Interactions of delta shock waves for the chromatography equations

Applied Mathematics Letters 26 (2013) 631–637

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Interactions of delta shock waves for the chromatography equations✩ Meina Sun ∗ School of Mathematics and Information, Ludong University, Yantai, Shandong Province, 264025, PR China

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Article history: Received 19 November 2012 Accepted 7 January 2013 Keywords: Chromatography equations Delta shock wave Riemann problem Wave interaction

abstract This paper is devoted to the interactions of the delta shock waves with the shock waves and the rarefaction waves for the simplified chromatography equations. The global structures of solutions are constructed completely if the delta shock waves are included when the initial data are taken three piece constants and then the stability of Riemann solutions is also analyzed with the vanishing middle region. In particular, the strength of delta shock wave is expressed explicitly and the delta contact discontinuity is discovered during the process of wave interaction. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction The chromatography model theory is the theoretical basis on the study of chromatographic separation process and has the important application in 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

   a1 u1   + ∂x u1 = 0, ∂t u1 + 1 − u1 + u2   a2 u2   + ∂x u2 = 0, ∂t u2 + 1 − u1 + u2

(1.1)

with certain Riemann initial data, in which u1 and u2 are the concentrations of the two absorbing species with u1 , u2 ≥ 0, 1 − u1 + u2 > 0 and a2 > a1 > 0. In order to make the detailed study on the delta shock wave for the nonlinear chromatography system, we are concerned with a simplified chromatography system

   v   = 0, v +  t 1 + v x w   = 0, w t + 1+v x

(1.2)

✩ This work is supported by National Natural Science Foundation of China (11271176), Shandong Provincial Doctoral Foundation (BS2010SF006) and the Project of Shandong Provincial Higher Educational Science and Technology Program (J12LI01). ∗ Tel.: +86 535 6697510; fax: +86 535 6681264. E-mail address: [email protected].

0893-9659/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2013.01.002

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M. Sun / Applied Mathematics Letters 26 (2013) 631–637

which can be derived from

   u1   = 0, (u1 )t +  1 + u1 + u2  x u2   = 0, (u2 )t + 1 + u1 + u2 x

(1.3)

by introducing the change of state variables

v = u1 + u2 ,

w = u1 − u2 .

(1.4)

The system (1.2) belongs to Temple class, i.e., the shock curves coincide with the rarefaction curves in the phase plane due to the special form [3,4]. Thanks to the above feature, well-posed results for Temple systems are available for a much larger class of initial data compared with general systems of conservation laws. Furthermore, the Riemann problem for these equations can be explicitly solved in the large and wave interactions (for example [5]) have a simplified structure. Recently, Ambrosio et al. [6] have introduced the change of state variables (1.4) and then studied (1.2) as an example by employing new well-posedness results for continuity and transport equations. The delta shock wave has been already obtained in the Riemann solutions to (1.2) by the method of vanishing viscosity in [7]. In addition, the wave interactions for (1.3) have been considered in [8] but not including the delta shock waves. Also see [9–11] for the related results of the delta shock wave for the other chromatography system. In this note, we are mainly concerned with the interactions of the delta shock waves with the shock waves and the rarefaction waves for (1.2). Thus, we consider the initial value problem of (1.2) with the following three pieces constant initial data:

 (vl , wl ), (v, w)(x, 0) = (vm , wm ), (vr , wr ),

− ∞ < x < −ε, − ε < x < ε, ε < x < +∞,

(1.5)

where ε > 0 is arbitrarily small. Here we restrict ourselves only to consider the situation vl,m,r ≥ 0. In fact, the initial data (1.5) can be regarded as a local perturbation of the corresponding Riemann initial data (2.1) below. In the following, we will also encounter the interesting question of determining whether the Riemann solutions of (1.2) and (2.1) are the limits of (vε , wε )(x, t ) as ε → 0 who are the solutions of (1.2) and (1.5) for given ε > 0. The novelty of this note mainly comes from the following three observations. At first, a delta shock wave will be decomposed into a shock wave and a delta contact discontinuity when it penetrates over a rarefaction wave in some certain situation. Secondly, by using the method of splitting delta function proposed by Nedeljkov [12–14], we discover that the strength of the delta shock wave only comes from the left-hand side and can be computed explicitly during the interaction process. Finally, compared with the other chromatography systems where the singular measure appears in multiple state variables and wave interactions have more complicated structures [9,1,2,11], here the singular measure only appears in the state variable w for (1.2) which is more easy to be dealt with, which enables us to see that the change of variables is a powerful tool for the chromatography system. 2. Preliminaries In this section, we restate the Riemann problem for (1.2) (see [7]). The Riemann initial data are

