The perturbed Riemann problem for the chromatography system of Langmuir isotherm with one inert component

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1 Aailable online at J. Nonlinear Sci. Appl , Research Article The perturbed Riemann problem for the chromatography system of Langmuir isotherm with one inert component Pengpeng Ji, Chun Shen School of Mathematics and Statistics Science, Ludong Uniersity, Yantai, Shandong , P. R. China. Communicated by D. Baleanu Abstract The solutions of the perturbed Riemann problem for the chromatography system of Langmuir isotherm with one inert component are constructed in completely explicit forms when the initial data are taken as three piecewise constant states. The wae interaction problem is inestigated in detail by using the method of characteristics. In addition, the generalized Riemann problem with the delta-type initial data is considered and the delta contact discontinuity is discoered. Moreoer, the strength of delta contact discontinuity decreases linearly at a constant rate and then the delta contact discontinuity degenerates to be the contact discontinuity when across the critical point. c 2016 all rights resered. Keywords: Chromatography system, Riemann problem, wae interaction, Temple class, hyperbolic conseration law MSC: 35L65, 35L67, 76N Introduction Chromatography is the terminology adopted by engineers and chemists to describe a process of separating two chemical components in a fluid phase [16]. Various mathematical models are used to understand and analyze dynamic composition front in chromatographic columns and thus it is necessary to deelop the theory of nonlinear chromatography system in order to inestigate the separation process of chromatography. It is well-known that the Langmuir model [12] is ery effectie to describe a ariety of real systems in the Corresponding author addresses: @qq.com Pengpeng Ji, shenchun3641@sina.com Chun Shen Receied

2 P. Ji, C. Shen, J. Nonlinear Sci. Appl , local equilibrium theory of chromatography [16]. The process of Langmuir isotherm can be described by the system of conseration laws to account for conection and exchange between the adsorbed phases and the fluid at the thermodynamic equilibrium [13]. Thus, it is ery necessary to look for the exact solutions of these models associated with suitable initial and boundary alue conditions to describe different chromatographic behaiors. Fortunately, it is amenable to gie an appropriate treatment in the theory of hyperbolic conseration laws. Recently, seeral generalizations of the Langmuir model within the time scale calculus hae been proposed in [2]. In addition, the Langmuir model was also modified in [3, 4] by taking into account some nonlinear effects such as diffusion and condensation. In this paper, we are concerned with the chromatography separation of two chemical species through a Langmuir isotherm reactor [15, 16] when one component is inert and the other component is actie, which can be described by the following hyperbolic system of conseration laws u t = 0, k t u x = 0, where u, are the non-negatie functions of the ariables x, t R R +, which stand for the concentrations of the species. The system 1.1 can be deried from the more general two-component chromatography system [16, 25] k 1 u u t u + x = 0, 1.2 k 2 t u + x = 0, by letting k 1 = 0 and k 2 = k, where k 1, k 2 [0, 1] are all known constants dependent on the nature of the Langmuir isotherm. If k 1 = k 2 is taken in 1.2, then it is called as the simplified chromatography system, which has been widely studied such as in [1, 19, 23] recently. It is well-known that a component with concentration u or is called as inert if k 1 = 0 or k 2 = 0 is taken [5]. Thus, the mass balances of the two chemical components in the process of chromatography separation goerned by the Langmuir isotherm can be described by the system 1.1 when the component u is inert and the component is actie. It is easy to see that the chromatography system 1.1 is strictly hyperbolic and the first characteristic field is linearly degenerated and while the second characteristic field is genuinely nonlinear. It is noteworthy that the system 1.1 belongs to the Temple class [25] for the shock cure coincides with the rarefaction one in the u, phase plane. One of the main purposes in this paper is to construct the global solutions of the particular Cauchy problem for the system 1.1 when the initial data are taken to be three piecewise constant states as u,, < x < ε, u, x, 0 = u m, m, ε < x < ε, 1.3 u +, +, ε < x < +, in which ε > 0 is arbitrarily small. It is worthwhile to notice that the initial data 1.3 may be iewed as the perturbation of the corresponding Riemann initial data u, x, 0 = { u,, < x < 0, u +, +, 0 < x < Thus, the particular Cauchy problem 1.1 and 1.3 is often called as the perturbed Riemann problem or the double Riemann problems below. The Riemann problem is of great importance in the theory of chromatography and needs to be considered when a stream of a gien composition is fed to a column initially saturated at a different composition or the saturation of an initially clean column with a feedstream has constant concentrations of the two solutions [16]. It can be seen that the solutions of Riemann problem 1.1 and 1.4 consist of constant states separated by elementary waes including rarefaction wae, shock wae and contact discontinuity. To study the perturbed Riemann problem 1.1 and 1.3, it is essential to study

