Lack of BV Bounds for Approximate Solutions to the p-system with Large Data

Transcription

1 Lack of BV Bounds for Approimae Soluions o he p-sysem wih Large Daa Albero Bressan (, Geng Chen (, and Qingian Zhang ( (* Deparmen of Mahemaics, Penn Sae Universiy Universiy Park, Pa. 680, U.S.A. (** School of Mahemaics Georgia Insiue of Technology Alana, Ga. 0, U.S.A.. s: bressan@mah.psu.edu, gchen7@mah.gaech.edu, zhang q@mah.psu.edu January, 04 Absrac We consider fron racking approimae soluions o he p-sysem of isenropic gas dynamics. A ineracion imes, he ougoing wave frons have he same srengh as in he eac soluion of he Riemann problem, bu some error is allowed in heir speed. For large BV iniial daa, we consruc eamples showing ha he oal variaion of hese approimae soluions can become arbirarily large, or even blow up in finie ime. This happens even if he densiy of he gas remains uniformly posiive. Inroducion For hyperbolic sysems of conservaion laws in one space dimension, a saisfacory eisenceuniqueness heory is currenly available for enropy weak soluions wih small oal variaion [, 5]. The well-posedness of he Cauchy problem holds also in he case of large daa, as long as he oal variaion remains bounded [, 4]. A major remaining open problem is wheher, for large BV iniial daa, he oal variaion remains uniformly bounded or can blow up in finie ime. Eamples of finie ime blow up have been consruced in [, ]. However, hese sysems do no come from physical models and do no admi a sricly conve enropy. For iniial daa having small oal variaion, regardless of he order in which wave frons cross each oher, he Glimm ineracion esimaes [9] show ha he oal srengh of waves remains small for all imes. There are few eamples of hyperbolic sysems where uniform BV esimaes hold also for soluions wih large daa [7, 0]. In he presen paper we sudy BV bounds for he p-sysem of isenropic gas dynamics in Lagrangean variables: { v u = 0, (. u + p(v = 0. Here u is he velociy, ρ is he densiy, v = is specific volume, while p = p(v is he pressure. Our main concern is wheher, for fron racking approimae soluions o he p- sysem, uniform BV esimaes can be esablished. More precisely, we sudy he following quesion:

2 (Q Consider a fron racking approimaion for (. wih large BV iniial daa. Assume ha, a each ineracion ime, he ougoing wave frons have he same srengh as in he eac soluion of he Riemann problem bu some error is allowed in heir speed. Can he oal variaion of such approimae soluion become arbirarily large? In his paper, some eamples will be consruced, showing ha he oal srengh of waves in a fron racking approimaion can indeed approach infiniy in finie ime. This confirms he non-eisence of a Lyapunov funcional which decreases a every wave-fron ineracion, as proved in [8]. For ineracions occurring near vacuum, i was already noiced in [6] ha uniform Glimmype esimaes were no longer valid. I hus comes as no surprise ha an approimae fron racking soluion can be consruced, where he oal srengh of waves (measured by he change in Riemann invarians blows up in finie ime. Remarkably, our las wo eamples show ha an arbirarily large amplificaion of he oal variaion is sill possible even if he gas densiy remains uniformly posiive. I should be clear ha he presen counereamples do no prove ha, for large BV soluions of he p-sysem, he oal variaion can blow up in finie ime. Indeed, we sill conjecure ha global BV bounds do hold. Our analysis only shows ha such BV bounds canno be proved by wave ineracion esimaes alone, and addiional properies of soluions mus be aken ino accoun. Apparenly, he decay of rarefacion waves due o genuine nonlineariy [, 4, 6, 0] should be used in a crucial way. In he las secion we revisi wo of he earlier eamples and show ha, if such decay is aken ino accoun, hese specific ineracion paerns do no produce blow up. Wave ineracions for he p-sysem We review here some sandard properies of characerisic curves and of shock curves. For deails we refer o [8, 9]. To simplify he compuaions, we assume ha in (. he pressure has he special form p(v = v = ρ, p (v = v 4. Smooh soluions of (. saisfy he quasilinear sysem ρ + ρ u = 0, u + ρ ρ = 0, (. wih characerisic speeds The variables λ = ± ρ, (. w = u, w = ρ + u, (. provide a coordinae sysem of Riemann invarians, in he u-ρ plane.

