ k ⩽ m . We impose that Y k ( 0 , t ) = 0 for any 0 ⩽ k ⩽ m . To get the asymptotic matching conditions we write that, for any fixed t and large z, Y 0 ( z , t ) + ε Y 1 ( z , t ) + ε 2 Y 2 ( z , t ) + ⋯ + ε m Y m ( z , t ) = y 0 ( x , t ) + ε y 1 ( x , t ) + ε 2 y 2 ( x , t ) + ⋯ + ε m y m ( x , t ) + O ( ε m + 1 ) . Rewriting the right-hand side of the above equality in terms of z, t and using Taylor expansions we have Y 0 ( z , t ) + ε Y 1 ( z , t ) + ε 2 Y 2 ( z , t ) + ⋯ + ε m Y m ( z , t ) = y 0 ( 1 − ε z , t ) + ε y 1 ( 1 − ε z , t ) + ε 2 y 2 ( 1 − ε z , t ) + ⋯ + ε m y m ( 1 − ε z , t ) + O ( ε m + 1 ) = y 0 ( 1 , t ) + y x 0 ( 1 , t ) ( − ε z ) + 1 2 y x x 0 ( 1 , t ) ( ε z ) 2 + ⋯ + 1 m ! ( y 0 ) x ( m ) ( 1 , t ) ( − ε z ) m + ε ( y 1 ( 1 , t ) + y x 1 ( 1 , t ) ( − ε z ) + 1 2 y x x 1 ( 1 , t ) ( ε z ) 2 + ⋯ + 1 ( m − 1 ) ! ( y 0 ) x ( m − 1 ) ( 1 , t ) ( − ε z ) m − 1 ) + ⋯ + ε m y m ( 1 , t ) + O ( ε m + 1 ) . Therefore the matching conditions read Y 0 ( z , t ) ∼ Q 0 ( z , t ) : = y 0 ( 1 , t ) , as z → + ∞ , Y 1 ( z , t ) ∼ Q 1 ( z , t ) : = y 1 ( 1 , t ) − y x 0 ( 1 , t ) z , as z → + ∞ , Y 2 ( z , t ) ∼ Q 2 ( z , t ) : = y 2 ( 1 , t ) − y x 1 ( 1 , t ) z + 1 2 y x x 0 ( 1 , t ) z 2 , as z → + ∞ , ⋯ Y m ( z , t ) ∼ Q m ( z , t ) : = y m ( 1 , t ) − y x m − 1 ( 1 , t ) z + 1 2 y x x m − 2 ( 1 , t ) z 2 + ⋯ + 1 m ! ( y 0 ) x ( m ) ( 1 , t ) ( − z ) m , Y m ( z , t ) ∼ as z → + ∞ . We thus define Y 0 as a solution of (15) Y z z 0 ( z , t ) + M Y z 0 ( z , t ) = 0 , ( z , t ) ∈ ( 0 , + ∞ ) × ( 0 , T ) , Y 0 ( 0 , t ) = 0 , t ∈ ( 0 , T ) , lim z → + ∞ Y 0 ( z , t ) = lim x → 1 y 0 ( x , t ) , t ∈ ( 0 , T ) . The last condition in (15) is the matching asymptotic condition. The general solution of (15)₁, (15)₂ is Y 0 ( z , t ) = C ( t ) ( 1 − e − M z ) , where C ( t ) is an arbitrary constant. The matching condition allows to find C ( t ) = y 0 ( 1 , t ) , therefore the solution of (15) is (16) Y 0 ( z , t ) = y 0 ( 1 , t ) ( 1 − e − M z ) , ( z , t ) ∈ ( 0 , + ∞ ) × ( 0 , T ) . Next we determine the general solution of Y z z 1 ( z , t ) + M Y z 1 ( z , t ) = y t 0 ( 1 , t ) ( 1 − e − M z ) , ( z , t ) ∈ ( 0 , + ∞ ) × ( 0 , T ) , Y 1 ( 0 , t ) = 0 , t ∈ ( 0 , T ) . We find Y 1 ( z , t ) = ( C ( t ) − y x 0 ( 1 , t ) z ) + e − M z ( − C ( t ) − y x 0 ( 1 , t ) z ) , where C ( t ) is an arbitrary constant. We determine C ( t ) by using the matching asymptotic condition lim z → + ∞ [ Y 1 ( z , t ) − Q 1 ( z , t ) ] = 0 , t ∈ ( 0 , T ) , which gives (17) Y 1 ( z , t ) = ( y 1 ( 1 , t ) − y x 0 ( 1 , t ) z ) + e − M z ( − y 1 ( 1 , t ) − y x 0 ( 1 , t ) z ) . The function Y 2 is defined as a solution of Y z z 2 ( z , t ) + M Y z 2 ( z , t ) = Y t 1 ( z , t ) , ( z , t ) ∈ ( 0 , + ∞ ) × ( 0 , T ) , Y 2 ( 0 , t ) = 0 , t ∈ ( 0 , T ) , lim z → + ∞ [ Y 2 ( z , t ) − Q 2 ( z , t ) ] = 0 , t ∈ ( 0 , T ) . We obtain (18) Y 2 ( z , t ) = ( y 2 ( 1 , t ) − y x 1 ( 1 , t ) z + y x x 0 ( 1 , t ) z 2 2 ) + e − M z ( − y 2 ( 1 , t ) − y x 1 ( 1 , t ) z − y x x 0 ( 1 , t ) z 2 2 ) . For 1 ⩽ k ⩽ m , the function Y k is defined iteratively as the solution of (19) Y z z k ( z , t ) + M Y z k ( z , t ) = Y t k − 1 ( z , t ) , ( z , t ) ∈ ( 0 , + ∞ ) × ( 0 , T ) , Y k ( 0 , t ) = 0 , t ∈ ( 0 , T ) , lim z → + ∞ [ Y k ( z , t ) − Q k ( z , t ) ] = 0 , t ∈ ( 0 , T ) .