3 P. Ji, C. Shen, J. Nonlinear Sci. Appl , the wae interaction problem for the system 1.1. The method adopted in this paper is to first construct the solutions of the perturbed Riemann problem 1.1 and 1.3 in the u, phase plane and consequently map them onto the x, t physical plane. In fact, this type of initial data 1.3 has been widely used to study all kinds of chromatography systems such as in [10, 20, 24, 27] for the reason that the wae interaction problem is one of the most basic problems [16] in the study of chromatography separation process. More precisely, the three piecewise constant initial states 1.3 should be taken when we deal with the problem of multi-component separation by the chromatographic cycle [16]. The perturbed Riemann problem 1.1 and 1.3 is one of the most important questions in the theory of chromatography and the fundamental features of wae interactions for the system 1.1 can be examined thoroughly by studying the perturbed Riemann problem 1.1 and 1.3. In this paper, the wae interaction problem for the system 1.1 has been inestigated in detail by using the method of characteristics and then the global solutions of the perturbed Riemann problem 1.1 and 1.3 hae been constructed in completely explicit forms for all the possible situations. During the process of wae interaction, the propagation speeds of shock and rarefaction waes are deliered and then the explicit expressions of shock cures are gien. It is interesting to obsere that the propagation speeds of shock and rarefaction waes increase or decrease when across the contact discontinuity which depends on the choice of initial data. Furthermore, the stability of solutions to the Riemann problem 1.1 and 1.4 can also be analyzed under the particular small perturbation 1.3 of the Riemann initial data 1.4 which is summarized in the theorem below. Theorem 1.1. The limits of the global solutions to the perturbed Riemann problem 1.1 and 1.3 are identical with the corresponding Riemann solutions of 1.1 and 1.4 when the limit ε 0 is taken. Thus, the Riemann solutions are stable with respect to the particular small perturbations 1.3 of the Riemann initial data 1.4. Recently, the delta shock wae has been obsered experimentally in [11, 14] for the local equilibrium model of two-component nonlinear chromatography attributed to a mixed competitie-cooperatie generalized Langmuir isotherm. In fact, the delta shock wae is a nonclassical and singular transition front between two constant composition states that may occur in the theory of chromatography due to the competitiecooperatie interaction between two chemical components. The delta shock wae may be regarded as a traeling spike superposed on a discontinuity to separate the initial and feed states [13]. Thus, the delta shock wae can also be seen as a reasonable supplement of classical waes inoling the rarefaction wae, shock wae and contact discontinuity in the theory of chromatography. Consequently, the study of delta shock wae for all kinds of chromatography systems has attracted extensie attention such as in [8, 18, 20, 23, 26]. Thus, it is natural to study the generalized Riemann problem for the chromatography system 1.1 with the delta-type initial data u, x, 0 = u,, < x < 0, u 0, mδx, x = 0, u +, +, 0 < x < +, where the symbol δ indicates the standard Dirac delta function. In other words, it may be assumed that the occurrence of a spike with infinitely high concentration [22] appears initially without loss of generality and then the spike propagation can be obsered. During the construction of solutions to the generalized Riemann problem 1.1 and 1.5, the delta contact discontinuity is captured which is the Dirac delta function supported on the line x = 0 of contact discontinuity. It is interesting to discoer that the strength of delta contact discontinuity decreases linearly at a constant rate and then becomes zero at the critical point, such that the delta contact discontinuity will degenerate to be the contact discontinuity when across the critical point. That is to say, the position of spike keeps inariant and the height of spike decreases linearly with respect to the time t such that the spike disappears in finite time for the generalized Riemann problem 1.1 and 1.5. The paper is organized as follows: In Section 2, we obtain the solutions of the Riemann problem 1.1 and 1.4. In Section 3, we mainly discuss all kinds of wae interactions when the initial data are taken to 1.5