3 ρ Q Q η(ε σ ε P P ε P Q P η(ε Q Figure : Ineracion of a -shock and a -rarefacion u A shock wih lef sae (u, and righ sae (u +, ρ +, raveling wih speed λ, saisfies he Rankine-Hugonio equaions ( λ = u u +, ρ + λ(u + u = p(v + p(v = ρ +. The La admissibiliy condiion here yields u + < u. Hence u + u = ( ρ + ( ρ ρ +, (.4 Seing from (.4 i follows λ = ± ρ ρ +. (.5 ρ + s = u u +, θ = ρ +, ( θ( θ s = ρ. (.6 θ For he p-sysem, he ineracion of wave frons has been horoughly sudied [9, 8]. For reader s convenience, we derive here wo elemenary esimaes which will be used in he sequel. Consider a -shock wih lef and righ saes (see Fig. P = (u,, Q = (u +, ρ +, wih srengh σ = (ρ + u + ( u

4 measured by he change in he -Riemann invarian. Assume ha his shock crosses a small -wave (compression or rarefacion of srengh σ = ε. We seek an esimae on he size of he ougoing waves, up o leading order. Le ( P = (u ε, ε, Q = u + η(ε, ρ + η(ε be he lef and righ saes across he -shock afer he ineracion. Seing s(ε = s ε + η(ε, θ(ε = θ η(ε ε, ψ(θ = ( θ( θ θ (.7 from (.6 i follows Differeniaing w.r.. ε, a ε = 0 we obain Two cases are relevan o our analysis. s(ε = ( ε ψ(θ(ε. (.8 η = ψ(θ + ψ(θ ψ (θθ = s + s θ + θ 4 θ θ η. (.9 CASE : The -shock has small ampliude. In his case, since shock and rarefacion curves coincide up o second order, for s small we have he epansion Insering (.0 in (.9 we obain θ = + s + O(s. (.0 ( s + O(s 4 (. η = + CASE : The -shock has a fied srengh σ, while he densiy approaches zero. By definiion, he srengh is compued by θ σ = (θ + s = (θ + θ + θ. (. As 0, from (. i follows ha θ. Indeed, dropping lower order erms we find θ / σ, θ ( / σ. (. Insering (. in (.9 and reaining only leading order erms, we obain η σ + σ θ (θ η, 4

5 η σ + σ + σ ( σ 4/ / ( σ /, η as 0. (.4 In oher words, when an infiniesimal -wave crosses a -shock of fied srengh σ, is size is amplified by a facor η κ / which becomes arbirarily large as 0, i.e. as he densiy approaches vacuum. As a special case, if a -shock of srengh σ = crosses a small -wave (eiher compression or rarefacion of srengh ε 0, he srengh of he shock remains consan, while he srengh of he -wave is amplified by a facor η(ε ε > ρ /. (.5 The above esimae is valid as soon as he densiy of he lef sae (i.e., ahead of he -shock is sufficienly small. One more case will be of relevance. Consider a small -shock wih righ sae (u +, ρ + = (0,. By (.4 is lef sae (u, saisfies ( u = ρ. (.6 Taking = s, we obain ( ( u = s s + s4 ( + s + s + = s + s 6 + s 6 + o(s. (.7 Ne, consider a small -shock, again wih righ sae (ũ +, ρ + = (0,. Le (ũ, be he lef sae. Taking = + r, by (.6 we now have ( ( ũ = r + r + r4 ( r + r = r + r 6 r 6 + o(r. (.8 Imposing u = ũ yields r = s + s4 + o(s4. Referring o Fig. 5, consider he poins and le B = A = (0,, C = (u,, D = (u,, ( u +, + Then he slope of he segmen A B is = s o(s4 s + s 6 + s4 + o(s4 ( s + s 6 + s4 + o(s4, + s4 6 + o(s4 5 = s + o(s. (.9

