/ Sprite 1984 - 1993 / Sprite 1984 - 1993.iso / lib / tex / inputs / amstexsiam-sample.tex

< prev

next >

MacOS 8.1

Win98

DOS

browse contents

view JSON data

view as text

+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 69 6e 70 75 74 20 61 | 6d 73 74 65 78 0a 5c 64 |\input a|mstex.\d|
|00000010| 6f 63 75 6d 65 6e 74 73 | 74 79 6c 65 7b 61 6d 73 |ocuments|tyle{ams|
|00000020| 74 65 78 73 69 61 6d 7d | 0a 25 25 25 25 25 25 25 |texsiam}|.%%%%%%%|
|00000030| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|00000040| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|00000050| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|00000060| 25 25 25 25 25 25 25 25 | 25 25 25 25 0a 25 20 4d |%%%%%%%%|%%%%.% M|
|00000070| 61 63 72 6f 20 64 65 66 | 69 6e 69 74 69 6f 6e 73 |acro def|initions|
|00000080| 20 66 6f 72 20 72 75 6e | 6e 69 6e 67 20 68 65 61 | for run|ning hea|
|00000090| 64 73 20 61 6e 64 20 66 | 69 72 73 74 20 70 61 67 |ds and f|irst pag|
|000000a0| 65 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 25 |e       |       %|
|000000b0| 0a 25 20 55 73 65 20 5c | 64 65 66 20 6f 72 20 5c |.% Use \|def or \|
|000000c0| 72 65 64 65 66 69 6e 65 | 20 66 6f 72 20 74 68 65 |redefine| for the|
|000000d0| 73 65 20 64 65 66 69 6e | 69 74 69 6f 6e 73 20 20 |se defin|itions  |
|000000e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000000f0| 20 20 20 25 0a 5c 61 63 | 63 65 70 74 65 64 20 20 |   %.\ac|cepted  |
|00000100| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000110| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000120| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000130| 20 20 20 20 20 20 20 25 | 0a 5c 70 61 67 65 6e 6f |       %|.\pageno|
|00000140| 3d 31 30 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |=10     |        |
|00000150| 20 20 20 20 20 20 20 20 | 20 20 20 25 20 73 74 61 |        |   % sta|
|00000160| 72 74 69 6e 67 20 70 61 | 67 65 20 6f 66 20 70 61 |rting pa|ge of pa|
|00000170| 70 65 72 20 20 20 20 20 | 20 20 20 25 0a 5c 64 65 |per     |   %.\de|
|00000180| 66 5c 6a 6f 75 72 6e 61 | 6c 6e 61 6d 65 7b 7b 5c |f\journa|lname{{\|
|00000190| 73 69 78 72 6d 20 53 49 | 41 4d 20 4a 2e 20 4d 7b |sixrm SI|AM J. M{|
|000001a0| 5c 66 69 76 65 72 6d 20 | 41 54 48 2e 7d 20 46 7b |\fiverm |ATH.} F{|
|000001b0| 5c 66 69 76 65 72 6d 20 | 4f 4f 4c 2e 7d 7d 7d 25 |\fiverm |OOL.}}}%|
|000001c0| 0a 5c 64 65 66 5c 69 73 | 73 75 65 76 6f 6c 75 6d |.\def\is|suevolum|
|000001d0| 65 7b 31 7d 20 20 20 20 | 20 20 20 20 20 20 20 20 |e{1}    |        |
|000001e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000001f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000200| 20 20 20 25 0a 5c 64 65 | 66 5c 69 73 73 75 65 6e |   %.\de|f\issuen|
|00000210| 75 6d 62 65 72 7b 32 7d | 20 20 20 20 20 20 20 20 |umber{2}|        |
|00000220| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000230| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000240| 20 20 20 20 20 20 20 25 | 0a 5c 64 65 66 5c 69 73 |       %|.\def\is|
|00000250| 73 75 65 6d 6f 6e 74 68 | 7b 46 65 62 72 75 61 72 |suemonth|{Februar|
|00000260| 79 7d 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |y}      |        |
|00000270| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000280| 20 20 20 20 20 20 20 20 | 20 20 20 25 0a 5c 64 65 |        |   %.\de|
|00000290| 66 5c 70 6c 61 63 65 6e | 75 6d 62 65 72 7b 30 30 |f\placen|umber{00|
|000002a0| 32 7d 20 20 20 20 20 20 | 20 20 20 20 20 20 20 25 |2}      |       %|
|000002b0| 20 70 6c 61 63 65 20 6f | 66 20 70 61 70 65 72 20 | place o|f paper |
|000002c0| 69 6e 20 74 68 69 73 20 | 69 73 73 75 65 20 20 25 |in this |issue  %|
|000002d0| 0a 5c 64 65 66 5c 69 73 | 73 75 65 79 65 61 72 7b |.\def\is|sueyear{|
|000002e0| 31 39 38 38 7d 20 20 20 | 20 20 20 20 20 20 20 20 |1988}   |        |
|000002f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000300| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000310| 20 20 20 25 0a 5c 64 65 | 66 5c 6f 64 64 68 65 61 |   %.\de|f\oddhea|
|00000320| 64 7b 61 20 73 61 6d 70 | 6c 65 20 70 61 70 65 72 |d{a samp|le paper|
|00000330| 7d 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |}       |        |
|00000340| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000350| 20 20 20 20 20 20 20 25 | 0a 5c 64 65 66 5c 65 76 |       %|.\def\ev|
|00000360| 65 6e 68 65 61 64 7b 62 | 72 61 64 6c 65 79 20 6a |enhead{b|radley j|
|00000370| 5c 2e 20 6c 75 63 69 65 | 72 20 61 6e 64 20 64 6f |\. lucie|r and do|
|00000380| 75 67 6c 61 73 20 6e 5c | 2e 20 61 72 6e 6f 6c 64 |uglas n\|. arnold|
|00000390| 7d 20 20 20 20 20 20 20 | 20 20 20 25 0a 25 25 25 |}       |   %.%%%|