4 P. Ji, C. Shen, J. Nonlinear Sci. Appl , be three piecewise constant states and then the global solutions of the perturbed Riemann problem 1.1 and 1.3 are constructed completely. In Section 4, the generalized Riemann problem with delta-type initial data is considered and the solutions are constructed. At the end, the conclusion is drawn in Section The Riemann problem In this section, we need to sole the Riemann problem 1.1 and 1.4 by using the standard technique such as in the classical books [6, 7, 9, 17, 21]. The eigenalues of 1.1 are λ 1 = 0, λ 2 = k1 + u 1 + u + 2, such that λ 1 < λ 2 holds for arbitrary u and. Thus, the system 1.1 is strictly hyperbolic in the quarter of u, phase plane. The corresponding right eigenectors for the system 1.1 are r 1 = u + 1, T and r2 = 0, 1 T, respectiely. Let us use the notation = u, to stand for the gradient operator. We hae λ 1 r 1 = 0 for the first characteristic field λ 1, therefore the first characteristic field λ 1 is linearly degenerated. Thus, the wae associated with λ 1 is the contact discontinuity denoted by J. On the other hand, we hae λ 2 r 2 = 2ku+1 0 for the second characteristic field λ 1+u+ 3 2, such that the second characteristic field λ 2 is genuinely nonlinear. Thus, the wae associated with λ 2 will be either the shock wae denoted by S or the rarefaction wae denoted by R. Let us first consider the smooth solutions. If ux, t is a solution of the Riemann problem 1.1 and 1.4, then uαx, αt is also a solution of the Riemann problem 1.1 and 1.4 for any α > 0. Thus, it is natural to consider the solution of the Riemann problem 1.1 and 1.4 which only depends on ξ = x t. Therefore, the Riemann problem 1.1 and 1.4 is reduced to the following boundary alue problem for the ordinary differential equations ξu ξ = 0, k ξ ξ u + ξ = 0, 2.1 associated with the boundary condition u, ± = u ±, ±. Let us rewrite 2.1 into the following form ξ 0 k k1 + u 1 + u u + 2 ξ uξ 0 =. ξ 0 Thus, for the gien left state u,, the contact discontinuity cure which is a wae of the first characteristic family can be expressed by { ξ = λ1 = 0, Ju, : 1 + u =, 1 + u and the rarefaction wae cure which is a wae of second characteristic family can be expressed by k1 + u ξ = λ Ru, : 2 = 1 + u + 2, u = u, <. Let us turn to the discontinuous solutions. For a discontinuous cure x = xt, the Rankine-Hugoniot relation σ[u] = 0, [ k ] σ[] =, 1 + u +

5 P. Ji, C. Shen, J. Nonlinear Sci. Appl , should hold, where σ = dx dt is the propagation speed of the discontinuity and [u] = u r u l is the jump across the discontinuity with u l = uxt 0, t and u r = uxt + 0, t, etc. By a simply calculation, we can also obtain the contact discontinuity which is a wae of the first characteristic family Ju, : { τ = 0, 1 + u = 1 + u, and the shock wae which is a wae of the second characteristic family k1 + u σ = Su, : 1 + u u +, u = u, >. S J u, -1 0 R u Figure 1: For the gien left state u,, the u, phase plane is shown for the Riemann problem 1.1 and 1.4. Clearly, the system 1.1 is attributed to the so-called Temple class [25] for the reason that the shock cure coincides with the rarefaction one in the u, phase plane. Let us draw Figure 1 to illustrate this situation. In summary, for the gien left state u,, there exist two kinds of Riemann solutions to 1.1 and 1.4 described below. 1 When + u + +1 > in which σ = 2 When + u + +1 < u +1, the Riemann solution is a contact discontinuity J followed by a shock wae S k1+u 1+u + 1+u u,, < x < 0, u, x, t = u +, u + + 1, 0 < x < σt, u + 1 u +, +, σt < x < +, is the propagation speed of the shock wae. u +1, the Riemann solution is a contact discontinuity J followed by a rarefaction wae R u, x, t = u,, < x < 0, u +, u + + 1, 0 < x < λ 2 u +, u t, u + 1 u + 1 u +,, λ 2 u +, u u + 1 u +, +, λ 2 u +, + t < x < +, in which the state ariable in R aries from u ++1 u +1 to +. t < x < λ 2 u +, + t,

6 P. Ji, C. Shen, J. Nonlinear Sci. Appl , Construction of global solutions to the perturbed Riemann problem 1.1 and 1.3 In this section, we are planing to construct the global solutions of the perturbed Riemann problem 1.1 and 1.3 for all kinds of situations. In other words, we need to study all the possible wae interactions for the system 1.1 by employing the method of characteristics. Case 1. J + S and J + S. First of all, we need to consider the case that both a contact discontinuity followed by a shock wae emitting from the initial points ε, 0 and ε, 0 see Figure 2. Obiously, the occurrence of this case depends on the condition + u > m u m + 1 > u + 1. For the sufficiently small time t, the solution may be represented succinctly as: u, + J 1 + u 1, 1 + S 1 + u m, m + J 2 + u 2, 2 + S 2 + u +, +, in which the states u 1, 1 and u 2, 2 are gien respectiely by u 1, 1 = u m, u m + 1 u + 1, u 2, 2 = u +, u m u m k1+u The propagation speed of S 1 is σ 1 = 1 1+u m+ m1+u > 0 and that of J 2 is τ 2 = 0, such that S 1 will interact with J 2 at a finite time t 1 and the interaction point is gien by { x1 = ε, 3.2 x 1 + ε = σ 1 t 1, which means that x 1, t 1 = ε, 2ε1 + u u m + m k1 + u At the point x 1, t 1, a new local Riemann problem will be formulated where the initial data are taken to be { u1, u, x, 0 = 1, x < ε, u 2, 2, x > ε. Furthermore, the solution of the new local Riemann problem at the point x 1, t 1 is a contact discontinuity J 2 followed by a shock wae S 3. Analogously, the intermediate state u 3, 3 between J 2 and S 3 can also be obtained by u 3, 3 = u 2, u u in which u 1, 1 and u 2, 2 are gien by 3.1. Then, we use the following lemma to describe the interaction between S 2 and S 3 see Figure 2. Lemma 3.1. If u + > u m, then we hae σ 1 > σ 3, namely the shock wae S 1 decelerates when it passes through J 2. Otherwise if u + < u m, then we hae σ 1 < σ 3, namely the shock wae S 1 accelerates when it passes through J 2. Proof. The propagation speeds of S 1 and S 3 can be computed respectiely by σ 1 = k1 + u u m + m 1 + u 1 + 1, σ 3 =, k1 + u u u

