PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY
CONTENTS A HYPERBOLIC MODEL OF GRANULAR ?OW HILBERTIAN APPROACHES TO SOME NON-LINEAR CONSERVATION LAWS ON THE ASYMPTOTIC BEHAVIOR OF THE GRADIENT ?OW OF A POLYCONVEX FUNCTIONAL ON DEGENERATE PARTIAL DI?ERENTIAL EQUATIONS SYMMETRIC SOLUTIONS TO MULTI-DIMENSIONAL CONSERVATION LAWS PRODUCT ESTIMATES FOR WAVE-SOBOLEV SPACES IN 2 + 1 AND 1 + 1 DIMENSIONS ON THE CAUCHY PROBLEM FOR THE MODI?ED KORTEWEG-DE VRIES EQUATION WITH STEPLIKE ?NITE-GAP INITIAL DATA ASYMPTOTIC ANALYSIS IN THERMODYNAMICS OF VISCOUS ?UIDS WELL-POSEDNESS AND BLOW-UP PHENOMENA FOR A MODI?ED TWO-COMPONENT CAMASSA-HOLM EQUATION INSTABILITY OF SOLITARY WAVES FOR A NONLINEARLY DISPERSIVE EQUATION KINETIC RELATIONS FOR UNDERCOMPRESSIVE SHOCK WAVES PHYSICAL, MATHEMATICAL, AND NUMERICAL ISSUES GLOBAL REGULARITY, AND WAVE BREAKING PHENOMENA IN A CLASS OF NONLOCAL DISPERSIVE EQUATIONS POTENTIAL BASED, CONSTRAINT PRESERVING, GENUINELY MULTI-DIMENSIONAL SCHEMES FOR SYSTEMS OF CONSERVATION LAWS A LOCAL AND LOW-ORDER NAVIER-STOKES-KORTEWEG SYSTEM FINITE DI?ERENCE METHODS FOR DISCRETIZING SINGULAR SOURCE TERMS IN A POISSON INTERFACE PROBLEM A HYPERBOLIC MODEL OF GRANULAR FLOW ABSTRACT IN THIS PAPER WE REVIEW SOME RECENT RESULTS FOR A MODEL FOR GRANULAR ?OW THAT WAS PROPOSED BY HADELER & KUTTLER IN [20] IN ONE SPACE DIMENSION, THIS MODEL CAN BE WRITTEN AS A 2 × 2 HYPERBOLIC SYSTEM OF BALANCE LAWS, IN WHICH THE UNKNOWNS REPRESENT THE THICKNESS OF THE MOVING LAYER AND THE ONE OF THE RESTING LAYER IF THE SLOPE DOES NOT CHANGE ITS SIGN, THE RESULTING SYSTEM CAN BE ANALYZED BY MEANS OF THE KNOWN THEORY, AS FOR INSTANCE IN THE CONTEXT OF SMALL C1 DATA OR SMALL BV DATA MOREOVER, DUE TO THE SPECIAL HYPERBOLICITY PROPERTIES OF THE SYSTEM AND OF THE SPECIAL FORM OF THE SOURCE TERM, IT IS POSSIBLE TO ENLARGE THE CLASS OF INITIAL DATA FOR WHICH GLOBAL IN TIME SOLUTIONS EXIST SEE [2, 29] FURTHER, WE STUDY THE “SLOW EROSION/DEPOSITION LIMIT”, [3], WHERE THE THICKNESS OF THE MOVING LAYER VANISHES BUT THE TOTAL MASS OF ?OWING DOWN MATERIAL REMAINS POSITIVE THE LIMITING BEHAVIOR FOR THE SLOPE OF THE MOUNTAIN PRO?LE PROVIDES AN ENTROPY SOLUTION TO A SCALAR INTEGRO-DI?ERENTIAL CONSERVATION LAW A WELL-POSEDNESS ANALYSIS OF THIS INTEGRO-DI?ERENTIAL EQUATION IS PRESENTED THEREFORE, THE SOLUTION FOUND IN THE LIMIT TURNS OUT TO BE UNIQUE CONTENTS 1 THE MODEL OF GRANULAR ?OW 2 GLOBAL SMOOTH SOLUTIONS 3 GLOBAL EXISTENCE OF LARGE BV SOLUTIONS 4 GLOBAL BV SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM 5 SLOW EROSION LIMIT REFERENCES H FOR H ?0 AND P > 0, ONE ?NDS TWO REAL DISTINCT EIGENVALUES ?1 < 0 ??2, AS ?1,2 = P H + O(H2) , ?2 = H P + O(H2) PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY ??2 H P2 , WHERE R1, R2 ARE THE CORRESPONDING EIGENVECTORS, AND THE “ ” STANDS FOR THE DIREC- TIONAL DERIVATIVE THIS SHOWS THE FACT THAT THE ?RST CHARACTERISTIC ?ELD IS GENUINELY NONLINEAR AWAY FROM THE LINE P = 1 AND THE SECOND ?ELD IS GENUINELY NONLINEAR AWAY FROM THE LINE H = 0, THEREFORE THE SYSTEM IS WEAKLY LINEARLY DEGENERATE AT THE POINT (H, P) = (0, 1) ALSO, THE DIRECTION OF INCREASING EIGENVALUES, FOR THE ?RST FAMILY, CHANGES WITH THE SIGN OF P ?1 THE LINES P = 1, H = 0 ARE CHARACTERISTIC CURVES OF THE ?RST, SECOND FAMILY RESPECTIVELY, ALONG WHICH THE SYSTEM BECOMES LINEAR: P = 1, HT ?HX = 0 ; H = 0, PT = 0 SEE FIGURE 2 FOR THE CHARACTERISTIC CURVES FIGURE 2 CHARACTERISTIC CURVES OF THE TWO FAMILIES IN THE H-P PLANE THE ARROWS POINT IN THE DIRECTION OF INCREASING EIGENVALUES IN THIS PAPER WE REVIEW SOME RECENT RESULTS ABOUT THE EXISTENCE OF SOLUTIONS FOR SYSTEM (1 2), THAT ARE SHOWN TO EXIST GLOBALLY IN TIME FOR SUITABLE CLASSES OF INITIAL DATA FOR SYSTEMS OF CONSERVATION LAWS WITH