|000003a0| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|000003b0| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|000003c0| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|000003d0| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|000003e0| 0a 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |.%%%%%%%|%%%%%%%%|
|000003f0| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|00000400| 25 25 25 25 0a 25 20 4d | 61 63 72 6f 73 20 73 70 |%%%%.% M|acros sp|
|00000410| 65 63 69 66 69 63 20 74 | 6f 20 74 68 69 73 20 70 |ecific t|o this p|
|00000420| 61 70 65 72 20 20 20 25 | 0a 5c 64 65 66 69 6e 65 |aper   %|.\define|
|00000430| 5c 42 62 62 52 7b 7b 5c | 42 62 62 20 52 7d 7d 20 |\BbbR{{\|Bbb R}} |
|00000440| 20 20 20 20 20 20 20 20 | 20 20 20 25 0a 5c 64 65 |        |   %.\de|
|00000450| 66 69 6e 65 5c 6c 6f 6e | 65 72 7b 7b 4c 5e 31 28 |fine\lon|er{{L^1(|
|00000460| 5c 42 62 62 52 29 7d 7d | 20 20 20 20 20 20 20 25 |\BbbR)}}|       %|
|00000470| 0a 5c 64 65 66 69 6e 65 | 5c 6c 69 6e 66 72 7b 7b |.\define|\linfr{{|
|00000480| 4c 5e 5c 69 6e 66 74 79 | 28 5c 42 62 62 52 29 7d |L^\infty|(\BbbR)}|
|00000490| 7d 20 20 25 0a 5c 64 65 | 66 69 6e 65 5c 62 76 72 |}  %.\de|fine\bvr|
|000004a0| 7b 7b 5c 72 6f 6d 61 6e | 7b 42 56 7d 28 5c 42 62 |{{\roman|{BV}(\Bb|
|000004b0| 62 52 29 7d 7d 20 20 25 | 0a 5c 64 65 66 69 6e 65 |bR)}}  %|.\define|
|000004c0| 5c 54 56 7b 7b 5c 72 6f | 6d 61 6e 20 7b 54 56 7d |\TV{{\ro|man {TV}|
|000004d0| 7d 7d 20 20 20 20 20 20 | 20 20 20 25 0a 5c 64 65 |}}      |   %.\de|
|000004e0| 66 69 6e 65 5c 73 64 6f | 74 7b 5c 2c 5c 63 64 6f |fine\sdo|t{\,\cdo|
|000004f0| 74 5c 2c 7d 20 20 20 20 | 20 20 20 20 20 20 20 25 |t\,}    |       %|
|00000500| 0a 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |.%%%%%%%|%%%%%%%%|
|00000510| 25 25 25 25 25 25 25 25 | 25 25 25 25 25 25 25 25 |%%%%%%%%|%%%%%%%%|
|00000520| 25 25 25 25 0a 5c 74 6f | 70 6d 61 74 74 65 72 0a |%%%%.\to|pmatter.|
|00000530| 5c 74 69 74 6c 65 0a 41 | 20 53 41 4d 50 4c 45 20 |\title.A| SAMPLE |
|00000540| 50 41 50 45 52 20 54 4f | 20 49 4c 4c 55 53 54 52 |PAPER TO| ILLUSTR|
|00000550| 41 54 45 20 54 48 45 20 | 0a 5c 41 6d 53 54 65 58 |ATE THE |.\AmSTeX|
|00000560| 5c 20 53 49 41 4d 20 53 | 54 59 4c 45 5c 66 6f 6f |\ SIAM S|TYLE\foo|
|00000570| 74 6e 6f 74 65 22 2a 22 | 7b 55 6e 6c 69 6b 65 6c |tnote"*"|{Unlikel|
|00000580| 79 20 74 6f 20 61 70 70 | 65 61 72 2e 7d 0a 5c 65 |y to app|ear.}.\e|
|00000590| 6e 64 74 69 74 6c 65 0a | 5c 61 75 74 68 6f 72 0a |ndtitle.|\author.|
|000005a0| 42 52 41 44 4c 45 59 20 | 4a 2e 20 4c 55 43 49 45 |BRADLEY |J. LUCIE|
|000005b0| 52 5c 66 6f 6f 74 6e 6f | 74 65 22 5c 64 61 67 22 |R\footno|te"\dag"|
|000005c0| 7b 44 65 70 61 72 74 6d | 65 6e 74 20 6f 66 20 4d |{Departm|ent of M|
|000005d0| 61 74 68 65 6d 61 74 69 | 63 73 2c 20 50 75 72 64 |athemati|cs, Purd|
|000005e0| 75 65 20 55 6e 69 76 65 | 72 73 69 74 79 2c 0a 57 |ue Unive|rsity,.W|
|000005f0| 65 73 74 20 4c 61 66 61 | 79 65 74 74 65 2c 20 49 |est Lafa|yette, I|
|00000600| 6e 64 69 61 6e 61 20 34 | 37 39 30 37 2e 0a 54 68 |ndiana 4|7907..Th|
|00000610| 65 20 77 6f 72 6b 20 6f | 66 20 74 68 65 20 66 69 |e work o|f the fi|
|00000620| 72 73 74 20 61 75 74 68 | 6f 72 20 77 61 73 20 6e |rst auth|or was n|
|00000630| 6f 74 20 73 75 70 70 6f | 72 74 65 64 20 62 79 20 |ot suppo|rted by |
|00000640| 74 68 65 0a 57 6f 6c 66 | 20 46 6f 75 6e 64 61 74 |the.Wolf| Foundat|
|00000650| 69 6f 6e 2e 0a 7d 20 61 | 6e 64 0a 44 4f 55 47 4c |ion..} a|nd.DOUGL|
|00000660| 41 53 20 4e 2e 20 41 52 | 4e 4f 4c 44 5c 66 6f 6f |AS N. AR|NOLD\foo|
|00000670| 74 6e 6f 74 65 22 5c 64 | 64 61 67 22 7b 44 65 70 |tnote"\d|dag"{Dep|
|00000680| 61 72 74 6d 65 6e 74 20 | 6f 66 20 4d 61 74 68 65 |artment |of Mathe|
|00000690| 6d 61 74 69 63 73 2c 20 | 55 6e 69 76 65 72 73 69 |matics, |Universi|
|000006a0| 74 79 0a 6f 66 20 4d 61 | 72 79 6c 61 6e 64 2c 20 |ty.of Ma|ryland, |
|000006b0| 43 6f 6c 6c 65 67 65 20 | 50 61 72 6b 2c 20 4d 61 |College |Park, Ma|
|000006c0| 72 79 6c 61 6e 64 20 32 | 30 37 34 32 2e 7d 0a 5c |ryland 2|0742.}.\|
|000006d0| 65 6e 64 61 75 74 68 6f | 72 0a 5c 61 62 73 74 72 |endautho|r.\abstr|
|000006e0| 61 63 74 7b 54 68 69 73 | 20 73 61 6d 70 6c 65 20 |act{This| sample |
|000006f0| 70 61 70 65 72 20 69 6c | 6c 75 73 74 72 61 74 65 |paper il|lustrate|
|00000700| 73 20 6d 61 6e 79 20 6f | 66 20 74 68 65 20 61 6d |s many o|f the am|
|00000710| 73 74 65 78 0a 6d 61 63 | 72 6f 73 20 61 73 20 75 |stex.mac|ros as u|
|00000720| 73 65 64 20 77 69 74 68 | 20 74 68 65 20 5c 41 6d |sed with| the \Am|
|00000730| 53 54 65 58 5c 20 53 49 | 41 4d 20 73 74 79 6c 65 |STeX\ SI|AM style|