6 Head-on ineracions 0 δ 0 Figure : The head-on ineracion of a -shock wih a rain of smooh -waves, in Eample. Eample. Consider an iniial daa ( ρ, ū consising of a -shock of srengh σ =, approaching a rain of smooh -waves, near vacuum (Fig.. In erms of Riemann invarians (w, w, assume ha he iniial daa is given by { w (0, = w ( = 0, w (0, = w ( = α ( + sin β for [0, δ], (. for suiable consans α, β > 0 and some δ > 0 suiably small. In addiion, assume ha his iniial daa has a -shock of size σ = a =, and is consan on he half lines where < 0 and >. For his iniial daa we consruc an approimae soluion such ha, a every ineracion, he srengh of ougoing waves is he same as in an eac soluion. However, insead of (. or (.5, we le he waves ravel wih consan speeds, say, c for he -shock and c for he -waves, for some consan c > 0. Then, as T = δ/c, all he -waves cross he shock and he oal variaion of he soluion approaches infiniy. Indeed, by (.5, choosing δ > 0 sufficienly small he following holds. A any ime 0 < < T we have w, (, + c ρ( / w,( for all [0, 0 ] such ha + c > c. (. ( α / / α α ( + sin β β α β cos β, If we now choose 0 < α/ < β < α <, hen he iniial daa has bounded variaion, because 0 w, ( d = 0 α α ( + sin β β α β cos β d <. On he oher hand, as T he oal variaion blows up, because 0 lim /c c w, (, + c d = ( 0 α (α/ ( + sin β β (α/ β cos β d =

7 In he previous eample he iniial daa conains vacuum. Moreover, in he erminal profile he large variaion is achieved a very low gas densiy. The ne eample shows ha his arbirarily large amplificaion of he oal variaion can be achieved also wih an iniial daum having uniformly posiive densiy. As before, we require ha a each ineracion he srengh of ougoing waves is he same as in an eac soluion, bu we allow a small error in he wave speed. Namely, in our approimae soluion all -waves ravel wih speed c while -waves ravel wih speed c. ρ = 0 ρ = ρ = ρ = a b Figure : Evoluion of he densiy profile in Eample. Top lef: a ime = 0 he iniial densiy is uniformly posiive. Top righ: afer wo rarefacion waves of he opposie families cross, in he middle secion he same configuraion in Eample is recreaed. Boom lef: when he -shock crosses he rain of -waves a low densiy, he oal variaion grows wihou bounds. Boom righ: afer crossing a -compression, an arbirarily large amoun of oal variaion occurs wihin he inerval [a, b], a uniformly posiive densiy. Eample. As shown in Fig., consider an iniial daa similar o (., bu wih he inserion of wo addiional rarefacion waves, and a compression. A ime, afer crossing rarefacion waves of he opposie family, a ime = he rain of -waves and he -shock recreae he iniial daa in Eample. A ime, he oal variaion of he second Riemann invarian w (, becomes infinie. A ime, afer crossing a -compression, his infinie oal variaion occurs a uniformly posiive densiy. In he - plane, he soluion is described in Fig. 4. Remark. Eample shows ha here eiss an iniial profile (ū, ρ BV wih ρ( ρ 0 > 0, and an approimae soluion wih frons moving a consan speed ±c such ha he following holds. If a each ineracion he srengh of ougoing waves is he same as in he eac soluion of he Riemann problem, hen he oal variaion blows up in finie ime. More 7

8 precisely, a some ime { } ρ(, for 0, To.Var. ρ(, ; [0, [ =. (. In paricular, is is no possible o pu a coninuous weigh on wave srenghs, possibly approaching infiniy as ρ 0, in order o conrol he oal variaion. Indeed, for any choice of he weighs, coninuous on he region where ρ > 0, he iniial weighed oal variaion will be finie, while he final weighed oal variaion will be infinie. To.Var.= + 8 ρ > ρ ~ 0 compression small waves raref. raref. shock Figure 4: The wave paern described in Eample. The bo shows he region where he ineracions in Eample ake place. 4 An eample wih uniformly posiive densiy In he previous eamples, he blow up of he oal variaion was achieved because waves crossing a shock of uni srengh were amplified by an arbirarily large facor as he gas densiy approached vacuum. The following eample shows ha he oal variaion can become arbirarily large even if he gas densiy remains uniformly bounded away from zero, a all imes. Eample. STEP. We begin by consrucing a symmeric ineracion paern conaining four wave frons, as shown in Fig. 5. We choose he srenghs of he wo large shocks S, S and of he wo inermediae waves in such a way ha, afer a whole round of ineracions, hese srenghs are he same as a he iniial ime. Working in he u-ρ plane, his is done as follows. (i Sar by consrucing wo symmeric shocks: he -shock A C and he -shock A C, approaching each oher. (ii Deermine he wo ougoing shocks DA and DA, resuling from he crossing of he above wo shocks. 8

