4 Weak solution A natural way to define eralized solution of the not require differentiability is to go back to the integral form of the co nservation law, and say that u(a, t) is a generalized solution of the equation if the integral form of the equation is satisfied for all r and ar An alternative to the above approach is to write a weak statement in a form which requires less smoothness from the solut Multiply ut+f2=0byo(x,t)∈C(R×R+) co is the space of continuous functions aith continuous first derivative that have lot u +pr f] dadt P(, Ou(a, Odr=0 The above statement will be modified accordingly for a bounded spatial domain ting the appropriate boundar If above statement is satisfied for all E Co(IR x R+) then u(a, t)is a weak Weak solutions are essentially solutions that satisfy the diff chere the solution is smooth, and the jump condition at dis continuities. A solutions of the differential equation are also weak solutions; the reverse usly not true. Unfortunately there is a price to pay by enlarging the class Weak solutions to conservation laws are often non unique❒ ❮❰❅Ï➭ÐÒÑ❚Ó❱Ô✾Õ×Ö➦Ø➥Ó✙Ù❞Ú Û Ü✢Ý✰Þ✁ß✒à✥á âäã✦å❵æèç✎é✐å❵ê①ëìåîí❃æ➌ïñð❳ò✡óìã✦ò➸å➭ô✢ò✾ã✦ò✾é✐åîêöõ❖÷✏òtð❱ø✾ïîêùç✎æèõ✡ï❵ã✗ï➨ú➡æ✰û✝ò❣õ✰ã✝ü✏õ✻ø✾ý➥õ✡ð×òtþ➥ç✝å❵æèõ✡ï❵ã▲æ✰û✝åîæ✩ð❳ï✔ò➥ø ã✦ï❵æ➦é✐òtþ➥ç✎õ✰é❜ò➼ð❵õÿ➇ò✾é✐ò➥ã✚æèõ✡å✁￾➥õ✰êùõ✰æèíñõ✻ø➇æ➵ï➭ô✢ï✂￾tå❳ý☎✄❱æ➵ï✙æ✰û✝ò➼õ✰ã✝æ➵ò➌ô❵é✐å❵êîú➥ïîé✝✆ ï✹ú➇æ➐û✥ò➭ýtïîã✝ø✾ò➥é↔üîåîæõ✡ïîã ê❖å❵ë✟✞❐åîã①ð❃ø✾åîí❚æ✰û✝åîæ✟✠☛✡✌☞✎✍✑✏✓✒➸õ✻ø➩å❱ô❳ò✾ã✦ò✾é✐åîêùõ❖÷✔òtð❚ø✾ïîêöç✎æõ✡ïîã ï➨ú➸æ✰û✝ò✙òtþ✾ç✝åîæõ✡ïîã✒õú➭æ✰û✝ò✙õ✰ã✝æ➵òô✫é✐å❵ê ú➥ïîé✝✆ ï➨ú➸æ✰û✝ò✙òtþ✾ç✝åîæõ✡ïîã♦õ✻ø❣ø✾åîæõ✻ø➐ó✩òtð❅ú➥ï❵é✙åîê✰ê✔☞✖✕❬åîã①ð✗☞✙✘✛✚ â➡ã★å❵êöæ➵ò➥é↔ã①åîæõ✰ü✤ò■æ➵ï✗æ✰û✝ò■å✁￾tïîüîò➺å☎✜✁✜✦é✐ï✏å✫ýtû❺õ✻ø❚æ➵ï➺ë➦é↔õ✰æ➵ò▲å✣✢✥✤✧✦✩★✫✪✭✬✮✦✯✬✰✤✝✱✲✤✝✳✴✬✙õ✰ã❬å✙ú➥ïîé✝✆ ë✟û❯õ✡ýtû➺é✐òtþ➥ç✎õ✰é✐ò➥ø➭ê❖ò➥ø❜ø❣ø✝✆ñï✔ï❵æ✰û✎ã✦ò➥ø❜ø➜úté✐ï✩✆ æ➐û✥ò➸ø✾ïîêöç✎æõ✡ïîã✴✚ ✵✷✶✯✸✺✹✑✻✽✼✯✸✽✾ ✠✙✿✴❀❂❁❄❃❆❅❈❇❊❉✾●❋ ✡✌☞✎✍✓✏✓✒■❍●❏▲❑▼ ✡❖◆P❘◗●◆P❚❙✥✒ ❏ ❑ ▼ õ✻ø❫æ➐û✥ò➇ø✮✜✥å❳ý❜ò❣ï✹ú➇ýtïîã✝æõ✰ã✝ç✝ï❵ç❯ø①útç✎ã①ý➥æõ✡ïîã✝ø❫ë➦õ✰æ✰û×ýtïîã✝æõ✰ã✝ç✝ï❵ç❯ø⑧ó➜étø↔æ☞ð❳ò➥é↔õ✰üîåîæõ✰ü✤ò➭æ✰û✝åîæ⑧û✥åîüîò ýtï❄✆❚✜✥å❳ý➥æ✩ø↔ç❯✜❱✜✝ïîé↔æ❳❲➜õ✌✚✰ò✭✚➼æ➐û✥å❵æ❐å❵é✐ò❣÷✔ò✾é✐ï×åîæ❡õ✰ã✔ó➜ã✚õ✰æèí❱✚ ❨❬❩▼ ❨❂❩❭ ❩ ❋ ✡✌✠✿ ❀❬❁❃ ✒❫❪✁☞✖❪✁✏✥❅❴❇ ◆❛❵✹✑❜✭❝❱❞❢❡❄✹✑✻❵❝ ❉✾✲✼✔❡✩❞✑✹☎❣ ❨❩ ▼ ❨❩ ❭ ❩✐❤❋✿ ✠❊❀ ❋❃ ❁✖❥❦❪❱☞✙❪❱✏▲❀ ❨❩ ❭ ❩ ❋ ✡❖☞▲✍✑❇❦✒❛✠✟✡❖☞✎✍☎❇✁✒✓❪❱☞❧❅❴❇ ♠û✥ò❃å❱￾tï❵ü✤ò➩ø↔æ➌å❵æ➌ò♥✆ñò➥ã✚æìë➦õ✰ê✰ê☛￾tò❆✆❃ï✏ðîõó❡òtð×å❳ý❜ýtï❵é✐ðîõ✰ã❯ô❵êöí➼ú➥ïîé➩å❧￾tïîç✎ã✦ð❳òtð❃ø✮✜✥å❵æèõ✡å❵ê☞ð❳ï❄✆ñåîõ✰ã✯✞ ￾✾íñõ✰ã✦ýtïîé✮✜✝ïîé✐å❵æèõ✰ã❯ô×æ✰û✝ò✙å❢✜❱✜✦é❜ï☎✜✦é↔õ✡å❵æ➌ò♦￾tï❵ç✎ã✦ð✫å❵é↔í➍æ➵ò➥é✝✆✙ø♥✚ Û Ü✢Ý✰Þ✁ß✒à✯♣ ◆✰q ❡❉✖r❄s❜✂❣✓✹☎❡✩✹✑❜✭t✂❜❵✹✉✻✽❣✗❣☎❡❄✹☎✻✽❣✓✈✔❜✭✇ q❖r❞✗❡❱✸✺✸①❋ ❍②❏ ❑ ▼ ✡❖◆P③◗④◆P❚❙✥✒ ✹✑⑤✯❜❵⑥✠✟✡❖☞▲✍✓✏✓✒ ✻⑦❣✗❡●⑧⑩⑨❦❶✯❷ ❸❯❹✖❺❳❻✎❼❯❽✌❹✙❾ ❿ò❜å❯✄ ø✾ïîêöç✎æõ✡ïîã✝ø✒åîé✐ò ò↔øtø✾ò➥ã✚æèõ✡å❵ê✰êöí❺ø➥ï❵êöç✎æõ✡ïîã✥ø✗æ✰û✝åîæ❃ø✾åîæõ✻ø➐útí❺æ➐û✥ò ð❵õÿ➇ò➥é❜ò➥ã✚æèõ✡å❵ê➩ò❜þ✾ç✝åîæõ✡ïîã ë✟û✥ò✾é✐ò➺æ✰û✝ò❞ø✾ïîêöç✎æõ✡ïîã❲õ✻ø×ø✝✆ñï✔ï❵æ✰û➀✞❃åîã①ð æ➐û✥ò➂➁↔ç➃✆❚✜➀ýtï❵ã✦ðîõ✰æõ✡ïîã åîæ❱ð❵õ✻ø✾ý❜ï❵ã✝æõ✰ã✝ç✎õ✰æõ✡ò↔ø♥✚ â➡ê✰ê ➄ø↔æèé✐ï❵ã✢ô✧➅➇ø✾ïîêùç✎æèõ✡ï❵ã✥ø➇ï➨ú❡æ➐û✥ò➼ðîõÿ➇ò✾é✐ò✾ã✝æõ✡åîê✚òtþ✾ç✝åîæõ✡ïîã×å❵é✐ò➡å❵êø✾ï➭ëìòtå✭✄➭ø✾ïîêöç✎æõ✡ïîã✝ø❢❲✁æ✰û✝ò❅é❜ò➥üîò➥étø✾ò õ✻ø❐ï✁￾➥ü✏õ✡ïîç❯ø↔êöí➭ã①ïîæ⑧æé↔ç✝ò❯✚⑩➆✟ã✔ú➥ï❵é↔æèç✎ã①åîæ➵ò➥êùí➇✞⑧æ✰û✝ò➥é✐ò❫õ✻ø❐å➈✜①é↔õ✡ý❜ò❅æ➌ï①✜✝åîí✉￾✾í❣ò✾ã✝ê◗åîéèô✫õ✰ã✢ô➸æ✰û✝ò❅ý➥ê◗å✤øtø ï➨ú➼ø✾ï❵êöç✎æèõ✡ï❵ã✥ø♥✚ ➉③⑨➀❶➃❷ ❸✭❹✙❺✌❻▲❼✭❽❳❹✖❾☛❸ ✹r♦➊✧r✁❵❣✓❜✭❞s❡✩✹✑✻r❱❵ ✸⑦❡➇➋✛❣■❶➃➌❄⑨ r❱q✹✑❜❵ ❾☛❹✙❾➍❻☛❾☛❽✌➎❫❻⑨✖➏ ➐♥➑