SOURCE TERM, SOME DISSIPATION CONDITIONS ARE KNOWN IN THE LITERATURE THAT ENSURE THE GLOBAL IN TIME EXISTENCE OF (SMOOTH OR WEAK) SOLUTIONS; WE REFER TO [22], TO KAWASHIMA–SHIZUTA CONDITION (SEE [21]) FOR SMOOTH SOLUTIONS AND TO [15, 24] FOR THE WEAK SOLUTIONS THESE CONDITIONS EXPLOIT A SUITABLE BALANCE BETWEEN THE DI?ERENTIAL TERMS AND THE SOURCE TERM THAT ENABLE TO CONTROL THE NONLINEARITY OF THE SYSTEM IT IS INTERESTING TO REMARK THAT SYSTEM (1 2) DOES NOT SATISFY ANY OF THESE CONDITIONS, NEVERTHELESS IT ADMITS GLOBAL IN TIME SOLUTIONS FOR A DERIVATION OF THE MODEL (1 1) OF GRANULAR ?OW WE REFER TO [20] THE PRESENT MODEL HAS RECENTLY RAISED A CERTAIN ATTENTION IN THE MATHEMATICAL COMMUNITY; IN- DEED, IT EXIBITS INTERESTING PROPERTIES AS A NONLINEAR MODEL, AND AT THE SAME TIME IT PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY IS CAPABLE TO DESCRIBE APPROXIMATELY A COMPLEX DYNAMICS LIKE THE EVOLUTION OF TWO LAYERS GRANULAR MATTER A MATHEMATICAL ANALYSIS OF STEADY STATE SOLUTIONS FOR (1 1) WAS CARRIED OUT IN [9, 10]; A NUMERICAL STUDY HAS BEEN PERFORMED IN [17] OTHER MODELS CAN BE FOUND IN [6, 16, 27] WE REFER ALSO TO THE RECENT PAPER [13] WHERE AN EXTENSION OF THE PRESENT MODEL IS PROPOSED; THERE, IT IS INTRODUCED AN ADDITIONAL EQUATION FOR THE VELOCITY OF THE SLIDING MATERIAL, LEADING TO A RE?NED DESCRIPTION OF THE COMPLETE DYNAMICS NOTE THAT, BESIDES [29], THE PAPERS [2, 3] PROVIDE THE ?RST ANALYTICAL STUDY OF TIME DEPENDENT SOLUTIONS TO THIS SYSTEM 2 GLOBAL SMOOTH SOLUTIONS THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS IS ESTABLISHED IN [29], UNDER SUITABLE ASSUMPTIONS ON THE INITIAL DATA BY A DECOUPLED INITIAL DATA WE MEAN A SET OF INITIAL CONDITIONS OF THE FORM (2 1) H(0, X) = ?(X) P(0, X) = 1 + ?(X) WITH ?, ? SATISFYING ?(X) = IF X /?[A, B] , ?(X) = IF X /?[C, D] THE INTERVALS ARE DISJOINT, I E , A < B < C < D MOREOVER WE ASSUME ?(X) > ?1 FOR ALL X FOR DECOUPLED INITIAL DATA, A GLOBAL SOLUTION OF THE CAUCHY PROBLEM CAN BE EXPLICITLY GIVEN, NAMELY H(T, X) = ?(X + T) , P(T, X) = 1 + ?(X) , X ?IR , T ?0 OUR ?RST RESULT PROVIDES THE STABILITY OF THESE DECOUPLED SOLUTIONS MORE PRE- CISELY, EVERY SU?CIENTLY SMALL, COMPACTLY SUPPORTED PERTURBATION OF A LIPSCHITZ CONTINUOUS DECOUPLED SOLUTION EVENTUALLY BECOMES DECOUPLED MOREOVER, NO GRADI- ENT CATASTROPHE OCCURS, I E , SOLUTIONS REMAIN SMOOTH FOR ALL TIME THEOREM 2 1 (GLOBAL SMOOTH SOLUTIONS) LET A < B < C < D BE GIVEN, TOGETHER WITH LIPSCHITZ CONTINUOUS, DECOUPLED INITIAL DATA AS IN (2 1) THEN THERE EXISTS ? > SUCH THAT THE FOLLOWING HOLDS FOR EVERY PERTURBATIONS ˜?, ˜?, SATISFYING (2 2) ˜?(X) = ˜?(X) = IF X /?[A, D] , |˜??(X)| ?? , | ˜??(X)| ?? , THE CAUCHY PROBLEM FOR (1 2) WITH INITIAL DATA (2 3) H(0, X) = ?(X) + ˜?(X) , P(0, X) = 1 + ?(X) + ˜?(X) , HAS A UNIQUE SOLUTION, DE?NED FOR ALL T ?0 AND GLOBALLY LIPSCHITZ CONTINUOUS MORE- OVER, THIS SOLUTION BECOMES DECOUPLED IN ?NITE TIME THE PROOF RELIES ON THE METHOD OF CHARACTERISTICS [22] ONE MUST BOUND THE L?AND L1 NORMS OF HX AND PX SINCE WE ARE LOOKING FOR CONTINUOUS SOLUTIONS, IT IS CONVENIENT TO WORK IN A SET OF RIEMANN COORDINATES LET (H, P) ?(W, Z) BE THE COORDINATE TRANSFORMATION SUCH THAT (W, Z)(H, 1) = (H, 0) FOR ALL H , (W, Z)(0, P) = (0, P ?1) FOR ALL P > 0 , AND R1 Z ?0 , R2 W = 0 IN THESE NEW VARIABLES, THE SYSTEM (1 2) TAKES THE FORM WT + ?1(W, Z) WX = F(W, Z) , ZT + ?2(W, Z)ZX = G(W, Z) , PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY 0 ?Q(0) ?C2C1(D ?A)? FOR SOME CONSTANTS C0, C1 BY A COMPARISON ARGUMENT ONE CAN DEDUCE (2 5) V (T) ?V (0) + ? , Q(T) ?? FOR ALL T ?0 , FOR ANY ? > 0, PROVIDED THAT ? > 0 IS CHOSEN SMALL ENOUGH THE ?RST ESTIMATE IN (2 5) PROVIDES AN A-PRIORI BOUND ON THE TOTAL VARIATION OF THE SOLUTION, VALID AS