7 P. Ji, C. Shen, J. Nonlinear Sci. Appl , t + J 1 J J m 2 2 S 1 J 1 1 S 2 u 3 S 3 S S S m ε ε Figure 2: The interaction between J + S and J + S is displayed when + u + +1 > m > u m+1 x u and u+ > um. +1 Then, we hae σ 1 σ 3 = k 1 + u 11 + u u u u u m + m 1 + u m + m 1 + u u u = k [1 + u 11 + u u 1 ]1 + u [1 + u u u 3 ]1 + u m + m 1 + u m + m 1 + u u u = k[1 + u 11 + u u 1 ]u u m m 1 + u m + m 1 + u u u k1 + u 1 u u m m = 1 + u m + m 1 + u u 2 + 2, in which u 1 = u 3 has been used. If u + > u m, then we hae 2 > m and u 2 > u m, such that σ 1 > σ 3. Otherwise if u + < u m, then σ 1 < σ 3 can be achieed similarly. The proof is completed. Finally, we consider the coalescence of two shock waes belonging to the same family shown below. Lemma 3.2. The two shock waes S 3 and S 2 belonging to the second family coalesce into a new shock wae of the second family. Proof. The propagation speed of S 2 is which, together with 3.4, yields σ 3 σ 2 = σ 2 = k1 + u u u + + +, k1 + u u u u > 0, in which u + = u 2 = u 3 and + > 2 > 3 hae been used. Hence, S 3 catches up with S 2 in finite time and the intersection x 2, t 2 is determined by { x2 ε = σ 2 t 2, which yields x 2, t 2 = x 2 ε = σ 3 t 2 t 1, ε + 2ε1 + u 21 + u u m + m 1 + u u , 2ε1 + u u m + m 1 + u k1 + u It can be seen from u 3 = u + that the two states u +, + and u 3, 3 can also be connected by a shock wae directly. Thus, after t 2, S 2 and S 3 coalesce into a new shock wae S 4 whose propagation speed is σ 4 = It is easy to get σ 3 > σ 4 > σ 2. The proof is completed. k1 + u u u

8 P. Ji, C. Shen, J. Nonlinear Sci. Appl , Case 2. J + S and J + R. For this case, we need to cope with the situation that the Riemann solution at ε, 0 is J 1 + S 1 and at ε, 0 is J 2 + R 2. On this occasion, the initial data 1.3 should satisfy the condition m u m + 1 > max u + 1, +. u The solution of 1.1 and 1.3 for sufficiently small t may be indicated as u, + J 1 + u 1, 1 + S 1 + u m, m + J 2 + u 2, 2 + R 2 + u +, +. Here and below the states u 1, 1, u 2, 2, and u 3, 3 hae the same presentations as those in Case 1. As in Case 1, S 1 collides with J 2 at the point x 1, t 1 which has the same expression with 3.3. After the time t 1, the new local Riemann problem whose left state is u 1, 1 and right state is u 2, 2 can also be soled by a contact discontinuity J 2 and a shock wae denoted by S 3. The result of Lemma 3.1 can also be obtained here in contrast with the two propagation speeds of S 1 and S 3. Therefore, we are now in a position to consider the situation that the shock wae S 3 penetrates the rarefaction wae R 2 which can be summarized below. Lemma 3.3. The shock wae S 3 catches up with the wae back of the rarefaction wae R 2 in finite time and consequently begins to penetrate R 2. More precisely, if + u + +1 > u +1, then S 3 is able to cancel the whole R 2 thoroughly. Otherwise, if + u + +1 < u +1, then S 3 penetrates R 2 partially and finally has the line x = ε + k1+u 3t 1+u in R 2 2 as its asymptotic line. Proof. One can see that the propagation speed of S 3 is gien by 3.4 and that of the wae back of R 2 is calculated by ξ 2 u 2, 2 = k1 + u u By irtue of u 2 = u 3 and 2 > 3, we hae σ 3 ξ 2 u 2, 2 = k1 + u u u > 0, thus S 3 keeps up with the wae back of R 2 at the point x 2, t 2 which is computed by { x2 ε = σ 3 t 2 t 1, x 2 ε = ξ 2 u 2, 2 t 2, 3.5 in which t 1 is gien by 3.3. Thus, we hae x 2, t 2 = ε + 2ε1 + u 21 + u u m + m 1 + u u , 2ε1 + u u m + m 1 + u k1 + u Consequently, S 3 begins to penetrate R 2 after the time t 2. During the process of penetration, we denote it with S 4 whose propagation speed is determined by dx dt = k1 + u u u +, k1 + ut x ε = 1 + u + 2, 3.7 xt 2 = x 2, where u, changes from u 2, 2 to u +, +. Taking into account u = u 2 = u 3 = u +, differentiating 3.7

