Formation and Transition of Delta Shock in the Limits of Riemann Solutions to the Perturbed Chromatography Equations

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1 Jornal of Mathematical Research with Applications Jan., 29, Vol.39, No., pp.6 74 DOI:.377/j.issn: Formation and Transition of Delta Shock in the Limits of Riemann Soltions to the Pertrbed Chromatography Eqations Xinli HAN, Lijn PAN 2,. School of Science, Nanjing Uniersity of Posts and Telecommnications, Jiangs 223, P. R. China; 2. Department of Mathematics, Nanjing Uniersity of Aeronatics and Astronatics, Jiangs 26, P. R. China Abstract This paper is concerned with the formation and transition of delta shock soltions to the pertrbed chromatography eqations. We discss the Riemann problem for the pertrbed chromatography eqations. By stdying the its of the Riemann soltions as the pertrbation parameter tends to zero, we can obsere two important phenomena. One is that a shock and a contact discontinity coincide to form a delta shock. The second is that the transition from one kind of delta shock on which two state ariables simltaneosly contain the Dirac delta fnction, to another kind of delta shock on which only one state ariable contains the Dirac delta fnction. Keywords chromatography eqations; pertrbation; delta shock; formation; transition MR(2 Sbject Classification 35L65; 35L67; 76L5; 76N. Introdction The nonlinear chromatography eqations can be expressed as x {( }, (. x {( }, where and are non-negatie fnctions of the ariables (x,t R R, which express the concentrations of the two absorbing species, and >. Eq.(. are a common analytical tool to stdy the preparatie separations in the pharmacetical, food, and agrochemical indstries. Yang and Zhang [], Cheng and Yang [2] stdied the Riemann problem of Eq.(. and proed the existence and niqeness of the soltion in recent years. Receied Janary 23, 28; Accepted Jly 7, 28 Spported by the National Natral Science Fondation of China (Grant No. 3264, China Scholarship Concil, China Postdoctoral Science Fondation (Grant No. 23M53343, the Fndamental Research Fnds for the Central Uniersities(Grant No. NZ247, Jiangs Oerseas Research & Training Program for Uniersity Prominent Yong & Middle-Aged Teachers and Presidents, the Natral Science of Jiangs Proince (Grant No. BK * Corresponding athor address: xinlihan@26.com (Xinli HAN; 9849@63.com (Lijn PAN

2 62 Xinli HAN and Lijn PAN Eq.(. can be deried from a more general nonlinear chromatography system x t {( a }, x (.2 t {( a 2 }, where a and a 2 are constants with a 2 > a > ; and are non-negatie fnctions of the ariables (x,t R R, and where >. The difference between (. and (.2 is that system (.2 is hyperbolic in the region (a (a 2 ( 2 4a a 2 ( > and elliptic in the complementary part of it in the (, plane, while (. is always hyperbolic in the whole composition space. A distinctie featre of (. and (.2 is that the delta shock with Dirac delta fnction will appear in both and (see [2]. This fact was also captred nmerically and experimentally by Mazzotti et al. [3, 4] for (.2. This delta shock phenomenon originates in the synergisticcompetitie behaior of the two species as described in [2]. Another system of nonlinear chromatography eqations is introdced in [5 7]. The model reads x (, (.3 x (, where (x,t, (x,t which express transformations of the concentrations of two soltes. System (.3 is widely sed by chemists and engineers to stdy the separation of two chemical components in a flid phase. Different from (. and (.2, the delta shock does not deelop in the soltions of (.3. Recently, Ambrosio et al. [5] hae introdced the change of ariables θ, η, (.4 then the system (.3 can be changed to θ x (θ θ η, η x (η η (.5 η, where η. Becase of the conditions and, the change of ariables (.4 is not one-on-one, which implies that system (.3 and system (.5 are not eqialent. The existence and niqeness of soltions to (.5 are proen by employing the self-similar iscosity anishing approach in [8]. The delta shock appears in the Riemann soltion of (.5. Howeer, for this kind of delta shock, only one state ariable θ contains the Dirac delta fnction and the other η has bonded ariation. From the aboe discssion, one can obsere that the essential difference among the nonlinear chromatography eqations (., (.3 and (.5 is the coefficient ale in front of the absorbing species. The strctres of soltions for these nonlinear chromatography eqations are qite different, in which the delta shock plays an important role.