(v, w)(x, 0) =



(vl , wl ), (vr , wr ),

x < 0, x > 0,

(2.1)

where vl,r and wl,r are all given constants. 1 1 The characteristic roots of (1.2) are λ1 = (1+v) 2 and λ2 = 1+v . Thus, (1.2) is strictly hyperbolic in the half-plane of the phase space v > 0 and non-strictly hyperbolic on the borderline v = 0. The corresponding right characteristic vectors are − → − → r1 = (v, w)T and r2 = (0, 1)T respectively. Thus λ1 is genuinely nonlinear for v ̸= 0 and λ2 is always linearly degenerate. Therefore, the associated waves are rarefaction waves R or shock waves S for the first family and contact discontinuities J for the second family. The Riemann invariants along the characteristic fields are wv and v . If vl > vr , then the Riemann solution of (1.2) and (2.1) can be expressed as R + J:

 (vl , wl ),         t w t l   − 1, −1 ,  v x (v, w)(x, t ) =  x  l  vr w l   ,  vr ,   vl  (vr , wr ),

x < λ1 (vl )t ,

λ1 (vl )t ≤ x ≤ λ1 (vr )t , (2.2)

λ1 (vr )t < x < τ t , x > τ t.

M. Sun / Applied Mathematics Letters 26 (2013) 631–637

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Fig. 1. Interaction of R + J and δ S, left for 0 = vm < vl < vr and right for 0 = vm < vr < vl .

If 0 < vl < vr , then the Riemann solution of (1.2) and (2.1) can be expressed as S + J:

 (v , w ),   l vl w  r l (v, w)(x, t ) = vr , ,  vl  (vr , wr ),

x < σ t,

σ t < x < τ t,

(2.3)

x > τ t,

1 in which the propagation speed of S is σ = (1+v )(1 1+v ) and that of J is τ = 1+v . r r l In particular, if 0 = vl < vr , then S and J coincide to form a delta shock wave (see such as [15–24] for the related study). Hence the Riemann solution containing a weighted δ -measure supported on a line should be constructed. In order to define the measure solutions, for any test function ψ(x, t ) ∈ C0∞ (R × R+ ), the two-dimensional weighted δ -measure β(s)δΓ supported on a smooth curve Γ = {(x(s), t (s)) : a < s < b} can be defined by

⟨β(s)δΓ , ψ(x, t )⟩ =

b



β(s)ψ(x(s), t (s))ds.

(2.4)

a

Lemma 1 ([7]). If 0 = vl < vr , then the Riemann problem (1.2) and (2.1) has the piecewise smooth solution as

(0, wl ), (v, w)(x, t ) = (vδ , β(t )δ(x − σδ t )), (vr , wr ), 

x < σδ t , x = σδ t , x > σδ t ,

(2.5)

in which

vδ = vr ,

σδ =

1 1 + vr

,

β(t ) =

vr wl t 1 + vr

.

(2.6)

In addition, the measure solution (2.5) with (2.6) satisfies the generalized Rankine–Hugoniot condition dx dt

= σδ ,

dβ(t ) dt

= σδ [w] −



 w , 1+v

σδ [v] =



 v , 1+v

(2.7)

where [w] = w(x(t ) + 0, t ) − w(x(t ) − 0, t ) denotes the jump of w across the discontinuity x = x(t ), etc. In order to ensure uniqueness, the measure solution should also satisfy the δ− entropy condition: λ1r < λ2r = σδ < λ1l = λ2l . 3. Interactions of the delta shock waves In order to cover all the cases, our discussion should be divided into two parts according to the appearance of the delta shock wave or not. If the delta shock wave does not appear in the solutions of Cauchy problem (1.2) and (1.5), then the interactions are classical and well known hence will not be pursued here. On the other hand, our discussion is divided into the following three cases according to the different wave combinations from (−ε, 0) and (ε, 0): 1. R + J and δ S, 2. δ S and S + J, 3. δ S and R + J. Case 1. R + J and δ S In this case, we consider the interaction of R + J starting from (−ε, 0) and δ S starting from (ε, 0). The occurrence of this case depends on the condition vm = 0 and vl , vr > 0 (see Fig. 1). For vl > vm = 0, R + J is a composite wave, namely J is coincident with the wavefront of R since both of them are w supported on a line x + ε = t. The states on the rarefaction wave R should satisfy wv = v l , thus the left state of J should be l

