We consider the more general viscous solution, and take the limit as E- This is the vanis hing viscosity solution discussed earlier. This condition is not particularly easy to work We will examine below some equivalent forms LL, S, LV of expressing this entropy condition 2.1 Convex(concave)fluxes SLIdE 35 When f(a) is convex i.e. f"(u)>0(or concave i.e. f"(u)<0 for all u, the entropy condition can be written as a(ur)=f()>s>f(ur)=a() Characteristics must run into the shock for increasing t, not emerge from it. SLIDE 36 at VIOLATING 4.2.2 Example sLide 3 Solution A a(uL) Entropy condition is violated -char acteristics emerge from the shock Solution b No shock→OK 4.2.3 Oleinik's Condition slide 38 u(a, t)is the entropy satisfying solution if all discontinuities sat isfy the property that f(u) -f(ul)❞✧❡✧❢❤❣❥✐✟❦✴❧✌♠✏❡✒♥♣♦rqs❡♣t✁❣✙♥✝❡✁✉●❡✕✐❊❡✕♥❤✈✙✇②①✮❧③❦✒❢④❣✙⑤✼❦⑥❦✒❣✙✇✰⑤⑦♦✛❧✌❣✙✐s⑧⑨✈✙✐❊♠⑩♦✶✈❝❶✏❡◗♦❷q✟❡♣✇❸❧rt❹❧r♦❹✈✙❦⑨❺⑥❻❽❼✄❾ ❿❊q✼❧③❦➀❧③❦❹♦rqs❡➀①✙✈✙✐✱❧③❦✝q✼❧r✐●✉✧①✮❧③❦✒❢④❣➁❦✴❧r♦✛➂➃❦✒❣✙✇✰⑤⑦♦✛❧✌❣✙✐➄♠❥❧③❦✕❢✒⑤✼❦④❦✒❡❤♠♣❡④✈✙♥✴✇❸❧✌❡✕♥✒❾✧❿❊q✼❧③❦⑨❢④❣❥✐✄♠✙❧r♦✛❧✌❣✙✐➅❧③❦➆✐❊❣✙♦ ➇✈✙♥✴♦✛❧✌❢✕⑤⑦✇➈✈❥♥✴✇✰➂➉❡④✈➁❦✴➂➊♦❄❣✧➋➌❣❥♥✝❶✗➋➍❧r♦rqs❾➎❞♣❡➏➋➍❧r✇r✇②❡❤➐✼✈✙t⑨❧r✐✄❡◗➑④❡✕✇➈❣❥➋➒❦✒❣✙t✁❡♣❡❤➓✒⑤⑦❧r①✙✈✙✇➈❡✒✐s♦■➔✕❣✙♥✴t❹❦ →❷➣⑧r↔✱⑧ ➣➌↕➛➙ ❣➜➔❹❡✝➐➇♥❤❡✴❦④❦✴❧r✐●✉➏♦rq⑦❧③❦➀❡✕✐✱♦✸♥✝❣➇➂➏❢④❣✙✐❊♠✙❧r♦✛❧✌❣✙✐❊❾ ➝➟➞③➠P➞✌➡ ➢❹➤✄➥❊➦✱➧●➨➫➩❚➭✏➤✄➥➟➭➛➯✏➦✄➧✟➲⑨➳➸➵➟➨P➧✼➺ ➻✄➼●➽r➾➪➚⑩➶✟➹ ➘➷➴✟➬✮➮⑩➱■✃❷❐❪❒❰❮③ÏÑÐ✕Ò✏➮✼Ó➛➬✕Ô♣❮✸Õ ➬✏Õ▼➱❊Ö Ö✸✃✌❐❊❒❃×Ø❼➉✃➌Ò➛Ù❃Ð✒Ò✏➮sÐ✮Ú➁Ó✏➬➆❮✛Õ ➬➛Õ❨➱❊Ö Ö✛✃❷❐❪❒❃ÛØ❼➛❒❰Ü❷Ò✏Ù❨Ú❥Ý❘Ý➟❐➸Þ✄ß✝➴✟➬ ➬✒➮●ß❤Ù✝Ò➛à●á➃Ð✒Ò✏➮sâ✟❮❸ß❤❮➈Ò➛➮◗Ð✮Ú❥➮♣ã✱➬▼ä❰Ù❤❮➈ß❚ß✝➬✮➮◗Ú➛Ï å ✃❷❐❊æ❪❒➌çè➱ Ö ✃❷❐❊æP❒✬é➷ê▼é★➱ Ö ✃❷❐❊ë➍❒➌ç å ✃✌❐✄ë✾❒ ì➴sÚ✏Ù❤Ú➛Ð✴ß✝➬✮Ù✝❮③Ï❚ß✝❮③Ð✒Ï☛í➀îsÏ❚ßïÙ✝îs➮♣ð➥❊ñ✮➤ ß❤➴✟➬➆Ï❚➴sÒ✼Ð④ò➃Ü❷Ò✏Ù❰❮❘➮sÐ✕Ù❤➬✮Ú➛Ï❚❮❘➮✟ó⑨ô✴Þ✟➮✟Ò❥ß❰➬✮í⑨➬✮Ù✝ó➛➬ÑÜ❷Ù✝Ò➛íõ❮❸ß❝Õ ➻✄➼●➽r➾➪➚⑩➶✟ö ENTROPY SATISFYING ENTROPY VIOLATING ➝➟➞③➠P➞③➠ ÷❃➨P➯✟ø➅ù■ú✌➧ ➻✄➼●➽r➾➪➚⑩➶✱û üÒ➛Ý➈î⑦ß❤❮➈Ò➛➮◗ý å ✃❷❐æ ❒✾ç➷❐æ çÿþ✁￾ ê❃ç✄✂ ☎ ✃❷❐ë✝✆ ❐æ ❒✾ç❴❼ å ✃❷❐ë ❒➌ç ❐ë ç✞￾ ✟✾➮●ß❤Ù✝Ò➛à●á➃Ð✒Ò✏➮sâ✟❮❸ß❤❮➈Ò➛➮⑥❮③Ï❰Ó✼❮➈Ò➛Ý❘Ú❥ß✝➬❝â✡✠☛Ð④➴sÚ✏Ù❤Ú➛Ð✴ß❤➬✒Ù❤❮❘Ï❚ß✝❮③Ð✒Ï☛➬✒í➬✒Ù❤ó✏➬ÑÜ❷Ù❤Ò✏í ß❤➴✟➬➆Ï✝➴✟Ò⑦Ð④ò üÒ➛Ý➈î⑦ß❤❮➈Ò➛➮☞☛ ✌Ò Ï✝➴✟Ò⑦Ð④ò✎✍ ✏✒✑ ➝➟➞③➠P➞✔✓ ✕➃ú✌➧ ð➥ð✗✖✙✘ ➺ ➢❹➤✄➥✛✚ðñ ð➤✄➥ ➻✄➼●➽r➾➪➚⑩➶✢✜ ✣àsà✟Ý➈❮③Ð✒Ú✏ã✟Ý❘➬❨ß✝Ò ó✏➬✮➮✟➬✒Ù④Ú❥Ý✥✤sî⑦Ô⑥Ü❷î✟➮sÐ✕ß✝❮❘Ò✏➮sÏ ❐■✃✗✦★✧✝ô❚❒➸❮❘Ï➍ß✝➴✟➬❰➬✮➮➛ß❤Ù✝Ò➛à✼á➆Ï✝Ú❥ß✝❮③Ï➜Ü❷á✼❮❘➮✟ó▼Ï❚Ò➛Ý➈î⑦ß❤❮➈Ò➛➮ ❮❸Ü❪Ú❥Ý❘Ý✟â⑦❮③Ï✝Ð✒Ò✏➮●ß✝❮❘➮✼î✟❮❸ß❤❮➈➬❝Ï✾Ï❤Ú✙ß✝❮③Ï❚Ü❷á▼ß✝➴✟➬❰àsÙ✝Ò➛à✱➬✮Ù❚ß➜á ß✝➴✱Ú✙ß ➱■✃❷❐❪❒➸þ➅➱■✃❷❐❊æ❊❒ ❐➃þ⑩❐✄æ ×➷ê▼× ➱■✃❷❐❪❒➸þ➅➱■✃❷❐❊ë■❒ ❐➃þ⑩❐✄ë Ü❷Ò✏ÙïÚ✏Ý➈Ý❪❐✧ã✄➬✕ß➜ä✬➬✒➬✮➮♣❐æ Ú❥➮sâ⑥❐ë Õ ￾✪✩