9 P. Ji, C. Shen, J. Nonlinear Sci. Appl , with respect to t leads to d 2 x dt 2 = k1 + u u u d dt, dx dt = k1 + u u k1 + u 3t 1 + u d dt. 3.8 t J 1 J 2 3 S J2 2 R 2 + J 1 m S 1 1 u S 1 1 m ε S 3 2 ε R 2 + x Figure 3: The interaction between J + S and J + R is shown when u + < u m and + u + +1 < u +1 < m. u m+1 It yields d dt < 0 by substituting the first equation of 3.7 into 3.8. Furthermore, we hae d2 x > 0, dt 2 which means that S 3 accelerates during the process of penetration. On the other hand, it follows from 3.6 and 3.7 that the cure of the shock wae S 4 is determined by x ε = k1 + u3 t 1 + u u Therefore, there exist two possible solutions as follows: 2ε1 + u u u m + m u u a If + u + +1 > u +1, then S 4 is able to cross the whole R 2 completely and terminates at the point x 3, t 3 which is gien by x 3 ε = ξ 2 u +, + t 3, k1 + u3 t 3 1 2ε1 + u u u m + m 2 3 x3 ε =, 1 + u u u u such that we hae x 3, t 3 = ε + 2ε1 + u +1 + u u m + m u u , 2ε1 + u u m + m u k1 + u u After the penetration, we denote the shock wae with S 5 whose propagation speed is b If + u + +1 < σ 5 = k1 + u u u u +1, then S 4 cannot cancel the entire R 2 thoroughly and ultimately has the characteristic line x = ε + k1+u 3t 1+u in R 2 as the asymptotic line see Figure 3.

10 P. Ji, C. Shen, J. Nonlinear Sci. Appl , Case 3. J + R and J + S. In this case, we consider that the Riemann solution at ε, 0 is J 1 + R 1 and at ε, 0 is J 2 + S 2. This case happens if and only if m u m + 1 < min u + 1, + u is satisfied. When t is small enough, the solution of 1.1 and 1.3 is u, + J 1 + u 1, 1 + R 1 + u m, m + J 2 + u 2, 2 + S 2 + u +, +. Let us first consider the interaction between R 1 and J 2 and use the following lemma to depict it. Lemma 3.4. The rarefaction wae R 1 passes through J 2 and then a transmitted rarefaction wae denoted by R 2 is generated during the process of penetration. If u + > u m, then the rarefaction wae slows down across J 2. On the contrary, if u + < u m, then it speeds up across J 2. Proof. The propagation speed of J 2 is τ 2 = 0 and those of the characteristic lines in R 1 are ξ 1 u, = k1 + u 1 + u + 2 > 0, where the states u, in R 1 ary from u 1, 1 to u m, m. It is obious that the rarefaction wae R 1 can across J 2 absolutely. In addition, J 2 interacts with the wae front of R 1 at the point which is determined by { x 1 = ε, which yields x 1 + ε = ξ 1 u m, m t 1, x 1, t 1 = ε, 2ε1 + u m + m k1 + u m On the other hand, the intersection of J 2 and the wae back of R 1 can also be calculated by x 2, t 2 = ε, 2ε1 + u k1 + u 1 Now, let us compare the propagation speeds of rarefaction waes before and after penetration when across J 2. The state u, in R 1 becomes the matched one u +, + in R 2 when acrosses J 2, which should satisfy + u = u The propagation speeds of the matched characteristic lines in R 1 and R 2 can be calculated respectiely by Then, we hae ξ 1 u, ξ 2 u +, + ξ 1 u, = k1 + u 1 + u + 2, ξ 2u +, + = k1 + u+ 1 + u = k 1 + u 1 + u u u u u = k [1 + u 1 + u u ]1 + u [1 + u u u + ]1 + u u u = k[1 + u 1 + u u ]u u 1 + u u