3 Formation and transition of delta shock 63 Adeltashockisageneralizationofanordinaryshock. Itismorecompressiethananordinary shock in the sense that more characteristics enter the discontinity line. Mathematically, the delta shocks are new type singlar soltions sch that their components contain delta fnctions and their deriaties. Physically, they are interpreted as the process of formation of the galaxies in the nierse, or the process of concentration of particles []. The theory of the delta shock has been intensiely deeloped in the last 2 years. The delta shock soltion and the corresponding Rankine-Hgoniot condition were presented by Zeldoich and Myshkis [9] in the case of the continity eqation. In 999, Sheng and Zhang [] discssed the Riemann problem for the zero-pressre gas dynamics, in which the delta shocks appear. For the preios delta shock, the inestigations hae mostly been focsed on the case that only one state ariable deelops the Dirac delta fnction and the others hae bonded ariations. In 22, Yang and Zhang[] established a new theory of delta shock with Dirac delta fnctions deeloping in two state ariables for a class of nonstrictly hyperbolic systems of conseration laws. As for delta shock, there are nmeros excellent papers, for the related references we can see [,8, 8] and the references cited therein. In the delta shock theories, there are still many open and complicated problems. Stdy of this area gies a new perspectie in the theory of conseration law systems [9,2]. In this paper, we are interested in the research of internal mechanism of the aboe two different kinds of delta shocks, i.e., the formation and transition of two different kinds of delta shocks. For this prpose, we stdy the Riemann problem of the pertrbed nonlinear chromatography eqations x {( }, k (.6 x {( }, k with initial ale (,(x, ( ±, ±, ±x >. (.7 Here k is a pertrbation parameter, and are the non-negatie fnctions of the ariables (x,t R R and k >. ± and ± are constants with (, (,. System (.6 can also be sed to analyze the stability property of the nonlinear chromatography eqations de to the pertrbation on the absorbing species. We show that, as the pertrbation parameter k anishes, there appear three phenomena: ( The transition from one kind of delta shock on which two state ariables and simltaneosly contain the Dirac delta fnction, to another kind of delta shock on which only one state ariable contains the Dirac delta fnction. (2 The formation of delta shock. That is, a shock and a contact discontinity coincide to form a delta shock. (3 The transition from a rarefaction wae to a left contact discontinity. The paper is organized as follows. In Section 2, we present some preinary knowledge abot the pertrbed chromatography Eq.(.6. Then in Section 3, we discss the Riemann problem

4 64 Xinli HAN and Lijn PAN of (.6 and deelop the it behaior of Riemann soltions as the pertrbation parameter k anishes. 2. Preinaries The eigenales of (.6 are λ (,;k with the corresponding left eigenectors and the corresponding right eigenectors By a direct calclation, we hae k, λ 2(,;k (k 2, (2. l (,;k (,, l 2 (,;k (k,, (2.2 r (,;k (, k T, r 2 (,;k (, T. (2.3 λ r, λ 2 r 2 2(k (k 3. (2.4 Therefore, λ is always linearly degenerate, λ 2 is geninely nonlinear if k, and linearly degenerate if k. The Riemann inariants of system (.6 along the characteristic fields are ζ(,;k k, ς(,;k. (2.5 Definition 2. A pair (, constittes a soltion of (.6 in the sense of distribtions if it satisfies (φ t ( k φ xdxdt (φ t ( k φ xdxdt for all test fnctions φ C ((, [,. (x,φ(x,dx, (x,φ(x,dx, Definition 2.2 A two-dimensional weighted delta fnction w(sδ l, spported on a smooth cre l parameterized as x x(s,t t(s (c s d, is defined by w(sδ l,φ d for all the test fnctions φ C ((, [,. c (2.6 w(sφ(x(s, t(sds (2.7 Besidesthe constantsoltion, it iseasytocheckthat there areself-similarwaes. Set ξ x/t. For a gien left state (,, all possible states (, which can be connected to (, on the right by a rarefaction wae mst be located on the following cre ξ λ R(, 2 (,;k : (k 2,, k < k. (2.8

