GEOMETRIC STURMIAN THEORY OF NONLINEAR PARABOLIC EQUATIONS AND APPLICATIONS

By Y. GINAO

CONTENTS INTRODUCTION: STURM THEOREMS AND NONLINEAR SINGULAR PARABOLIC EQUA- TIONS 1 STURM THEOREMS FOR LINEAR PARABOLIC EQUATIONS AND INTERSECTION COMPARISON B -EQUATIONS 1 1 FIRST STURM THEOREM: NONINCREASE OF THE NUMBER OF SIGN CHANGES 1 2 SECOND STURM THEOREM: EVOLUTION FORMATION AND COLLAPSE OF MULTIPLE ZEROS 1 3 FIRST ASPECTS OF INTERSECTION COMPARISON OF SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS 1 4 GEOMETRICALLY ORDERED ?OWS: TRANSVERSALITY AND CONCAVITY TECH- NIQUES 1 5 EVOLUTION B-EQUATIONS PRESERVING STURMIAN PROPERTIES REMARKS AND COMMENTS ON THE LITERATURE SURVEY ON STURM’S PDE THEORY ON SPECTRA OF MULTIPLE ZEROS IN LINEAR AND QUASILINEAR PARABOLIC EQUATIONS 2 TRANSVERSALITY, CONCAVITY AND SIGN-INVARIANTS SOLUTIONS ON LINEAR INVARIANT SUBSPACES 2 1 INTRODUCTION: FILTRATION EQUATION AND CONCAVITY PROPERTIES 2 2 PROOFS OF TRANSVERSALITY AND CONCAVITY ESTIMATES BY INTERSECTION COMPARISON WITH TRAVELLING WAVES 2 3 EVENTUAL CONCAVITY FOR THE ?LTRATION EQUATION 2 4 CONCAVITY FOR ?LTRATION EQUATIONS WITH LOWER-ORDER TERMS 2 5 SINGULAR EQUATIONS WITH THE P-LAPLACIAN OPERATOR PRESERVING CONCAVITY 2 6 CONCEPTS OF B-CONCAVITY AND B-CONVEXITY FIRST EXAMPLE OF SIGN-INVARIANTS 2 7 VARIOUS B-CONCAVITY PROPERTIES FOR THE POROUS MEDIUM EQUATION AND SIGN-INVARIANTS B-CONCAVITY AND SIGN-INVARIANTS FOR THE HEAT EQUATION B-CONCAVITY AND TRANSVERSALITY FOR THE POROUS MEDIUM EQUATION WITH SOURCE B-CONVEXITY FOR EQUATIONS WITH EXPONENTIAL NONLINEARITIES 2 11 SINGULAR PARABOLIC DIFFUSION EQUATIONS IN THE RADIAL N -DIMENSIONAL GEOMETRY 2 12 ON GENERAL B-CONCAVITY VIA SOLUTIONS ON LINEAR INVARIANT SUB- SPACES REMARKS AND COMMENTS ON THE LITERATURE 3 B-CONCAVITY AND TRANSVERSALITY ON NONLINEAR SUBSETS FOR QUASILIN- EAR HEAT EQUATIONS 3 1 INTRODUCTION: BASIC EQUATIONS AND CONCAVITY ESTIMATES 3 2 LOCAL CONCAVITY ANALYSIS VIA TRAVELLING WAVE SOLUTIONS 3 3 CONCAVITY FOR THE P-LAPLACIAN EQUATION WITH ABSORPTION B-CONCAVITY RELATIVE TO TRAVELLING WAVES B-CONCAVITY FOR THE ?LTRATION EQUATION B-CONCAVITY RELATIVE TO INCOMPLETE FUNCTIONAL SUBSETS 3 7 EVENTUAL B-CONCAVITY 4 EVENTUAL B-CONVEXITY: A CRITERION OF COMPLETE BLOW-UP AND EXTINC- TION FOR QUASILINEAR HEAT EQUATIONS 4 1 INTRODUCTION: THE BLOW-UP PROBLEM 4 2 EXISTENCE AND NONEXISTENCE OF SINGULAR BLOW-UP TRAVELLING WAVES 4 3 DISCUSSION OF THE BLOW-UP CONDITIONS PATHOLOGICAL EQUATIONS 4 4 PROOF OF COMPLETE AND INCOMPLETE BLOW-UP 4 5 THE EXTINCTION PROBLEM 4 6 COMPLETE AND INCOMPLETE EXTINCTION VIA SINGULAR TRAVELLING WAVES 5 BLOW-UP INTERFACES FOR QUASILINEAR HEAT EQUATIONS 5 1 INTRODUCTION: FIRST PROPERTIES OF INCOMPLETE BLOW-UP 5 2 EXPLICIT PROPER BLOW-UP TRAVELLING WAVES AND ?RST ESTIMATES OF BLOW-UP PROPAGATION 5 3 EXPLICIT BLOW-UP SOLUTIONS ON AN INVARIANT SUBSPACE 5 4 LOWER SPEED ESTIMATE OF BLOW-UP INTERFACES 5 5 DYNAMICAL EQUATION OF BLOW-UP INTERFACES 5 6 BLOW-UP INTERFACES ARE NOT C 2 FUNCTIONS 5 7 LARGE TIME BEHAVIOUR OF PROPER BLOW-UP SOLUTIONS 5 8 BLOW-UP INTERFACES FOR THE P-LAPLACIAN EQUATION WITH SOURCE 5 9 BLOW-UP INTERFACES FOR EQUATIONS WITH GENERAL NONLINEARITIES 5 10 EXAMPLES OF BLOW-UP SURFACES IN I R N 6 COMPLETE AND INCOMPLETE BLOW-UP IN SEVERAL SPACE DIMENSIONS 6 1 INTRODUCTION: THE BLOW-UP PROBLEM IN I R N AND CRITICAL EXPONENTS 6 2 CONSTRUCTION OF THE PROPER BLOW-UP SOLUTION: EXTENSION OF MONO- TONE SEMIGROUPS 6 3 GLOBAL CONTINUATION OF NONTRIVIAL PROPER SOLUTIONS 6 4 ON BLOW-UP SET IN THE LIMIT CASE P = 6 5 COMPLETE BLOW-UP UP TO CRITICAL SOBOLEV EXPONENT 6 6 COMPLETE BLOW-UP OF FOCUSED SOLUTIONS IN THE SUBCRITICAL CASE 6 7 COMPLETE BLOW-UP IN THE CRITICAL SOBOLEV CASE 6 8 COMPLETE BLOW-UP OF UNFOCUSED SOLUTIONS 6 9 COMPLETE BLOW-UP IN THE SUPERCRITICAL CASE 6 10 COMPLETE AND INCOMPLETE BLOW-UP FOR THE EQUATION WITH THE