2.2. Second order approximation

Here we take m = 2 . The outer expansion is ∑ k = 0 2 ε k y k ( x , t ) , where y 0 and y k ( k = 1 , 2 ) are given by (11) and (12), respectively, and the inner expansion is ∑ k = 0 2 ε k Y k ( z , t ) , where Y 0 , Y 1 and Y 2 are given by (16), (17) and (18), respectively. We introduce a C ∞ cut-off function X : R → [ 0 , 1 ] such that (20) X ( s ) = 1 , s ⩾ 2 , 0 , s ⩽ 1 , and define, for γ ∈ ( 0 , 1 ) , the function X ε : [ 0 , 1 ] → [ 0 , 1 ] , plotted on Fig. 1, by (21) X ε ( x ) = X ( 1 − x ε γ ) .

OPEN IN VIEWER

Fig. 1.

The function X ε .

Then we introduce the function w 2 ε by (22) w 2 ε ( x , t ) = X ε ( x ) ∑ k = 0 2 ε k y k ( x , t ) + ( 1 − X ε ( x ) ) ∑ k = 0 2 ε k Y k ( 1 − x ε , t ) , ( x , t ) ∈ Q T , defined to be the second order asymptotic approximation of the solution y ε of (8). To justify all the computations we will perform we need some regularity assumptions on the data y 0 , v 0 , v 1 and v 2 . We have the following result.

Lemma 2.1.

(i) Assume that y 0 ∈ C 5 [ 0 , 1 ] , v 0 ∈ C 5 [ 0 , T ] and the following C 5 -matching conditions are satisfied (23) M p ( y 0 ) ( p ) ( 0 ) + ( − 1 ) p + 1 ( v 0 ) ( p ) ( 0 ) = 0 , 0 ⩽ p ⩽ 5 . Then the function y 0 defined by ( 11 ) belongs to C 5 ( Q T ‾ ) .