11 P. Ji, C. Shen, J. Nonlinear Sci. Appl , = k1 + u u u 1 + u u + + +, in which 3.12 has been used. If u + > u m, then we hae u + > u and + >, such that ξ 1 u, > ξ 2 u +, +, which means that R 1 decelerates when it passes through J 2. Otherwise, if u + < u m, then we hae u + < u and + <, such that ξ 1 u, < ξ 2 u +, +, which means that R 1 accelerates when it passes through J 2. Thus, the conclusion of the lemma can be drawn J 1 R 1 m 2 J 2 S 2 u t J 1 J 2 R 1 3 R S 2 + m ε ε Figure 4: The interaction between J + R and J + S is shown when u + < u m and + u + +1 > S 3 S 4 u +1 > x m. u m+1 On the other hand, the rarefaction wae R 2 continues to moe forwards and penetrates the shock wae S 2 which can be summarized in the following lemma. Lemma 3.5. If + u + +1 < + u + +1 > u +1, then S 2 has the ability to cancel the whole R 2 thoroughly. Otherwise, if u +1, then S 2 penetrates R 2 partially and eentually takes the characteristic line x = ε+ in R 2 as its asymptotic line. Proof. The propagation speed of S 2 and that of the wae front in R 2 are computed respectiely by σ 2 = k1 + u u u + + +, ξ 2u 2, 2 = k1 + u u Noticing that u 2 = u + and + > 2, it is easy to know ξ 2 u 2, 2 σ 2 = k1 + u u u > 0. k1+u +t 1+u Equialently, the wae back of R 2 catches up with the shock wae S 2 in finite time. In fact, the intersection is determined by { x3 ε = ξ 2 u 2, 2 t 3 t 1, x 3 ε = σ 2 t 3, in which x 1, t 1 is gien by 3.11, which enables us to hae x 3, t 3 = ε + 2ε1 + u m + m, 2ε1 + u u m + m k1 + u m + 2 After the time t 3, the shock wae starts to penetrate R 2 with arying propagation speeds and is labeled by S 3 during the process of penetration. The cure of S 3 may be determined by dx dt = k1 + u u u + + +, x ε = k1 + u+ 1 + u t t, 2 xt 3 = x 3,

12 P. Ji, C. Shen, J. Nonlinear Sci. Appl , in which t changes from t 1 to t 2 and u +, + is the corresponding state that the characteristic line starting from the point ε, t in R 2 arries at S 3. For our knowledge, it is impossible to calculate the explicit form for the cure of S 3 due to the fact that R 2 is a non-centered rarefaction wae. Depending on the relation between + u + +1 and + u +1, there are two possible situations as follows: a If u + +1 > u +1, then S 3 is able to cancel the entire R 2 thoroughly see Figure 4. The shock wae is denoted with S 4 after penetration whose propagation speed is gien by b If + u + +1 < k1+u +t 1+u line x = ε + Case 4. J + R and J + R. σ 4 = k1 + u 1 + u u u +1, then S 2 cannot penetrate the whole R 2 completely and at last has the characteristic associated with the state u 3, 3 in R 2 as its asymptotic line. In the end, we consider the situation that there are both J + R originating from ε, 0 and ε, 0. This case arises when + u < m u m + 1 < u + 1 is satisfied. The solution of 1.1 and 1.3 for sufficiently small t may be symbolized as u, + J 1 + u 1, 1 + R 1 + u m, m + J 2 + u 2, 2 + R 2 + u +, +. t J 1 J 2 3 R3 1 0 J R R 2 1 J 2 + m u 1 R 1 2 R 2 m + 0 x 0 x Figure 5: The interaction between J + R and J + R is shown when + u + +1 < m < u m+1 u and u+ > um. +1 Similar to that in Case 3, the forward rarefaction wae R 1 penetrates J 2 at a time. This penetration also gies rise to a transmitted rarefaction wae R 3. In addition, the propagation speed will change and obey the same rule in Lemma 3.4 when the rarefaction wae R 1 crosses the contact discontinuity J 2. On the other hand, the wae front of R 3 is parallel to the wae back of R 2, such that these two waes R 2 and R 3 cannot interact with each other see Figure The generalized Riemann problem with delta-type initial data In this section, we draw our attention on the generalized Riemann problem for the system 1.1 with the delta-type initial data 1.5. In order to construct the solution of the generalized Riemann problem 1.1 and 1.5, we should also consider the particular Cauchy problem for the system 1.1 with the following perturbed Riemann initial data u, x, 0 = u,, < x < ε, u 0, m, 2ε ε < x < ε, u +, +, ε < x < +, 4.1