5 Formation and transition of delta shock 65 In particlar, we call it a backward rarefaction wae symbolized by R when < k < k, and a forward rarefaction wae symbolized by R when k < k <. The state which can be connected to a gien left state (, on the right by a contact discontinity mst be located on the cre J(, : σ k k. (2.9 All possible states which can be connected to (, on the right by a shock mst be located on the cre S(, : σ (k(k,. (2. Here we notice that the shock cre coincides with the rarefaction wae cre in the phase plane, i.e., (.6 belongs to Temple class [2]. In particlar, we call it a backward shock symbolized by S when < k < k, and a forward shock symbolized by S when k < k <. 3. Formation and transition In this section, we stdy the formation and transition of two different kinds of delta shocks. We can captre this phenomenon by stdying the it behaior of Riemann soltions of (.6 as the pertrbation parameter k. First, we gie some reslts on the Rieamnn soltions to system (.6. Let the left state (, be fixed, and allow the right state (, to ary. If (, lies on any of the aboe cres (2.8 (2., we hae soled the problem. Assme that (, is off the aboe cres in the rest of the paper. We pt all of these cres together in the plane. Then the plane is diided into different disjoint regions. According to the right state (, in the different region, one can constrct the niqe global Riemann soltion to the problem (.6 and (.7. Lemma 3. The Riemann soltions to (.6 and (.7 are selected throgh the following conditions: ( If k k, the soltion is a delta shock δs; (2 If k < k, the soltion is R J; (3 If < k < k, the soltion is S J; (4 If k < < k, the soltion is R R 2 ; (5 If k < k, the soltion is J R; (6 If k < k <, the soltion is J S. 3.. Formation of delta shock Lemma 3.2 Let <. For arbitrarily small k >, the Riemann soltion of (.6 is

6 66 Xinli HAN and Lijn PAN S J: (,(x,t (,, x < σ(kt, (,, σ(kt < x < λ (, ;kt, (,, x > λ (, ;kt, (3. where the intermediate state can be calclated as (, ( k,, and σ(k k (k (k is the propagation speed of the shock S. Proof De to <, there is a k >, sch that if < k < k, then < k < k. This, together with (2.9, (2. and Lemma 3. yield the lemma. Theorem 3.3 Let <. As k, the Riemann soltion (3. conerges to (,, x < σ δ t, (θ,η(x,t (,(x,t ( ω(tδ(x x(t, k, x σ δ t, (,, x > σ δ t, (3.2 in the sense of distribtions, which forms a δ-shock soltion of (.5 with the same initial data ( ±, ±. Here δ( is the standard Dirac measre, ω(t t and σ δ are strength and elocity of the delta shock x x(t, respectiely (see Figre. t S t δs (, (, J (, (ˆω(tδ(x ˆx(t, (, (, O x O (a k > (b k Figre Formation of delta shock x Proof Letting k, it is obios to see that the intermediate state (, satisfies At the same time, we hae k k k k, k k. (3.3 σ(k λ (, ;k k k. (3.4 That is, the propagation speed of shock S tends to that of contact discontinity J. Combining (3.3 and (3.4, we dedce S and J coincide to form a new type of nonlinear hyperbolic wae, which is called delta shock in [8]. The propagation speed of the delta shock is σ δ.

7 Formation and transition of delta shock 67 It is not hard to proe that the delta shock satisfies the δ-entropy condition λ 2 (, ; λ (, ; σ δ λ (,; λ 2 (,;, which means none of the for characteristic lines on both side of the delta shock x σ δ t is otgoing. Now let s calclate the total qantities of between S and J as k. Since both (.6 and (.7 are inariant nder the transformation (x,t (αx,αt with the constant α >, we consider the soltion of the form (,(x,t (U,V(ξ (U,V( x t. Sbstitting the aboe eqation into (.6, one can see that the system (.6 becomes U ξu ξ (U ku V ξ, V ξv ξ (V ku V ξ. We define the qantities a k and b (k (k. From the first eqation of (3.5, it follows ξa (ξu An easy comptation leads to ξdu d(u ξb ξa ξa ξb ξb k ξa ξb U ku V Udξ (U (3.5 U ku V ξa ξb. (3.6 U(ξdξ, (3.7 which shows that (x,t U(ξ has the same singlarity as a weighted Dirac delta fnction at ξ. In iew of (3.4 and (3.7, we erify that the Riemann soltion (3. conerges to (3.2 as k. In the following, we proe that the delta shock defined by (3.2 satisfies (.5 in the sense of distribtions. For any test fnctions φ C ((, [,, since, we hae ( ( (θφ t (θ θ η φ xdxdt x(t 2 x(t ((θφ t ((θ θ ((θ θ η φdt θφdx η φ xdxdt ωdφ φ( ( σ δ dt φ( ( σ δ d ω dt dt φd ω (φ t σ δ φ x ωdt