P-LAPLACIAN OPERATOR 6 11 EXTINCTION PROBLEMS IN I R N AND THE CRITERIA OF COMPLETE AND INCOMPLETE SINGULARITIES 7 GEOMETRIC THEORY OF NONLINEAR SINGULAR PARABOLIC EQUATIONS MAX- IMAL SOLUTIONS 7 1 INTRODUCTION: MAIN STEPS AND CONCEPTS OF THE GEOMETRIC THEORY 7 2 SET B OF SINGULAR TRAVELLING WAVES AND RELATED GEOMETRIC NOTIONS: PRESSURE, SLOPES, INTERFACE OPERATORS, TW-DIAGRAM 7 3 ON CONSTRUCTION OF PROPER MAXIMAL SOLUTIONS 7 4 EXISTENCE: INCOMPLETE SINGULARITIES IN I R AND I R N 7 5 COMPLETE SINGULARITIES IN I R AND I R N IN?NITE PROPAGATION AND PATHOLOGICAL EQUATIONS 7 6 FURTHER GEOMETRIC NOTIONS: B-CONCAVITY, SIGN-INVARIANTS, B- NUMBER 7 7 REGULARITY IN B-CLASSES BY TRANSVERSALITY: GRADIENT ESTIMATES, INSTANTANEOUS SMOOTHING, LIPSCHITZ INTERFACES, OPTIMAL MODULI OF CONTINUITY 7 8 TRANSVERSALITY AND SMOOTHING IN THE RADIAL GEOMETRY IN I R N B-CONCAVITY IN THE RADIAL GEOMETRY IN I R N 7 10 INTERFACE OPERATORS AND EQUATIONS, UNIQUENESS 7 11 APPLICATIONS TO VARIOUS NONLINEAR MODELS WITH EXTINCTION AND BLOW-UP SINGULARITIES IN I R AND I R N 8 GEOMETRIC THEORY OF GENERALIZED FREE-BOUNDARY PROBLEMS NON- MAXIMAL SOLUTIONS 8 1 INTRODUCTION: ONE-PHASE FREE-BOUNDARY STEFAN AND FLORIN PROB- LEMS 8 2 CLASSI?CATION OF FREE-BOUNDARY PROBLEMS FOR THE HEAT EQUATION 8 3 CLASSI?CATION OF FREE-BOUNDARY PROBLEMS FOR THE QUADRATIC POROUS MEDIUM EQUATION 8 4 ON GENERAL ONE-PHASE FREE-BOUNDARY PROBLEMS 8 5 HIGHER-ORDER FREE-BOUNDARY PROBLEMS FOR THE POROUS MEDIUM EQUATION WITH ABSORPTION 8 6 HIGHER-ORDER FREE-BOUNDARY PROBLEMS FOR THE DUAL POROUS MEDIUM EQUATION WITH SINGULAR ABSORPTION 8 7 ON GENERALIZED TWO-PHASE FREE-BOUNDARY PROBLEMS 9 REGULARITY OF SOLUTIONS OF CHANGING SIGN 9 1 INTRODUCTION: SOLUTIONS OF CHANGING SIGN AND THE PHENOMENON OF SINGULAR PROPAGATION 9 2 APPLICATION: THE SIGN POROUS MEDIUM EQUATION WITH SINGULAR ABSORPTION 9 3 ON PROPAGATION OF SINGULARITY CURVES 10 DISCONTINUOUS LIMIT SEMIGROUPS FOR THE SINGULAR ZHANG EQUATION 10 1 INTRODUCTION: NEW NONLINEAR MODELS WITH DISCONTINUOUS SEMI- GROUPS 10 2 EXISTENCE AND NONEXISTENCE RESULTS FOR THE HYDRODYNAMIC VERSION 10 3 A GENERALIZED MODEL WITH COMPLETE AND INCOMPLETE SINGULARITIES 10 4 COMPLETE SINGULARITY IN THE CAUCHY PROBLEM FOR THE ZHANG EQUATION 10 5 INSTANTANEOUS SHAPE SIMPLI?CATION IN THE DIRICHLET PROBLEM FOR THE ZHANG EQUATION IN ONE DIMENSION 10 6 DISCONTINUOUS LIMIT SEMIGROUPS AND OPERATOR OF SHAPE SIMPLI?CA- TION FOR SINGULAR EQUATIONS IN I R N 11 FURTHER EXAMPLES OF DISCONTINUOUS AND CONTINUOUS LIMIT SEMI- GROUPS 11 1 EQUATIONS IN I R N WITH BLOW-UP AND SPATIAL SINGULARITIES: DISCON- TINUOUS SEMIGROUPS AND SINGULAR INITIAL LAYERS 11 2 WHEN DO SINGULAR INTERFACES NOT MOVE? REMARKS AND COMMENTS ON THE LITERATURE ON LIMIT MINIMAL SEMIGROUPS FOR SINGULAR INITIAL DATA REFERENCES LIST OF FREQUENTLY USED ABBREVIATIONS CHAPTER STURM THEOREMS FOR LINEAR PARABOLIC EQUATIONS AND INTERSECTION COMPARISON B-EQUATIONS IN THIS CHAPTER WE STATE AND PROVE TWO FUNDAMENTAL STURM THEOREMS ON ZERO SETS FOR LINEAR PARABOLIC EQUATIONS TRYING, WHENEVER POSSIBLE, TO KEEP STURM’S ORIGINAL NOTATIONS AND CALCULATIONS WE NEXT DESCRIBE NECESSARY EXTENSIONS OF STURM’S THE- OREMS TO WIDER CLASSES OF EQUATIONS AND BOUNDARY-VALUE PROBLEMS FURTHERMORE, WE PRESENT A GEOMETRIC INTERPRETATION OF ?OWS SATISFYING STURMIAN INTERSECTION PRINCIPLES AND INTRODUCE A CLASS OF THE SO-CALLED B-EQUATIONS PRESERVING STURMIAN INTERSECTION PROPERTIES WITH RESPECT TO A GIVEN COMPLETE FUNCTIONAL SUBSET B PLAY- ING THE ROLE OF EXACT CHARACTERISTIC SOLUTIONS THE LAST SECTION CONTAINS A SURVEY ON APPLICATIONS OF STURM THEOREMS AND IDEAS IN VARIOUS AREAS OF THE ODES AND PDES, AS WELL AS IN GEOMETRIC AND OTHER PROB- LEMS 1 1 FIRST STURM THEOREM: NONINCREASE OF THE NUMBER OF SIGN CHANGES LET D AND J BE OPEN BOUNDED INTERVALS IN I R CONSIDER IN S = D J A LINEAR PARABOLIC EQUATION U T = A(X; T)U XX + B(X; T)U X + C(X; T)U: (1 1) GIVEN A CONSTANT J, LET @ S BE THE PARABOLIC BOUNDARY OF THE DOMAIN S = S \ FT < G, I E , THE LATERAL BOUNDARY AND THE BOTTOM OF THE BOUNDARY OF S GIVEN A SOLUTION U = U(X; T) DE?NED ON S , THE POSITIVE AND NEGATIVE SETS OF U ARE DE?NED AS FOLLOWS: U + = F(X; T) S : U(X; T) > 0G; U = F(X; T) S : U(X; T) < 0G: (1 2) A COMPONENT OF U + (OR U ) IS A MAXIMAL OPEN CONNECTED SUBSET OF U + (OR U ) GIVEN A T J, THE NUMBER (?NITE OR IN?NITE) OF COMPONENTS OF FX D : U(X; T) 6= 0G MINUS ONE IS CALLED THE NUMBER OF SIGN CHANGES OF U(X; T) AND IS DENOTED BY Z (T; U) ALTERNATIVELY, Z (T; U) = SUP FK G; WHERE THE SUBSET FK G CONSISTS OF INTEGER K 0 SUCH THAT THERE EXIST K + 1 POINTS FROM D, X < X ; T) < FOR ALL J = 0; 1; :::; K 1: THEOREM 1 1 (FIRST STURM THEOREM: SIGN CHANGES) LET A, B, C BE CONTINUOUS, GEOMETRIC STURMIAN THEORY BOUNDED AND A > 0 IN S LET U(X; T) BE A SOLUTION OF (1:1) IN S THAT IS CONTINUOUS ON S (I) SUPPOSE THAT ON @ S THERE ARE PRECISELY N (RESPECTIVELY M) DISJOINT INTERVALS WHERE U IS POSITIVE (RESP NEGATIVE ) THEN U + (RESP U ) HAS AT MOST N (RESP M) COMPONENTS IN S AND THE CLOSURE OF EACH COMPONENT MUST INTERSECT @ S IN AT LEAST ONE INTERVAL (II) THE NUMBER OF SIGN CHANGES Z ( ; U) OF U(X; ) ON D IS NOT GREATER THAN THE NUMBER OF SIGN CHANGES OF U ON @ S PROOF THE PROOF IS BASED ON THE MAXIMUM PRINCIPLE (THE MP FOR SHORT) (I) LET I @ S BE THE MAXIMAL INTERVAL WHERE U > 0 SUPPOSE THAT TWO OPEN CONNECTED SUBSETS F ; F U + INTERSECT @ S IN DISJOINT OPEN INTERVALS I ; I I SINCE U IS CONTINUOUS IN S , THERE EXISTS AN OPEN NEIGHBOURHOOD G U + WHOSE CLOSURE IN S CONTAINS I THEN G MUST CONTAIN POINTS OF BOTH F AND F 2, SO THAT THESE MUST BELONG TO THE SAME OPEN COMPONENT OF U + THUS AT MOST ONE COMPONENT OF U + INTERSECTS EACH OF THE N OPEN INTERVALS ON @ S WHERE U > 0 THE SAME RESULT HOLDS FOR THE COMPONENTS OF U THEREFORE, IT SUF?CES TO SHOW THAT EVERY COMPONENT OF U + (OR U ) INTERSECTS @ S IN AN INTERVAL WE CAN ASSUME THAT C 0 IN S OTHERWISE, WE SET U = E T V (THEN U STAY THE SAME FOR V), WHERE V THEN SOLVES EQUATION (1 1) WITH THE LAST COEF?CIENT C ON THE RIGHT-HAND SIDE REPLACED BY C AND WE CAN CHOOSE THE CONSTANT SUP C LET F U + BE A COMPONENT IN S SINCE U IS CONTINUOUS, IT MUST ATTAIN A POSITIVE MAXIMUM ON F THEN C 0 IMPLIES U T AU XX + BU X IN F ; AND, BY CONTINUITY, U = 0 AT ANY BOUNDARY POINT OF F THAT IS INTERIOR TO S BY THE MP, U CANNOT ATTAIN ITS MAXIMUM AT AN INTERIOR POINT OF F OR ON THE LINE FT = G HENCE, F MUST HAVE A BOUNDARY POINT Q @ S SUCH THAT U(Q) > 0 AND BY CONTINUITY U IS POSITIVE IN AN INTERVAL OF @ S ABOUT Q (II) IS A STRAIGHTFORWARD CONSEQUENCE OF (I) THE STURM THEOREM IS TRUE FOR WIDER CLASSES OF LINEAR PARABOLIC EQUATIONS, WHICH ARE REGULAR ENOUGH TO HAVE THE STRONG MP APPLIED IN PARTICULAR, AN IM- PORTANT EXAMPLE IS THE RADIAL PARABOLIC EQUATION IN I R N WITH CONTINUOUS BOUNDED COEF?CIENTS AND A > 0, U T = A(R ; T)U + B(R ; T)U R + C(R ; T)U; (1 3) WHERE R = JXJ 0 DENOTES THE SINGLE SPATIAL VARIABLE THE RADIAL LAPLACE OPERATOR U = U R R + N R U R IS FORMALLY SINGULAR AT R = 0 BEARING IN MIND THAT WE CONSIDER SMOOTH BOUNDED SOLUTIONS SATISFYING THE SYMMETRY CONDITION AT THE ORIGIN, U R (0; T) = FOR T J ; 1 STURM THEOREMS THE MP APPLIES TO EQUATION (1 3) IN S = D J, WHERE D = FR < R G IS A BALL IN I R N, AND THE ?RST STURM THEOREM HOLDS 1 2 SECOND STURM THEOREM: EVOLUTION FORMATION AND COLLAPSE OF MULTIPLE ZEROS RESULTS IN THE CLASS OF C 1 FUNCTIONS BEFORE STATING THE RESULT, WE NOTE THAT PARABOLIC EQUATIONS WITH ANALYTIC COEF?