LONG AS IT REMAINS CONTINUOUS FOR A MORE DETAILED PROOF, WE REFER TO [29] 3 GLOBAL EXISTENCE OF LARGE BV SOLUTIONS FOR MORE GENERAL INITIAL DATA, DUE TO THE NONLINEARITY OF THE ?UX, THE SOLUTIONS WILL DEVELOP DISCONTINUITIES (SHOCKS) IN ?NITE TIME SOLUTIONS SHOULD BE DE?NED IN THE SPACE OF BV FUNCTIONS ASSUMING THE HEIGHT OF THE MOVING LAYER H SU?CIENTLY SMALL, IN [2] WE PROVE THE GLOBAL EXISTENCE OF LARGE BV SOLUTIONS, FOR A CLASS OF INITIAL DATA WITH BOUNDED BUT POSSIBLY LARGE TOTAL VARIATION MORE PRECISELY, CONSIDER INITIAL DATA OF THE FORM (3 1) H(0, X) = ¯H(X) , P(0, X) = ¯P(X) WE ASSUME THAT ¯H : R ?R? + AND ¯P : R ?R? + ARE NON-NEGATIVE FUNCTIONS WITH BOUNDED VARIATION, SUCH THAT TOT VAR {¯P} ?M , TOT VAR {¯H} ?M , (3 2) ?¯H?L1 ?M , ?¯P ?1?L1 ?M , ¯P(X) ?P0 > 0 , (3 3) FOR SOME CONSTANTS M (BOUNDED BUT POSSIBLY LARGE) AND P0 THE FOLLOWING THEOREM IS PROVED IN [2] THEOREM 3 1 (GLOBAL EXISTENCE OF LARGE BV SOLUTIONS) FOR ANY CONSTANTS M, P0 > 0, THERE EXISTS ? > 0 SMALL ENOUGH SUCH THAT, IF (3 2)–(3 3) HOLD TOGETHER WITH (3 4) ?¯H?L??? , THEN THE CAUCHY PROBLEM (1 2), (3 1) HAS AN ENTROPY WEAK SOLUTION, DE?NED FOR ALL T ?0, WITH UNIFORMLY BOUNDED TOTAL VARIATION PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY COMPARED WITH PREVIOUS LITERATURE, THE MAIN NOVELTY OF THE PRESENT RESULT STEMS FROM THE FACT THAT I THE SYSTEM (1 2) CONTAINS SOURCE TERMS; II WE ASSUME A SMALL L?BOUND ON H( ), BUT NOT ON THE COMPONENT P( ); III WE HAVE ARBITRARILY LARGE BV DATA; IV THE SYSTEM IS STRICTLY HYPERBOLIC, BUT ONE OF THE CHARACTERISTIC ?ELDS IS NEITHER GENUINELY NONLINEAR NOR LINEARLY DEGENERATE IN THE LITERATURE, FOR SYSTEMS WITHOUT SOURCE TERMS AND SMALL BV DATA, THE GLOBAL EXISTENCE AND UNIQUENESS OF ENTROPY-WEAK SOLUTIONS TO THE CAUCHY PROBLEM ARE WELL KNOWN EVEN IN THE GENERAL CASE OF SYSTEMS WHICH ARE NEITHER GENUINELY NONLINEAR NOR LINEARLY DEGENERATE, GLOBAL SOLUTIONS HAVE BEEN CONSTRUCTED BY THE GLIMM SCHEME [18, 23, 25, 30], BY FRONT TRACKING APPROXIMATIONS [8, 4, 5], AND BY VANISHING VISCOSITY APPROXIMATIONS [7] IN SOME SPECIAL CASES, EXISTENCE AND UNIQUENESS OF GLOBAL SOLUTIONS IN THE PRESENCE OF A SOURCE TERM WERE PROVED IN [15, 24, 11] AND IN [14, 1, 12], RESPECTIVELY HOWEVER, GLOBAL EXISTENCE OF SOLUTIONS TO HYPERBOLIC SYSTEMS WITH LARGE BV DATA IS A MORE DI?CULT, STILL LARGELY OPEN PROBLEM IN ADDITION TO THE SPECIAL SYSTEM [26], TWO MAIN CASES ARE KNOWN IN THE LITERATURE, WHERE GLOBAL EXISTENCE OF LARGE BV SOLUTIONS IS ACHIEVED ONE IS THE CASE OF TEMPLE CLASS SYSTEMS [28] HERE ONE CAN MEASURE THE WAVE STRENGTHS IN TERMS OF RIEMANN INVARIANTS, SO THAT THE TOTAL STRENGTH OF ALL WAVE FRONTS DOES NOT INCREASE IN TIME, ACROSS EACH INTERACTION A SECOND MAJOR RESULT [19] REFERS TO GENERAL 2 × 2 SYSTEMS, WHERE AGAIN WE CAN MEASURE WAVE STRENGTHS IN TERMS OF RIEMANN COORDINATES TO SEE WHAT HAPPENS AT AN INTERACTION, LET ?1 AND ?2 BE THE STRENGTHS OF THE INCOMING WAVES OF DI?ERENT FAMILIES, AND LET ?? 1 AND ?? 2 BE THE STRENGTHS OF THE OUTGOING WAVES ONE THEN HAS A CUBIC INTERACTION ESTIMATE OF THE FORM (3 5) |?? 1 ??1| + |?? 2 ??2| = O(1) |?1| |?2| (|?1| + |?2|) THANKS TO THE ADDITIONAL TERM |?1| + |?2|, IF THE L?NORM OF THE SOLUTION IS SUF- ?CIENTLY SMALL, THE INCREASE OF TOTAL VARIATION PRODUCED BY THE INTERACTION IS VERY SMALL, AND A GLOBAL EXISTENCE RESULT OF LARGE BV SOLUTIONS CAN THEN BE ESTABLISHED WE REMARK THAT THE CUBIC ESTIMATE IN (3 5) IS USELESS IN OUR CASE, SINCE THE L?NORM OF THE COMPONENT P IN (1 2) CAN BE LARGE, AND SO IS THE ADDITIONAL TERM |?1| + |?2| IN (3 5) THE VALIDITY OF THEOREM 3 1 RELIES HEAVILY ON SOME SPECIAL PROPERTIES OF THE HYPERBOLIC SYSTEM (1 2) FIRST, THE SYSTEM IS LINEARLY DEGENERATE ALONG THE STRAIGHT LINE WHERE H = 0 IN THE REGION WHERE H IS VERY SMALL, THE SYSTEM IS “ALMOST-TEMPLE CLASS” AND ALMOST LINEARLY DEGENERATE RAREFACTION CURVE AND SHOCK CURVE THROUGH THE SAME POINT ARE VERY CLOSE TO EACH OTHER MORE PRECISELY, LET UO BE A POINT ON THE RAREFACTION CURVE OF THE ?RST FAMILY THROUGH THE POINT U = (H, P) THEN, THERE EXISTS A POINT U ?ON THE SHOCK CURVE THROUGH U, WHICH IS VERY CLOSE TO UO, SUCH THAT |U ??UO| = O(1) H2 THIS ALLOWS US TO REPLACE THE ESTIMATE (3 5) WITH (3 6) |?? 