|00000740| 20 66 69 6c 65 2e 0a 54 | 68 65 20 5c 41 6d 53 54 | file..T|he \AmST|
|00000750| 65 58 5c 20 53 49 41 4d | 20 73 74 79 6c 65 20 66 |eX\ SIAM| style f|
|00000760| 69 6c 65 20 28 61 6d 73 | 74 65 78 73 69 61 6d 2e |ile (ams|texsiam.|
|00000770| 73 74 79 29 2c 20 77 68 | 69 63 68 20 69 73 20 61 |sty), wh|ich is a|
|00000780| 20 6d 6f 64 69 66 69 63 | 61 74 69 6f 6e 20 6f 66 | modific|ation of|
|00000790| 0a 74 68 65 20 61 6d 73 | 70 70 74 20 73 74 79 6c |.the ams|ppt styl|
|000007a0| 65 20 6f 66 20 4d 69 63 | 68 61 65 6c 20 53 70 69 |e of Mic|hael Spi|
|000007b0| 76 61 6b 2c 20 67 69 76 | 65 73 20 61 75 74 68 6f |vak, giv|es autho|
|000007c0| 72 73 20 65 61 73 79 0a | 61 63 63 65 73 73 20 74 |rs easy.|access t|
|000007d0| 6f 20 6d 6f 73 74 20 6f | 66 20 74 68 65 20 74 79 |o most o|f the ty|
|000007e0| 70 6f 67 72 61 70 68 69 | 63 61 6c 20 63 6f 6e 73 |pographi|cal cons|
|000007f0| 74 72 75 63 74 69 6f 6e | 73 20 75 73 65 64 20 69 |truction|s used i|
|00000800| 6e 20 53 49 41 4d 20 6a | 6f 75 72 6e 61 6c 73 2e |n SIAM j|ournals.|
|00000810| 0a 49 74 20 64 6f 65 73 | 20 6e 6f 74 20 61 64 64 |.It does| not add|
|00000820| 72 65 73 73 20 74 68 65 | 20 69 73 73 75 65 73 20 |ress the| issues |
|00000830| 6f 66 20 74 68 65 20 74 | 61 62 6c 65 20 6f 66 20 |of the t|able of |
|00000840| 63 6f 6e 74 65 6e 74 73 | 0a 6f 72 20 74 61 62 6c |contents|.or tabl|
|00000850| 65 73 2c 20 77 68 69 63 | 68 20 6d 75 73 74 20 62 |es, whic|h must b|
|00000860| 65 20 73 65 74 20 75 73 | 69 6e 67 20 6d 6f 72 65 |e set us|ing more|
|00000870| 20 70 72 69 6d 69 74 69 | 76 65 20 5c 54 65 58 5c | primiti|ve \TeX\|
|00000880| 20 6d 61 63 72 6f 73 2e | 7d 0a 5c 6b 65 79 77 6f | macros.|}.\keywo|
|00000890| 72 64 73 7b 70 6f 72 6f | 75 73 20 6d 65 64 69 75 |rds{poro|us mediu|
|000008a0| 6d 2c 20 69 6e 74 65 72 | 66 61 63 65 20 63 75 72 |m, inter|face cur|
|000008b0| 76 65 73 7d 0a 5c 73 75 | 62 6a 63 6c 61 73 73 7b |ves}.\su|bjclass{|
|000008c0| 36 35 4e 36 30 7d 0a 5c | 65 6e 64 74 6f 70 6d 61 |65N60}.\|endtopma|
|000008d0| 74 74 65 72 0a 5c 64 6f | 63 75 6d 65 6e 74 0a 5c |tter.\do|cument.\|
|000008e0| 73 75 62 68 65 61 64 69 | 6e 67 7b 31 2e 20 49 6e |subheadi|ng{1. In|
|000008f0| 74 72 6f 64 75 63 74 69 | 6f 6e 7d 0a 57 65 20 61 |troducti|on}.We a|
|00000900| 72 65 20 63 6f 6e 63 65 | 72 6e 65 64 20 77 69 74 |re conce|rned wit|
|00000910| 68 20 6e 75 6d 65 72 69 | 63 61 6c 20 61 70 70 72 |h numeri|cal appr|
|00000920| 6f 78 69 6d 61 74 69 6f | 6e 73 20 74 6f 20 74 68 |oximatio|ns to th|
|00000930| 65 20 73 6f 2d 63 61 6c | 6c 65 64 0a 70 6f 72 6f |e so-cal|led.poro|
|00000940| 75 73 2d 6d 65 64 69 75 | 6d 20 65 71 75 61 74 69 |us-mediu|m equati|
|00000950| 6f 6e 20 5c 63 69 74 65 | 7b 36 7d 2c 0a 24 24 0a |on \cite|{6},.$$.|
|00000960| 5c 61 6c 69 67 6e 65 64 | 61 74 32 0a 20 20 26 75 |\aligned|at2.  &u|
|00000970| 5f 74 3d 5c 70 68 69 28 | 75 29 5f 7b 78 78 7d 2c |_t=\phi(|u)_{xx},|
|00000980| 26 26 5c 71 71 75 61 64 | 20 78 5c 69 6e 5c 42 62 |&&\qquad| x\in\Bb|
|00000990| 62 52 2c 5c 71 75 61 64 | 20 74 3e 30 2c 5c 71 75 |bR,\quad| t>0,\qu|
|000009a0| 61 64 5c 70 68 69 28 75 | 29 3d 75 5e 6d 2c 5c 71 |ad\phi(u|)=u^m,\q|
|000009b0| 75 61 64 20 6d 3e 31 2c | 0a 5c 5c 0a 20 20 26 75 |uad m>1,|.\\.  &u|
|000009c0| 28 78 2c 30 29 3d 75 5f | 30 28 78 29 2c 26 26 5c |(x,0)=u_|0(x),&&\|
|000009d0| 71 71 75 61 64 20 78 5c | 69 6e 5c 42 62 62 52 2e |qquad x\|in\BbbR.|
|000009e0| 0a 5c 65 6e 64 61 6c 69 | 67 6e 65 64 61 74 0a 5c |.\endali|gnedat.\|
|000009f0| 74 61 67 20 31 2e 31 0a | 24 24 0a 57 65 20 61 73 |tag 1.1.|$$.We as|
|00000a00| 73 75 6d 65 20 74 68 61 | 74 20 74 68 65 20 69 6e |sume tha|t the in|
|00000a10| 69 74 69 61 6c 20 64 61 | 74 61 20 24 75 5f 30 28 |itial da|ta $u_0(|
|00000a20| 78 29 24 20 68 61 73 20 | 62 6f 75 6e 64 65 64 20 |x)$ has |bounded |
|00000a30| 73 75 70 70 6f 72 74 2c | 20 74 68 61 74 0a 24 30 |support,| that.$0|
|00000a40| 5c 6c 65 71 20 75 5f 30 | 5c 6c 65 71 20 4d 24 2c |\leq u_0|\leq M$,|
|00000a50| 20 61 6e 64 20 74 68 61 | 74 20 24 5c 70 68 69 28 | and tha|t $\phi(|
|00000a60| 75 5f 30 29 5f 78 5c 69 | 6e 5c 62 76 72 24 2e 0a |u_0)_x\i|n\bvr$..|
|00000a70| 49 74 20 69 73 20 77 65 | 6c 6c 20 6b 6e 6f 77 6e |It is we|ll known|
|00000a80| 20 74 68 61 74 20 61 20 | 75 6e 69 71 75 65 20 73 | that a |unique s|
|00000a90| 6f 6c 75 74 69 6f 6e 20 | 24 75 28 78 2c 74 29 24 |olution |$u(x,t)$|
|00000aa0| 20 6f 66 20 28 31 2e 31 | 29 20 65 78 69 73 74 73 | of (1.1|) exists|
|00000ab0| 2c 0a 61 6e 64 20 74 68 | 61 74 20 24 75 24 20 73 |,.and th|at $u$ s|
|00000ac0| 61 74 69 73 66 69 65 73 | 0a 24 24 0a 20 20 30 5c |atisfies|.$$.  0\|
|00000ad0| 6c 65 71 20 75 5c 6c 65 | 71 20 4d 5c 74 65 78 74 |leq u\le|q M\text|