- CIENTS ADMIT ANALYTIC SOLUTIONS AND THEN ANY ZERO OF U(X; T) 0 HAS ?NITE MULTI- PLICITY THEOREM 1 2 (SECOND STURM THEOREM: MULTIPLE ZEROS) LET O = (0; 0) S AND U (S ) \ C (S ) BE A SOLUTION OF EQUATION (1:1), WITH C 1-COEF?CIENTS A; B, C, WHERE A > 0 IN S ASSUME THAT U(X; T) DOES NOT HAVE SIGN CHANGES ON THE LATERAL BOUNDARY OF S, AND LET U(X; 0) HAVE A ZERO OF MULTIPLICITY M 2 AT THE ORIGIN X = 0, I E , D K X U(0; 0) = FOR K = 0; 1; :::; M 1 AND D M X U(0; 0) = AM! 6= 0: (1 4) THEN Z (T; U) DROPS AT T = 0 AND, FOR ANY T < T 2 NEAR T = 0, THERE HOLDS Z (T ; U) Z (T ): (1 6) BY TAYLOR’S EXPANSION IN T, U(X; T) = U(X; 0) + U T (X; 0)T + 2! U TT (X; 0)T , IF M IS ODD LET US ESTIMATE THE COEF?CIENTS IN (1 7) DENOTE D J = M! (M 2J )! ; J = 0; 1; :::; N: IT FOLLOWS FROM THE PARABOLIC EQUATION (1 1) AND (1 6) THAT U T (X; 0) HAS THE FOLLOW- ING TAYLOR EXPANSION: U T (X; 0) = A(X; 0)U XX (X; 0) + B(X; 0)U X (X; 0) + C(X; 0)U(X; 0) = A ); WHERE A = A(0; 0) AND A(X; 0) = A + O (X): DIFFERENTIATING THE EQUATION AND USING EXPANSION (1 6) AGAIN WE OBTAIN, KEEPING THE LEADING TERMS ONLY, U TT (X; 0) = AU TXX + ::: = A T A U XX (0; 0) + ::: = A N GEOMETRIC STURMIAN THEORY TAYLOR’S EXPANSION IN BOTH INDEPENDENT VARIABLES X AND T IS NOW U(X; T) = A(X M + A T + 2! A N! A N ) (I) BACKWARD CONTINUATION CONSIDER THE BEHAVIOUR FOR T THE DIMENSIONAL STRUCTURE OF THE RIGHT-HAND SIDE OF (1 8) SUGGESTS REWRITING THIS EXPANSION IN TERMS OF THE RESCALED STURM BACKWARD CONTINUATION VARIABLE Z = X= P A (T) FOR T < 0: (1 9) SUBSTITUTING X = Z P A (T) YIELDS A (1) J D J J ! Z M2J : THE FUNCTION P M (Z ) IS INDEED THE MTH ORDER HERMITE POLYNOMIAL H M (Z ) (UP TO A CONSTANT MULTIPLIER, WHICH WE OMIT IN WHAT FOLLOWS) EACH ORTHOGONAL POLYNOMIAL H M (Z ) HAS EXACTLY M SIMPLE ZEROS FZ I ; I = 1; :::; MG, WITH H M (Z I ) 6= 0, I E , THE INTERSECTIONS WITH ZERO ARE ALWAYS TRANSVERSAL STURM PROVED THIS SEPARATELY ON P 426 THIS IS NOWADAYS A WELL-KNOWN FACT IN THE CLASSICAL THEORY OF ORTHOGONAL POLYNOMIALS; SEE G SZEG¨O’S BOOK [328, CHAPTER 6] A SIMILAR EXPANSION FOR THE DERIVATIVE U X (X; T) SHOWS THAT EXPANSION (1 10) CAN BE DIFFERENTIATED IN X, YIELDING THE DERIVATIVE P M (Z ) ON THE RIGHT-HAND SIDE IT FOLLOWS FROM THE EXPANSIONS OF U(X; T) AND U X (X; T) NEAR THE MULTIPLE ZERO THAT, FOR ANY T , THE SOLUTION U(X; T) HAS M SIMPLE ZEROS FX I (T); I = 1; :::; MG SUCH THAT U X (X I (T); T) 6= 0, WITH THE FOLLOWING ASYMPTOTIC BEHAVIOUR: X I (T) = Z I P T + O (T) ! AS T ! ; SO EXACTLY M SMOOTH ZERO CURVES INTERSECT EACH OTHER AT THE ORIGIN (0; 0) FOR INSTANCE, ON FIGURE 1 1, U(X; T) FOR X 0, T HAS THE STRUCTURE OF THE ?FTH HERMITE POLYNOMIAL AS IN (1 10) (II) FORWARD CONTINUATION FOLLOWING STURM’S ANALYSIS, WE NOW CONSIDER THE BE- HAVIOUR OF THE SOLUTION U(X; T) AS T ! + INTRODUCING THE HEAT KERNEL RESCALED VARIABLE OF THE FORWARD CONTINUATION Z = X= P A T FOR T > 0; (1 11) INSTEAD OF (1 10) WE OBTAIN ANOTHER POLYNOMIAL ON THE RIGHT-HAND SIDE A D J J ! Z M2J : THE MTH ORDER POLYNOMIAL Q M (Z ) HAS POSITIVE COEF?CIENTS IF M AND N ARE ODD, THEN Q M (Z ) IS STRICTLY INCREASING AND Q M (0) = 0 IF M AND N ARE EVEN, THIS POLYNOMIAL HAS A SINGLE POSITIVE MINIMUM AT Z = 0 THEREFORE, (1 12) IMPLIES 1 STURM THEOREMS U(X; 1) U(X; ) U(X; 0) X U FIGURE 1 1 FORMATION AS T ! OF A MULTIPLE ZERO (A HIGHER-ORDER IN?ECTION) OF U(X; 0) FROM ?VE TRANSVERSAL ZEROS OF U(X; T) FOR T < 0 THAT, FOR SMALL T > 0 ON COMPACT SUBSETS FJXJ C P TG WITH ANY C > 0, THE SOLUTION U(X; T) HAS A UNIQUE SIMPLE ZERO ~ X (T) = O (T) IF M IS ODD, AND NO ZEROS IF M IS EVEN THIS IS STURM’S ANALYSIS ON P 423 IN ORDER TO COMPLETE THE PROOF, IT SUF?CES TO OBSERVE THAT, IF M IS EVEN AND, SAY, A > 0, THERE EXISTS A SMALL INTERVAL ("; ") SUCH THAT U(X; T) BECOMES STRICTLY POSITIVE ON ("; ") FOR ALL SMALL T > 0 THIS FOLLOWS FROM THE STRONG MP HENCE, AT LEAST M ZERO CURVES DISAPPEAR AT (0; 0) IF M IS ODD AND A > 0, THEN APPLYING THEOREM 1 1 TO THE DOMAIN S = ("; ") (0; ") WE HAVE THAT ON ("; ") FOR T > THERE EXISTS A UNIQUE CONTINUOUS CURVE OF SIMPLE ZEROS ~ X (T) STARTING FROM (0; 0) IN THIS CASE AT LEAST M 1 ZERO CURVES DISAPPEAR AT THE ORIGIN AS T ! SINCE, BY THE