10.1090/mmono/027 TRANSLATIONS OF MATHEMATICAL MONOGRAPHS Volume 27 THE STEFAN PROBLEM by L I. RUBENSTEIN AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 02904 1971 nPOBJIEMA CTE$AH A JI. M . PYBWHU1TEMH JIaTBMMCKMM rocyAapcTBeHHH M yHMBepcMTe T MMeHw nETPA CTY^IK M BbHMCJIMTEJlbHblM UEHT P M3AaTejibCTB0 „3Baiir3He " Pural967 Translated fro m the Russian by A. D. Solomon Library o f Congress Card Number 75-168253 International Standard Book Number 0-8218-1577- 6 Copyright© 197 1 by the American Mathematical Society Printed in the United States of America All Rights Reserved May not be reproduced in any form without permission of the publishers PREFACE In recen t year s w e hav e witnesse d a n intensiv e developmen t o f th e Stefan problem , wit h whic h th e autho r ha s lon g bee n concerned . This monograp h wa s originall y intende d t o examin e th e importan t results concerned wit h th e classica l Stefa n proble m an d it s generaliza - tions, up to the present time, but thi s progra m coul d no t b e carrie d ou t completely, first of all because the literature concerned with this problem is continuousl y increasing , an d ver y importan t work , i n whic h th e Stefan problem i s treated a s a problem o f numerica l analysis , ha s bee n only partiall y publishe d an d partiall y announce d i n publication s an d at scientifi c conferences; 1* a t th e sam e tim e th e wor k planne d fo r ou r monograph was to a large extent complete . I n thi s regar d th e numerica l solution o f problem s o f Stefa n typ e i s usuall y discusse d i n monograph s in an extremely concis e manner. We will limit ourselve s t o brie f descrip - tions o f finite differenc e algorithm s whic h hav e bee n propose d fo r th e solution o f th e Stefa n proble m (Par t 2 , Chapte r VIII) ; a s a rul e w e will not give a justification o f th e algorith m bu t refe r th e reade r t o th e original literature. The autho r is , o n th e on e hand , mainl y intereste d i n the physical formulatio n o f a problem which reduces to some form of the Stefan problem (Par t 1 , Chapters I—IV) and , on the other hand , i n th e examination of general theoretical problems, i.e. problems of a qualitative character (Par t 2 , Chapter s I—VII) . I n Supplemen t 3 w e presen t illustrative examples o f th e numerica l solutio n t o problem s examine d i n the first part . I n th e first tw o supplement s w e presen t certai n know n results whic h ar e use d i n th e mai n part s o f th e book . At th e en d o f th e boo k w e includ e a bibliograph y o f th e problem . Unfortunately, man y source s wer e no t availabl e t o th e author . I t i s probable tha t man y work s exis t whic h hav e excape d hi s attention . Therefore w e mak e n o clai m o f completenes s fo r th e bibliography . The autho r take s thi s opportunit y t o expres s hi s appreciatio n t o his colleague s N . Avdoijins , M . Antimirov , A . Kuikis , H . German , We have in mind the joint work of B. M . Budak, F. P. VasiTev and A. B. Uspenskil, as presented in [15 ] and announced in their articles in a number of reports of conference s on numerica l analysi s (Mosco w Stat e Univ. , 1965) . in iv PREFACE E. Enikeeva, M. Zavorina, B. Martuzans, E. Pi^enkova and A. Skroman, who checked the accuracy o f th e numerica l solution s o f th e example s o f Supplement 3 and aide d th e autho r i n eliminatin g man y errors , a s wel l as t o E . Riekstiij s an d V . Abolinja , wh o rea d th e manuscrip t o f th e monograph. Th e autho r expresse s specia l thank s t o th e Recto r o f th e Stucki Institut