NON-LINEAR WAVES IN DISPERSIVE MEDIA
CONTENTS BLOWUP RATE FOR MASS CRITICAL ROTATIONAL NONLINEAR SCHR¨ODINGER EQUATIONS GRADIENT ESTIMATES FOR WEAK SOLUTIONS OF LINEAR ELLIPTIC SYSTEMS WITH SINGULAR-DEGENERATE COE?CIENTS VIRIAL ESTIMATES FOR HARD SPHERES THE ALMOST GLOBAL EXISTENCE TO CLASSICAL SOLUTION FOR A 3-D WAVE EQUATION OF NEMATIC LIQUID-CRYSTALS INSTABILITY OF SOLITONS–REVISITED, I: THE CRITICAL GENERALIZED KDV EQUATION INSTABILITY OF SOLITONS–REVISITED, II: THE SUPERCRITICAL ZAKHAROV-KUZNETSOV EQUATION STABILIZATION OF DISPERSION-GENERALIZED BENJAMIN-ONO UNIQUENESS OF STANDING-WAVES FOR A NON-LINEAR SCHR¨ODINGER EQUATION WITH THREE PURE-POWER COMBINATIONS IN DIMENSION ONE A REGULARITY UPGRADE OF PRESSURE ON LARGE FUTURE-GLOBAL-IN-TIME SOLUTIONS TO ENERGY-SUPERCRITICAL NONLINEAR WAVE EQUATION THE NONLINEAR SCHR¨ODINGER EQUATION WITH AN INVERSE-SQUARE POTENTIAL THE HARMONIC MAP HEAT ?OW ON CONIC MANIFOLDS ON THE GLOBAL REGULARITY ISSUE OF THE TWO-DIMENSIONAL MAGNETOHYDRODYNAMICS SYSTEM WITH MAGNETIC DI?USION WEAKER THAN A LAPLACIAN ORBITAL STABILITY OF STANDING WAVES FOR FRACTIONAL HARTREE EQUATION WITH UNBOUNDED POTENTIALS BLOWUP RATE FOR MASS CRITICAL ROTATIONAL NONLINEAR SCHR¨ODINGER EQUATIONS ABSTRACT WE CONSIDER THE BLOWUP RATE FOR BLOWUP SOLUTIONS TO L2-CRITICAL, FOCUSING NLS WITH A HARMONIC POTENTIAL AND A ROTATION TERM UNDER A SUITABLE SPECTRAL CONDITION WE PROVE THAT THERE HOLDS THE “LOG-LOG LAW” WHEN THE INI- TIAL DATA IS SLIGHTLY ABOVE THE GROUND STATE WE ALSO CONSTRUCT MINIMAL MASS BLOWUP SOLUTIONS NEAR THE GROUND STATE LEVEL WITH DISTINCT BLOWUP RATES 1 INTRODUCTION CONSIDER THE FOCUSING NONLINEAR SCHR¨ODINGER EQUATION (NLS) WITH AN ANGULAR MOMENTUM TERM ON R1+N: IUT = ??U + V U ?|U|P?1U + IA ?U U(0, X) = U0 ?H 1, (1) WHERE U = U(T, X): R × RN ?C DENOTES THE WAVE FUNCTION, V (X) = ?2|X|2, ? > 0 IS A TRAPPING HARMONIC POTENTIAL THAT CON?NES THE MOVEMENT OF PARTICLES, AND A(X) = MX, WITH M = ?M T BEING AN N × N REAL-VALUED SKEW-SYMMETRIC MATRIX THE LINEAR HAMILTONIAN HA,V := ?? + V + IA ?IS ESSENTIALLY SELF-ADJOINT IN L2, WHOSE EIGENVALUES ARE ASSOCIATED TO THE LANDAU LEVELS AS QUANTUM NUMBERS THE ANGULAR MOMENTUM OPERATOR LAU := IA ?U GENERATES THE ROTATION ETA ?U = U(ETMX) IN RN THE SPACE H 1 = H 1,2 DENOTES THE WEIGHTED SOBOLEV SPACE GIVEN BY: FOR R ?