8 68 Xinli HAN and Lijn PAN and ( x(t ( (ηφ t (η η η φ xdxdt 2 x(t ((η η η φdt ηφdx φ( σ δ dt Transition of two different kinds of delta shocks ((ηφ t ((η η η φ xdxdt Lemma 3.4 Let <. For k < with k arbitrarily small, the Riemann soltion of (.6 and (.7 is a delta shock δs: (,, x < σ δ (kt, (,(x,t (ω(t;kδ(x σ δ (kt, kω(t;kδ(x σ δ (kt, x σ δ (kt, (,, x > σ δ (kt, where ω(t;k (k (k t and σ δ(k (k (k are strength and elocity of the delta shock x x(t, respectiely. The strength ω(t;k and the elocity σ δ (k of the delta shock hae the following properties: (3.8 k ω(t;k ω(t and k σ δ(k σ δ. (3.9 Proof De to <, there exists a k < sch that k k for k < k <. Comparing this with Lemma 3., we get the Riemann soltion of (.6 and (.7 is a delta shock. An easy calclation shows and k ω(t;k k (k (k t t ω(t k σ δ(k k ( (k (k σ δ. Theorem 3.5 Let <. As k, the δ-shock soltion (3.8 conerges to (3.2, which is a δ-shock soltion of the Riemann problem (.5 with the same initial data ( ±, ± ; it shows how the Dirac delta fnction in disappears when the parameter k anishes (see Figre 2. Proof Since the delta shock soltion (3.8 satisfies (.6 and (.7 in the sense of distribtions, for all the test fnctions φ C ((, [,, we hae (φ t ( k φ xdxdt (3.

9 Formation and transition of delta shock 69 and (φ t ( k φ xdxdt. (3. For the left-hand side of (3., sing Green s formlation and integration by parts, we calclate ( ( (φ t ( k φ xdxdt x(t 2 φ([ φ( t ( (( φ(([ φ([ φ([ x(t ((φ t (( k xdxdt k φdt φdx k ] []σ δdt k ] []σ δdt dω k ] []σ δ dω dt dt k φ xdxdt φdω (φ t σ δ φ x ωdt ωdφ k ] []σ δ (k (k dt. Here and below, we se the sal notation [] r l with l and r the ales of the fnction on the left-hand and right-hand sides of a discontinity, etc. Taking the it in (3., in iew of (3.9 and the aboe expression, we hae k k (φ t ( φ([ k φ xdxdt k ] []σ δ (k (k dt φ([θ θ η ] [θ] σ δ dt φ([θ θ η ] [θ] σ δ d ω dt. (3.2 dt As for the left-hand side of (3., with the same reason as before, we arrie at ( x(t ( (φ t ( k φ xdxdt 2 φ( t ( (( x(t ((φ t (( k xdxdt k φdt φdx k φ xdxdt k(φ t σ δ φ x ωdt kωdφ

10 7 Xinli HAN and Lijn PAN φ([ k ] []σ δdt kφdω φ(([ k ] []σ δdt kdω φ([ k ] []σ δ k dω dt dt φ([ k ] []σ k δ (k (k dt and k k (φ t ( φ([ k φ xdxdt k ] []σ k δ (k (k dt φ([η η η ] [η] σ δdt. (3.3 De to (3.9, (3.2 and (3.3, as k, we erify that the it of (3.8 is (3.2, which is the Riemann soltion of the problem (.5 with the same initial data ( ±, ±. It is interesting to see from (3.2 that the Dirac delta fnction in remains as k. Howeer, from (3.3, we can see that the Dirac delta fnction in disappears. t t δs δs (, kω(t;kδ(x σ δ (kt (, O (a k < (, (, (, ( ω(tδ(x x(t, x O (b k Figre 2 Transition of delta shock x Remark 3.6 From the proof of Theorem 3.5, we can see the transition between two different kinds of delta shocks as k. That is, the transition from one kind of delta shock on which both state ariables and simltaneosly contain the Dirac delta fnction, to another kind of delta shock on which only one state ariable θ contains the Dirac delta fnction Transition from rarefaction wae to contact discontinity Lemma 3.7 Let k < with k arbitrarily small. If > >, the Riemann soltion of (.6 and (.7 is R J:

11 Formation and transition of delta shock 7 (,, x < λ 2 (, ;kt, R, λ2 (, ;kt x λ 2 (, ;kt, (,(x,t (,, λ 2 (, ;kt < x < λ (, ;kt, (,, x > λ (, ;kt, (3.4 wheretheintermediatestate(, canbecalclatedas(, ( k k, k k, and the rarefaction wae R is gien by (,(x,t ( k ( ξ, k ( ξ, ξ x t. (3.5 If >, the Riemann soltion is either (3.4 or R R 2 : (,, x < λ 2 (, ;kt, R, λ 2 (, ;kt x < 2t, (,(x,t (,, x 2t, R2, 2t < x < λ 2 (,;kt, (,, x > λ 2 (,;kt, (3.6 where the rarefaction wae R can be gien by (3.5 and the rarefaction wae R 2 is defined by Frthermore, (,(x,t ( k ( ξ,, ξ x t. (3.7 and λ (,;k λ (,;, λ 2 (,;k λ 2 (,;, (3.8 k k k k k k k û, k k k ˆ (3.9 hold tre. Proof If > >, then there exists k < sch that k < k for all < k < k. This together with Lemma 3. implies that the Riemann soltion of (.6 and (.7 is R J. If >, then there exists k < sch that k < k or k < < k for all < k < k. From Lemma 3., it follows that the Riemann soltion of (.6 and (.7 is R J or R R 2. The calclations for (3.8 and (3.9 are straightforward, and we omit them. Theorem 3.8 Let >. As k, both the Riemann soltion (3.4 and (3.6

12 72 Xinli HAN and Lijn PAN conerge to (,, x < λ 2 (, ;t, R, λ2 (, ;t x λ 2 (û,ˆ ;t, (θ,η(x,t (û,ˆ, λ 2 (û,ˆ ;t < x < λ (, ;t, (,, x > λ (, ;t, (3.2 where the intermediate state (û,ˆ is defined by (3.9, and the rarefaction wae R is gien by (θ,η(x,t ( ( ξ, ξ, ξ x t ; (3.2 it shows how a rarefaction wae degenerates into a left contact discontinity (see Figre 3. Frthermore, the it (3.2 is a soltion of Riemann problem (.5 with the same initial data ( ±, ±. t R t R x 2t R2 J (, (, (, (, O x O (a k < (b k < (, x t R (, J (, O x (ck Figre 3 Transition from rarefaction wae to contact discontinity Proof If > >, for k < with k arbitrarily small, we dedce from Lemma 3.7 that the Riemann soltion of (.6 is gien by (3.4. Taking the it k in (3.5, we hae k k ( ξ ( ξ, (3.22 k k ( ξ ξ.

13 Formation and transition of delta shock 73 From (3.8, (3.9 and (3.22, it yields that the Riemann soltion (3.4 conerges to (3.2 as k. If >, for k < with k arbitrarily small, the Riemann soltion of (.6 is either (3.4 or (3.6. First, assme that the Riemann soltion of (.6 is gien by (3.4. As in the proof of sbcase > >, we can proe that the Riemann soltion (3.4 conerges to (3.2 when the parameter k anishes. Secondly, assme that the Riemann soltion of (.6 is (3.6. Sbstitting the left bondary ξ 2 of rarefaction wae R 2 into (3.7 and taking the it in it, we hae k (x,t k k (. ( Similarly, sbstitting the right bondary ξ λ 2 (,;k of rarefaction wae R 2 into (3.7 and taking the it in it, we get Frthermore, (x,t k k k ( (k 2. (3.24 k λ 2(,;k 2 λ (,;. (3.25 From (3.23 (3.25, it is easily seen that the left bondary and the right bondary of the rarefaction wae R 2 in (3.6, coincide to form a left contact discontinity with speed λ (,; 2 as k. Howeer, the rarefaction wae R in (3.6 can retain its form after the it. So we erify that the it of (3.6 is (3.2. For any test fnction φ C ((, [,, it is not hard to show that (3.2 implies the following eqations: ( λ 2(, ; λ2(û,ˆ ; λ(, ; ( ξθ ξ (θ θ λ 2(, ; λ 2(û,ˆ ; λ (, ; η ξφdξ, (3.26 ( λ 2(, ; λ2(û,ˆ ; λ(, ; ( ξη ξ (η η λ 2(, ; λ 2(û,ˆ ; λ (, ; η ξφdξ. (3.27 In iew of (3.26 and (3.27, we conclde that the it fnction (3.2 is a soltion to the Riemann problem (.5 with the same initial data ( ±, ±. Theorems 3.3 and 3.5 proide a detailed description of formation and transition of different kinds of delta shocks, which allow s to better inestigate instability and internal mechanism of delta shocks. Acknowledgements The athors are gratefl to the anonymos referee for her/his helpfl comments which improe both the mathematical reslts and the way to present them. Xinli HAN and Lijn PAN wold also like to thank the hospitality of Professors Tong LI and Lihe WANG, and the spport of Department of Mathematics at Uniersity of Iowa, dring their isit in

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