1 ??1| + |?? 2 ??2| = O(1) |?1| |?2| ?H?L? BESIDES (3 6), INTERACTION ESTIMATES OF WAVES FROM THE SAME FAMILY ARE ALSO IM- PROVED AS FOLLOWS IF TWO 2-WAVES OF STRENGTH ?2 AND ˜?2 INTERACT, THEN THE STRENGTHS PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY ?+ 1 AND ?+ 2 OF THE OUTGOING WAVES SATISFY (3 7) |?+ 1 | + ?+ 2 ?(?2 + ˜?2) = O(1) HL |?2 ˜?2| IF TWO 1-WAVES OF SIZE ?1 AND ˜?1 INTERACT, THEN THE STRENGTHS ?+ 1 AND ?+ =? ?+ = ?1 + O(1) ?T |PL ?1| ?1 , ?+ = O(1) ?T HL ?2 , ?+ = ?2 + O(1) ?T HL ?2 THE GLOBAL EXISTENCE OF LARGE BV SOLUTIONS IS ESTABLISHED BY DERIVING THE FOL- LOWING GLOBAL A PRIORI BOUNDS FOR THE APPROXIMATE SOLUTIONS: THE NORMS ?H?(T, )?L1 AND ?P?(T, ) ?1?L1 ; THE LOWER BOUND ON P?, I E , THE QUANTITY INFX P?(T, X) ; THE UNIFORM BOUNDS ?H?(T, )?L?AND ?P?(T, )?L?; THE TOTAL VARIATIONS TOT VAR {H?(T, )} AND TOT VAR {P?(T, )} MOST OF THE A PRIORI BOUNDS ARE OBTAINED BY DE?NING SUITABLE WEIGHTED FUNC- TIONALS THAT ARE NON-INCREASING IN TIME THE WEIGHTS ARE CHOSEN IN A WAY SUCH THAT THEY ACCOUNT FOR THE MASS TO BE CROSSED IN FUTURE DUE TO THE STRICTLY DI?ERENT WAVE SPEEDS, THE WEIGHTS WILL BE NON-INCREASING IN TIME, ACHIEVING THE DESIRED ESTIMATES PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY WE NOW STUDY IN MORE DETAIL THE ?RST POINT ABOVE: HOW TO OBTAIN A PRIORI BOUNDS ON ?H?(T, )?L1, ?P?(T, )?1?L1 WE DERIVE THEM IN A FORMAL WAY, ASSUMING THAT (H, P)(T, X) IS A WEAK SOLUTION OF THE SYSTEM (1 2), SATISFYING H(T, X) ?0 AND P(T, X) > 0 BOUND ON THE L1 NORM OF P ?1 WE REWRITE THE SECOND EQUATION OF (1 2) AS (P ?1)T + (P ?1)H X = 0 THEN THE FOLLOWING INEQUALITY IS SATIS?ED IN D?: |P ?1|T + |P ?1|H X ?0 , (3 13) HENCE THE L1-NORM OF (P ?1) IS NON-INCREASING IN TIME: ?P(T, ) ?1?L1 ??¯P ?1?L1 , FOR ALL T ?0 BOUND ON THE L1 NORM OF H FROM THE ?RST EQUATION HT ?(HP)X = (P ?1)H, WE GET ?H(T)?L1 ? T H(?, X) DXD? THAT MAY INCREASE IN TIME TO CONTROL THE POSSIBLE INCREMENT, WE INTRODUCE THE QUANTITY W(T, X) = EXP X ?? |P(T, Y) ?1| DY THIS TERM ACCOUNTS FOR THE EROSION/DEPOSITION THAT THE MASS H AT THE POINT X WILL ENCOUNTER IN THE FUTURE NOTE THAT W IS GLOBALLY BOUNDED, THANKS TO THE BOUND ON ?P(T, ) ?1?L1: 1 ?W(T, X) ?EXP {?¯P ?1?L1} MOREOVER, USING THE INEQUALITY (3 13), WE GET D DT X ?? |P(T, Y) ?1| DY ? ?|P(T, X) ?1| H(T, X) = ?D DX X ?? |P(T, Y) ?1| DY H(T, X) , SO THAT WT + HWX ?0 THEN WE INTRODUCE THE QUANTITY IH(T) = W(T, X)H(T, X) DX WE WANT TO GIVE AN A-PRIORI BOUND ON IH(T) THIS WILL LEAD TO AN A-PRIORI BOUND ON THE L1 NORM OF H: INDEED, RECALLING THAT W ?1, H ?0 ONE GETS ?H(T, )?L1 = H(T, X) DX ?IH(T) LEMMA 3 2 THE QUANTITY IH(T) IS NON-INCREASING IN TIME PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY AND P > 0 THEN WE ACHIEVE THE DESIRED INEQUALITY (3 14) THE PROPERTY STATED IN LEMMA 3 2 LEADS TO AN A-PRIORI BOUND ON THE L1 NORM OF H: ?H(T, )?L1 ?IH(T) ?IH(0) ?EXP (?¯P ?1?L1) ?