(1, ?), H 1,R(RN) := {F ?LR : ?F, XF ?LR} , WHICH IS ENDOWED WITH THE NORM |F|H 1,R = |?F|R + |XF|R + |F|R, HERE | |R := | |LR DENOTING THE LR-NORM WHEN N = 3, EQUATION (1) IS ALSO KNOWN AS GROSS-PITAEVSKII EQUATION WHICH MODELS ROTATING BOSE-EINSTEIN CONDENSATION (BEC) WITH ATTRACTIVE PARTICLE INTER- ACTIONS IN A DILUTE GASEOUS ULTRA-COLD SUPER?UID THE OPERATOR LA ARE USUALLY DENOTED BY ?? L, WHERE ? = (?1, ?2, ?3) ?R3 IS A GIVEN ANGULAR VELOCITY VECTOR AND L := ?IX ??, IN WHICH CASE M = ? ? ? ? KEY WORDS AND PHRASES BLOWUP RATE, HARMONIC POTENTIAL, ANGULAR MOMENTUM SUCH A SYSTEM DESCRIBING ROTATING PARTICLES IN A HARMONIC TRAP HAS ACQUIRED SIG- NI?CANCE IN CONNECTION WITH OPTICS AND ATOMIC PHYSICS IN THEORETICAL AND EXPERIMEN- TAL PHYSICS [1,3,4,13,18,25] MEANWHILE, IT DEMANDS RIGOROUS MATHEMATICAL ANAL- YSIS ON THE EVOLUTION AND DYNAMICS OF THE QUANTIZED ?OW FOR P ?(1, 1+4/(N?2)), THE LOCAL IN TIME EXISTENCE AND UNIQUENESS OF (1) HAS BEEN ESTABLISHED IN [2,6,14,15], SEE ALSO [8, 11, 28] FOR THE TREATMENT IN A GENERAL MAGNETIC SETTING THE PURPOSE OF THIS ARTICLE IS TO STUDY HOW THE ROTATION A?ECTS THE WAVE COLLAPSE AS WELL AS EN- ERGY CONCENTRATION UNDER A TRAPPING POTENTIAL IN PARTICULAR, WE WILL ADDRESS THE BLOWUP RATE FOR THE BLOWUP SOLUTION OF THE L2-CRITICAL FOCUSING EQUATION IN (1) WHERE P = 1 + 4/N LET H1 = {U ?L2 : ?U ?L2} BE THE USUAL SOBOLEV SPACE RECALL THAT THE STANDARD NLS READS I?T = ??? ?|?|P?1?, ?(0, X) = ?0 ?H1, (2) AND THE WELL-POSEDNESS PROBLEM FOR P ?(1, 1 + 4/(N ?2)] HAS BEEN STUDIED FOR A FEW DECADES AND IS QUITE WELL UNDERSTOOD IN THE EUCLIDEAN SPACE LET Q ?H1(RN) BE THE UNIQUE POSITIVE RADIAL FUNCTION THAT SATIS?ES ([17,26]) ?Q = ??Q ?Q1+ N (3) WHEN P = 1+ N AND |?0|2 = |Q|2, MERLE [19] WAS ABLE TO DETERMINE THE PRO?LE FOR ALL BLOWUP SOLUTIONS WITH MINIMAL MASS AT THE GROUND STATE LEVEL, WHICH ARE OBTAINED FROM PSEUDO-CONFORMAL TRANSFORM HENCE ALL BLOWUP SOLUTIONS HAVE BLOWUP RATE (T ?T)?1 IN THE MASS CRITICAL AND SUPERCRITICAL CASE P ?