(ii) Additionally, assume that v 1 ∈ C 3 [ 0 , T ] , v 2 ∈ C 1 [ 0 , T ] and the following C 3 and C 1 -matching conditions are satisfied, respectively, (24) v 1 ( 0 ) = 0 , ( v 1 ) ( 1 ) ( 0 ) = M − 2 ( v 0 ) ( 2 ) ( 0 ) = y 0 ( 2 ) ( 0 ) , ( v 1 ) ( 2 ) ( 0 ) = 2 M − 2 ( v 0 ) ( 3 ) ( 0 ) = − 2 M y 0 ( 3 ) ( 0 ) , ( v 1 ) ( 3 ) ( 0 ) = 3 M − 2 ( v 0 ) ( 4 ) ( 0 ) = 3 M 2 y 0 ( 4 ) ( 0 ) , (25) v 2 ( 0 ) = 0 , ( v 2 ) ( 1 ) ( 0 ) = 0 . Then the function y 1 defined by ( 12 ) (with k = 1 ) belongs to C 3 ( Q T ‾ ) , and the function y 2 defined by ( 12 ) (with k = 2 ) belongs to C 1 ( Q T ‾ ) .

Proof.

(i) According to the explicit form (11), it suffices to match the partial derivative of y 0 on the characteristic line { ( x , t ) , x − M t = 0 } . Differentiating (11) p times ( p ⩽ 5 ) with respect to x we have ∂ p y 0 ∂ x p ( x , t ) = y 0 ( p ) ( x − M t ) , x > M t , ( − 1 ) p M p ( v 0 ) ( p ) ( t − x M ) , x < M t . Matching the expressions of ∂ p y 0 ∂ x p upper and under the characteristic line { ( x , t ) , x − M t = 0 } gives (23) and ensures the continuity of ∂ p y 0 ∂ x p in Q T ‾ . Differentiating (11) p times with respect to t we have ∂ p y 0 ∂ t p ( x , t ) = ( − 1 ) p M p y 0 ( p ) ( x − M t ) , x > M t , ( v 0 ) ( p ) ( t − x M ) , x < M t , then we see that the continuity of ∂ p y 0 ∂ t p holds under condition (23). Using equation (9) we easily verify that the mixed partial derivatives, of order p ⩽ 5 , of y 0 are continuous under condition (23).

(ii) Arguing as previously, using formula (13) and equation (10) (with k = 1 ) we find the matching conditions (24). Then, using formula (14) and equation (10) (with k = 2 ) we find the matching conditions (25). □

We have the following result.

Lemma 2.2.

Let w 2 ε be the function defined by ( 22 ). Assume that the assumptions of Lemma  2.1 hold true. Then there is a constant c independent of ε such that (26) ‖ L ε ( w 2 ε ) ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ c ε 5 γ 2 .

Proof.