¯H?L1 THE PROOF OF THE A PRIORI BOUNDS ON THE TOTAL VARIATION RELY ON THE FOLLOWING KEY OBSERVATION: IF ALL WAVE STRENGTHS ARE MEASURED IN TERMS OF RIEMANN COORDINATES, THEN ALL THE INTERACTION ESTIMATES (3 6)-(3 8) CONTAIN THE ADDITIONAL FACTOR ?H?L? THEREFORE, IF THE NORM ?H?L?REMAINS SU?CIENTLY SMALL, WE CAN ASSUME THAT THE TOTAL STRENGTH OF ALL NEW WAVES PRODUCED BY INTERACTIONS IS AS SMALL AS WE LIKE IN ESSENCE, THE CHANGE IN THE TOTAL VARIATION AND IN THE L?NORMS OF H, P IS THUS DETERMINED ONLY BY THE SOURCE TERM IN THE ?RST EQUATION OF (1 2) FOR DETAILS OF THE PROOF OF THEOREM 3 1 WE REFER TO [2] 4 GLOBAL BV SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM NEXT WE STUDY HOW THE MOUNTAIN PRO?LE EVOLVES WHEN THE THICKNESS OF THE MOVING LAYER APPROACHES ZERO, BUT THE TOTAL MASS OF SLIDING MATERIAL REMAINS POS- ITIVE THE LIMITING BEHAVIOR OF THE SLOPE P( ), WHEN THE NORMS ?H?L?AND ?F?L? APPROACH ZERO, IS OF PRACTICAL INTERESTS THIS DESCRIBES HOW THE MOUNTAIN PRO?LE EVOLVES, WHEN THE GRANULAR MATERIAL IS POURED DOWN AT A VERY SLOW RATE THIS RESULT IS BEST FORMULATED IN CONNECTION WITH AN INITIAL-BOUNDARY VALUE PROBLEM BY A TRANSLATION OF COORDINATES, IT IS NOT RESTRICTIVE TO CONSIDER THE DOMAIN R? = {X < 0} ON R?, CONSIDER THE INITIAL-BOUNDARY VALUE PROBLEM FOR (1 2), WITH INITIAL DATA (3 1) AND THE FOLLOWING BOUNDARY CONDITION AT X = (4 1) P(T, 0)H(T, 0) = F(T) THE CONDITION (4 1) PRESCRIBES THE INCOMING ?UX F(T) OF GRANULAR MATERIAL THROUGH THE POINT X = 0, SEE THE ?RST EQUATION IN (1 2) WE ASSUME THAT (4 2) F(T) ?0 , TOT VAR {F} ?M , 0 < M ? < ? F(?) D? ?M PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY AS A PARTIAL STEP TOWARD THE SLOW EROSION LIMIT, WE PROVE IN [3] NEXT THEOREM ON THE GLOBAL EXISTENCE OF LARGE BV SOLUTIONS TO THIS INITIAL-BOUNDARY VALUE PROBLEM, PROVIDED THAT ?¯H?L?AND ?F?L?ARE SU?CIENTLY SMALL THEOREM 4 1 (GLOBAL EXISTENCE OF LARGE BV SOLUTIONS FOR THE INITIAL-BOUNDARY VALUE PROBLEM) GIVEN M, P0 > 0 (AND FOR M ? > 0 ARBITRARY) THERE EXISTS ? > SUCH THAT THE ASSUMPTIONS (3 2), (3 3) AND (4 2), TOGETHER WITH (4 3) ?¯H?L??? , ?F?L??? , IMPLY THAT THE INITIAL-BOUNDARY VALUE PROBLEM (1 2), (3 1), (4 1) HAS A GLOBAL IN TIME SOLUTION, WITH UNIFORMLY BOUNDED TOTAL VARIATION FOR ALL T ?0 THE MAIN STEPS IN THE PROOF OF THEOREM 4 1 ARE THE SAME AS FOR THEOREM 3 1 WE APPROXIMATE THE INITIAL DATA AND THE BOUNDARY DATA WITH PIECEWISE CONSTANT FUNCTIONS THE APPROXIMATED ?UX ON THE BOUNDARY IS SET TO BE CONSTANT ON EACH TIME INTERVAL (TK?1, TK) ON THE TIME INTERVALS (TK?1, TK) AN APPROXIMATE SOLUTION OF THE CONSERVATION LAWS (3 9) IS CONSTRUCTED BY FRONT TRACKING, WITH CONSTANT ?OW AT THE BOUNDARY X = 0 AT TIME T = TK THE SOLUTION IS UPDATED BY MEANS OF (3 10) AND DUE TO THE POSSIBLY NEW VALUE FOR F THE MAIN DI?ERENCE IS CAUSED BY THE BOUNDARY CONDITION AT X = 0 WE HAVE THE FOLLOWING BOUNDARY ESTIMATES: (I) AT A TIME ? WHERE A 2-WAVE OF STRENGTH ?2 HITS THE BOUNDARY AT X = 0, A NEW RE?ECTED FRONT OF THE ?RST FAMILY IS CREATED CALLING HL THE STATE TO THE LEFT OF THE JUMP ?2 AND ?+ 1 THE SIZE OF THE NEW JUMP, WE HAVE THE ESTIMATE (4 4) |?+ 1 | = O(1) HL |?2| (II) AT A TIME ? = TK, THE INDUCTIVE STEP (3 10) IS PERFORMED AND THE ?UX F = HP HAS A JUMP THEN A FRONT OF THE ?RST FAMILY, ENTERING THE DOMAIN, IS CREATED ITS STRENGTH |?+ 1 | SATIS?ES (4 5) |?+ 1 | = O(1) ?T H?+ O(1) |F(?+) ?F(??)| WHERE H?IS THE VALUE AT X = 0 BEFORE THE TIME STEP WE NOTE THAT THE ESTIMATES (4 4), (4 5) CONTAIN EITHER THE TERM ?H?L?OR THE TERM ?F?L?, WHICH ARE ARBITRARILY SMALL SAME GLOBAL A PRIORI ESTIMATES AS FOR THEOREM 3 1 CAN BE ESTABLISHED, PROVING THE GLOBAL EXISTENCE OF LARGE BV SOLUTION FOR DETAILS, SEE [3] 5 SLOW EROSION LIMIT WE NOW STUDY THE SLOW EROSION/DEPOSITION LIMIT FOR THE REST OF THE PAPER, WE WILL SIMPLY CALL IT THE “SLOW EROSION LIMIT” NUMERICAL SIMULATIONS IN [29] SHOW THE FOLLOWING OBSERVATION: WHEN THE HEIGHT OF THE MOVING LAYER H IS VERY SMALL, THE PRO?LE OF THE STANDING LAYER DEPENDS ONLY ON THE TOTAL MASS OF THE AVALANCHE ?OWING DOWNHILL, NOT ON THE TIME-LAW DESCRIBING AT WHICH RATE THE MATERIAL SLIDES DOWN THIS OBSERVATION IS PROVED RIGOROUSLY IN [3] IN MORE DETAIL, WE CONSIDER THE INITIAL-BOUNDARY VALUE PROBLEM FOR (1 2), WITH INITIAL DATA ¯H AND BOUNDARY DATA F SATISFYING THE ASSUMPTIONS OF THEOREM 4 1 WE ARE INTERESTED IN THE LIMIT AS ?H?L?, ?F?L?TEND TO ZERO PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY WE DE?NE A NEW VARIABLE WHICH MEASURES THE TOTAL MASS OF AVALANCHE ?OWING DOWN, AND USE IT IN PLACE OF THE TIME VARIABLE, AS FOLLOWS RECALLING THAT F ?0, SEE (4 2), THE MAP (5 1) ?