[1 + 4/N, ?), THE WAVE COLLAPSE DYNAMICS APPEARS VERY SUBTLE ISSUE FOR |U0|2 = |Q|2 WITHIN AN ARBITRARILY SMALL NEIGHBORHOOD OF Q, THERE ALWAYS EXIST ?0 AND ?0 SUCH THAT THE ?OW ?0 ?? BLOWUPS IN ?NITE TIME, AND, ?0 ?? EXISTS GLOBAL IN TIME AND SCATTERS AS T ??IN H1 WHEN P = 1+ N AND |?0|2 > |Q|2, BOURGAIN AND WANG [5] CONSTRUCTED SOLUTIONS OF POSITIVE ENERGY HAVING BLOWUP RATE (T ?T)?1 IN DIMENSIONS N = 1, 2, WHICH WAS LATER SHOWN UNSTABLE HOWEVER PERELMAN [23] GAVE THE ?RST RIGOROUS DEMONSTRATION OF THE EXISTENCE AND STABILITY FOR THE LOG-LOG SPEED FOR GENERIC BLOWUP SOLUTIONS IN 1D MORE RECENTLY, UNDER CERTAIN SPECTRAL CONDITION IN THE SPECTRAL PROPERTY (SECTION 3), MERLE AND RAPHA¨EL [21] PROVED THE SHARP BLOWUP RATE OF THE SOLUTIONS FOR (2), I E , THERE EXISTS A UNIVERSAL CONSTANT ??SUCH THAT IF ?0 ?B??WITH NEGATIVE ENERGY, THEN ?(T, X) IS A BLOWUP SOLUTION TO (2) WITH MAXIMAL INTERVAL OF EXISTENCE [0, T) SATISFYING THE LOG-LOG BLOWUP RATE AS T ?T |??(T)|2 ? LOG | LOG(T ?T)| T ?T , WHERE B? := ? ?H1 : |Q|2DX < |?|2DX < |Q|2DX + ? , SEE THEOREM 3 1 SUCH A BLOWUP RATE IS ALSO KNOWN TO BE STABLE IN H1 IN THE PRESENCE OF A ROTATIONAL TERM, WE WILL SHOW HOW TO PROVE SUCH A “LOG-LOG LAW” FOR THE NLS (1) OUR PROOF IS BASED ON A VIRIAL IDENTITY FOR (1), THE R-TRANSFORM (7) THAT MAPS SOLUTIONS OF (2) TO SOLUTIONS OF (1), AND AN APPLICATION OF THE ABOVE RESULT OF MERLE AND RAPHA¨EL’S THIS TREATMENT IS MOTIVATED BY [29], WHERE THE NON LINEAR WAVES IN DISPERSIVE MEDIA ANALOGOUS RESULT IS OBTAINED FOR THE CASE A = 0 AND V BEING A HARMONIC POTENTIAL OUR MAIN RESULT IS STATED AS FOLLOWS LET ??BE THE ABOVE-MENTIONED UNIVERSAL CONSTANT THEOREM 1 1 LET P = 1 + N IN (1), 1 ?N ?5 SUPPOSE U0 ?B??SATI?ES |?U0|2 ? N N + |U0|2+ T ?T LOG |LOG(T ?T)| = ? 2? , WHERE Q IS THE UNIQUE SOLUTION OF (3) 2 LOCAL WELLPOSEDNESS FOR (1) FOR ?0 ?H 1, THE LOCAL WELL-POSEDNESS OF (1) WAS PROVED FOR 1 ?P < 1 + N?2, SEE E G IN [11, 28] THE PROOF FOR THE LOCAL WELL-POSEDNESS RELIES ON LOCAL IN TIME DISPERSIVE ESTIMATES FOR U(T) = E?ITHA,V , THE FUNDAMENTAL SOLUTION ON [0, ?) (FOR SOME SMALL ? > 0) CONSTRUCTED IN [27] THE VECTORIAL FUNCTION A REPRESENTS A MAGNETIC POTENTIAL THAT INDUCES THE CORIOLIS E?ECT OR CENTRIFUGAL FORCE FOR THE SPINOR PARTICLES ALTERNATIVELY, THIS CAN ALSO BE DONE BY MEANS OF THE EXPLICIT FORMULA IN (12) FOR U(T) DE?NED IN (0, ?