|00000ae0| 7b 20 61 6e 64 20 7d 5c | 54 56 5c 70 68 69 28 75 |{ and }\|TV\phi(u|
|00000af0| 28 5c 2c 5c 63 64 6f 74 | 5c 2c 2c 74 29 29 5f 78 |(\,\cdot|\,,t))_x|
|00000b00| 5c 6c 65 71 5c 54 56 5c | 70 68 69 28 75 5f 30 29 |\leq\TV\|phi(u_0)|
|00000b10| 5f 78 2e 0a 5c 74 61 67 | 20 31 2e 32 0a 24 24 0a |_x..\tag| 1.2.$$.|
|00000b20| 49 66 20 74 68 65 20 64 | 61 74 61 20 68 61 73 20 |If the d|ata has |
|00000b30| 73 6c 69 67 68 74 6c 79 | 20 6d 6f 72 65 20 72 65 |slightly| more re|
|00000b40| 67 75 6c 61 72 69 74 79 | 2c 20 74 68 65 6e 20 74 |gularity|, then t|
|00000b50| 68 69 73 20 74 6f 6f 20 | 69 73 20 73 61 74 69 73 |his too |is satis|
|00000b60| 66 69 65 64 0a 62 79 20 | 74 68 65 20 73 6f 6c 75 |fied.by |the solu|
|00000b70| 74 69 6f 6e 2e 20 53 70 | 65 63 69 66 69 63 61 6c |tion. Sp|ecifical|
|00000b80| 6c 79 2c 20 69 66 20 24 | 6d 24 20 69 73 20 6e 6f |ly, if $|m$ is no|
|00000b90| 20 67 72 65 61 74 65 72 | 20 74 68 61 6e 20 74 77 | greater| than tw|
|00000ba0| 6f 20 61 6e 64 0a 24 75 | 5f 30 24 20 69 73 20 4c |o and.$u|_0$ is L|
|00000bb0| 69 70 73 63 68 69 74 7a | 20 63 6f 6e 74 69 6e 75 |ipschitz| continu|
|00000bc0| 6f 75 73 2c 20 74 68 65 | 6e 20 24 75 28 5c 2c 5c |ous, the|n $u(\,\|
|00000bd0| 63 64 6f 74 5c 2c 2c 74 | 29 24 20 69 73 20 61 6c |cdot\,,t|)$ is al|
|00000be0| 73 6f 20 4c 69 70 73 63 | 68 69 74 7a 3b 0a 69 66 |so Lipsc|hitz;.if|
|00000bf0| 20 24 6d 24 20 69 73 20 | 67 72 65 61 74 65 72 20 | $m$ is |greater |
|00000c00| 74 68 61 6e 20 74 77 6f | 20 61 6e 64 20 24 28 75 |than two| and $(u|
|00000c10| 5f 30 5e 7b 6d 2d 31 7d | 29 5f 78 5c 69 6e 5c 6c |_0^{m-1}|)_x\in\l|
|00000c20| 69 6e 66 72 24 2c 20 74 | 68 65 6e 0a 24 28 75 28 |infr$, t|hen.$(u(|
|00000c30| 5c 2c 5c 63 64 6f 74 5c | 2c 2c 74 29 5e 7b 6d 2d |\,\cdot\|,,t)^{m-|
|00000c40| 31 7d 29 5f 78 5c 69 6e | 5c 6c 69 6e 66 72 24 20 |1})_x\in|\linfr$ |
|00000c50| 0a 28 73 65 65 20 5b 33 | 5d 29 2e 20 28 54 68 69 |.(see [3|]). (Thi|
|00000c60| 73 20 77 69 6c 6c 20 66 | 6f 6c 6c 6f 77 20 66 72 |s will f|ollow fr|
|00000c70| 6f 6d 20 72 65 73 75 6c | 74 73 20 70 72 65 73 65 |om resul|ts prese|
|00000c80| 6e 74 65 64 20 68 65 72 | 65 2c 20 61 6c 73 6f 2e |nted her|e, also.|
|00000c90| 29 0a 57 65 20 61 6c 73 | 6f 20 75 73 65 20 74 68 |).We als|o use th|
|00000ca0| 65 20 66 61 63 74 20 74 | 68 61 74 20 74 68 65 20 |e fact t|hat the |
|00000cb0| 73 6f 6c 75 74 69 6f 6e | 20 24 75 24 20 69 73 20 |solution| $u$ is |
|00000cc0| 48 5c 22 6f 6c 64 65 72 | 20 63 6f 6e 74 69 6e 75 |H\"older| continu|
|00000cd0| 6f 75 73 20 69 6e 20 24 | 74 24 2e 0a 0a 5c 73 75 |ous in $|t$...\su|
|00000ce0| 62 68 65 61 64 69 6e 67 | 7b 32 2e 20 24 5c 6c 69 |bheading|{2. $\li|
|00000cf0| 6e 66 72 24 20 65 72 72 | 6f 72 20 62 6f 75 6e 64 |nfr$ err|or bound|
|00000d00| 73 7d 0a 41 66 74 65 72 | 20 61 20 73 69 6d 70 6c |s}.After| a simpl|
|00000d10| 65 20 64 65 66 69 6e 69 | 74 69 6f 6e 2c 20 77 65 |e defini|tion, we|
|00000d20| 20 73 74 61 74 65 20 61 | 20 74 68 65 6f 72 65 6d | state a| theorem|
|00000d30| 0a 74 68 61 74 20 65 78 | 70 72 65 73 73 65 73 20 |.that ex|presses |
|00000d40| 74 68 65 20 65 72 72 6f | 72 20 6f 66 20 61 70 70 |the erro|r of app|
|00000d50| 72 6f 78 69 6d 61 74 69 | 6f 6e 73 20 24 75 5e 68 |roximati|ons $u^h|
|00000d60| 24 20 69 6e 0a 74 65 72 | 6d 73 20 6f 66 20 74 68 |$ in.ter|ms of th|
|00000d70| 65 20 77 65 61 6b 20 74 | 72 75 6e 63 61 74 69 6f |e weak t|runcatio|
|00000d80| 6e 20 65 72 72 6f 72 20 | 24 45 24 2e 0a 5c 70 72 |n error |$E$..\pr|
|00000d90| 6f 63 6c 61 69 6d 7b 44 | 65 66 69 6e 69 74 69 6f |oclaim{D|efinitio|
|00000da0| 6e 20 32 2e 31 7d 5c 72 | 6d 20 41 20 7b 5c 69 74 |n 2.1}\r|m A {\it|
|00000db0| 20 64 65 66 69 6e 69 74 | 69 6f 6e 7d 0a 69 73 20 | definit|ion}.is |
|00000dc0| 74 68 65 20 73 61 6d 65 | 20 61 73 20 61 20 74 68 |the same| as a th|
|00000dd0| 65 6f 72 65 6d 20 73 65 | 74 20 69 6e 20 72 6f 6d |eorem se|t in rom|
|00000de0| 61 6e 0a 74 79 70 65 2e | 0a 5c 65 6e 64 70 72 6f |an.type.|.\endpro|
|00000df0| 63 6c 61 69 6d 0a 5c 70 | 72 6f 63 6c 61 69 6d 7b |claim.\p|roclaim{|
|00000e00| 54 68 65 6f 72 65 6d 20 | 32 2e 31 7d 0a 4c 65 74 |Theorem |2.1}.Let|
|00000e10| 20 24 5c 7b 75 5e 68 5c | 7d 24 20 62 65 20 61 20 | $\{u^h\|}$ be a |
|00000e20| 66 61 6d 69 6c 79 20 6f | 66 20 61 70 70 72 6f 78 |family o|f approx|
|00000e30| 69 6d 61 74 65 20 73 6f | 6c 75 74 69 6f 6e 73 20 |imate so|lutions |
|00000e40| 73 61 74 69 73 66 79 69 | 6e 67 0a 74 68 65 20 66 |satisfyi|ng.the f|
|00000e50| 6f 6c 6c 6f 77 69 6e 67 | 20 63 6f 6e 64 69 74 69 |ollowing| conditi|
|00000e60| 6f 6e 73 20 66 6f 72 20 | 24 30 5c 6c 65 71 20 74 |ons for |$0\leq t|
