TV(u(a, T) b-ual+ Juc -ue+luf -e By inspection we see that ub-ua+luc -ubl+lud -ucl +lue -udl+luf -uel If there are shocks: TV((a, r))<TV(u(a, 0)) u The total variation of an entropy satisfying solution is a non-increasing function of time REFERENCES [S/P K. Sweby, Theory of Scalar One-dimensional Hyperbolic Conservation Laws, 2nd ERCOFTAC Summer school on Advanced Flow Simulation, Septem LLV R.J. Le Venue, Numerical Methods for Conservation Laws, Lecture Notes n Mathematics, ETH Zurich, Birkhauser, 1992 [/ P D. Lar, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM 1973 /W G. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974❄❆❅❈❇✕❉❊❇✕❋✦●✣❄❆❍■❍❑❏ ▲ ❉✁▼❊◆❖❉✆P✠▲✴◗❘▲ ❉✆❙❊◆❖❉✁❚✹▲❯◗❱▲ ❉✁❲✽◆❳❉✆❚✑▲ ❨❬❩✒❭✓❪✄❫✍❴✄❵✞❛✴❜❝❭✏❞✹❪❢❡❣❵❤❫✴❵✞❵✽❜✕✐✄❥✑❜ ❄❆❅✒❇✏❉❦❇✏❋✦●✣❄❆❍■❍❑❏ ▲ ❉ ▼ ◆❖❉P ▲❯◗❱▲ ❉▼ ◆❖❉ ❚ ▲✴◗❧▲ ❉ ❲ ◆❖❉❚ ▲ ♠ ▲ ❉ ▼ ◆❖❉P ▲❯◗❱▲ ❉❙ ◆❳❉ ▼ ▲✴◗❱▲ ❉✆♥♦◆❖❉❙ ▲✴◗❧▲ ❉ ❚ ◆❳❉✆♥✪▲❯◗❧▲ ❉ ❲ ◆❖❉❚ ▲ ❏ ❄❆❅✒❇✕❉❊❇✕❋✛●✣♣✪❍■❍ q✼r✪s✓t✦✉❖✈✖✇ ①③②❦④✣⑤✄⑥❯⑦✣⑥✽⑧✹⑦★⑥✽⑨■⑤✳⑩✖❶✞❷✠⑨❯❸❊❄❆❅✔❇✕❉❊❇✕❋✛●■❄❆❍✣❍ ♠ ❄❆❅❈❇✕❉❊❇✕❋✦●★♣✫❍✣❍ ①✷❹❻❺✜⑥❯❹✄⑥✴⑦✞⑧✹❼✍❽ ❾ ❾✫❿ ➀✼➁➃➂➂ ➂ ➂❝➄❉ ➄ ❋ ➂ ➂ ➂ ➂ ❾❋✳➅ ♠ ♣ ➆✿➇❦➈➊➉❯➋✱➉❯➌✄➍✱➎✄➌✠➏✑➐✏➌✄➉❯➐✏➋✼➑➒➋✼➓✆➌✄➑➒➈✖➑✁➉✴➏✑➋✳➔✆→↔➣❯➌✄➉❯➐✏➣↕➓✍→➙➐✏➑❦➛➜➣↕➋✼➍✏➝✛➉❯➐✏➋✼➑↔➐✏➣✤➌➜➑❦➋✱➑✦➞★➐❝➑✛➟✜➏✑➈✪➌✳➣❯➐❝➑✛➛ ➓③➝✛➑❦➟✹➉↕➐✏➋✼➑➠➋✱➓✶➉↕➐✏➡➢➈✼➤ ➥➧➦✾➨❊➦➩➥❆➦➩➫✒➭✦➦✥➯ ➲➯↕➳➒➵♦➸➻➺✔➸❤➯✱❡❣❵↕➼☞❩✲➽➢➾✁✐✄❵✞❞✑➚✺❩➪❞✻➶❢➯✁❛✞❥✹➹➘❥✑➚➷➴✶❪✼❵✴➬■➮✑❭✓➱✒❵☞❪✳❫✺❭✏❞✑❪✼❥✹➹❬✃✽❩★❴✳❵☞➚✴➼✞❞✑➹❐❭✏❛➪➭✾❞✑❪✳❫✴❵☞➚✺❒✑❥✑❜✍❭✏❞✑❪ ❮✛❥✑❡❊❫✞➽✛❰✫❪✼➮❆➦✾➥❤➭✶➴❣➨➊➾✫Ï➜➭Ð➯✼Ñ✠➱➜➱✒❵✴➚✥➯✆❛★✐✳❞❯❞✹➹✼❞✹❪✒Ï❆➮✑❒✑❥✑❪✁❛★❵✞➮Ò➨✾➹Ó❞✑❡Ô➯✼❭✓➱➜Ñ✠➹➘❥✹❜❝❭✏❞✹❪✄➽✛➯✁❵✷❴✼❜✷❵☞➱❀➬ ➼✞❵☞➚❤Õ✜Ö✜Ö✪×✜➸ ➲❮❣Ø✫➳✽➥✿➸✘Ù✠➸✾❮✛❵✄Ø✱❵✞Ú✴Ñ✳❵✺➽✥➫✿Ñ✠➱✔❵✴➚✺❭✏❛✞❥✑➹➩Û❢❵✴❜✓✐✳❞❯➮✑❫➧➶☞❞✑➚➒➭✾❞✑❪✳❫✴❵☞➚✺❒✑❥✑❜✍❭✏❞✑❪➢❮✦❥✹❡❊❫✞➽♦❮✛❵★❛✴❜✍Ñ✠➚✣❵✿➫❤❞✹❜③❵☞❫ ❭✓❪ÜÛ❢❥✹❜✓✐✳❵☞➱✒❥✑❜✍❭✏❛✺❫✞➽❊➦❤➾✱✃ÞÝ✥ß❉❦➚✺❭✏❛★✐✖➽❊❨♦❭✓➚✣à↕✐✳ßá Ñ✖❫☞❵✴➚✞➽❣Õ✹Ö✫Ö✹❰✪➸ ➲❮✠➳✽➵♦➸➻â❀➸❦❮✛❥✴ã✜➽✾✃✽❩★❴✳❵☞➚✴➼✞❞✹➹✮❭✏❛✔➯✼❩✲❫✞❜✷❵✴➱❤❫❀❞■➶ä➭✾❞✹❪✄❫✴❵☞➚✺❒✑❥✑❜✍❭✏❞✑❪❳❮✛❥✑❡❊❫✒❥✑❪✁➮↔❜✓✐✳❵➜ÛÜ❥✹❜✓✐✳❵☞➱✒❥✑❜✍❭✏❛✞❥✑➹ ➾✁✐✄❵✞❞✑➚✺❩↔❞✻➶➊➯✄✐✄❞↕❛★à➢å❻❥✹❒✲❵☞❫✞➽✾➯✖æ★ÏÒÛçÕ✹Ö✳è❯é✫➸ ➲å➊➳✔ê♦➸✽å✒✐✠❭✓❜✓✐✳❥✑➱➜➽❊❮❊❭✓❪✁❵★❥✹➚❤❥✑❪✁➮➜➫➜❞✑❪✳➹❐❭✓❪✼❵✞❥✑➚➒å➒❥✑❒✑❵✺❫✞➽❤å↔❭✓➹➘❵✴❩✑➬✏æ✺❪✱❜③❵✴➚✞❫✴❛☞❭✏❵✴❪✼❛✞❵☞➽✤Õ✹Ö✳è✣ë✱➸ ì✑í