Viscous Regularization of Delta Shock Wave Solution for a Simplified Chromatography System
The delta shock wave for a simplified chromatography system is obtained in the Riemann solution when . In fact, the result in this paper is the reasonable generalization of the result in Sun (2011) with which is exactly the critical situation for . The self-similar viscosity vanishing approach is also used to check the delta shock wave solution.
The theory of nonlinear chromatography system is the foundation of chromatography separation process and also plays an important role in the modern industry. Recently, the delta shock wave has been captured numerically and experimentally by Mazzotti et al. [1, 2] in the Riemann solutions for the local equilibrium model of two-component nonlinear chromatography. Since then, extensive attention has been drawn on the formation of delta shock wave solution to all kinds of chromatography systems [3–6].
In , Sun considered the Riemann problem for a simplified chromatography system: with the initial data where and are all given constants. She discovered that a delta shock wave solution also appears in the situation by employing the self-similar viscosity vanishing approach.
In the present paper, we extend the result from the critical situation in  to the general situation and discover that the delta shock wave also appears in the Riemann solution to (1) and (2). Actually, in the situation , the Riemann solution to (1) and (2) is a delta shock wave connecting two constant states whose propagation speed and strength are, respectively, On the other hand, in the situation in , the propagation speed and strength of delta shock wave are, respectively, If we let in the situation , then the result is identical with that for the situation in . Thus, one can see that the situation is the critical one, where the delta shock wave can be obtained, and the result in this paper is the reasonable generalization of the result in .
In addition, we emphasize that our result here is also reasonable in the sense of chromatography engineer since a simplified chromatography system can also be derived from the other nonlinear chromatography system: by introducing the change of variables which is different from the derivation in . Here , are the nonnegative functions of the variables to denote the Langmuir-like and the anti-Langmuir-like components, respectively, which express transformations of the concentrations of two solutes. It is clear that the delta measure appears in for the system (1) is equivalent to saying that the delta measures appear in both the state variables and for the system (6). It is easily derived that the strengths of the delta measures in and are equal to each other. In other words, the phenomena of singular concentrations in the two components and appear simultaneously for the system (6).
The system (1) is the simplest example for systems of Temple class . In other words, the shock curves coincide with the rarefaction curves in the phase plane. Compared with general systems of conservation laws, well-posed results for systems of Temple class  are available for a much larger class of initial data due to the above feature. Furthermore, thanks to the particular structure, wave interactions for the systems of Temple class can also be dealt with completely [9–12], which enables us to solve the Cauchy problem for arbitrary data of bounded variation.
The self-similar viscosity vanishing approach was first proposed by Dafermos  to construct Riemann solutions for broad classes of systems that include the equations of isentropic, Lagrangian, and gas dynamics. This method was widely used in  for the admissibility of weak solutions of the Riemann problem for hyperbolic systems of conservation laws, and the admissibility criterion was called the wave-fan criterion in . Special attention was also paid in [6, 16–22] to the formation of the delta shock waves in the Riemann solutions for some systems of hyperbolic conservation laws.
There exist numerous excellent papers for the related equations and results about the measure-valued solutions for hyperbolic systems of conservation laws. The well-known examples are the transport equations [21, 23–27] (or called the pressureless Euler equations) and the Chaplygin gas dynamics equations [28, 29] which have been extensively studied. For the other related equations and results, we can refer to [30–35] and the related references therein.
The paper is organized as follows. In Section 2, the Riemann problem of (1) and (2) is considered. In particular, the delta shock wave does occur in the Riemann solution when . In Section 3, we show that there exists a smooth solution for the regularized system (5) with the initial data (2). Then, by letting , we prove that the limit of the regularized solution is exactly the corresponding Riemann solution to (1) and (2) when .