|00000e70| 5c 6c 65 71 20 54 24 3a | 0a 5c 72 6f 73 74 65 72 |\leq T$:|.\roster|
|00000e80| 0a 5c 69 74 65 6d 20 46 | 6f 72 20 61 6c 6c 20 24 |.\item F|or all $|
|00000e90| 78 5c 69 6e 5c 42 62 62 | 52 24 20 61 6e 64 20 70 |x\in\Bbb|R$ and p|
|00000ea0| 6f 73 69 74 69 76 65 20 | 24 74 24 2c 20 24 30 5c |ositive |$t$, $0\|
|00000eb0| 6c 65 71 20 75 5e 68 28 | 78 2c 74 29 5c 6c 65 71 |leq u^h(|x,t)\leq|
|00000ec0| 20 4d 24 3b 0a 5c 69 74 | 65 6d 20 42 6f 74 68 20 | M$;.\it|em Both |
|00000ed0| 24 75 24 20 61 6e 64 20 | 24 75 5e 68 24 20 61 72 |$u$ and |$u^h$ ar|
|00000ee0| 65 20 48 5c 22 6f 6c 64 | 65 72 2d 2d 24 5c 61 6c |e H\"old|er--$\al|
|00000ef0| 70 68 61 24 20 69 6e 20 | 24 78 24 0a 66 6f 72 20 |pha$ in |$x$.for |
|00000f00| 73 6f 6d 65 20 24 5c 61 | 6c 70 68 61 5c 69 6e 28 |some $\a|lpha\in(|
|00000f10| 30 2c 31 5c 77 65 64 67 | 65 20 31 2f 28 6d 2d 31 |0,1\wedg|e 1/(m-1|
|00000f20| 29 29 24 3b 20 24 75 5e | 68 24 20 69 73 20 72 69 |))$; $u^|h$ is ri|
|00000f30| 67 68 74 20 63 6f 6e 74 | 69 6e 75 6f 75 73 20 69 |ght cont|inuous i|
|00000f40| 6e 20 24 74 24 3b 0a 61 | 6e 64 20 24 75 5e 68 24 |n $t$;.a|nd $u^h$|
|00000f50| 20 69 73 20 48 5c 22 6f | 6c 64 65 72 20 63 6f 6e | is H\"o|lder con|
|00000f60| 74 69 6e 75 6f 75 73 20 | 69 6e 20 24 74 24 20 6f |tinuous |in $t$ o|
|00000f70| 6e 0a 73 74 72 69 70 73 | 20 24 5c 42 62 62 52 5c |n.strips| $\BbbR\|
|00000f80| 74 69 6d 65 73 28 74 5e | 6e 2c 74 5e 7b 6e 2b 31 |times(t^|n,t^{n+1|
|00000f90| 7d 29 24 2c 20 77 69 74 | 68 20 74 68 65 20 73 65 |})$, wit|h the se|
|00000fa0| 74 20 24 5c 7b 74 5e 6e | 5c 7d 24 20 68 61 76 69 |t $\{t^n|\}$ havi|
|00000fb0| 6e 67 20 6e 6f 0a 6c 69 | 6d 69 74 20 70 6f 69 6e |ng no.li|mit poin|
|00000fc0| 74 73 3b 20 61 6e 64 0a | 5c 69 74 65 6d 20 54 68 |ts; and.|\item Th|
|00000fd0| 65 72 65 20 65 78 69 73 | 74 73 20 61 20 70 6f 73 |ere exis|ts a pos|
|00000fe0| 69 74 69 76 65 20 66 75 | 6e 63 74 69 6f 6e 20 24 |itive fu|nction $|
|00000ff0| 5c 6f 6d 65 67 61 28 68 | 2c 5c 65 70 73 69 6c 6f |\omega(h|,\epsilo|
|00001000| 6e 29 24 20 73 75 63 68 | 20 74 68 61 74 3a 0a 77 |n)$ such| that:.w|
|00001010| 68 65 6e 65 76 65 72 20 | 24 5c 7b 77 5e 5c 65 70 |henever |$\{w^\ep|
|00001020| 73 69 6c 6f 6e 5c 7d 5f | 7b 30 3c 5c 65 70 73 69 |silon\}_|{0<\epsi|
|00001030| 6c 6f 6e 5c 6c 65 71 5c | 65 70 73 69 6c 6f 6e 5f |lon\leq\|epsilon_|
|00001040| 30 7d 24 20 69 73 20 61 | 20 66 61 6d 69 6c 79 20 |0}$ is a| family |
|00001050| 6f 66 20 66 75 6e 63 74 | 69 6f 6e 73 0a 69 6e 20 |of funct|ions.in |
|00001060| 24 5c 62 6f 6c 64 20 58 | 24 20 66 6f 72 20 77 68 |$\bold X|$ for wh|
|00001070| 69 63 68 0a 7b 5c 72 6f | 73 74 65 72 0a 5c 69 74 |ich.{\ro|ster.\it|
|00001080| 65 6d 22 28 61 29 22 20 | 74 68 65 72 65 20 69 73 |em"(a)" |there is|
|00001090| 20 61 20 73 65 71 75 65 | 6e 63 65 20 6f 66 20 70 | a seque|nce of p|
|000010a0| 6f 73 69 74 69 76 65 20 | 6e 75 6d 62 65 72 73 20 |ositive |numbers |
|000010b0| 24 5c 65 70 73 69 6c 6f | 6e 24 20 74 65 6e 64 69 |$\epsilo|n$ tendi|
|000010c0| 6e 67 0a 74 6f 20 7a 65 | 72 6f 2c 20 73 75 63 68 |ng.to ze|ro, such|
|000010d0| 20 74 68 61 74 20 66 6f | 72 20 74 68 65 73 65 20 | that fo|r these |
|000010e0| 20 76 61 6c 75 65 73 20 | 6f 66 0a 24 5c 65 70 73 | values |of.$\eps|
|000010f0| 69 6c 6f 6e 24 2c 20 24 | 5c 7c 77 5e 5c 65 70 73 |ilon$, $|\|w^\eps|
|00001100| 69 6c 6f 6e 5c 7c 5f 5c | 69 6e 66 74 79 5c 6c 65 |ilon\|_\|infty\le|
|00001110| 71 20 31 2f 5c 65 70 73 | 69 6c 6f 6e 24 2c 0a 5c |q 1/\eps|ilon$,.\|
|00001120| 69 74 65 6d 22 28 62 29 | 22 20 66 6f 72 20 61 6c |item"(b)|" for al|
|00001130| 6c 20 70 6f 73 69 74 69 | 76 65 0a 24 5c 65 70 73 |l positi|ve.$\eps|
|00001140| 69 6c 6f 6e 24 2c 20 24 | 5c 7c 77 5f 78 5e 5c 65 |ilon$, $|\|w_x^\e|
|00001150| 70 73 69 6c 6f 6e 28 5c | 73 64 6f 74 2c 74 29 5c |psilon(\|sdot,t)\|
|00001160| 7c 5f 5c 6c 6f 6e 65 72 | 5c 6c 65 71 20 31 2f 5c ||_\loner|\leq 1/\|
|00001170| 65 70 73 69 6c 6f 6e 5e | 32 24 2c 20 61 6e 64 0a |epsilon^|2$, and.|
|00001180| 5c 69 74 65 6d 22 28 63 | 29 22 20 66 6f 72 20 61 |\item"(c|)" for a|
|00001190| 6c 6c 20 24 5c 65 70 73 | 69 6c 6f 6e 3e 30 24 2c |ll $\eps|ilon>0$,|
|000011a0| 20 0a 24 24 0a 5c 73 75 | 70 5c 53 62 0a 78 5c 69 | .$$.\su|p\Sb.x\i|
|000011b0| 6e 5c 42 62 62 52 5c 5c | 30 5c 6c 65 71 20 74 5f |n\BbbR\\|0\leq t_|
|000011c0| 31 2c 74 5f 32 5c 6c 65 | 71 20 54 5c 65 6e 64 53 |1,t_2\le|q T\endS|
|000011d0| 62 0a 5c 64 66 72 61 63 | 7b 7c 77 5e 5c 65 70 73 |b.\dfrac|{|w^\eps|
|000011e0| 69 6c 6f 6e 28 78 2c 74 | 5f 32 29 2d 77 5e 5c 65 |ilon(x,t|_2)-w^\e|
|000011f0| 70 73 69 6c 6f 6e 28 78 | 2c 74 5f 31 29 7c 7d 7b |psilon(x|,t_1)|}{|
|00001200| 7c 74 5f 32 2d 74 5f 31 | 7c 5e 70 7d 5c 6c 65 71 ||t_2-t_1||^p}\leq|