/2?) THIS FORMULA IS OBTAINED FROM THE SO-CALLED R-TRANSFORM IN (7), A TYPE OF PSEUDO-CONFORMAL TRANSFORM PROPOSITION 2 1 LET 1 + N ?P < 1 + N?2 SUPPOSE U0 ?H 1 (A) [BLOWUP ALTERNATIVE] (I) IF 1 ?P < 1 + (ENERGY) E(U) = |?U|2 + V |U|2 ? P + 1|U|P+1 + I¯UA ?U (ANGULAR MOMENTUM) ?A(U) = I¯UA ?U IN THE CRITICAL CASE P = 1+4/N, FROM [6] WE KNOW THAT |Q|2 IS THE SHARP THRESHOLD SUCH THAT: (A) IF |U0| < |Q|2, THEN (1) HAS A UNIQUE GLOBAL IN TIME SOLUTION (B) FOR ALL C ?|Q|2, THERE EXISTS U0 WITH |U0|2 = C SO THAT U IS A ?NITE TIME BLOWUP SOLUTION OF (1) NON LINEAR WAVES IN DISPERSIVE MEDIA AS WE MENTIONED IN THE INTRODUCTION SECTION, IF |U0| = |Q|2, FROM MERLE’S CHARAC- TERIZATION FOR THE BLOWUP PRO?LE OF (2), ALL SUCH BLOWUP SOLUTIONS HAVE THE BLOWUP RATE |?U(T)|2 ? C T ??T AS T ?T ?, SEE PROPOSITION 4 5 3 A SPECTRAL PROPERTY AND THE LOG-LOG LAW TO SHOW THE BLOWUP RATE FOR INITIAL DATA ABOVE THE GROUND STATE Q AS STATED IN THEOREM 1 1, WE NEED THE FOLLOWING SPECTRAL PROPERTY LET Y DENOTE THE SPATIAL VARIABLE IN RN SPECTRAL PROPERTY CONSIDER THE TWO SCHR¨ODINGER OPERATORS L1 := ?? + N + N ?1Y ?Q, L2 := ?? + |??|2DY + |?|2E?|Y|DY THE PROOF OF THE SPECTRAL PROPERTY IN ANY DIMENSION IS NOT COMPLETE IT HAS BEEN PROVED IN [20] FOR DIMENSION N = 1 BY USING THE EXPLICIT SOLUTION Q(X) = COSH2(2X) 4 TO (3) ONE CAN ALSO ?ND A COMPUTER ASSISTED PROOF OF THE SPECTRAL PROPERTY IN DIMENSIONS N = 2, 3, 4 IN [12] FOR DIMENSION 5 AND HIGHER SEE REMARK 3 2 THE SPECTRAL PROPERTY IS EQUIVALENT TO THE COERCIVITY FOR L1 AND L2 ON QUADRATIC FORMS, THE STUDY OF WHICH INVOLVING THE GROUND STATE SOLUTION Q NATURALLY APPEARS IN A PERTURBATION SETTING WHEN DEALING WITH STABILITY PROBLEM THESE TWO OPERATORS ARE RELATED TO THE LYAPOUNOV FUNCTIONALS L±, WHERE L+ = ?? + 1 ?(1 + N)Q N AND L?= ?? + 1 ?Q N , SEE [12] USING THE SPECTRAL PROPERTY, MERLE AND RAPHA¨EL OBTAINED THE FOLLOWING BLOWUP RATE FOR (2) IN THE ABSENCE OF POTENTIALS, SEE [12,21,22] THEOREM 3 1 LET P = 1 + N IN (2) LET 1 ?N ?5 THERE EXISTS A UNIVERSAL CONSTANT ??