ASSUMPTION, NEW ZEROS OF U(X; T) DO NOT APPEAR ON THE LATERAL BOUNDARY OF S AND CANNOT OCCUR IN S BY THEOREM 1 1, THIS COMPLETES THE PROOF OF (1 5) SUCH A COMPLETE ANALYSIS OF THE EVOLUTION OF MULTIPLE ZEROS IN ONE DIMENSION APPLIES TO MORE GENERAL PARABOLIC EQUATIONS IN PARTICULAR, IN THE N-DIMENSIONAL GEOMETRY SIMILAR RESULTS ARE TRUE FOR RADIAL SOLUTIONS U = U(R ; T) OF PARABOLIC EQUA- TIONS (1 3) WITH C 1 (OR ANALYTIC) COEF?CIENTS, WHERE THE LAGUERRE POLYNOMIALS OCCUR IN ASYMPTOTIC EXPANSIONS INSTEAD OF THE HERMITE ONES; SEE REMARKS COMMENTS ON STURM’S EVOLUTION ANALYSIS OF ZEROS STURM’S PROOF CONSISTING OF PARTS (I) AND (II) EXHIBITS TYPICAL FEATURES OF A DETAILED ASYMPTOTIC EVOLUTION ANALYSIS FOR GENERAL LINEAR UNIFORMLY PARABOLIC EQUATIONS, WHICH CAN BE CLASSI?ED AS FOLLOWS: (I) A ?NITE-TIME CREATION OF A MULTIPLE ZERO AS T ! AS A SINGULARITY FORMATION (SINGLE POINT BLOW-UP SELF-FOCUSING OF ZERO CURVES), AND (II) DISAPPEARANCE OF MULTIPLE ZEROS AT T = +, I E , INSTANTANEOUS COLLAPSE OF GEOMETRIC STURMIAN THEORY SINGULARITY AND A UNIQUE CONTINUATION (EXTENSION) OF THE SOLUTION BEYOND SINGULAR- ITY DISCUSSING THIS PART OF STURM’S ANALYSIS, FOR CONVENIENCE, WE PRESENT THE ABOVE RESULTS SEPARATELY AS FOLLOWS, STATING THE PRECISE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS COROLLARY 1 3 UNDER THE ASSUMPTIONS OF THEOREM 1:2, THERE HOLDS: (I) AS T ! , THE RESCALED SOLUTION CONVERGES TO THE MTH ORDER HERMITE POLY- NOMIAL, A U(X; T) ! H M (Z ) (1 13) UNIFORMLY ON COMPACT SUBSETS IN THE VARIABLE Z GIVEN BY (1:9) (II) AS T ! +, THE RESCALED SOLUTION CONVERGES TO THE NON-OSCILLATING MTH ORDER POLYNOMIAL, A U(X; T) ! Q M (Z ) (1 14) UNIFORMLY ON COMPACT SUBSETS IN Z GIVEN BY (1:11) PHENOMENA OF SINGULARITY BLOW-UP FORMATION, COLLAPSE AND SOLUTION EXTENSIONS BEYOND SINGULARITIES ARE IMPORTANT SUBJECTS OF THE GENERAL PDE THEORY IN APPLICA- TIONS TO SEMILINEAR AND QUASILINEAR PARABOLIC EQUATIONS OF REACTION-DIFFUSION TYPE, THE PERTURBATION TECHNIQUES FOR IN?NITE-DIMENSIONAL DYNAMICAL SYSTEMS OFTEN PLAY A KEY ROLE WE NOW BRIE?Y COMMENT ON STURM’S ANALYSIS USING THE STANDARD PER- TURBATION THEORY OF LINEAR OPERATORS REVEALING ITS OTHER ASPECTS THAT ARE IMPORTANT FOR FURTHER EXTENSIONS (I) FORMATION OF MULTIPLE ZEROS: BACKWARD CONTINUATION USING STURM’S BACK- WARD RESCALED VARIABLE (1 9), WE INTRODUCE THE RESCALED SOLUTION U(X; T) = (Z ; ); Z = Z = P A HENCE, IN THIS VARIABLE, STURM’S MULTIPLE ZEROS FORMATION ANALYSIS AS T ! REDUCES TO THE IN?NITE TIME, AS ! 1, ASYMPTOTIC BEHAVIOUR PROBLEM FOR THE RESCALED SOLUTION (Z ; ) SUBSTITUTING (1 15) INTO EQUATION (1 1) YIELDS THE FOLLOW- ING RESCALED EQUATION: = B + C( ) ; (1 16) WHERE B IS THE LINEAR SYMMETRIC OPERATOR B = D Z D DZ D DZ D DZ WITH WEIGHT (Z ) = E Z (1 17) THE NON-AUTONOMOUS PERTURBATION IN (1 16) HAS THE FORM C( ) = AA B P A Z + E C ; WHERE, FOR THE REGULAR COEF?CIENT A(X; T) ON COMPACT SUBSETS IN Z, (A(X; T) A (A(Z [A ; T) A H Z Z O (1) + B P A C I 1 STURM THEOREMS IS EXPONENTIALLY SMALL AS ! 1 HENCE, THE EQUATION (1 16) IS AN EXPONENTIAL SMALL PERTURBATION OF THE AUTONOMOUS EQUATION = B : (1 18) THE SYMMETRIC OPERATOR B IS THE CLASSICAL SINGULAR STURM-LIOUVILLE OPERATOR, [328], [57] IT IS KNOWN TO BE SELF-ADJOINT IN THE WEIGHTED SPACE L (I R ) WITH THE INNER PRODUCT (V ; W ) = Z (Z )V (Z )W (Z ) DZ : ITS DOMAIN D (B) = H (I R ) IS A HILBERT SPACE OF FUNCTIONS V SATISFYING V ; V ; V LO C (I R ) WITH THE INNER PRODUCT HV ; W I = (V ; W ) + (V ; W ) + (V ) AND THE INDUCED NORM KV K = HV ; V I B HAS COMPACT RESOLVENT AND ITS SPECTRUM CONSISTS OF EIGENVALUES (B) = F K = K ; K = 0; 1; :::G: THE EIGENFUNCTIONS ARE ORTHONORMAL HERMITE POLYNOMIALS ~ H K (Z ) = C K H K (Z ), WHERE C K ARE NORMALIZATION CONSTANTS, AND THE EIGENFUNCTION SUBSET FH K G IS COM- PLETE AND CLOSED IN L (I R ) THESE ARE CLASSICAL RESULTS OF THE THEORY OF LINEAR SELF-ADJOINT OPERATORS IN HILBERT SPACES; SEE ?RST CHAPTERS OF M S BIRMAN AND M Z SOLOMJAK’S BOOK [57] USING EIGENFUNCTION EXPANSIONS AND SEMIGROUP ES- TIMATES (SEE REMARKS) YIELDS THAT THE EXPONENTIALLY PERTURBED DYNAMICAL SYSTEM (1 16) ON L (I R N ) ADMITS A DISCRETE SUBSET OF DIFFERENT ASYMPTOTIC PATTERNS THEY COINCIDE WITH THOSE FOR THE UNPERTURBED EQUATION (1 18) AND EXHIBIT THE ASYMPTOTIC BEHAVIOUR ON TANGENT STABLE ( M < 0) EIGENSPACES OF B THIS IMPLIES (1 13) AND AS ! 