|00001210| 20 31 2f 5c 65 70 73 69 | 6c 6f 6e 5e 32 2c 0a 24 | 1/\epsi|lon^2,.$|
|00001220| 24 0a 77 68 65 72 65 20 | 24 70 24 20 69 73 20 73 |$.where |$p$ is s|
|00001230| 6f 6d 65 20 6e 75 6d 62 | 65 72 20 6e 6f 74 20 65 |ome numb|er not e|
|00001240| 78 63 65 65 64 69 6e 67 | 20 24 31 24 2c 0a 5c 65 |xceeding| $1$,.\e|
|00001250| 6e 64 72 6f 73 74 65 72 | 7d 0a 74 68 65 6e 20 24 |ndroster|}.then $|
|00001260| 7c 45 20 28 75 5e 68 2c | 77 5e 5c 65 70 73 69 6c ||E (u^h,|w^\epsil|
|00001270| 6f 6e 2c 54 29 7c 5c 6c | 65 71 5c 6f 6d 65 67 61 |on,T)|\l|eq\omega|
|00001280| 28 68 2c 5c 65 70 73 69 | 6c 6f 6e 29 2e 24 0a 5c |(h,\epsi|lon).$.\|
|00001290| 65 6e 64 72 6f 73 74 65 | 72 0a 54 68 65 6e 2c 20 |endroste|r.Then, |
|000012a0| 74 68 65 72 65 20 69 73 | 20 61 20 63 6f 6e 73 74 |there is| a const|
|000012b0| 61 6e 74 20 24 43 3d 43 | 28 6d 2c 4d 2c 54 29 24 |ant $C=C|(m,M,T)$|
|000012c0| 20 73 75 63 68 20 74 68 | 61 74 0a 24 24 0a 5c 7c | such th|at.$$.\||
|000012d0| 75 2d 75 5e 68 5c 7c 5f | 7b 5c 69 6e 66 74 79 2c |u-u^h\|_|{\infty,|
|000012e0| 5c 42 62 62 52 5c 74 69 | 6d 65 73 5b 30 2c 54 5d |\BbbR\ti|mes[0,T]|
|000012f0| 7d 5c 6c 65 71 0a 43 5c | 6c 65 66 74 5b 0a 5c 73 |}\leq.C\|left[.\s|
|00001300| 75 70 20 5c 6c 65 66 74 | 20 7c 5c 69 6e 74 5f 5c |up \left| |\int_\|
|00001310| 42 62 62 52 28 75 5f 30 | 28 78 29 2d 75 5e 68 28 |BbbR(u_0|(x)-u^h(|
|00001320| 78 2c 30 29 29 20 20 77 | 28 78 2c 30 29 20 5c 2c |x,0))  w|(x,0) \,|
|00001330| 64 78 5c 72 69 67 68 74 | 7c 2b 0a 5c 6f 6d 65 67 |dx\right||+.\omeg|
|00001340| 61 28 68 2c 5c 65 70 73 | 69 6c 6f 6e 29 2b 5c 65 |a(h,\eps|ilon)+\e|
|00001350| 70 73 69 6c 6f 6e 5e 5c | 61 6c 70 68 61 5c 72 69 |psilon^\|alpha\ri|
|00001360| 67 68 74 5d 2c 0a 5c 74 | 61 67 20 32 2e 31 0a 24 |ght],.\t|ag 2.1.$|
|00001370| 24 0a 77 68 65 72 65 20 | 74 68 65 20 73 75 70 72 |$.where |the supr|
|00001380| 65 6d 75 6d 20 69 73 20 | 74 61 6b 65 6e 20 6f 76 |emum is |taken ov|
|00001390| 65 72 20 61 6c 6c 20 24 | 77 5c 69 6e 5c 62 6f 6c |er all $|w\in\bol|
|000013a0| 64 20 58 24 2e 0a 5c 65 | 6e 64 70 72 6f 63 6c 61 |d X$..\e|ndprocla|
|000013b0| 69 6d 0a 5c 64 65 6d 6f | 7b 50 72 6f 6f 66 7d 4c |im.\demo|{Proof}L|
|000013c0| 65 74 20 24 7a 24 20 62 | 65 20 69 6e 20 24 5c 62 |et $z$ b|e in $\b|
|000013d0| 6f 6c 64 20 58 24 2e 20 | 42 65 63 61 75 73 65 20 |old X$. |Because |
|000013e0| 24 45 28 75 2c 5c 73 64 | 6f 74 2c 5c 73 64 6f 74 |$E(u,\sd|ot,\sdot|
|000013f0| 29 5c 65 71 75 69 76 30 | 24 2c 0a 45 71 75 61 74 |)\equiv0|$,.Equat|
|00001400| 69 6f 6e 20 28 31 2e 35 | 29 20 69 6d 70 6c 69 65 |ion (1.5|) implie|
|00001410| 73 20 74 68 61 74 0a 24 | 24 0a 5c 69 6e 74 5f 5c |s that.$|$.\int_\|
|00001420| 42 62 62 52 5c 44 65 6c | 74 61 20 75 7a 7c 5e 54 |BbbR\Del|ta uz|^T|
|00001430| 5f 30 64 78 3d 5c 69 6e | 74 5f 30 5e 54 5c 69 6e |_0dx=\in|t_0^T\in|
|00001440| 74 5f 5c 42 62 62 52 0a | 5c 44 65 6c 74 61 20 75 |t_\BbbR.|\Delta u|
|00001450| 28 7a 5f 74 2b 5c 70 68 | 69 5b 75 2c 75 5e 68 5d |(z_t+\ph|i[u,u^h]|
|00001460| 7a 5f 7b 78 78 7d 29 5c | 2c 64 78 5c 2c 64 74 2d |z_{xx})\|,dx\,dt-|
|00001470| 0a 45 28 75 5e 68 2c 7a | 2c 74 29 2c 0a 5c 74 61 |.E(u^h,z|,t),.\ta|
|00001480| 67 20 32 2e 32 0a 24 24 | 0a 77 68 65 72 65 20 24 |g 2.2.$$|.where $|
|00001490| 5c 44 65 6c 74 61 20 75 | 3d 75 2d 75 5e 68 24 20 |\Delta u|=u-u^h$ |
|000014a0| 61 6e 64 20 0a 24 24 0a | 5c 70 68 69 5b 75 2c 75 |and .$$.|\phi[u,u|
|000014b0| 5e 68 5d 3d 5c 64 66 72 | 61 63 7b 5c 70 68 69 28 |^h]=\dfr|ac{\phi(|
|000014c0| 75 29 2d 5c 70 68 69 28 | 75 5e 68 29 7d 7b 75 2d |u)-\phi(|u^h)}{u-|
|000014d0| 75 5e 68 7d 2e 0a 24 24 | 0a 45 78 74 65 6e 64 20 |u^h}..$$|.Extend |
|000014e0| 24 5c 70 68 69 5b 75 2c | 75 5e 68 5d 28 5c 63 64 |$\phi[u,|u^h](\cd|
|000014f0| 6f 74 2c 74 29 3d 5c 70 | 68 69 5b 75 2c 75 5e 68 |ot,t)=\p|hi[u,u^h|
|00001500| 5d 28 5c 63 64 6f 74 2c | 30 29 24 20 66 6f 72 20 |](\cdot,|0)$ for |
|00001510| 6e 65 67 61 74 69 76 65 | 20 24 74 24 2c 20 61 6e |negative| $t$, an|
|00001520| 64 0a 24 5c 70 68 69 5b | 75 2c 75 5e 68 5d 28 5c |d.$\phi[|u,u^h](\|
|00001530| 63 64 6f 74 2c 74 29 3d | 5c 70 68 69 5b 75 2c 75 |cdot,t)=|\phi[u,u|
|00001540| 5e 68 5d 28 5c 63 64 6f | 74 2c 54 29 24 0a 66 6f |^h](\cdo|t,T)$.fo|
|00001550| 72 20 24 74 3e 54 24 2e | 5c 66 6f 6f 74 6e 6f 74 |r $t>T$.|\footnot|
|00001560| 65 7b 54 68 69 73 20 69 | 73 20 61 6e 20 6f 62 76 |e{This i|s an obv|
|00001570| 69 6f 75 73 20 70 6c 6f | 79 2c 20 62 75 74 20 77 |ious plo|y, but w|
|00001580| 65 20 6e 65 65 64 20 61 | 20 66 6f 6f 74 6e 6f 74 |e need a| footnot|
|00001590| 65 2e 7d 0a 46 69 78 20 | 61 20 70 6f 69 6e 74 20 |e.}.Fix |a point |