13 P. Ji, C. Shen, J. Nonlinear Sci. Appl , in which ε> 0 is sufficiently small. Then, the solutions of 1.1 and 1.5 can be obtained by letting ε 0 in the solutions of 1.1 and 4.1. Obiously, is much bigger than u +1 as well as + u + +1 for ε sufficiently small. Proided that m 2ε1+u 0 ε is a sufficiently small positie number, the Riemann solution emitting from ε, 0 is always a contact discontinuity J 1 followed by a shock wae S 1 and the Riemann solution emitting from ε, 0 is always a contact discontinuity J 2 followed by a rarefaction wae R 2, respectiely. As in Case 2, S 1 interacts with J 2 at the point determined by 3.2, which yields x 1, t 1 = ε, 2ε1 + u u 0 + m 2ε. 4.2 k1 + u At the point x 1, t 1, a new Riemann problem with the initial data u 1, 1 = u 0, 1 + u 0, u 2, 2 = 1 + u u +, m1 + u + 2ε1 + u 0, 4.3 is formed. Analogously, the Riemann solution is also a contact discontinuity J 2 followed by a shock wae S 3, in which the intermediate state u 3, 3 is gien by u 3, 3 = u +, 1 + u u After that, S 3 begins to penetrate the rarefaction wae R 2 at the point which is determined by 3.5, which together with 4.2 gies x 2, t 2 = k1 + u 2 t 1 ε u , 1 + u t Making use of the relation between u +1 and + u + +1, there are also two possibilities which are similar as that in Lemma 3.3. Besides, when u +1 < + u + +1, S 4 is able to cancel R 2 entirely and the process ends at the point gien by 3.9, where the intermediate states are gien by 4.3 and 4.4. It follows from 4.2 and 4.5 that lim x 1, t 1 = 0, m1 + u +, limx 2 = ε 0 k1 + u ε 0 By making use of 3.5, we hae t 1 σ 3 ξ 2 u 2, 2 lim = lim ε 0t 2 ε 0 σ 3 = lim u 01 + u + ε u 1 + u 0 + m 2ε = On the other hand, taking into account 4.3 and 4.4, it follows from 3.4 that lim σ k1 + u 1 + u 0 3 = lim ε 0 ε u u u 0 + m = ε For conenience, let us denote t = m1+u + k1+u. It is clear to see from 4.6 and 4.7 that both the points x 1, t 1 and x 2, t 2 will tend to the same point 0, t in the limit ε 0 situation. Furthermore, it is obsered from 4.8 that the shock wae S 3 is also compressed at the point 0, t in the limit ε 0 situation. Thus, the shock wae S 4 starts to propagate from the point 0, t in the limit ε 0 situation. On the other hand, it follows from 3.9 that m1 + lim x u+ 1 + u 1 + u + 3, t 3 = ε u + u + 2, m1 + u 1 + u u k + + u + u + 2.

14 P. Ji, C. Shen, J. Nonlinear Sci. Appl , It follows from 3.7 that the tangent slope of S 4 at the point x 2, t 2 can be calculated by dx dt = k1 + u 3 x ε x2,t u t x2,t2 = k1 + u 1 + u u u u 0 + m 2ε, dx such that we hae lim ε 0 dt x 2,t 2 = 0. That is to say, the shock cure S 4 is tangent to the t axis at the point 0, t in the limit ε 0 situation. Finally, it can be seen from 3.10 that the propagation speed of S 5 is unchanged in the limit ε 0 situation. t t J 1 J 2 3 S 5 S 4 J 3 S 4 S 5 S 3 R 2 0, t 1 2 S 1 + δj m x ε ε 0 Figure 6: When < + u +1 u +, the solution of the particular Cauchy problem 1.1 and 4.1 is shown for the gien sufficiently +1 small ε on the left-hand side and the solution of the generalized Riemann problem 1.1 and 1.5 is shown which is the limit ε 0 of the solution of 1.1 and 4.1 on the right-hand side. R 2 + x Now, let us consider the limit ε 0 situation for the rarefaction wae R 2. First of all, the propagation speed of the wae front of R 2 is also ξ 2 u +, + = k1+u + 1+u which remains unchanged after taking the 2 limit ε 0. On the other hand, about the wae back of R 2, we hae lim ξ k1 + u 0 2 2u 2, 2 = lim ε 0 ε u u 0 + m = 0, 2ε 2 which means that the wae back of R 2 coincides with the t axis. In the end, let us turn our attention on the mass accumulation on the t axis in the limit ε 0 situation. Let us use x 1 t and x 2 t to denote the cures of S 1 and the wae back of R 2, which are expressed respectiely by x 1 t = k1 + u t 1 + u u 0 + m 2ε ε, x 2t = In what follows, let us calculate the mass of in the region ε, ε as below β ε t = = x1 t ε x1 t ε 1 dx + ε x 1 t 1 + u u dx + m dx + ε x 1 t x2 t ε m 2ε dx + 2 dx x2 t ε 1 + u + m 1 + u 0 2ε dx k1 + u 0 2 t 1 + u u 0 + m 2ε + ε. = 1 + u 0 x 1 t + ε + m 1 + u 2ε ε x 1t u + m 1 + u 0 2ε x 2t ε = 1 + u 0 k1 + u t 1 + u 1 + u u 0 + m 2ε + m 2ε k1 + u t 2ε 1 + u u 0 + m 2ε u + m 1 + u 0 2ε k1 + u 0 2 t 1 + u u 0 + m 2ε, which enables us to hae lim ε 0 β εt = m k1 + u t 1 + u +.