9 D S S 6 Ur A B B C A U l 5 4 Figure 5: A periodic ineracion paern. (iii Consruc a square having wo opposie verices a C and D. Call B, B he remaining wo verices. (iv Choose U l so ha he wo poins B and A are on he same -shock curve wih lef sae U l. Symmerically, choose U r so ha he wo poins B and A are on he same -shock curve wih righ sae U r. The eisence of saes U l, U r saisfying he condiions (iv is now proved (see Fig. 6. Lemma. In he u-ρ plane, consider wo poins B = (u, ρ and A = (u, ρ. Assume ha (i u < u, and ρ > ρ. (ii Calling A = (u, ρ he poin on he -shock curve wih righ sae B having he same ρ-componen as A, one has u < u. Then here eiss a sae U l = (u l, ρ l, wih 0 < ρ l < ρ, such ha boh B and A lie on he -shock curve wih lef sae sae U l. Proof. We shall use (.4 wih (u, = (u l, ρ l while (u +, ρ + = (u, ρ or (u +, ρ + = (u, ρ. To prove he lemma we need o find (u l, ρ l such ha ( u l u = ( ρ l ρ ρ l ρ (, u l u = ( ρ l ρ ρ l ρ. (4. This will be achieved if we can find ρ l such ha u u = G(ρ l, (4. where G is he funcion defined as. ( G(ρ = ( ρ ρ ρ ρ ( ρ ρ ( ρ ρ. The assumpion (ii implies G(ρ = ( ρ ( ρ ρ ρ < u u. 9

10 Moreover, a direc compuaion yields ρ l G(ρ l < 0, lim G(ρ ρl l = +. 0 Therefore here eiss a unique value of ρ l such ha G(ρ l = u u. ρ ρ B 0 u A U A u U = (u, ρ l l l u Figure 6: By moving he poin U along he -shock curve wih righ sae B, we evenually reach a lef sae U l such ha he -shock curve hrough U l conains A as well. Having consruced he above wave curves, consider he following ineracion paern, shown in Fig. 5, righ: A ime he iniial daum consising of four shocks, connecing he saes U l, A, C, A, U r. A ime he profile sill consiss of four shocks (he wo middle ones have crossed each oher, connecing he saes U l, A, D, A, U r. A ime he profile consiss of wo large shocks and wo rarefacions (he wo middle shocks have joined he big ones, generaing wo rarefacions, connecing he saes U l, B, D, B, U r. A ime 4 he profile sill consiss of wo large shocks and wo rarefacions (he wo rarefacions have crossed each oher, connecing he saes U l, B, C, B, U r. A ime 5 he iniial daum consising of four shocks (he wo rarefacions have impinged on he big shocks, generaing wo inermediae shocks of he opposie families, connecing he saes U l, A, C, A, U r, eacly he same as a ime. The paern hus repeas iself. Remark. I is clear ha for he above eample i is essenial o have large oal variaion. Indeed, if we choose he middle shock A C small, hen he line A B will be almos horizonal and he poin U r mus be far away, wih densiy close o zero. On he oher hand, in his soluion obained by fron racking he densiy rivially remains uniformly bounded away from zero. Since he ineracion paern is periodic, we conclude ha Even under he assumpion ha he densiy remains uniformly posiive, here is no way o consruc a Lyapunov funcional conrolling he oal variaion, which is sricly decreasing a every ineracion. 0