In this section, we generalize the Riemann problem for (1) and (2) from the phase space in  to the phase space . Compared with the result in  that there are three kinds of the Riemann solutions to (1) and (2) in the phase space , there exist six kinds of the Riemann solutions to (1) and (2) in the phase space (Figure 1). In what follows, we first depict the properties of (1) and then list the Riemann solutions to (1) and (2) in the phase space case by case. Finally, the generalized Rankine-Hugoniot relations of delta shock wave are derived in detail. We can also refer to [3, 4, 12] for the related results about the other nonlinear chromatography systems.
The characteristic eigenvalues of (1) are and , respectively. Thus, it is clear to see that (1) is strictly hyperbolic in the phase space and and is nonstrictly hyperbolic on the line . The corresponding right characteristic vectors are and , respectively. Thus is genuinely nonlinear for , and is always linearly degenerate. Therefore, the associated waves are rarefaction waves or shock waves for the characteristic field and contact discontinuities for the characteristic field . The Riemann invariants along the characteristic fields are and , respectively.
Now we are in a position to deliver the Riemann solutions to (1) and (2) according to the value of as follows.(1)If , then the Riemann solution of (1) and (2) can be expressed as :in which the propagation speed of is for from to such that and that of is .(2)If , then the Riemann solution of (1) and (2) can be expressed as : in which the propagation speed of is and that of is . It is clear to see that .(3)If , then the Riemann solution of (1) and (2) can be expressed as : It should be emphasized that the state in is determined by which is on the left of the line ; otherwise, the state in is determined by which is on the right of the line .(4)If , then the Riemann solution of (1) and (2) can be expressed as : in which the propagation speed of is and that of is . Here we notice that .(5)If , then the Riemann solution of (1) and (2) can be expressed as :in which the propagation speed of is and that of is for from to .(6)For the case , motivated by the result in , we consider that the Riemann solution to (1) and (2) is a delta shock wave connecting the two constant states and . In order to deal with it, we need the following definition of a delta shock wave solution in the framework introduced in  and used in [12, 37].
Let us suppose that is a graph in the closed upper half-plane , containing Lipschitz continuous arcs with , in which is a finite index set. Let us also suppose that is the subset of containing all indices of arcs linking to the -axis and is the set of initial points of the arcs with .
Definition 1. Let be a pair of distributions, where is represented in the form in which and the singular part should be defined by Let us consider the initial data of the form in which ; then the earlier pair of distributions are called a generalized delta shock wave solution of the simplified chromatography system (1) with the initial data (15) if the following integral identities hold for all test functions , in which denotes the tangential derivative of a function on the graph and denotes the line integral along the arc .
Theorem 2. For the case , the Riemann problem (1) and (2) has the piecewise smooth Riemann solution in the form where The measure solution (17) with (18) also satisfies the generalized Rankine-Hugoniot condition as follows:
Proof. Let us check that the -measure solution (17) with (18) obeys (1) in the sense of distributions. In other words, we only need to check that (17) with (18) satisfies
Actually, we can verify the first equation in (20) as follows: in which the fact that is compactly supported in the region has been used.
Consequently, for the second equation of (20), we have in which the changes of variables and integration by parts have been used.
Through the earlier verification, we can conclude that (17) with (18) is indeed the piecewise smooth Riemann solution to (1) and (2) in the sense of distributions.
Remark 3. If we take the limit in the Riemann solution (17) with (18), then we can obtain the result which is identical with that in . In other words, the Riemann solution to (1) and (2) for can be seen as the critical situation for . Thus, the result in  can be seen as the special situation in this paper.
Remark 4. If we take the limit in the Riemann solution (17) with (18), then both the propagation speed and strength of the delta shock wave will tend to . In fact, both and tend to if we let on the right. Thus, the line in the phase space can be seen as a singular curve.
Remark 5. The Riemann solution to (1) and (2) can be constructed simply if we take the Riemann initial data (2) in the phase space . However, it is difficult to construct the global Riemann solution to (1) and (2) if we require the Riemann initial data (2) in the phase space . In this situation, we at least need to make some criterions on the line .