A straightforward calculation gives L ε ( w 2 ε ) ( x , t ) = ∑ i = 1 5 I ε i ( x , t ) , with I ε 1 ( x , t ) = − ε 3 y x x 2 ( x , t ) X ε ( x ) , I ε 2 ( x , t ) = ε 2 ( 1 − X ε ( x ) ) Y t 2 ( 1 − x ε , t ) , I ε 3 ( x , t ) = M X ′ ( 1 − x ε γ ) ε − γ ( ∑ k = 0 2 ε k Y k ( 1 − x ε , t ) − ∑ k = 0 2 ε k y k ( x , t ) ) , I ε 4 ( x , t ) = X ″ ( 1 − x ε γ ) ε 1 − 2 γ ( ∑ k = 0 2 ε k Y k ( 1 − x ε , t ) − ∑ k = 0 2 ε k y k ( x , t ) ) , I ε 5 ( x , t ) = 2 X ′ ( 1 − x ε γ ) ε 1 − γ ( ε − 1 ∑ k = 0 2 ε k Y z k ( 1 − x ε , t ) + ∑ k = 0 2 ε k y x k ( x , t ) ) . Clearly, (27) ‖ I ε 1 ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ ε 3 ‖ y x x 2 ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ c ε 3 , and ‖ I ε 2 ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ ε 2 ‖ ( 1 − X ε ( x ) ) Y t 2 ( 1 − x ε , t ) ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ ε 2 max t ∈ [ 0 , T ] ( ∫ 1 − 2 ε γ 1 | Y t 2 ( 1 − x ε , t ) | 2 d x ) 1 / 2 . Using a change of variable we have ( ∫ 1 − 2 ε γ 1 | Y t 2 ( 1 − x ε , t ) | 2 d x ) 1 / 2 = ( ε ∫ 0 2 ε γ ε | Y t 2 ( z , t ) | 2 d z ) 1 / 2 . Thanks to the explicit form (18) we have, for 0 < ε ⩽ ε 0 small enough, max t ∈ [ 0 , T ] ( ε ∫ 0 2 ε γ ε | Y t 2 ( z , t ) | 2 d z ) 1 / 2 ⩽ c ‖ y x x t 0 ‖ C ( [ 0 , 1 ] × [ 0 , T ] ) ( ε ∫ 0 2 ε γ ε z 4 d z ) 1 / 2 ⩽ c ε − 2 ε 5 γ 2 . It results that (28) ‖ I ε 2 ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ c ε 5 γ 2 . Using Taylor expansions, for 1 − x = ε z → 0 , we have ∑ k = 0 2 ε k y k ( x , t ) = ∑ k = 0 2 ε k y k ( 1 − ε z , t ) = ∑ k = 0 2 ε k ( ∑ i = 0 2 − k 1 i ! ∂ i y k ∂ x i ( 1 , t ) ( − ε z ) i ) + ε 2 O ( ( ε z ) ) . Since Y k ( z , t ) = Q k ( z , t ) + e − M z P k ( z , t ) , ( z , t ) ∈ ( 0 , + ∞ ) × ( 0 , t ) , with P k ( z , t ) = − ∑ i = 0 k ∂ i y k ∂ x i ( 1 , t ) z i , Q k ( z , t ) = ∑ i = 0 k ( − 1 ) i ∂ i y k ∂ x i ( 1 , t ) z i , we have (29) ∑ k = 0 2 ε k Y k ( z , t ) − ∑ k = 0 2 ε k y k ( 1 − ε z , t ) = ε 2 O ( ( ε z ) ) + e − M z ∑ k = 0 2 ε k P k ( z , t ) . Using the previous estimate we have ‖ I ε 3 ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) = M ε − γ ‖ X ′ ( 1 − x ε γ ) ( ∑ k = 0 2 ε k Y k ( 1 − x ε , t ) − ∑ k = 0 2 ε k y k ( 1 − ε z , t ) ) ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) (30) ⩽ c ε 2 − γ ( ∫ 1 − 2 ε γ 1 − ε γ ( 1 − x ) 2 d x ) 1 / 2 ⩽ c ε 2 + γ 2 . Similarly we have (31) ‖ I ε 4 ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ c ε 3 − γ 2 . It results from (29) that ε − 1 ∑ k = 0 2 ε k Y z k ( z , t ) + ∑ k = 0 2 ε k y x k ( 1 − ε z , t ) = ε O ( ( ε z ) ) + ε − 1 e − M z ∑ k = 0 2 ε k P z k ( z , t ) − ε − 1 M e − M z ∑ k = 0 2 ε k P k ( z , t ) . Arguing as for I ε 3 we find that (32) ‖ I ε 5 ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ c ε 2 + γ 2 . Collecting estimates (27), (28), (30)–(32) we obtain (26). The proof of the lemma is complete. □

Let us now consider the initial layer corrector θ ε defined as the solution of (33) θ t ε − ε θ x x ε + M θ x ε = 0 , ( x , t ) ∈ Q T , θ ε ( 0 , t ) = θ ε ( 1 , t ) = 0 , t ∈ ( 0 , T ) , θ ε ( x , 0 ) = θ 0 ε ( x ) , x ∈ ( 0 , 1 ) , with (34) θ 0 ε ( x ) = : y 0 ( x ) − w 2 ε ( x , 0 ) = ( 1 − X ε ( x ) ) ( y 0 ( x ) − ∑ k = 0 2 ε k Y k ( 1 − x ε , 0 ) ) , x ∈ ( 0 , 1 ) . The following lemma gives an exponential decay property of θ ε .