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M. Sun / Applied Mathematics Letters 26 (2013) 631–637

(0, 0). The propagation speed of J (or the wavefront of R) is τ = 1 and that of δ S is σδ = overtakes δ S in finite time, whose intersection point (x1 , t1 ) is determined by x1 + ε = τ t1

1 1+vr

. Thus, it is easy to see that J

and x1 − ε = σδ t1 ,

(3.1)

which implies that

     2 1 (x1 , t1 ) = ε 1 + , 2ε 1 + . vr vr

(3.2)

The strength of δ S at (x1 , t1 ) can be calculated by

β(t1 ) =

vr wm t1 = 2εwm . 1 + vr

(3.3)

A new local Riemann problem will be formed at the intersection point (x1 , t1 ) as

v|t =t1

 η, = vr ,

 ηwl

x < x1 , x > x1 ,

w|t =t1 =

vl wr ,

,

x < x1 x > x1

 + β(t1 )δ(x1 ,t1 ) ,

(3.4)

1 where we suppose that η > 0 is sufficiently small and satisfies x+ε = (1+η) 2 , namely we assume that R is approximated by t a set of non-physical shock waves [25,26]. In fact, we can construct the solution of the initial value problem (1.2) and (3.4) in the form:

v(x, t ) =

 η, vr ,

x < x(t ), x > x(t ),

 ηwl  ,    l  vvr w l , w(x, t ) =  vl    wr ,

x < x( t ) x(t ) < x < ε + x>ε+

t 1 + vr

t

     

1 + vr  

 + β(t1 )δ x − ε −

t 1 + vr



,

(3.5)

  

where x = x(t ) is the shock wave curve and can be expressed as x(t ) = x1 + (1+η)(11+v ) in the local neighborhood of (x1 , t1 ) r for sufficiently small η > 0. Now we check that (3.5) is indeed the weak solution of the initial value problem (1.2) and (3.4). For every ϕ ∈ C0∞ (R × R+ ), it is obviously true t −t

   v   , ϕ⟩ = 0, ⟨vt + 1 + v x w   , ϕ⟩ = 0, ⟨wt + 1+v x

(3.6)

t if suppϕ ∩ {(x, t )|x = ε + 1+v , t > t1 } = ∅. Otherwise we can see that (3.5) is also the weak solution of (1.2) and (3.4) r near the support of the delta function. The first equation of (1.2) does not contain w and the result is obviously true. Then, by substituting (3.5) into the second equation of (1.2), we can see that

wt +



w 1+v



     vr wl wr vr w l β(t1 ) ′ ′ =− wr − δ + β(t1 )δ + − δ+ δ = 0, 1 + vr vl 1 + vr vl (1 + vr ) 1 + vr 1

x

(3.7)

t holds near the line x = ε + 1+v for t ≥ t1 in the weak (or distributional) sense. r t Thus, we can see that the Dirac delta function is now supported on the contact discontinuity line x = ε + 1+v and r be named as the delta contact discontinuity δ J in [14,27]. When t > t1 , δ S is decomposed into δ J and S and the state 1 (v∗ , w∗ ) = (vr vrvwl l ) lies between them. Consequently, δ J will continue to move forwards with the constant speed 1+v and r the invariant strength β(t1 ). On the other hand, S will continue to penetrate R whose varying states (v, w) are determined by

x+ε =

t

(1 + v)2

,

w wl = , v vl

(0 ≤ v ≤ vl ).

(3.8)

Thus the shock wave curve can be calculated by dx dt

=

1

(1 + v)(1 + vr )

=

1

(1 + vr )



x+ε t

,

(3.9)

M. Sun / Applied Mathematics Letters 26 (2013) 631–637

635

with the initial condition (3.2), which implies that

 √



t

Γ : x(t ) =

1 + vr

+

2

2εvr

− ε.