3. Viscous Regularization of Delta Shock Wave Solution
Performing the self-similar transformation , we get the boundary value problem: with the boundary conditions
With the similar deduction and calculation in Theorem 4.1 in , we can also obtain the same conclusion as follows.
Theorem 6. For each fixed , there exists a unique and monotonic smooth solution of the boundary value problem (25) for .
Now, we turn our attention to the second equation in (23) and consider the existence of the solution . Based on Theorem 6, also with the similar deduction and calculation in Theorem 4.2 in , we can obtain the following theorem.
Theorem 7. For each fixed , assume that is a weak solution of the following boundary value problem:
where is the solution of (25).
Then, one has which can be expressed aswhere is the unique fixed point of .
It follows from Theorem 6 that is strictly monotonically increasing with respect to when , which enables us to see that both and are strictly monotonically decreasing with respect to when . Thus, there exist unique fixed points of the following equations, and we use and to express as respectively. Consequently, we take the limits and , which will play very critical roles in the following discussion.
Lemma 8. For any , draws the conclusion that is in uniform in the earlier intervals.
Proof. It is easy to take sufficiently small such that for . If we take , then it is clear that . From Theorem 6, we know that if . Then (25) can be written as
By integrating (31) over , we arrive at
Integrating (32) over , we get
Because is monotonically increasing in the real axis, we have and . Thus, one can conclude thatfor and is a constant independent of , in which the change of variables has been used. So, we can get . Together with (32), we know that In view of (29), we have noticing that, by applying the mean value theorem, and then there exists such that So, we get the following inequality: which means that uniformly on .
For any , we have Obviously, we know that uniformly on in view of (38).
Now, we try to prove that uniformly on . Similarly, it is easy to take sufficiently small such that . If we take , then we also have . By integrating (25) over , we have Integrating (25) over , we have Thus, we get the following result:for and is a constant independent of , in which the change of variables has also been used. Thus we have , together with (40), and we also know that Similarly, by applying the mean value theorem, then there exists such that Combining (43) together with (44), we get which implies that uniformly on .
Thus, for any , we have Obviously, we can see that uniformly on in view of (45).
Lemma 9. Let and be defined as above. Then, one has
Proof. Take any where . It follows from (25) that
By employing integration by parts and applying the fact that is compactly supported in , we have Taking the limit in (49), it can be obtained from Lemma 8 that
Through simplifying (50), we get which leads to for the arbitrary of .
Let us turn back to calculate ; we have
It is clear that for . Furthermore, if we assume that , through taking here, then we can take sufficiently small such that for . It can be derived from Lemma 8 that . Thus, we arrive at by applying (52), which contradicts the assumption .
On the other hand, if we assume that , by taking now, then we can also take sufficiently small such that . It can be derived from Lemma 8 that . Thus, we arrive at by applying (52), which also contradicts the assumption .
From the above discussion, we can get the relation . The proof is completed.
Lemma 10. For any , one can obtain uniformly in the above intervals.
The process of this proof is similar to that of Lemma 5.3 in , and thus we omit it.
Proof. To prove it, we should consider the limit behavior of in the neighborhood of . We take with and satisfy on the neighborhood of such as for sufficiently small , which is called the sloping test functions in .
Then, by (26), we have
It follows from for that
By taking the limit above, we have in which denotes the Heaviside function.
Combining (54) together with (56), we have for all sloping test functions . With the same technique in [5, 22], one can see that the conclusion can also be drawn for any test function .
By taking the limits and in (57), we can obtain Thus the limit of is exactly the corresponding Riemann solution (17) with (18) for when .
Finally, the value of at the discontinuity point should be calculated. Then, (26) can be changed into the following form:
By substituting (58) into (59), through a simple calculation, one can arrive at namely, Since is arbitrary, one can obtain , which implies that .
This work is supported by the National Natural Science Foundation of China (11001116, 11271176) and the Project of Shandong Provincial Higher Educational Science and Technology Program (J11LA03, J12LI01) and Shandong Provincial Natural Science Foundation (ZR2010AL012).
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