11 In he above approimae soluion he wave srenghs follow a periodic paern. To achieve an arbirarily large amplificaion of he oal variaion, a furher consrucion is needed. B P P A Figure 7: A periodic paern ha amplifies a small wave fron. STEP : As shown in Fig. 7, righ, on op of he previous paern we add a very small wave fron. To fi he ideas, consider a -rarefacion of srengh ε > 0, locaed a A. Wihin a ime period, his fron will Cross he inermediae -shock. Inerac wih he large -shock a P producing a -compression. Cross he inermediae -shock. Cross he inermediae -rarefacion. Inerac wih he large -shock a P producing a -rarefacion. Cross he inermediae -rarefacion. We analyze he case where he wo middle shocks are small and he densiy of heir lef sae is. We claim ha, when he addiional fron reaches B, is size will be increased by a facor κ >. Indeed, when a small wave of srengh ε crosses a shock of he opposie family of srengh s a densiy =, by (. he srengh of he ougoing fron is ( ε + = + s + o(s ε. (4. When he fron crosses a rarefacion of he opposie family, is srengh does no change. Finally, when he small wave impinges on a large shock a P or a P, we need o esimae he relaive size of he refleced wave fron. Toward his goal, le ρ = ψ(u be he equaion of he shock curve wih righ sae U r, passing hrough boh A and B, as consruced in

12 Lemma. Calling ε, ε + he srenghs of he fron before and afer ineracion, o leading order we have ( ε + = ( ψ (uε = s 6 + o(s ε. (4.4 Indeed, by (.9 we have ψ (u = s / + o(s. Calling ε A and ε B respecively he srenghs of he small wave-fron a A and a B, we hus have ( ( ( ε B = + s + o(s s 6 + o(s ε A = + s + o(s ε A. (4.5 B σ ε ε P ε ε σ P A Figure 8: Righ: a periodic paern ha amplifies infiniesimal waves. Cener: if a fron of srengh σ crosses a compression (no a shock! of size ε and hen a rarefacion of size ε, is final srengh is no changed. The ougoing frons have srenghs ε, ε. Here ε > ε if he fron σ is a shock, oherwise ε = ε. Lef: o consruc a ineracion paern ha yields an arbirarily large oal variaion, we replace he single infiniesimally small fron in Fig. 7 by counably many pairs rarefacion + compression, of opposie size. The oal srengh of hese waves is finie, each fron having srengh ε. STEP. Consider a periodic paern ha amplifies an infiniesimal wave, as in Sep. By coninuiy, here eiss λ > and ε > 0 such ha any wave-fron (rarefacion or compression of size < ε, raveling from A o B along he pah in Fig. 7, is amplified by a facor λ. We now consruc an iniial se of wave frons where he infiniesimal fron is replaced by counably many pairs rarefacion + compression, whose sizes eacly cancel each oher (Fig. 8, lef. A ime = 0, he oal srengh of all hese small frons can be aken o be =. The key observaion is ha each of hese pairs leaves no fooprin on he underlying soluion consruced in Sep. Indeed, if a fron of size σ crosses a rarefacion and a compression of eacly opposie sizes, afer he wo crossings he size of he fron is sill σ (Fig. 8, cener. As a resul, he paern of four large frons reains is periodiciy. By consrucion, afer each period each pair of opposie small wavefrons is enlarged by a facor λ. When a pair grows o size > ε, we can perform a parial cancellaion so ha is size remains [ε/, ε]. Since he oal number of small wave-frons is infinie, afer several periods a larger and larger number of pairs (compression + rarefacion reaches size > ε/. Hence, as, he oal variaion of his approimae soluion grows wihou bounds.

13 5 Blow up in finie ime The previous consrucion shows ha, if he oal variaion is iniially sufficienly large, hen here eiss an ineracion paern ha renders he oal variaion arbirarily large as. This can be achieved even wih a uniform lower bound on he densiy. The ne quesion is wheher one can arrange he order of wave-fron ineracions so ha he oal variaion blows up in finie ime. Noice ha his is no he case in he previous eample. Indeed, if -frons ravel wih speed ẋ [ C, 0[ and -frons ravel wih speed ẋ ]0, C], i akes a uniformly posiive amoun of ime for each inermediae fron o bounce back and forh beween wo large shocks. Hence he arbirarily large amplificaion of he oal variaion is only achieved in he limi as +. T 4 S S S S Figure 9: A paern yielding finie ime blow up of he oal variaion. A each sep he oal amoun of small waves bouncing back and forh beween he wo large shocks keeps increasing. In his secion we briefly indicae how he previous consrucion can be modified, providing finie ime blow up of he oal variaion. The main idea is illusraed in Fig. 9. We consider a sequence of imes 0 = 0 < < < < T. During each ime inerval J i = [ i, i ], a counable number of pairs of small waves (compression + rarefacion is amplified by a very large facor. Before ime i, he large -shock S i is compleely canceled by impinging -rarefacions, and a new -shock S (i+ of he same srengh is recreaed a a locaion closer o he large -shock S. Figures 0 and show how his can be achieved, saring wih very many pairs of small waves (compression + rarefacion. By leing each compression fron collapse o a shock, and hen canceling his shock wih a rarefacion fron of he same family, we obain a rain of pairs of small waves (compression + rarefacion in he opposie family (Fig. 0, lef. By varying he locaions of hese ineracions, insead of many pairs of small waves we can achieve a large compression followed by a large rarefacion fron (Fig. 0, righ. The basic sep is illusraed in Fig.. A large number of small compression+rarefacion pairs produces a large -rarefacion, which sars depleing he -shock along he line AB, and a