1, UNIFORMLY ON COMPACT SUBSETS IN Z, THERE HOLDS (Z ; ) = C E M H M (Z ) + O(E M ) WITH A CONSTANT C 6= 0: (1 19) (II) COLLAPSE OF MULTIPLE ZEROS ON SPATIAL STRUCTURE OF ADJOINT POLYNOMIALS: FORWARD CONTINUATION FOR T > 0, THE FORWARD RESCALED VARIABLE (1 11) IS USED SIMILARLY, THE RESCALED FUNCTION U(X; T) = G (Z ; S), WHERE THE TIME VARIABLE IS NOW S = LOG T ! 1 AS T ! +, SOLVES THE EXPONENTIALLY PERTURBED EQUATION G S = (B I )G + C(S)G AS S ! 1; (1 20) WHERE B IS THE ADJOINT DIFFERENTIAL OPERATOR ( I DENOTES THE IDENTITY) B = D Z D DZ + D DZ ( D DZ ) + I WITH THE WEIGHT FUNCTION (Z ) = E Z AS S ! IS EXPONENTIALLY SMALL ON COMPACT SUBSETS FOR SMOOTH SOLUTIONS B IS ALSO A CLAS- SICAL SINGULAR STURM-LIOUVILLE OPERATOR AND IS SELF-ADJOINT IN L (I R ), D (B ) = H (I R ), WITH THE SAME POINT SPECTRUM (B ) = (B) AND A COMPLETE AND CLOSED SUBSET OF ORTHONORMAL EIGENFUNCTIONS, AND SO ON BUT AS OFTEN HAPPENS IN THE EVOLUTION SINGULARITY THEORY, UNLIKE THE PHENOMENON GEOMETRIC STURMIAN THEORY OF THE EVOLUTION BLOW-UP FORMATION OF MULTIPLE ZEROS, IN THE COMPLEMENTARY ASYMPTOTIC ANALYSIS BEYOND BLOW-UP, AS T ! + ( S ! 1), SPECTRAL PROPER- TIES AND EIGENFUNCTIONS OF B PLAY NO ROLE THE LIMIT T ! + CORRESPONDS TO THE COLLAPSE OF THE INITIAL “SINGULARITY” CREATED BY THE PRECEDING SINGULARITY FORMATION AS T ! OBVIOUSLY, FOR SUCH PARABOLIC EQUATIONS, THE BEHAVIOUR OF U(X; T) AS T ! < JXJ 1G; SEE REFERENCES IN REMARKS) THAT, AS T ! + O(1)); (1 21) WHERE THE RIGHT-HAND SIDE WILL BE DENOTED BY U(X; 0) THE SOLUTION G (Z ; S) OF THE RESCALED EQUATION (1 20) WITH INITIAL DATA CALCULATED IN (1 21) HAS THE EXPANSION G (Z ; S) = ~ C E M S Q M (Z ) + :::; ~ C 6= 0; (1 22) WHERE Q M IS THE POLYNOMIAL SOLUTION OF THE LINEAR EQUATION (B I )Q M = M Q M : THIS GIVES THE LINEAR PROBLEM FOR THE “ADJOINT” POLYNOMIALS FQ M G NOTICE THAT THESE HAVE NOTHING TO DO WITH THE ORTHOGONAL SUBSET OF EIGENFUNCTIONS FH M (Z )E Z G OF THE ADJOINT OPERATOR B MOREOVER Q M X M + ::: AS T ! + : BY MATCHING WITH (1 21), THIS UNIQUELY DETERMINES THE CONSTANT ~ C = C IN (1 22) AND COMPLETES THE ASYMPTOTIC ANALYSIS OF BOTH THE BACKWARD AND FORWARD EVOLUTION OF MULTIPLE ZEROS RESULTS IN CLASSES OF ?NITE REGULARITY FIX A ?NITE T > 0 AND SET J = (0; T ) AS WE MENTIONED, IF U(X; T) 0 IS A SOLUTION ANALYTIC IN X OF THE LINEAR PARABOLIC EQUATION (1 1) THAT HAS ANALYTIC COEF?CIENTS A; B; C, THEN, FOR ANY T (0; T ), ALL ZEROS OF U(X; T) ARE ISOLATED AND HENCE THE NUMBER OF SIGN CHANGES Z (T; U) IS ?NITE EVEN IF Z (0; U) = 1 IT TURNS OUT THAT A SIMILAR CONCLUSION REMAINS VALID IN CLASSES OF EQUATIONS AND SOLUTIONS OF ?NITE REGULARITY WE PRESENT HERE THE TWO MOST GENERAL RESULTS BY S ANGENENT [11]; MORE REFERENCES ARE GIVEN IN REMARKS WE BEGIN WITH INITIAL-BOUNDARY VALUE PROBLEMS THEOREM 1 4 LET U BE A BOUNDED SOLUTION OF (1:1) IN S = D (0; T ), WHICH DOES NOT CHANGE SIGN ON THE LATERAL BOUNDARY OF S ASSUME THAT THE COEF?CIENTS A, B AND C OF THE EQUATION ARE SUCH THAT A; A ; A X ; A XX ; B; B T ; B X ; C (S ): (1 23) 1 STURM THEOREMS THEN THE NUMBER OF SIGN CHANGES OF U(; T) SATIS?ES: (I) Z (T; U) IS ?NITE AND NONINCREASING ON (0; T ) (II) IF X = X D IS A MULTIPLE ZERO OF U(X; T ) FOR SOME T (0; T ), THEN, FOR ALL < T < T , THE STRICT INEQUALITY Z (T ; U) > Z (T ; U) HOLDS, SO Z (T; U) IS STRICTLY DECREASING AT T = T 0 AS A CONSEQUENCE, ANY GLOBAL SOLUTION U(X; T) DE?NED IN S = D I R + HAS ONLY SIMPLE ZEROS FOR ALL T 1 A SIMILAR RESULT IS VALID FOR PARABOLIC EQUATIONS IN UNBOUNDED DOMAINS IF WE RESTRICT THE ANALYSIS TO CLASSES OF FUNCTIONS WITH A ?XED GROWTH AT IN?NITY, SIMILAR TO TIKHONOV’S CLASSES OF UNIQUENESS LET D = I R, AND CONSIDER THE FOLLOWING LINEAR PARABOLIC EQUATION: U T = U XX + Q (X; T)U IN S = I R (0; T ): (1 24) THEOREM 1 5 LET Q (S ); (1 25) AND LET U(X; T) BE A SOLUTION OF (1:24) IN THE CLASS FJU(X; T)J AE B X 2 IN S G FOR SOME POSITIVE CONSTANTS A AND B THEN: (I) FOR EACH T (0; T ), THE ZERO SET OF THE SOLUTION FX I R : U(X; T) = 0G IS A DISCRETE SUBSET OF I R THE FOLLOWING CONCLUSION IS A CONSEQUENCE (II) IF X = 1 ARE NOT ACCUMULATION POINTS OF ZEROS OF U(X; 0), THEN STATEMENTS (I) AND (II) OF THEOREM 1:4 HOLD THIS THEOREM IS TRUE FOR MORE GENERAL EQUATIONS SUCH AS (1 1) IN UNBOUNDED DO- MAINS IN SUITABLE CLASSES OF UNIQUENESS INDEED, EQUATION (1 1) CAN BE REDUCED TO (1 24) BY THE LIOUVILLE TRANSFORMATION USING THE NEW SPATIAL COORDINATE Y = R X DS P A(S;T) ; WE HAVE THAT U = U(Y ; T) SATIS?ES THE EQUATION U T = U Y Y + ~ B (Y ; T)U Y + ~ C(Y ; T)U: SUBSTITUTING V (Y ; T) = EXP N R Y ~ B (S; T) DS O U(Y ; T); YIELDS EQUATION (1 24) FOR V (Y ; T) WITH A POTENTIAL ~ Q (Y ; T) CHECKING NECESSARY PROPERTIES OF ~ Q (Y ; T), WE DEDUCE THAT STURM’S RESULTS ARE VALID IN THE CORRESPONDING UNIQUENESS CLASSES 1 3 FIRST ASPECTS OF INTERSECTION COMPARISON OF SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS MAIN RESULTS OF THE G-THEORY ARE PROVED BY INTERSECTION COMPARISON OF SOLUTIONS V = V (X; T) OF NONLINEAR PARABOLIC EQUATIONS OF THE GENERAL FORM V T = F(V ) F (X; T; V ; V X ; V XX ) IN S; (1 26) GEOMETRIC STURMIAN THEORY WHERE F (X; T; P; Q ; R ) IS NONDECREASING RELATIVE TO THE LAST ARGUMENT R I R (THE PARABOLICITY CONDITION), WITH A SUBSET B = FV (X; T)G OF SOME PARTICULAR SOLU- TIONS OF THE SAME EQUATIONS IN GENERAL, THESE ARE NOT CLASSICAL SOLUTIONS BUT SOME WEAK OR PROPER, MINIMAL OR MAXIMAL ONES CONSTRUCTED BY SMOOTH AND, IF POSSIBLE, MONOTONE APPROXIMATIONS (THIS IS EXPLAINED IN DETAIL IN CHAPTER 6) WE ALWAYS DEAL WITH SOLUTIONS V (X; T) AND V (X; T) THAT ARE CONTINUOUS IN X, SO THAT WE CAN DE?NE THE NUMBER OF INTERSECTIONS AS THE NUMBER OF SIGN CHANGES (SECTION 1 1) OF THE DIFFERENCE W (X; T) = V (X; T) V (X; T): IN T(T; V ) = Z (T; W ) FOR T [0; T ): IF F IN (1 26) IS SUF?CIENTLY SMOOTH, THE DIFFERENCE W SATIS?ES A LINEAR PARABOLIC EQUATION W T = AW XX + BW X + CW ; (1 27) WHERE THE COEF?CIENTS ARE GIVEN BY HADAMARD’S FORMULAE A = R )V XX ) D 0; B = R )V X ; V XX ) D ; C = R )V ; V X ; V XX ) D : IF THESE COEF?CIENTS SATISFY CONDITIONS OF STURM THEOREMS FOR LINEAR EQUATIONS, THEN THE NUMBER OF INTERSECTIONS IN T(T; V ) OF TWO SOLUTIONS V AND V OF THE PARABOLIC EQUATION (1 26) OBEYS THE SAME PROPERTIES AS THE NUMBER OF SIGN CHANGES OF THE DIFFERENCE W SATISFYING (1 27) AT THIS STAGE THE INTERSECTION COMPARISON APPROACH USES THE FACT THAT THE NON- INCREASE WITH TIME OR COLLAPSE OF MULTIPLE INTERSECTIONS HOLD WITH RESPECT TO ANY ?XED SOLUTION V (X; T) V N OF A MONOTONE SEQUENCE FV N G OF CLASSICAL SOLUTIONS OF REGULARIZED UNIFORMLY PARABOLIC EQUATIONS V T = F N (X; T; V ; V X ; V XX ); WHERE (F N ) R N > 0: THE SEQUENCE FF N G IS ASSUMED TO CONVERGE TO F MONOTONICALLY AND UNIFORMLY 1 STURM THEOREMS ON COMPACT SUBSETS BOUNDED AWAY FROM THE SINGULARITY THE INITIAL AND BOUNDARY CONDITIONS ARE ASSUMED TO BE REGULARIZED IN A SUITABLE MONOTONE MANNER THEN SUCH A PROPER SOLUTION IS UNIQUE AND DOES NOT DEPEND ON MONOTONE APPROXIMATIONS (CHAPTER 6) PERFORMING A SIMILAR CONSTRUCTION FOR ANY CHARACTERISTIC SOLUTION V V N ; AND ORGANIZING A SUITABLE COMPARISON OF EACH PAIR V N AND V N ON THE PARABOLIC BOUNDARY, WE OBTAIN STURM’S RESULTS ON SIGN CHANGES Z (T; W N ) OF THE DIFFERENCE W N = V N V N SATISFYING A LINEAR PARABOLIC EQUATION WITH SUF?CIENTLY SMOOTH COEF?CIENTS PASSING TO THE LIMIT N ! 