|000015a0| 24 78 5f 30 24 20 61 6e | 64 20 61 20 6e 75 6d 62 |$x_0$ an|d a numb|
|000015b0| 65 72 20 24 5c 65 70 73 | 69 6c 6f 6e 3e 30 24 2e |er $\eps|ilon>0$.|
|000015c0| 20 4c 65 74 20 24 6a 5f | 5c 65 70 73 69 6c 6f 6e | Let $j_|\epsilon|
|000015d0| 24 0a 62 65 20 61 20 73 | 6d 6f 6f 74 68 20 66 75 |$.be a s|mooth fu|
|000015e0| 6e 63 74 69 6f 6e 20 6f | 66 20 24 78 24 20 77 69 |nction o|f $x$ wi|
|000015f0| 74 68 20 69 6e 74 65 67 | 72 61 6c 20 24 31 24 20 |th integ|ral $1$ |
|00001600| 61 6e 64 20 73 75 70 70 | 6f 72 74 20 69 6e 20 0a |and supp|ort in .|
|00001610| 24 5b 2d 5c 65 70 73 69 | 6c 6f 6e 2c 5c 65 70 73 |$[-\epsi|lon,\eps|
|00001620| 69 6c 6f 6e 5d 24 2c 0a | 61 6e 64 20 6c 65 74 20 |ilon]$,.|and let |
|00001630| 24 4a 5f 5c 64 65 6c 74 | 61 24 20 62 65 20 61 20 |$J_\delt|a$ be a |
|00001640| 73 6d 6f 6f 74 68 20 66 | 75 6e 63 74 69 6f 6e 20 |smooth f|unction |
|00001650| 6f 66 0a 24 78 24 20 61 | 6e 64 20 24 74 24 20 77 |of.$x$ a|nd $t$ w|
|00001660| 69 74 68 20 69 6e 74 65 | 67 72 61 6c 20 24 31 24 |ith inte|gral $1$|
|00001670| 20 61 6e 64 20 73 75 70 | 70 6f 72 74 20 69 6e 20 | and sup|port in |
|00001680| 0a 24 5b 2d 5c 64 65 6c | 74 61 2c 5c 64 65 6c 74 |.$[-\del|ta,\delt|
|00001690| 61 5d 5c 74 69 6d 65 73 | 5b 2d 5c 64 65 6c 74 61 |a]\times|[-\delta|
|000016a0| 2c 5c 64 65 6c 74 61 5d | 24 3b 20 24 5c 64 65 6c |,\delta]|$; $\del|
|000016b0| 74 61 24 20 61 6e 64 20 | 24 5c 65 70 73 69 6c 6f |ta$ and |$\epsilo|
|000016c0| 6e 24 20 61 72 65 0a 70 | 6f 73 69 74 69 76 65 20 |n$ are.p|ositive |
|000016d0| 6e 75 6d 62 65 72 73 20 | 74 6f 20 62 65 20 73 70 |numbers |to be sp|
|000016e0| 65 63 69 66 69 65 64 20 | 6c 61 74 65 72 2e 0a 57 |ecified |later..W|
|000016f0| 65 20 63 68 6f 6f 73 65 | 20 24 7a 3d 7a 5e 7b 5c |e choose| $z=z^{\|
|00001700| 65 70 73 69 6c 6f 6e 5c | 64 65 6c 74 61 7d 24 20 |epsilon\|delta}$ |
|00001710| 74 6f 20 73 61 74 69 73 | 66 79 0a 24 24 0a 5c 61 |to satis|fy.$$.\a|
|00001720| 6c 69 67 6e 65 64 0a 20 | 20 26 7a 5f 74 2b 28 5c |ligned. | &z_t+(\|
|00001730| 64 65 6c 74 61 2b 4a 5f | 5c 64 65 6c 74 61 2a 5c |delta+J_|\delta*\|
|00001740| 70 68 69 5b 75 2c 75 5e | 68 5d 29 7a 5f 7b 78 78 |phi[u,u^|h])z_{xx|
|00001750| 7d 3d 30 2c 5c 71 71 75 | 61 64 20 78 5c 69 6e 5c |}=0,\qqu|ad x\in\|
|00001760| 42 62 62 52 2c 5c 3b 30 | 5c 6c 65 71 20 74 5c 6c |BbbR,\;0|\leq t\l|
|00001770| 65 71 20 54 2c 0a 5c 5c | 0a 20 20 26 7a 28 78 2c |eq T,.\\|.  &z(x,|
|00001780| 54 29 3d 6a 5f 5c 65 70 | 73 69 6c 6f 6e 28 78 2d |T)=j_\ep|silon(x-|
|00001790| 78 5f 30 29 2e 0a 5c 65 | 6e 64 61 6c 69 67 6e 65 |x_0)..\e|ndaligne|
|000017a0| 64 0a 5c 74 61 67 20 32 | 2e 33 0a 24 24 0a 54 68 |d.\tag 2|.3.$$.Th|
|000017b0| 65 20 63 6f 6e 63 6c 75 | 73 69 6f 6e 20 6f 66 20 |e conclu|sion of |
|000017c0| 74 68 65 20 74 68 65 6f | 72 65 6d 20 6e 6f 77 20 |the theo|rem now |
|000017d0| 66 6f 6c 6c 6f 77 73 20 | 66 72 6f 6d 20 28 32 2e |follows |from (2.|
|000017e0| 31 29 20 61 6e 64 20 74 | 68 65 20 66 61 63 74 20 |1) and t|he fact |
|000017f0| 74 68 61 74 0a 24 24 0a | 7c 6a 5f 5c 65 70 73 69 |that.$$.||j_\epsi|
|00001800| 6c 6f 6e 2a 5c 44 65 6c | 74 61 20 75 28 78 5f 30 |lon*\Del|ta u(x_0|
|00001810| 2c 74 29 2d 5c 44 65 6c | 74 61 20 75 28 78 5f 30 |,t)-\Del|ta u(x_0|
|00001820| 2c 74 29 7c 5c 6c 65 71 | 20 43 5c 65 70 73 69 6c |,t)|\leq| C\epsil|
|00001830| 6f 6e 5e 5c 61 6c 70 68 | 61 2c 0a 24 24 0a 77 68 |on^\alph|a,.$$.wh|
|00001840| 69 63 68 20 66 6f 6c 6c | 6f 77 73 20 66 72 6f 6d |ich foll|ows from|
|00001850| 20 20 41 73 73 75 6d 70 | 74 69 6f 6e 20 32 2e 5c |  Assump|tion 2.\|
|00001860| 71 65 64 0a 5c 65 6e 64 | 64 65 6d 6f 0a 5c 52 65 |qed.\end|demo.\Re|
|00001870| 66 73 0a 5c 72 65 66 0a | 20 20 5c 6e 6f 20 31 0a |fs.\ref.|  \no 1.|
|00001880| 20 20 5c 62 79 20 4c 2e | 20 41 2e 20 43 61 66 66 |  \by L.| A. Caff|
|00001890| 61 72 65 6c 6c 69 20 61 | 6e 64 20 41 2e 20 46 72 |arelli a|nd A. Fr|
|000018a0| 69 65 64 6d 61 6e 0a 20 | 20 5c 70 61 70 65 72 20 |iedman. | \paper |
|000018b0| 52 65 67 75 6c 61 72 69 | 74 79 20 6f 66 20 74 68 |Regulari|ty of th|
|000018c0| 65 20 66 72 65 65 20 62 | 6f 75 6e 64 61 72 79 20 |e free b|oundary |
|000018d0| 6f 66 20 61 20 67 61 73 | 20 66 6c 6f 77 20 69 6e |of a gas| flow in|
|000018e0| 20 61 6e 20 0a 20 20 20 | 20 20 20 20 20 20 24 6e | an .   |      $n|
|000018f0| 24 2d 64 69 6d 65 6e 73 | 69 6f 6e 61 6c 20 70 6f |$-dimens|ional po|
|00001900| 72 6f 75 73 20 6d 65 64 | 69 75 6d 0a 20 20 5c 6a |rous med|ium.  \j|
|00001910| 6f 75 72 20 49 6e 64 69 | 61 6e 61 20 4d 61 74 68 |our Indi|ana Math|
|00001920| 2e 20 4a 2e 0a 20 20 5c | 76 6f 6c 20 32 39 0a 20 |. J..  \|vol 29. |
|00001930| 20 5c 79 72 20 31 39 38 | 30 0a 20 20 5c 70 61 67 | \yr 198|0.  \pag|