14 Figure 0: Saring wih a large number of pairs of small waves, one can generae a large number of similar pairs in he oher characerisic family (lef, or one single large pair of frons (righ. τ D B τ U l S ρ U U l τ C S U r U r U 4 U S S U u A Figure : Lef: beween A and B he shock S is compleely canceled by impinging -rarefacions. Beween C and D, he new -shock S is formed by impinging compression waves. Noice ha he -compression frons emerging from S are used o compleely cancel he -rarefacions ha would oherwise be produced by he ineracions of -compressions wih S. Righ: if he shock S is very large, i canno be canceled by one single large rarefacion fron. Therefore, we need o produce several small rarefacions a subsequen imes, so ha he densiy ρ remains uniformly posiive. 4

15 large compression, which builds up a new -shock along he line CD. Since we canno allow he densiy o become negaive, i may no be possible o cancel he large shock S wih one single large -rarefacion. For his reason, his cancellaion may be accomplished in several sages. For eample, he firs se of -rarefacions reduce he size of he -shock S from (U l, U r o (U, U r. A a ime τ, he profile u(τ, hus conains he -shock (U l, U, he -compression (U, U, and he new -shock (U, U r. A a laer ime τ, he profile u(τ, conains he (shrinking -shock (U l, U 4, he -compression (U 4, U, and he (growing -shock (U, U r. A a laer ime τ, he original -shock has been compleely depleed by -rarefacions. A -shock connecing eacly he same wo saes (U l, U r is formed a a differen locaion, as desired. By canceling he large -shock and reconsrucing i a a differen locaion, shifed o he lef, we can reproduce he paern in Fig. 9. Since a each sep he oal srengh of he small inermediae waves can be amplified by an arbirarily large facor, as T he oal variaion of our approimae soluion blows up o +. 6 Concluding remarks The eamples presened in his paper show ha, if he srengh of wave-frons is compued eacly bu some error is allowed heir speeds, hen he oal variaion of approimae soluions can blow up. I is ineresing o revisi some of he previous eamples, aking ino accoun he decay of rarefacion waves due o genuine nonlineariy. Looking a eac soluions, i becomes clear ha hese paricular ineracion paerns do no yield an arbirarily large amplificaion of he oal variaion. 6. Head-on ineracions, near vacuum. Consider an eac soluion of he sysem (., wih iniial daa as in Eample. We show ha here is no way o choose α, β in (. so ha he following requiremens are simulaneously saisfied: (i The -shock crosses all -waves in finie ime. (ii The -waves do no break before crossing he shock. (iii The sum of srenghs of he -waves is iniially finie, and becomes infinie as hey all cross he -shock. (iv The -waves do no break immediaely afer crossing he shock. For a -shock of uni srengh, assume ha he lef sae (ahead of he shock has densiy 0. Then by (.4 he righ sae (behind he shock has densiy ρ + saisfying ρ +. 5