(T) = T F(?) D? IS MONOTONE NON-DECREASING AND THEN ADMITS THE GENERALIZED INVERSE T(?) = MIN {? ?0 ; ??(?) = ?} , WHICH IS WELL DE?NED FOR ? ?[0, M ?] WE WILL USE THE NEW VARIABLE ? IN PLACE OF T AND RE-PARAMETRIZE THE PRO?LE P AS FOLLOWS: P(?, X) = P(T(?), X) (5 2) THE FORMAL LIMIT LET US DERIVE THE FORMAL SLOW EROSION LIMIT WITH INITIAL DATA ¯H ?0 AND THE FOLLOWING BOUNDARY DATA: (PH)(T, 0) = F ?(T) = ? ¯F(?T) > AS ? ?0 DE?NE ??(T) ?= T F ?(S) DS = ?T P? M? X = 0 PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY P M X = (5 5) THE ?RST EQUATION IN (5 5) IS THEN REDUCED TO A LINEAR DI?ERENTIAL EQUATION FOR M, WITH P AS A COE?CIENT AT X = 0 IT SATIS?ES THE BOUNDARY CONDITION M(0, ?) = F(?) , WITH F(?) AS IN (5 3) SOLVING THE EQUATION ON X < 0, WE ?ND M(X, ?) = F(?) EXP P EXP AND AFTER RENAMING THE TIME VARIABLE, IT CAN BE REWRITTEN AS QT + K(T, X) Q Q + X = 0 , (5 6) WITH K(T, X) = EXP X Q(T, ?) Q(T, ?) + 1 D? (5 7) THE ?UX IN (5 6) IS GIVEN BY THE PRODUCT OF THE NONLOCAL TERM K AND OF THE LOCAL TERM F(Q) = Q Q + 1 , (5 8) THAT SATIS?ES F ? > 0, F ?? < 0 ON OUR DOMAIN OF INTEREST, NAMELY Q +1 > 0 NOTE THAT THE CHARACTERISTIC SPEED OF THE EQUATION, KF ?(Q), IS POSITIVE SINCE THE ?UX FUNCTION IS CONCAVE DOWN AS A FUNCTION OF Q, THE ADMISSIBLE SHOCKS IN Q WILL BE THE ONES THAT JUMP UP THE EQUATION (5 6) IS SUPPLEMENTED BY THE INITIAL DATA Q(0, X) = Q0(X) = ¯P(X) ?1 , X < 0 (5 9) AT THE BOUNDARY X = 0 NO CONDITION IS PROVIDED, SINCE THE CHARACTERISTIC SPEED IS POSITIVE FOR Q + 1 > 0 PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A NATURAL DE?NITION OF AN ENTROPY WEAK SOLUTION OF (5 6)–(5 7), (5 9) CAN BE GIVEN AS FOLLOWS LET C, P0 BE SOME POSITIVE CONSTANTS AND DE?NE D = DC,P0 AS THE SET OF FUNCTIONS Q( ) SATISFYING THE FOLLOWING UNIFORM BOUNDS INF X<0 Q(X) + 1 ?P0 > 0, TV Q( ) ?C, ?Q?L1(R?) ?C (5 10) LET Q0 ?D WE SAY THAT Q PROVIDES AN ENTROPY WEAK SOLUTION TO (5 6)–(5 7), (5 9) ON THE DOMAIN [0, T] × R?IF THE FOLLOWING CONDITIONS (H1), (H2) ARE SATIS?ED: (H1) Q : [0, T] ?D; THE MAP [0, T] ?T ?Q(T) IS LIPSCHITZ IN L1(R?); (H2) Q IS A WEAK ENTROPY SOLUTION OF THE SCALAR CONSERVATION LAW (5 6), WITH K DE?NED BY (5 7), AND WITH Q(0, ) = Q0 WE REMARK THAT, THANKS TO (H1), THE FUNCTION K HAS THE FOLLOWING PROPERTIES: (I) IT IS DE?NED ON [0, T] × R?; (II) IT IS BOUNDED AND LIPSCHITZ CONTINUOUS, WITH INF K > 0; AND (III) THE TOTAL VARIATION OF K(T, ) AND KX(T, ) ARE BOUNDED UNIFORMLY IN TIME INDEED, ACCORDING TO THE ASSUMPTION (H1), THE INTEGRAL TERM X Q(T, ?) Q(T, ?) + 1 D? IS BOUNDED, SINCE X Q(T, ?) Q(T, ?) + 1 D? ? MOREOVER, IT IS LIPSCHITZ CONTINUOUS AS A FUNCTION OF T ABOUT KX, WE HAVE KX = ?K Q Q + TV (Q) ? M TV (Q) FOR A SUITABLE M > 0 WE CAN SHOW THAT THE INTEGRO-DI?ERENTIAL EQUATION (5 6)–(5 7) IS WELL POSED THIS IS NOT TRIVIAL BECAUSE THE ?UX IS A NONLOCAL FUNCTION MORE PRECISELY, WE PROVE THAT THE ?OW GENERATED BY THE INTEGRO-DI?ERENTIAL EQUATION (5 6)–(5 7) IS LIPSCHITZ CONTINUOUS RESTRICTED TO THE DOMAIN OF FUNCTIONS SATISFYING THE BOUNDS (5 10) INDEED, CONSIDER Q, ˜Q : [0, T] ?D SATISFYING (H1) AND (H2), WITH CORRESPOND- ING INITIAL DATA Q(0, X) = Q0(X) , ˜Q(0, X) = ˜Q0(X) X < 0 ONE CAN SHOW THAT ?Q(T, ) ?˜Q(T, )?L1(R?) ? ?Q0 ?˜Q0?L1 + L T ?Q(S, ) ?˜Q(S, )?L1 DS PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY FOR A SUITABLE CONSTANT L, FOR ALL T ?