1 YIELDS ESTIMATES ON THE NUMBER OF INTER- SECTIONS IN T(T; V ) = Z (T; W ) OF PROPER CONTINUOUS SOLUTIONS V AND ANY V B THEREFORE, THE INTERSECTION COMPARISON PRINCIPLES CAN BE APPLIED TO CLASSES OF “WEAK” (GENERALIZED) SOLUTIONS OF PARABOLIC PDES WITH, POSSIBLY, UNKNOWN PRECISE REGULARITY PROPERTIES 1 4 GEOMETRICALLY ORDERED ?OWS: TRANSVERSALITY AND CONCAVITY TECHNIQUES WE NOW FOCUS ON CRUCIAL IDEAS OF THE INTERSECTION REGULARITY ANALYSIS TECHNIQUES OF THE G-THEORY ARE BASED ON GENERAL CONCEPTS OF GEOMETRICALLY ORDERED EVOLU- TION OF CURVES AND ARE NOT NECESSARILY CONNECTED WITH NONLINEAR PARABOLIC PDES THE FUNCTIONS AND COEF?CIENTS INVOLVED MANIPULATIONS BELOW ARE ASSUMED TO BE SUF?CIENTLY SMOOTH REGULARITY IN ORDERED SMOOTH GEOMETRIC EVOLUTION LET B = FV G BE A ?NITE- PARAMETER SET OF GIVEN SUF?CIENTLY SMOOTH “CHARACTERISTIC” FUNCTIONS V = V (X; T) DE?NED IN S = I R I R +, WHERE X I R DENOTES A SINGLE SPACE VARIABLE AND T 0 IS THE EVOLUTION TIME VARIABLE WE SUPPOSE THAT THE FUNCTIONS FROM B SATISFY A HYPOTHESIS OF STRONG SEPARABILITY AND TRANSVERSALITY: GIVEN ANY PAIR OF DIFFERENT FUNCTIONS V ; ~ V B AND ANY ?XED MOMENT T 0, THERE HOLDS: (I) V (X; T) IN T(T; V ; ~ V ) SUP V ; ~ V 2B IN T(0; V ; ~ V ): LET US NOW DE?NE A SMOOTH GEOMETRICALLY ORDERED EVOLUTION BY MEANS OF THE CHARACTERISTIC FUNCTIONS B NAMELY, LET A SMOOTH FUNCTION V = V (X; T) B SAT- ISFYING THE STRONG SEPARABILITY PROPERTY, V (X; T) V (X; T) FOR ANY T 0, EVOLVE GEOMETRIC STURMIAN THEORY ON B WE ASSUME THAT THE STURMIAN PROPERTY (1 28) HOLDS FOR THE NUMBER OF INTER- SECTIONS IN T(T; V ; V ) BETWEEN V (X; T) AND ANY FUNCTION V (X; T) B IN FACT, THIS IS THE ONLY “EVOLUTION EQUATION” DESCRIBING THE CORRESPONDING GEOMETRICALLY ORDERED ?OW WE THEN ARRIVE AT THE FOLLOWING QUESTION CONCERNING THIS GEOMETRIC EVOLUTION: HOW MANY AND WHICH EVOLUTION PROPERTIES OF THE SET B COMPOSED OF PARTICULAR CHARACTERISTIC FUNCTIONS V CAN BE TRANSLATED TO ARBITRARY FUNCTIONS V? FIRST OF ALL, ONE CAN SEE THAT (1 28) ASSUMES THE COMPARISON PRINCIPLE (I) COMPARISON: A STANDARD ORDER-PRESERVING PROPERTY THE EVOLUTION ON B SATIS?ES THE COMPARISON THEOREM: FOR ANY FUNCTION V B SUCH THAT V (X; 0) V (X; 0), THERE HOLDS V (X; 0) V (X; 0) IN I R = ) V (X; T) V (X; T) IN S: (1 29) INDEED, STURMIAN PROPERTY (1 28) GIVES IN T(0; V ; V ) = 0, HENCE IN T (T; V ; V ) = FOR ALL T > 0 AND COMPARISON FOLLOWS BY SEPARABILITY A SIMILAR COMPARISON FROM BELOW IS ALSO TRUE AS USUAL IN THE PARABOLIC THEORY, COMPARISON GIVES L 1- ESTIMATES OF THE SOLUTIONS WE CAN OBTAIN MORE IF WE KNOW MORE ABOUT THE INTERNAL GEOMETRIC STRUCTURE OF THE SET B FURTHER REGULARITY OF THE GEOMETRIC EVOLUTION USES TWO GEOMETRIC PRINCIPLES OF TRANSVERSALITY AND TANGENCY WE BEGIN WITH AN EASY TRANSVERSALITY ANALYSIS FOR SIMPLICITY WE ASSUME THAT EACH V (X; T) B IS MONOTONE INCREASING WITH X AND IS UNBOUNDED AS JXJ ! (I E , V (X; T) ! 1 AS X ! 1), WHILE V (X; T) IS A UNIFORMLY BOUNDED FUNCTION WITH SMOOTH INITIAL DATA V (X; 0) SATISFYING JV (X; 0)J 1; JV X (X; 0)J 1 AND JV XX (X; 0)J IN I R : (1 30) THEN NEW INTERSECTIONS BETWEEN V (X; T) AND V (X; T) CANNOT APPEAR FOR JXJ 1 (II) TRANSVERSALITY: A ?RST-ORDER ESTIMATE GIVEN INITIAL DATA V (X; 0), WE CHOOSE A SUBSET B + B OF “STEEP” ENOUGH INCREASING WITH X FUNCTIONS INTERSECTING V (X; T) AT T = 0 EXACTLY ONCE AND TRANSVERSALLY FOR INSTANCE, IN VIEW OF THE SECOND BOUND IN (1 30), V B + 1 IF V X (X; 0) > 1 FOR ALL X I R THEN (1 28) IMPLIES THAT, FOR ANY V B + 1 , IN T(0; V ; V ) = ) IN T(T; V ; V ) 1 FOR T > 0: (1 31) ONE CAN SEE THAT THE TRANSVERSALITY OF ALL INTERSECTIONS GIVES AN ESTIMATE OF THE SPATIAL DERIVATIVE V X IN TERMS OF V, V X (X; T) V X (X; T) AT ANY INTERSECTION POINT OF V AND V B + ; (1 32) AS SHOWN ON FIGURE 1 2 WE ARGUE BY CONTRADICTION IF FOR SOME T > 0, V X (X; T) > V X (X