16 Hence he righ sae and he speed of he shock are given respecively by ρ + ( /, ẋ ρ + = / ρ /. (6. If he iniial profile is given by (., so ha w (0, = 0, w (0, = α ( + sin β [0, δ], (6. hen he requiremen (i will be saisfied provided ha α <. (6. Ne, o make sure ha he -waves do no break before crossing he -shock, we look a he evoluion of w, along characerisics. From ( w w, + w, = 0 i follows ( w w, + w, = w 4 w,. By a comparison argumen, we conclude ha he gradien w, will no blow up before ime T > 0 provided ha he iniial daa saisfy w, (0, w (0, < T. Recalling (6., he condiion (ii is hus saisfied if α + (α β > 0. (6.4 As shown in he discussion of Eample, condiion (iii is saisfied provided ha 0 < α < β < α <. (6.5 To check wheher (iv can be saisfied, le T be he ime when he -shock reaches he origin, crossing all -waves. Denoe by (, y he posiion of a -characerisic saring a (0, y = y, Calling τ(y he ime where his -characerisic crosses he -shock, we find T τ(y y α/. We consider he evoluion of w, (, (, y along his -characerisic. For < τ(y we have w, (, (, y y α β. When his characerisic crosses he -shock a ime τ = τ(y, by (.5, his gradien is amplified by a facor w, (τ+ w, (τ y α/. Moreover, w (τ+ y α/. To make sure ha his gradien remains bounded during he ime inerval [τ(y, T ], we need Therefore we should have y α/ w, (τ w (τ+ (T = O(. y α/ y α β y α/ y α/ = y β = O(. (6.6 This condiion is incompaible wih he requiremen (6.5 ha β > 0. 6

17 T γ 0 Figure : Lef: Before crossing he large -shock, he small -waves do no break, because when he gas densiy ρ > 0 is very small, he sysem is almos linearly degenerae. Cener: afer crossing he -shock, he -compression waves break and a large amoun of cancellaion beween -rarefacions and -shocks occurs. A he erminal ime T he oal srengh of waves is sill finie, due o hese cancellaions. Righ: in he - plane his paern produces an infinie oal variaion only along he (dashed ime-like curve γ. 6. Waves bouncing back and forh beween wo large shocks From our earlier analysis, his should be he paern ha achieves he greaes amplificaion of wave srenghs (Fig.. If he size of he shocks S, S is sufficienly large, he srengh of a refleced -fron σ is almos he same as he srengh of he impinging -fron σ. Aferwards, as his -fron crosses oher -shocks, is srengh increases by a large facor. Repeaing his paern, i may appear ha an arbirarily large amplificaion of wave srenghs can be achieved. The following analysis shows ha his is no he case, if we ake ino accoun he decay of rarefacion waves due o genuine nonlineariy. For some consan c 0, he wo large shocks will have speeds ẋ ( c 0 < 0 < c 0 ẋ (. (6.7 Consider a -rarefacion wave (Fig. emerging from he large -shock a some ime τ and impinging on he opposie -shock a ime τ. The upper and lower esimaes on he velociy yield an esimae of he form τ κτ. (6.8 By wave decay esimaes, he densiy of such -rarefacion a ime τ is C τ τ Cκ κ Therefore, he oal amoun of -rarefacions ha impinge on he large -shock wihin a ime inerval [T 0, T ] is T Cκ κ τ dτ. T 0 An enirely similar esimae holds for he -rarefacions impinging on he large -shock. Ne, fi any large ime T. As in Figure, consider he maimal backward -characerisic hrough he poin (T, (T. This will cross he large -shock a an earlier ime T 0. The oal τ. 7

18 S S T τ T 0 τ ( ( σ σ Figure : The oal amoun of compression waves (including shocks a ime T is bounded in erms of he oal amoun of rarefacion waves ha impinge on he large -shock during he inerval [T 0, T ]. amoun of -shocks (ogeher wih -compressions a a given ime T can be esimaed as: Toal amoun of -shocks a ime T conained in he inerval [ (T, (T ] = O( amoun of -rarefacions impinging on he -shock for [T 0, T ] = O( T T 0 Cκ κ τ dτ = O( ln T T 0 = O(. (6.9 Here O( denoes a quaniy ha remans uniformly bounded (provided ha some upper and lower bounds on he densiy ρ are given. An enirely similar esimae of course holds for -shocks (ogeher wih -compressions. In addiion, he rarefacion waves can be esimaed as oal amoun of -rarefacions a ime T conained in he inerval [ (T, (T ] = O( amoun of -shocks impinging on he -shock for [T 0, T ] oal amoun of -shocks a ime T 0 conained in he inerval [ (T 0, (T 0 ] cons. (6.0 Here he las inequaliy follows from (6.9, wih T replaced by T 0. This yields a uniform a priori bound on he oal srengh of waves produced by his paricular wave-ineracion paern. 8

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