15 P. Ji, C. Shen, J. Nonlinear Sci. Appl , Thus, we can see that the two contact discontinuities J 1 and J 2 coalesce into the delta contact discontinuity δj on the t axis before the time t in the limit ε 0 situation. But the strength of the delta contact discontinuity δj decreases linearly at the rate k1+u 1+u + and becomes zero at the point 0, t. Thus, the delta contact discontinuity δj degenerates to be the contact discontinuity J after the time t. 1+u < + If 1+u +, then the shock wae S 4 is able to penetrate R 2 completely in the limit ε 0 situation see Figure 6. Otherwise, if 1+u > + 1+u +, then the shock wae S 4 cannot penetrate R 2 completely and eentually takes the characteristic line x = k1+u +t 1+u in R 2 2 as its asymptotic line in the limit ε 0 situation. Thus, we can obtain the solutions of the generalized Riemann problem 1.1 and 1.5 by taking the limit ε 0 of the solutions to the particular Cauchy problem 1.1 and 4.1. Furthermore, it is easily seen that the solutions of the generalized Riemann problem 1.1 and 1.5 also conerge to the corresponding solutions of the Riemann problem 1.1 and 1.4 when the limit m 0 is taken. 5. Conclusion So far, we hae finished the discussion for all kinds of wae interactions and the global solutions of the perturbed Riemann problem 1.1 and 1.3 hae been constructed completely. It is clear to see that the large-time asymptotic states of the global solutions to the perturbed Riemann problem 1.1 and 1.3 are exactly the corresponding Riemann solutions of 1.1 and 1.4 and the asymptotic behaiors of the solutions to the perturbed Riemann problem 1.1 and 1.3 are goerned completely by the Riemann initial data u ±, ±. Thus, the Riemann solutions are stable with respect to the particular small perturbations 1.3 of the Riemann initial data 1.4 and the proof of Theorem 1.1 has been finished in iew of the aboe analysis. In addition, the generalized Riemann problem for the system 1.1 with the delta-type initial data 1.5 can also be considered by irtue of the solutions of the perturbed Riemann problem 1.1 and 4.1 and then the delta contact discontinuity is captured and obsered clearly. Acknowledgement This work is partially supported by National Natural Science Foundation of China , and Shandong Proincial Natural Science Foundation ZR2014AM024. The authors are ery grateful to the two anonymous referees for their suggestions and corrections which hae improed the original manuscript greatly. References [1] L. Ambrosio, G. Crippa, A. Figalli, L. V. Spinolo, Some new well-posedness results for continuity and transport equations, and applications to the chromatography system, SIAM J. Math. Anal., , [2] D. Băleanu, R. R. Nigmatullin, Linear discrete systems with memory: a generalization of the Langmuir model, Open Phys., , [3] M. C. Băleanu, R. R. Nigmatullin, S. Okur, K. Ocakoglu, New approach for consideration of adsorption/desorption data, Commun. Nonlinear Sci. Numer. Simul., , [4] D. Băleanu, Y. Y. Okur, S. Okur, K. Ocakoglu, Parameter identification of the Langmuir model for adsorption and desorption kinetic data, Nonlinear Complex Dyn., Springer, New York, 2011, [5] C. Bourdarias, M. Gisclon, S. Junca, Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases, J. Math. Anal. Appl., , [6] A. Bressan, Hyperbolic systems of conseration laws, The one-dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, Oxford Uniersity Press, Oxford, [7] T. Chang, L. Hsiao, The Riemann problem and interaction of waes in gas dynamics, Pitman Monographs and Sureys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, [8] H. J. Cheng, H. C. Yang, Delta shock waes in chromatography equations, J. Math. Anal. Appl., , [9] C. M. Dafermos, Hyperbolic conseration laws in continuum physics, Third edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin,

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