Lemma 2.3.

Let θ ε be the solution of problem ( 33 ), ( 34 ). Assume γ ∈ ( 0 , 1 / 2 ] . Then there exists a constant c independent of ε such that (35) ‖ θ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ c ( ε 1 2 + ε 7 γ 2 ) e − M 2 2 ε γ t , ∀ t ∈ [ 0 , T ] .

Proof.

Let α > 0 . We define ρ ε ( x , t ) = e − M α x 2 ε θ ε ( x , t ) then check that L ε θ ε = e M α x 2 ε ( ρ t ε − ε ρ x x ε + M ( 1 − α ) ρ x ε − M 2 4 ε ( α 2 − 2 α ) ρ ε ) in Q T . Consequently, ρ ε is a solution of (36) ρ t ε − ε ρ x x ε + M ( 1 − α ) ρ x ε − M 2 4 ε ( α 2 − 2 α ) ρ ε = 0 , ( x , t ) ∈ Q T , ρ ε ( 0 , t ) = ρ ε ( 1 , t ) = 0 , t ∈ ( 0 , T ) , ρ ε ( x , 0 ) = e − M α x 2 ε θ 0 ε ( x ) , x ∈ ( 0 , 1 ) . Multiplying the main equation of (36) by ρ ε and integrating over ( 0 , 1 ) then leads to d d t ‖ ρ ε ( · , t ) ‖ 2 + 2 ε ‖ ρ x ε ( · , t ) ‖ L 2 ( 0 , 1 ) 2 = M 2 2 ε ( α 2 − 2 α ) ‖ ρ ε ( · , t ) ‖ L 2 ( 0 , 1 ) 2 , and then to the estimate ‖ ρ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ ‖ ρ ε ( · , 0 ) ‖ L 2 ( 0 , 1 ) e M 2 4 ε ( α 2 − 2 α ) t , equivalently, to ‖ e − M α x 2 ε θ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ ‖ e − M α x 2 ε θ ε ( · , 0 ) ‖ L 2 ( 0 , 1 ) e M 2 4 ε ( α 2 − 2 α ) t . Consequently, ‖ θ ε ( · , t ) ‖ L 2 ( 0 , 1 ) = ‖ e M α x 2 ε e − M α x 2 ε θ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ ‖ e M α x 2 ε ‖ L ∞ ( 0 , 1 ) ‖ e − M α x 2 ε θ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ ‖ e M α x 2 ε ‖ L ∞ ( 0 , 1 ) ‖ e − M α x 2 ε θ ε ( · , 0 ) ‖ L 2 ( 0 , 1 ) e M 2 4 ε ( α 2 − 2 α ) t ⩽ ‖ e M α x 2 ε ‖ L ∞ ( 0 , 1 ) ‖ e − M α x 2 ε θ ε ( · , 0 ) ‖ L 2 ( 1 − 2 ε γ , 1 ) e M 2 4 ε ( α 2 − 2 α ) t ⩽ e M α 2 ε e − M α ( 1 − 2 ε γ ) 2 ε ‖ θ 0 ε ‖ L 2 ( 1 − 2 ε γ , 1 ) e M 2 4 ε ( α 2 − 2 α ) t = ‖ θ 0 ε ‖ L 2 ( 1 − 2 ε γ , 1 ) e − M α 2 ε ( − 2 ε γ + ( 1 − α 2 ) M t ) , using that (recall that α > 0 ) ‖ e M α x 2 ε ‖ L ∞ ( 0 , 1 ) = e M α 2 ε and ‖ e − M α x 2 ε ‖ L ∞ ( 1 − 2 ε γ , 1 ) = e − M α ( 1 − 2 ε γ ) 2 ε . The value α = ε 1 − γ then leads to ‖ θ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ ‖ θ 0 ε ‖ L 2 ( 1 − 2 ε γ , 1 ) e M e M 2 ε 1 − 2 γ 4 t e − M 2 2 ε γ t . Using the condition γ ∈ ( 0 , 1 / 2 ] it results that, for some c > 0 , (37) ‖ θ ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ c ‖ θ 0 ε ‖ L 2 ( 1 − 2 ε γ , 1 ) e − M 2 2 ε γ t , ∀ t ∈ [ 0 , T ] .