[0, T] BY GRONWALL’S LEMMA, THIS YIELDS THE LIPSCHITZ CONTINUOUS DEPENDENCE OF SOLUTIONS ON THE INITIAL DATA 5 2 CONVERGENCE TO THE LIMIT PRO?LE WE NOW STATE THE MAIN RESULT OF THIS SECTION THEOREM 5 1 (SLOW EROSION LIMIT) CONSIDER THE INITIAL BOUNDARY VALUE PROBLEM FOR (5 11) HT ?(HP)X = (P ?1)H , PT + (P ?1)H X = AS ? ?? (5 13) NOTE THAT THE INITIAL DATA FOR P IS NOT MODI?ED IN THE LIMIT DE?NE T?(?) = MIN T ?0 ; T F?(S) DS = ? ? ?[0, M ?] , AND CONSIDER THE RESCALED FUNCTIONS P?(?, X) = P?(T?(?), X) , OBTAINED BY USING ? AS NEW TIME VARIABLE THESE ARE WELL DE?NED FOR X ?0, ? ?[0, M ?] AND ? SU?CIENTLY LARGE NOTE THAT, WHILE THE MAP T ?P?(T, ) IS CONTINUOUS IN L1LOC, THE NOW DE?NED MAP ? ?P?(?, ) MAY FAIL TO BE CONTINUOUS IN L1LOC SINCE T?(?) IS NOT NECESSARILY CONTINUOUS STEP 1 WE ?RST ESTABLISH A LIPSCHITZ-TYPE DEPENDENCE OF P ON THE RESCALED TIME VARIABLE FOR THE SOLUTIONS OF (5 11) THE ESTIMATE IS UNIFORM IN ? MORE PRECISELY, LET (H, P) BE A SOLUTION OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR (5 11), WITH BOUNDARY CONDITION (4 1) (FOR BREVITY WE OMIT THE DEPENDENCE ON ?) FIX R > 0, ASSUME THAT T? < T?? AND SET G(X) = SUP T?[T?,T??] |P(T, X) ?P(T?, X)| (5 14) WE ESTABLISH THE ESTIMATE ?R G(X) DX ? L T?? T? F(T) DT + ?H?L? (5 15) PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY ¯P(X) ?CG(X) ?(X) ?2?H?L? DX WHERE ¯P(X) = P(T?, X) AND G IS DE?NED AT (5 14) BY USING A SUITABLE GRONWALL-TYPE ESTIMATE, ONE CAN OBTAIN THE ESTIMATES ?(?) ? (?(0) + 2M?H?L?) EXP ¯P(X) + CG(X) DX ?(?) ? (?(0) ?2M?H?L?) EXP ¯P(X) ?CG(X) DX PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY WHERE M IS INDEPENDENT ON ?, FOR ? ?[?R, 0] NOW INTRODUCE THE MAP K(T?, ?) = EXP C (R?), WE HAVE P(?, X) K(?, X) DX D? , (5 19) WHERE K(?, X) = EXP P(?, ?) D? WE START FROM THE WEAK FORMULATION OF THE SECOND EQUATION IN (5 11) ?R ?(X) P(T(?2), X) ?P(T(?1), X) DX (5 20) = T(?2) T(?1) P(?, X) (5 22) PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY P?(?, X) (P?H?)(?, X) ? DX D? BY USING (5 22), (5 25) AND (5 24) WE CONCLUDE THAT IT CONVERGES TO THE R H S OF (5 19) STEP 4 CHECK THE ENTROPY ADMISSIBILITY OF THE LIMIT FUNCTION ONE NEEDS TO CHECK THAT ALL SHOCKS JUMP IN THE CORRECT DIRECTION SINCE THE ?UX KF(Q) IS CONCAVE W R T Q, A DISCONTINUITY IS ENTROPIC IF Q JUMPS UP RECALLING FIGURE 2, ALL THE SHOCKS OF THE SECOND FAMILY JUMP UP IN THE P VARIABLE; THIS PROPERTY IS STILL SATIS?ED IN THE LIMIT FOR A DETAILED PROOF WE REFER TO [3] REFERENCES 1 AMADORI, D AND GOSSE, L AND GUERRA, G ; GLOBAL BV ENTROPY SOLUTIONS AND UNIQUENESS FOR HYPERBOLIC SYSTEMS OF BALANCE LAWS ARCH RATION MECH ANAL 162 (2002), 327–366 2 AMADORI, D AND SHEN, W ; GLOBAL EXISTENCE OF LARGE BV SOLUTIONS IN A MODEL OF GRANULAR FLOW COMM PARTIAL DI?ERENTIAL EQUATIONS 34 (2009), 1003–1040 3 AMADORI, D AND SHEN, W ; THE SLOW EROSION LIMIT FOR A MODEL OF GRANULAR FLOW ACCEPTED FOR PUBLICATION IN ARCH RATION MECH ANAL 4 ANCONA, F AND MARSON, A ; WELL-POSEDNESS FOR GENERAL 2 × 2 SYSTEMS OF CONSERVATION LAWS MEM AMER MATH SOC 169 (2004), NO 801 5 ANCONA, F AND MARSON, A ; EXISTENCE THEORY BY FRONT TRACKING FOR GENERAL NONLINEAR HY- PERBOLIC SYSTEMS ARCH RATION MECH ANAL 185 (2007), 287–340 6 BOUTREUX, T AND GENNES, P -G ; SURFACE ?OWS OF GRANULAR MIXTURES, I GENERAL PRINCIPLES AND MINIMAL MODEL J PHYS I FRANCE 6 (1996), 1295–1304 7 BIANCHINI, S AND BRESSAN, A ; VANISHING VISCOSITY SOLUTIONS TO NONLINEAR HYPERBOLIC SYSTEMS ANNALS OF MATHEMATICS 161 (2005), 223–342 8 BRESSAN, A ; HYPERBOLIC SYSTEMS OF CONSERVATION LAWS THE ONE DIMENSIONAL CAUCHY PROB- LEM, OXFORD UNIVERSITY PRESS, 2000 9 CANNARSA, P AND CARDALIAGUET, P ; REPRESENTATION OF EQUILIBRIUM SOLUTIONS TO THE TABLE PROBLEM