|00001940| 65 73 20 33 36 31 2d 2d | 33 39 31 0a 5c 65 6e 64 |es 361--|391.\end|
|00001950| 72 65 66 0a 5c 72 65 66 | 20 0a 20 20 5c 6e 6f 20 |ref.\ref| .  \no |
|00001960| 32 0a 20 20 5c 62 79 20 | 4b 2e 20 48 6f 6c 6c 69 |2.  \by |K. Holli|
|00001970| 67 20 61 6e 64 20 4d 2e | 20 50 69 6c 61 6e 74 0a |g and M.| Pilant.|
|00001980| 20 20 5c 70 61 70 65 72 | 20 52 65 67 75 6c 61 72 |  \paper| Regular|
|00001990| 69 74 79 20 6f 66 20 74 | 68 65 20 66 72 65 65 20 |ity of t|he free |
|000019a0| 62 6f 75 6e 64 61 72 79 | 20 66 6f 72 20 74 68 65 |boundary| for the|
|000019b0| 20 70 6f 72 6f 75 73 20 | 6d 65 64 69 75 6d 20 65 | porous |medium e|
|000019c0| 71 75 61 74 69 6f 6e 0a | 20 20 5c 70 61 70 65 72 |quation.|  \paper|
|000019d0| 69 6e 66 6f 20 4d 52 43 | 20 54 65 63 68 2e 20 52 |info MRC| Tech. R|
|000019e0| 65 70 2e 20 32 37 34 32 | 0a 5c 65 6e 64 72 65 66 |ep. 2742|.\endref|
|000019f0| 0a 5c 72 65 66 20 0a 20 | 20 5c 6e 6f 20 33 0a 20 |.\ref . | \no 3. |
|00001a00| 20 5c 62 79 20 4a 2e 20 | 4a 65 72 6f 6d 65 0a 20 | \by J. |Jerome. |
|00001a10| 20 5c 62 6f 6f 6b 20 41 | 70 70 72 6f 78 69 6d 61 | \book A|pproxima|
|00001a20| 74 69 6f 6e 20 6f 66 20 | 4e 6f 6e 6c 69 6e 65 61 |tion of |Nonlinea|
|00001a30| 72 20 45 76 6f 6c 75 74 | 69 6f 6e 20 53 79 73 74 |r Evolut|ion Syst|
|00001a40| 65 6d 73 20 0a 20 20 5c | 70 75 62 6c 20 41 63 61 |ems .  \|publ Aca|
|00001a50| 64 65 6d 69 63 20 50 72 | 65 73 73 20 0a 20 20 5c |demic Pr|ess .  \|
|00001a60| 70 75 62 6c 61 64 64 72 | 20 4e 65 77 20 59 6f 72 |publaddr| New Yor|
|00001a70| 6b 20 0a 20 20 5c 79 72 | 20 31 39 38 33 0a 5c 65 |k .  \yr| 1983.\e|
|00001a80| 6e 64 72 65 66 0a 5c 72 | 65 66 0a 20 20 5c 6e 6f |ndref.\r|ef.  \no|
|00001a90| 20 34 0a 20 20 5c 6d 61 | 6e 79 62 79 20 52 2e 20 | 4.  \ma|nyby R. |
|00001aa0| 4a 2e 20 4c 65 56 65 71 | 75 65 0a 20 20 5c 70 61 |J. LeVeq|ue.  \pa|
|00001ab0| 70 65 72 20 43 6f 6e 76 | 65 72 67 65 6e 63 65 20 |per Conv|ergence |
|00001ac0| 6f 66 20 61 20 6c 61 72 | 67 65 20 74 69 6d 65 20 |of a lar|ge time |
|00001ad0| 73 74 65 70 20 67 65 6e | 65 72 61 6c 69 7a 61 74 |step gen|eralizat|
|00001ae0| 69 6f 6e 20 6f 66 20 47 | 6f 64 75 6e 6f 76 27 73 |ion of G|odunov's|
|00001af0| 20 6d 65 74 68 6f 64 20 | 0a 20 20 20 20 20 20 20 | method |.       |
|00001b00| 20 20 66 6f 72 20 63 6f | 6e 73 65 72 76 61 74 69 |  for co|nservati|
|00001b10| 6f 6e 20 6c 61 77 73 0a | 20 20 5c 6a 6f 75 72 20 |on laws.|  \jour |
|00001b20| 43 6f 6d 6d 2e 20 50 75 | 72 65 20 41 70 70 6c 2e |Comm. Pu|re Appl.|
|00001b30| 20 4d 61 74 68 2e 0a 20 | 20 5c 76 6f 6c 20 33 37 | Math.. | \vol 37|
|00001b40| 20 0a 20 20 5c 79 72 20 | 31 39 38 34 0a 20 20 5c | .  \yr |1984.  \|
|00001b50| 70 61 67 65 73 20 34 36 | 33 2d 2d 34 37 38 0a 5c |pages 46|3--478.\|
|00001b60| 65 6e 64 72 65 66 0a 5c | 72 65 66 0a 20 20 5c 6e |endref.\|ref.  \n|
|00001b70| 6f 20 35 0a 20 20 5c 62 | 79 73 61 6d 65 0a 20 20 |o 5.  \b|ysame.  |
|00001b80| 5c 70 61 70 65 72 20 41 | 20 6c 61 72 67 65 20 74 |\paper A| large t|
|00001b90| 69 6d 65 20 73 74 65 70 | 20 67 65 6e 65 72 61 6c |ime step| general|
|00001ba0| 69 7a 61 74 69 6f 6e 20 | 6f 66 20 47 6f 64 75 6e |ization |of Godun|
|00001bb0| 6f 76 27 73 20 6d 65 74 | 68 6f 64 20 66 6f 72 20 |ov's met|hod for |
|00001bc0| 73 79 73 74 65 6d 73 0a | 20 20 20 20 20 20 20 20 |systems.|        |
|00001bd0| 20 6f 66 20 63 6f 6e 73 | 65 72 76 61 74 69 6f 6e | of cons|ervation|
|00001be0| 20 6c 61 77 73 0a 20 20 | 5c 6a 6f 75 72 0a 20 20 | laws.  |\jour.  |
|00001bf0| 5c 74 6f 61 70 70 65 61 | 72 0a 5c 65 6e 64 72 65 |\toappea|r.\endre|
|00001c00| 66 0a 5c 72 65 66 20 0a | 20 20 5c 6e 6f 20 36 0a |f.\ref .|  \no 6.|
|00001c10| 20 20 5c 62 79 20 42 2e | 20 4a 2e 20 4c 75 63 69 |  \by B.| J. Luci|
|00001c20| 65 72 0a 20 20 5c 70 61 | 70 65 72 20 4f 6e 20 6e |er.  \pa|per On n|
|00001c30| 6f 6e 6c 6f 63 61 6c 20 | 6d 6f 6e 6f 74 6f 6e 65 |onlocal |monotone|
|00001c40| 20 64 69 66 66 65 72 65 | 6e 63 65 20 6d 65 74 68 | differe|nce meth|
|00001c50| 6f 64 73 20 66 6f 72 20 | 73 63 61 6c 61 72 20 63 |ods for |scalar c|
|00001c60| 6f 6e 73 65 72 76 61 74 | 69 6f 6e 20 6c 61 77 73 |onservat|ion laws|
|00001c70| 0a 20 20 5c 6a 6f 75 72 | 20 4d 61 74 68 2e 20 43 |.  \jour| Math. C|
|00001c80| 6f 6d 70 2e 0a 20 20 5c | 76 6f 6c 20 34 37 0a 20 |omp..  \|vol 47. |
|00001c90| 20 5c 79 72 20 31 39 38 | 36 0a 20 20 5c 70 61 67 | \yr 198|6.  \pag|
|00001ca0| 65 73 20 31 39 2d 2d 33 | 36 0a 5c 65 6e 64 72 65 |es 19--3|6.\endre|
|00001cb0| 66 0a 5c 65 6e 64 64 6f | 63 75 6d 65 6e 74 0a    |f.\enddo|cument. |
+--------+-------------------------+-------------------------+--------+--------+