Let us now give an estimate of ‖ θ 0 ε ‖ L 2 ( 1 − 2 ε γ , 1 ) . Using (16)–(18) it holds that θ 0 ε = a ε + b ε , with a ε ( x ) = ( 1 − X ε ( x ) ) ( y 0 ( x ) − y 0 ( 1 ) + y 0 ( 1 ) ( 1 ) ( ε z ) − y 0 ( 2 ) ( 1 ) ( ε z ) 2 2 ) , b ε ( x ) = ( 1 − X ε ( x ) ) e − M z ( y 0 ( 1 ) + y 0 ( 1 ) ( 1 ) ( ε z ) + y 0 ( 2 ) ( 1 ) ( ε z ) 2 2 ) , ( z = 1 − x ε ) . Using Taylor’s expansion, for 1 − x = ε z → 0 , we have a ε ( x ) = ( 1 − X ε ( x ) ) ( 1 − x ) 3 6 y 0 ( 3 ) ( ζ ) , ζ ∈ ( x , 1 ) , hence ‖ a ε ‖ L 2 ( 0 , 1 ) ⩽ c ε 7 γ 2 . Since ‖ b ε ‖ L 2 ( 0 , 1 ) 2 ⩽ ∫ 1 − 2 ε γ 1 e − 2 M z ( y 0 ( 1 ) + y 0 ( 1 ) ( 1 ) ( ε z ) + y 0 ( 2 ) ( 1 ) ( ε z ) 2 2 ) 2 d x , we have ‖ b ε ‖ L 2 ( 0 , 1 ) ⩽ c ε 1 2 . It results that (38) ‖ θ 0 ε ‖ L 2 ( 0 , 1 ) ⩽ c ( ε 1 2 + ε 7 γ 2 ) , then estimate (35) results from (37) and (38). □

Let us now establish the following result.

Lemma 2.4.

Let y ε be the solution of problem ( 8 ), let w 2 ε be the function defined by ( 22 ) and let θ ε be the solution of problem ( 33 ), ( 34 ). Assume that the assumptions of Lemma  2.1 hold true. Then there exists a constant c > 0 , independent of ε, such that (39) ‖ y ε − w 2 ε − θ ε ‖ C ( [ 0 , T ] ; L 2 ( 0 , 1 ) ) ⩽ c ε 5 γ 2 .

Proof.

Let us consider the function z ε = y ε − w 2 ε − θ ε . By construction, it satisfies L ε ( z ε ) = − L ε ( w 2 ε ) , ( x , t ) ∈ Q T , z ε ( 0 , t ) = z ε ( 1 , t ) = 0 , t ∈ ( 0 , T ) , z ε ( x , 0 ) = 0 , x ∈ ( 0 , 1 ) , leading to the estimate ‖ z ε ( · , t ) ‖ L 2 ( 0 , 1 ) 2 ⩽ ‖ L ε ( w 2 ε ) ‖ L 2 ( Q T ) 2 e t , for all t ∈ ( 0 , T ] . Lemma 2.2 then implies (39). □

Using Lemmas 2.3 and 2.4 we immediately obtain the following result.

Theorem 2.1.

Let y ε be the solution of problem ( 8 ) and let w 2 ε be the function defined by ( 22 ). Assume that the assumptions of Lemma  2.1 hold true and γ ∈ ( 0 , 1 / 2 ] . Then there exist two positive constants c and ε 0 , c independent of ε, such that, for any 0 < ε < ε 0 , ‖ y ε ( · , t ) − w 2 ε ( · , t ) ‖ L 2 ( 0 , 1 ) ⩽ c ε 5 γ 2 + c ( ε 1 2 + ε 7 γ 2 ) e − M 2 2 ε γ t , ∀ t ∈ [ 0 , T ] .