FOR GROWING SANDPILES J EUR MATH SOC (JEMS) 6 (2004), 435–464 10 CANNARSA, P AND CARDALIAGUET, P AND CRASTA, G AND GIORGIERI, E ; A BOUNDARY VALUE PROBLEM FOR A PDE MODEL IN MASS TRANSFER THEORY: REPRESENTATION OF SOLUTIONS AND APPLICATIONS CALC VAR PARTIAL DI?ERENTIAL EQUATIONS 24 (2005), 431–457 PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY 11 CHRISTOFOROU, C ; HYPERBOLIC SYSTEMS OF BALANCE LAWS VIA VANISHING VISCOSITY J DI?ERENTIAL EQUATIONS 221 (2006), 470–541 12 CHRISTOFOROU, C ; UNIQUENESS AND SHARP ESTIMATES ON SOLUTIONS TO HYPERBOLIC SYSTEMS WITH DISSIPATIVE SOURCE COMM PARTIAL DI?ERENTIAL EQUATIONS 31 (2006), 1825–1839 13 COLOMBO, R AND GUERRA, G AND MONTI, F ; MODELING THE DYNAMICS OF GRANULAR MATTER, PREPRINT 2009 AVAILABLE AT HTTP://WWW MATH NTNU NO/CONSERVATION/2009/054 HTML 14 CRASTA, G AND PICCOLI, B ; VISCOSITY SOLUTIONS AND UNIQUENESS FOR SYSTEMS OF INHOMOGENEOUS BALANCE LAWS DISCRETE CONTIN DYNAM SYSTEMS 3 (1997), 477–502 15 DAFERMOS, C M AND HSIAO, L ; HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH INHOMOGENEITY AND DISSIPATION INDIANA U MATH J 31 (1982), 471–491 16 DURAN, J ; SANDS, POWDERS, AND GRAINS: AN INTRODUCTION TO THE PHYSICS OF GRANULAR MATE- RIALS, SPRINGER-VERLAG, 2000 17 FALCONE, M AND FINZI VITA, S ; A ?NITE-DI?ERENCE APPROXIMATION OF A TWO-LAYER SYSTEM FOR GROWING SANDPILES SIAM J SCI COMPUT 28 (2006), 1120–1132 18 GLIMM, J ; SOLUTIONS IN THE LARGE FOR NONLINEAR HYPERBOLIC SYSTEMS OF EQUATIONS COMM PURE APPL MATH 18 (1965), 697–715 19 GLIMM, J AND LAX, P ; DECAY OF SOLUTIONS OF SYSTEMS OF NONLINEAR HYPERBOLIC CONSERVATION LAWS, MEM AMER MATH SOC 101 (1970) 20 HADELER, K P AND KUTTLER, C ; DYNAMICAL MODELS FOR GRANULAR MATTER GRANULAR MATTER (1999), 9–18 21 HANOUZET, B AND NATALINI, R ; GLOBAL EXISTENCE OF SMOOTH SOLUTIONS FOR PARTIALLY DISSIPATIVE HYPERBOLIC SYSTEMS WITH A CONVEX ENTROPY ARCH RATION MECH ANAL 169 (2003), 89–117 22 LI, T T ; GLOBAL CLASSICAL SOLUTIONS FOR QUASILINEAR HYPERBOLIC SYSTEMS JOHN WILEY & SONS, CHICHESTER, 1994 23 LIU, T P ; ADMISSIBLE SOLUTIONS OF HYPERBOLIC CONSERVATION LAWS MEM AMER MATH SOC (1981), NO 240 24 LIU, T P ; QUASILINEAR HYPERBOLIC SYSTEMS COMMUN MATH PHYS 68 (1979), 141–172 25 LIU, T P AND YANG, T ; WEAK SOLUTIONS OF GENERAL SYSTEMS OF HYPERBOLIC CONSERVATION LAWS COMM MATH PHYS 230 (2002), 289–327 26 NISHIDA, T AND SMOLLER, J ; SOLUTIONS IN THE LARGE FOR SOME NONLINEAR CONSERVATION LAWS COMM PURE APPL MATH 26 (1973), 183–200 27 SAVAGE, S B AND HUTTER, K ; THE DYNAMICS OF AVALANCHES OF GRANULAR MATERIALS FROM INITIATION TO RUNOUT I ANALYSIS ACTA MECH 86 (1991), 201–223 28 SERRE, D ; SYSTEMS OF CONSERVATION LAWS I, II, CAMBRIDGE UNIVERSITY PRESS, 2000 29 SHEN, W ; ON THE SHAPE OF AVALANCHES J MATH ANAL APPL 339 (2008), 828–838 30 SMOLLER, J ; SHOCK WAVES AND REACTION-DI?USION EQUATIONS, SPRINGER-VERLAG, 1983 PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY ?I(GI(U)) = 0, WHERE U AND GI ARE SCALAR FUNCTIONS (DEPENDING ON F) WHEN U IS A STRICTLY CON- VEX FUNCTION, USUALLY CALLED ’ENTROPY FUNCTION’, THE SYSTEM AUTOMATICALLY GETS HYPER- BOLIC FOR MOST HYPERBOLIC SYSTEMS, SOLUTIONS ARE EXPECTED TO BECOME DISCONTINUOUS IN ?NITE TIME, EVEN FOR SMOOTH INITIAL CONDITIONS THERE IS NO THEORY AVAILABLE TO SOLVE THE INITIAL VALUE PROBLEM IN THE LARGE (SEE [18] FOR A MODERN REVIEW), EXCEPT IN TWO EXTREME SITUATIONS FIRST, FOR A SINGLE SPACE VARIABLE (D = 1) AND SMALL INITIAL CONDITIONS (IN TOTAL VARIATION), GLOBAL EXISTENCE AND UNIQUENESS OF ’ENTROPY SOLUTIONS’ HAVE BEEN ESTABLISHED THROUGH THE CELEBRATED RESULTS OF J GLIMM (EXIS- TENCE) AND A BRESSAN AND COLLABORATORS (WELL POSEDNESS) [22, 1] NEXT, FOR A SINGLE KEY WORDS AND PHRASES HYPERBOLIC CONSERVATION LAWS, HILBERTIAN FORMULATION