and letting E→0 gives, Ut+F≤0 in a weak sense The weak form of the entropy inequality is thus U+ PF]drdt+/(x,0)dx≤0 to be satisfied for all O E CO(R X R+)with o(a, t)>0 for all a and t Note that becasue we have now an inequality, we must require that the weighti functions o be positive 4.2.5 Example SLIDE 41 Burgers'equation f(u)=5u2 Take U(u)=u2 and F(u=2u' <0 5 Application Examples 5.1 Traffic Flow SLIDE 4 g equation: Pt +f(p)x=0 f()=pu=umax fle 門"(p) The fur is a concave function of the unknown p Umaxï❂ð✺ñ❏ò✔ó❙ô❞ô❞õ✔ð✢ö❩÷➫øúù❹ö◗õ✰û◗ó✪ü✪ý þ ÿ✁￾✄✂✆☎✞✝ ù õ✔ð✝ï✠✟➫ó✪ï☛✡❩ütó✪ð◆ütó✌☞ ✍✏✎ó✑✟➫ó✪ï✒✡✔✓✖✕☛✗✙✘✚✕✒✓❁ô✎ó✁ó✪ð❧ô✛✗✙✕☛✜✣✢✡õ✰ð◆ó✥✤✧✦◆ï❂ò✔õ✰ô★✢❩õ✔ü➹ô✎✦◆ü ✩✫✪✬ ✩✭✪✮ ✪✰✯✱ ÿ þ ￾ ✱ ☎ ✂✳✲✣✴☛✵✶✴☛✷✸￾ ✩✭✪✮ ✪✹✱✆✺✵✼✻ ù✌✽ ✴☛✵✾✝ ù ô✛✕✞✿✥ó✉ü❞ï❂ô❞õ✮ü★❀✺ó✪ñ❁✓✖✕☛✗➅ï❂ò✔ò ✱❃❂❅❄✼❆✬ ✺✖❇❈❊❉❅❇❈●❋ ✽❍✟➹õ✰ô✎ ✱✆✺✵✼✻■✷ ✽✏❏ ù❑✓✖✕☛✗➅ï❂ò✔ò ✵ ï❂ð✺ñ ✷ ☞ ▲✕❯ô❞ó✙ô✎ï✸ô✆✿✺ó◆▼✶ï◗ü■✦✢ó❖✟➮ó ✎ï❥û❯ó✙ðP✕◗✟➔ï❯ð✁õ✰ð✢ó◆✤✧✦◆ï❂ò✔õ➳ô★✢◗ý❘✟➮ó❙✘❑✦◆ütô✼✗❬ó✥✤✧✦✢õ❚✗❞ó➓ô✎ï✸ô✛ô✎ó❯✟➮ó✪õ✰ö✎ô❬õ✰ð◆ö ✓✖✦✢ð❱▼❙ô❞õ❚✕❯ð◆ü ✱ ✿✥ó❲✜❳✕❧ütõ✰ô❞õ✔û❯ó✌☞ ❨✼❩❭❬✁❩❭❪ ❫❵❴✁❛P❜❞❝❢❡❤❣ ✐✶❥✧❦♠❧✸♥♣♦✸q r✦P✗❬ö❯ó✥✗❬ü✥s❤ó✥✤✧✦◆ï✸ô❬õt✕◗ð✰✉ ✺❤✈✽❙✇ ❆① ✈ ① ✍ï✒✡❯ó þ ✺❤✈✽❙✇ ✈ ① ï❯ð◆ñ ✂ ✺✖✈ ✽❙✇ ①② ✈ ② ③ð❧ô✙✗✛✕✌✜✧✢✡õ✔ð✢ó◆✤✌✦✺ï❂ò✔õ➳ô★✢ ✺❤✈ ① ✽ ÿ✁￾✚④❖⑤⑥ ✈ ②⑧⑦ ☎ ✝ ù ⑨ ⑩❷❶❸❶❸❹⑧❺❼❻❾❽✳❿➀❺❼➁✠➂➄➃➆➅➇❽➉➈➊❶❸❹❼➋➍➌ ➎✏➏■➐ ➑✠➒P➓✆➔➣→↕↔➛➙★➜✼➝ ✐✶❥✧❦♠❧✸♥♣♦❳➞ ➟✕✸û❯ó✥✗❞ð◆õ✰ð✢ö❹ó✥✤✧✦◆ï❂ô❞õ❚✕❯ð➡➠❢➢ÿ✁￾ ✉ ✺ ➢✣✽ ☎ ✇➨ù ✉ ✺ ➢✣✽❙✇➆➢✈ ✇✹➢ ✈❳➤➉➥⑧➦ ④➨➧❾➩ ➢ ➢➤➉➥❼➦ ⑦ ✉➭➫ ✺➢➯✽❖✇ ✈✶➤➉➥⑧➦ ④ ➧❾➩ ⑤➢ ➢➤➉➥❼➦ ⑦ ✻ ✉➭➫ ➫ ✺➢✣✽❙✇ ➩ ⑤ ✈✶➤➉➥⑧➦ ➢➤➉➥⑧➦✹➲ ù ➳➭➵P➸❯➺❍➻☛➼✔➽❭➾➪➚❁➶➘➹✒➴✶➶➘➚◗➷◗➸❍➬➘➻➯➴✶➶⑧➮➱➽❤➹✒➴➡➹★➬✑➮✖➵P➸➪➻➯➴✣✃✒➴✶➹✒❐➀➴ ➢➨❒ ❮ ✇ ✯✉✲ ✯➢✲ ✇ ✈➤➉➥⑧➦ ④ ➧❾➩ ➢✣❰ ￾ ➢✧Ï ➢➤➉➥❼➦ ⑦ ➧✥Ð