> 0 SUCH THAT THE FOLLOWING IS TRUE SUPPOSE ?0 ?B??SATIS?ES |??0|2 ? N N + |?0|2+ ? 2? (5) NON LINEAR WAVES IN DISPERSIVE MEDIA REMARK 3 2 FROM [12] WE KNOW THAT THE SPECTRAL PROPERTY IS TRUE IN DIMEN- SIONS N = 1, 2, 3, 4 IF N = 5, THEOREM 3 1 CONTINUES TO HOLD TRUE AS SOON AS THE SPECTRAL PROPERTY VERI?ES THE ORTHOGONALITY CONDITION WITH (?1, Q1) = 0 REPLACED BY (?1, |Y|2Q) = 0, WHICH IS NUMERICALLY VERI?ED IN [12] FOR THE ABOVE-MENTIONED REASON, THEOREM 3 1 REMAINS OPEN IN DIMENSIONS N ?6 REMARK 3 3 IT IS WELL-KNOWN THAT THE LOG-LOG LAW IS A GENERIC BEHAVIOR FOR THOSE BLOWUP SOLUTIONS IN THE THEOREM, WHOSE PROOF RELIES ON ALGEBRAIC CANCELLATIONS RELATED TO THE TOPOLOGICAL DEGENERACY OF THE LINEAR OPERATORS L1 AND L2 AROUND Q SUCH BLOWUP RATE IS STABLE IN THE SENSE THAT THE SET U0 IN THE LOG-LOG REGIME IS OPEN IN H1, WHERE U0 DENOTES THE SET OF ALL INITIAL DATA ?0 IN B??SO THAT THE ?OW ?0 ??(T) OF (2) COLLAPSES IN ?NITE TIME T ?< ?WITH THE LOG-LOG SPEED GIVEN IN (5), SEE [12,24] 4 THE BLOWUP RATE FOR THE ROTATIONAL NLS IN THIS SECTION WE PROVE THEOREM 1 1 WE WILL ALWAYS ASSUME P = 1+ N IN BOTH (1) AND (2) WE WILL NEED A VIRIAL IDENTITY FOR (2) AND THE R-TRANSFORM INTRODUCED IN PROPOSITION 4 3 THIS TRANSFORM GIVES A RELATION BETWEEN THE TWO SOLUTIONS OF (1) AND (2), WHICH IS COINED AS A COMBINATION OF THE LENS TRANSFORM AND THE ROTATION ETA ? ONE CAN VIEW IT AS CERTAIN PSEUDO-CONFORMAL SYMMETRY IN THE ROTATIONAL CASE, SEE [7,29] IN THE PRESENCE OF A QUADRATIC POTENTIAL ONLY, I E , M = 0 AND ? ?= 0 THE FOLLOWING IS A STANDARD VIRIAL IDENTITY FOR (2) IN THE WEIGHTED SOBOLEV SPACE H 1, WHICH CAN BE PROVED BY A DIRECT CALCULATION LEMMA 4 1 LET ? BE A SOLUTION TO THE PROBLEM (2) IN C([0, T), H 1) DE?NE J(T) := |X|2|?|2DX THEN J?(T) = 4? X? ??DX, J??(T) = 8E(?0), WHERE E(?) = |??|2 ? N N + 2|?| DX LEMMA 4 2 (A) GIVEN ANY REAL N BY N MATRIX M AND ANY FUNCTION F IN C? 0 ?L2(RN), WE HAVE ET(MX) ?F(X) = F(ETMX) (6) (B) IF M IS A REAL SKEW-SYMMETRIC MATRIX, THEN ETM ?SO(N) FOR ALL T, WHERE SO(N) IS THE GROUP OF N BY N ORTHOGONAL MATRICES WITH DETERMINANT 1 MOREOVER, (MX) ?AND ? COMMUTE, I E , [(MX) ?, ?] = 0 PART (A) CAN BE PROVEN BY SHOWING THAT BOTH SIDES OF (6) OBEY THE ODE ?TF(T, X) = (MX) ?F(T, X), F(0, X) = F(X) PART (B) FOLLOWS FROM A STRAIGHTFORWARD CALCULATION NON LINEAR WAVES IN DISPERSIVE MEDIA PROPOSITION 4 3 LET ?