1 + vr

(3.10)

By differentiating (3.9) with respect to t, in view of (3.10), we have d2 x dt 2

=



1

(1 + vr )

x+ε t

 √

√

− 12 ·

x+ε 2t 2

·

t

1 + vr

√ −

x+ε

 =−



1

(1 + vr )t

εvr < 0. 2(1 + vr )t

(3.11)

It follows from (3.9) and (3.11) that S has the same speed with δ J at (x1 , t1 ) and consequently decelerates during the process of penetration, which implies that S and δ J are tangential at (x1 , t1 ). If vl > vr , then S cannot penetrate R completely and finally takes the line x = (1+vt )2 − ε as its asymptote (see the right r

of Fig. 1). If vl < vr , then S will penetrate the whole R completely in finite time (see the left of Fig. 1) and the intersection of S and the waveback of R is determined by (3.8) and (3.10) where v = vl . After penetration, the shock wave propagates with the invariant speed (1+v )(1 1+v ) . It is clear that β(t1 ) → 0 as ε → 0. Thus, in the limit situation, the delta contact r l discontinuity δ J will degenerate the contact discontinuity J. Thus, we can see that the limit of the solution of (1.2) and (1.5) is R + J if vl > vr or S + J if vl < vr , which is exactly the corresponding Riemann solution of (1.2) and (2.1) in the same situation. Case 2. δ S and S + J In this case, we consider the interaction of δ S emitting from (−ε, 0) and S + J emitting from (ε, 0) when 0 = vl < 1 vm < vr (see the left of Fig. 2). The propagation speed of δ S1 is σδ1 = 1+v and that of S is σ = (1+v 1)(1+v ) . With m m r the same reason as before, δ S1 catches   up with S in finite time. Similarly, we can conclude that the intersection point is

), 2ε(1+vvmr)(1+vr ) and the strength of δ S1 at (x1 , t1 ) is β(t1 ) = A new initial data at (x1 , t1 ) will be formulated as:    v , x < x1 , wl , x < x1 v|t =t1 = l w|t =t1 = + β(t1 )δ(x1 ,t1 ) v∗ , x > x1 , w∗ , x > x1

(x1 , t1 ) = ε(1 +

2

vr

2ε(1+vr )vm wl

vr

.

(3.12)

where (v∗ , w∗ ) = (vr , vrvwm ) is the state between S and J. m A new delta shock wave will be generated after the interaction of δ S1 and S, which will be denoted with δ S2 which can be expressed as



v(x, t ) = vl + (v∗ − vl )H , w(x, t ) = wl + (w∗ − wl )H + β− (t )D− + β+ (t )D+ ,

(3.13)

where H is the Heaviside function and β(t )D = β− (t )D− + β+ (t )D+ is a split delta function. All of them are supported on 1 the same line x = x1 +σδ2 (t − t1 ), namely they are the functions of x − x1 −σδ2 (t − t1 ), in which σδ2 = 1+v is the propagation r

speed of δ S2 . It is remarkable that D− is the delta measure on the set R2+ ∩ {(x, t )|x ≤ x1 + σδ2 (t − t1 )} and while D+ is the delta measure on the set R2+ ∩ {(x, t )|x ≥ x1 + σδ2 (t − t1 )} respectively. It follows from (3.13) that

wt (x, t ) = (−σδ2 (w∗ − wl ) + β−′ (t ) + β+′ (t ))δ − σδ2 (β− (t ) + β+ (t ))δ ′ ,       w∗ wl β− (t ) β+ (t ) ′ w (x, t ) = − δ+ + δ. 1+v x 1 + v∗ 1 + vl 1 + vl 1 + v∗

(3.14) (3.15)

Substituting (3.14) and (3.15) into the second equation of (1.2) and comparing the coefficients of δ and δ ′ , we have

w∗ wl − = 0, 1 + v∗ 1 + vl β− (t ) β+ (t ) −σδ2 (β− (t ) + β+ (t )) + + = 0. 1 + vl 1 + v∗ −σδ2 (w∗ − wl ) + β−′ (t ) + β+′ (t ) +

(3.16) (3.17)

Noticing vl = 0, it is derived directly from (3.16) that

β(t ) = β− (t ) + β+ (t ) = β(t1 ) +

vr wl 2ε(vm − vr )wl vr w l t (t − t1 ) = + , 1 + vr vr 1 + vr

(3.18)

where β(t ) denotes the strength of δ S2 when t > t1 . It is clear to see that β− (t ) = 0 and β+ (t ) = β(t ) from (3.17) to (3.18). For the singular measure only appears in w , if v and w are regarded as the velocity and density in intuition respectively,

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M. Sun / Applied Mathematics Letters 26 (2013) 631–637

Fig. 2. Left: interaction of δ S and S + J for 0 = vl < vm < vr ; Right: interaction of δ S and R + J for 0 = vl < vr < vm .