e o f th e Latvia n Stat e University , Professo r V . A . Steinberg, an d t o th e directo r o f th e Universit y computin g center , Docent E . I . Arin , withou t whos e efficien t ai d th e monograp h coul d not hav e bee n brough t t o completion . L. RUBINSTEI N TABLE O F CONTENT S PREFACE ii i INTRODUCTION 1 1. Historica l survey 1 2. Notatio n and terminolog y 1 5 PART ONE . Physica l Problems Reducing to Problems of the Stefan Typ e CHAPTER I . DIFFUSIO N O F HEA T B Y CONDUCTIO N I N A MEDIU M WITH A CHANGE O F PHASE STAT E 1 8 1. Th e Stefan condition s 1 8 2. Freezin g o f the ground 2 0 3. Crystallizatio n o f a melt when a plate is immersed i n it . 2 5 4. Formatio n o f a continuous ingot 3 2 5. Zona l noncrucible melting o f a cylindrical rod 3 7 CHAPTER II . THERMA L DIFFUSIO N PROCESSE S I N A MEDIU M WITH VARIABL E PHAS E STAT E 3 9 1. Dissolutio n o f a gas bubble in a liquid 3 9 2. Dynamic s o f one-dimensiona l nonisotherma l evaporatio n of an ideal liquid mixture 4 3 3. Crystallizatio n o f a binary allo y 5 2 CHAPTER III . PROBLEM S O F THE THEOR Y O F FILTRATION 6 1 1. Forcin g o f a hydrauli c solutio n int o th e groun d (proble m ofVerigin) 6 1 2. Advanc e o f a water-oil interface i n an elastic regime 6 3 3. Invers e problems 7 1 CHAPTER IV . SOM E PROBLEM S O F MECHANIC S O F CONTINUOU S MEDIA THA T REDUCE T O PROBLEMS OF STEFAN TYP E ... 8 1 1. A proble m o f convectio n arisin g fro m crystallizatio n o f a supercooled melt 8 1 2. Hig h speed flow o f a soli d bod y i n a viscou s incompressibl e fluid 8 5 3. Nonstationar y flows o f a viscous plastic medium 8 9 v VI CONTENTS PART TWO . Th e Classical Stefan Proble m and its Generalizations CHAPTER I . TH E SINGLE-PHAS E STEFA N PROBLE M WIT H STRON G NONLINEARITY 9 4 1. Statemen t o f the problem. Formulation o f the basic results. 9 4 2. Reductio n to an integral equation. Theorem o f equivalence. 9 7 3. Existenc e o f the solution in the small 10 4 4. Uniquenes s of the solution 11 4 5. Stabilit y o f the solution 13 0 6. Furthe r observations 13 5 CHAPTER II . TH E CLASSICA L TWO-PHAS E STEFA N PROBLEM . TH E CASE O F A TWO-PHAS E INITIA L STAT E WIT H CONTINUOU S AGREEMENT O F BOUNDARY AN D INITIA L CONDITION S 14 1 1. Statemen t o f the problem 14 1 2. Th e first boundar y proble m o n a segmen t fo r a two-phas e initial stat e an d continuou s agreemen t o f boundar y an d initial conditions. Existence o f the solution in the large . 14 3 3. Asymptoti c behavior o f the solution to Problem B x 15 5 4. Degeneratio n o f one of the phases for t > t 0 16 1 5. Th e thir d boundar y problem . Existenc e o f th e solutio n in the large 16 4 6. Th e thir d boundar y problem . Asymptoti c behavio r o f the solution 17 0 7. Th e Cauchy-Stefan proble m 18 1 CHAPTER III . TH E INITIA L VELOCIT Y O F ADVANC E O F TH E INTER - PHASE BOUNDAR Y 190 1. Th e integral equation s o f Problem s B x an d A x withou t th e continuous agreement o f boundary and initial data 190 2. Initia l spee d o f th e interphas e boundary . Formulatio n o f the result 19 5 3. ProofofTheoremll.Thecase/xCO ) ^ 0 19 9 4. ProofofTheoremll.CaseACO ) = 0 20 5 5. Proo f o f Theorem 1 2 21 3 CHAPTER IV . SOLUTIO N O F PROBLEMS A X AND A 3 21 7 1. Th e first boundary problem (A x). Subsidiary function s . 21 7 2. Existenc e o f a solution to problem Ai 22 1 3. Th e third boundary problem (A 3) 23 1 CONTENTS vi i CHAPTER V . TH E DOUBLE-LAYE R STEFA N PROBLE M WIT H DE - GENERATION O F ON E O F TH E PHASE S A T TH E INITIA L MOMENT AN D A DISCONTINUIT Y I N TH E BOUNDAR Y AN D INITIAL CONDITION S 23 8 1.