then the particles on the left will move towards and across the line of the delta shock wave and then gather tightly on the 1 right of the line. It is obvious to see that δ S2 and J have the same propagation speed 1+v and then no interaction happens. It r

v wt

follows from (3.18) that limε→0 β(t ) = 1r+vl which is identical with (2.6). Thus, in the limit situation, δ S2 and J will coalesce r a single delta shock wave and the assert is obviously true. Case 3. δ S and R + J This case is dedicated to the interaction of δ S emitting from (−ε, 0) and R + J emitting from (ε, 0) under the condition 1 0 = vl < vr < vm (see the right of Fig. 2). The propagation speed of δ S1 is σδ1 = 1+v and that of the waveback of R is

ξb =

m

1

(1+vm )2

, so δ S1 keeps up with the waveback of R in finite time. Similarly, we can conclude that the intersection point is

(x1 , t1 ) = ε(1 +

m) ), 2ε(1v+v m

2

2

vm



and the strength of δ S1 at (x1 , t1 ) is β(t1 ) = 2ε(1 + vm )wl .

After the interaction happens, δ S1 enters R and begins to penetrate it. During the penetration process,  we denote it with 

δ S2 and use Γ : {x = x(t )} to express, whose left state is (0, wl ) and right state is (

t x−ε

− 1, wvmm (

t x−ε

− 1)). It follows

from (2.7) that

σδ2 =

dx dt

=

v [ 1+v ]

 =

[v]

x−ε t

,

x(t1 ) = x1 ,

(3.19)

which implies that

Γ : x(t ) =

√

t−



2εvm

2

+ ε.

(3.20)

2

Similar to that in Case 1, we obtain ddt 2x > 0 which means that δ S2 accelerates during the process of penetration. Now, we can construct the delta shock wave supported on the curve Γ : {x = x(t )} in the form

v(x, t ) =

 0,  

   wl ,  t w(x, t ) = wm −1 ,   vm x−ε

x < x(t ), t x−ε

− 1,

x > x(t ),

x < x(t ) 



x > x(t ) 

+ + β− (t )D− Γ + β+ (t )DΓ ,

(3.21)

β− (t )D− Γ

β+ (t )D+ Γ

in which β(t )DΓ = + is a spit delta function supported on Γ and β(t ) = β− (t ) + β+ (t ) denotes the strength of δ S2 . Then, it follows from (3.21) that

      wm t wm 2 ′ ′ wt (x, t ) = H + −σδ − 1 − wl + β− (t ) + β+ (t ) δ − σδ2 (β− (t ) + β+ (t ))δ ′ , √ vm x−ε 2vm (x − ε)t (3.22)



w



1+v

(x, t ) = x

−wm H+ √ 2vm (x − ε)t



wm vm



 1−

x−ε



t





− wl δ + β− (t ) +



x−ε t

 · β+ (t ) δ ′ ,

(3.23)

where all H , δ, δ ′ are the functions of x − x(t ) and the expression of x(t ) is given in (3.20). Similarly, we have



wm −σδ (t ) vm 2



t x−ε







− 1 − wl +

−σδ2 (t )(β− (t ) + β+ (t )) + β− (t ) +



β−′ (t )

x−ε t

+

β+′ (t )

+

· β+ (t ) = 0.

wm vm



 1−

x−ε t



 − wl

= 0,

(3.24)

(3.25)

M. Sun / Applied Mathematics Letters 26 (2013) 631–637

637

When t ≥ t1 , by virtue of (3.19) and (3.20), it follows from (3.24) that

 β(t ) = β− (t ) + β+ (t ) = 2wl 2εvm t − 2ε(1 + vm )wl .

(3.26)

Based on (3.25) and (3.26) together, we also  have β− (t ) = 0 and β+ (t ) = β(t ).

 t − 1)) of δ S2 arrives at (vr , vrvwmm ), which is equivalent to x−ε − 1 = vr ,   2 2εvm 2εvm (1+vr ) and the strength of δ S2 at (x2 , t2 ) can be then δ S2 will across the whole R completely at (x2 , t2 ) = ε + 2 , v v2 When the right state (

t x−ε

− 1, wvmm (

t x−ε

r

r

1 calculated by (3.26). Consequently, it will be denoted with δ S3 which also has the invariant propagation speed 1+v and the r vr wl (t −t2 ) strength can be calculated by β(t ) = β(t2 ) + 1+v . Thus, in the limit ε → 0 situation is also a single delta shock wave. r By letting ε → 0, we see that the Riemann solutions are stable under the local small perturbation of the Riemann initial data (1.2) in the above three cases when the delta shock wave is included. Otherwise, the conclusion is classical when the delta shock wave is not involved. In a word, we can summarize our results in the following theorem.

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