(T, X) ?C([0, T), H1) BE A SOLUTION TO (2) WHERE T > 0 DE?NE THE R TRANSFORM ? ?R(?) TO BE R?(T, X) := COS N 2 (2?T)E?I ? 2 |X|2 TAN(2?T)? TAN(2?T) 2? , ETMX COS(2?T) (7) THEN U = R? IS A SOLUTION TO (1) IN C([0, ARCTAN(2?T ) 2? ), H 1) CONVERSELY, LET U(T, X) ?C([0, T ?), H 1) BE A SOLUTION TO (1) WHERE T ??(0, ? 4? ] THEN ? = R?1U, GIVEN BY ?(T, X) := COS N 2 +2(2?T)E?I ? 2 |X|2 TAN(2?T)? TAN(2?T) 2? , ETMX COS(2?T) + COS N 2 +2(2?T)E?I ? 2 |X|2 TAN(2?T)?T TAN(2?T) 2? , ETMX COS(2?T) + COS N 2 +1(2?T)E?I ? 2 |X|2 TAN(2?T)(ETMMX) ?? TAN(2?T) 2? , ETMX COS(2?T) + 2? SIN(2?T) COS N 2 +2(2?T)E?I ? 2 |X|2 TAN(2?T)(ETMX) ?? TAN(2?T) 2? , ETMX COS(2?T) (9) TO COMPUTE ?U AND I(MX) ?U, ?RST NOTE THAT ? ? TAN(2?T) 2? , ETMX COS(2?T) = COS(2?T)(ETM)T ?? TAN(2?T) 2? , ETMX COS(2?T) , ? ? TAN(2?T) 2? , ETMX COS(2?T) = COS2(2?T)?? TAN(2?T) 2? , ETMX COS(2?T) , WHERE WE USED (A) IF B ?MN×N IS A CONSTANT MATRIX, THEN ?(?(BX)) = BT (??)(BX); (B) ? = DIV(?), DIV F = TR( ?(F1 ,FN) ?(X1, ,XN)) = TRACE OF THE JACOBIAN OF F; (C) IF C IS A CONSTANT SQUARE MATRIX, W = [W1, , WN]T IS A VECTOR-VALUED FUNCTION OF X ?RN, THEN DIV(CW) = TR([?W1, , ?WN]CT ) (D) TR(U ??U) = TR(?) IF U IS A UNITARY MATRIX AND ? ?MN×N NON LINEAR WAVES IN DISPERSIVE MEDIA SIN2(2?T) COS N 2 +2(2?T)E?I ? 2 |X|2 TAN(2?T)? TAN(2?T) 2? , ETMX COS(2?T) ?I2? SIN(2?T) COS N 2 +2(2?T)E?I ? 2 |X|2 TAN(2?T)(ETMX) ?? TAN(2?T) 2? , ETMX COS(2?T) + COS N 2 +2(2?T)E?I ? 2 |X|2 TAN(2?T)?? TAN(2?T) 2? , ETMX COS(2?T) , (10) AND, NOTING THAT (MX) X = 0, I(MX) ?U = I COS N 2 +1(2?T)E?I ? 2 |X|2 TAN(2?T)(METMX) ?? TAN(2?T) 2? , ETMX COS(2?T) (11) BRING (7), (9), (10), AND (11) INTO CAUCHY PROBLEM (1), AND RECALL THAT ? SATIS?ES (2) WITH P = 1+4/N, HENCE, WE SEE THAT U IS A SOLUTION TO (1) WITH U(0, X) = ?(0, X) THE ABOVE VIRTUALLY SHOWS THAT UNDER THE RELATION U = R? ??? = R?1U, U SATIS?ES (1) IF AND ONLY IF ? SATIS?ES (2) THEREFORE THE SECOND PART OF THE PROPOSITION IS ALSO TRUE ? REMARK 4 4 THE R TRANSFORM ALSO ALLOWS US TO SOLVE THE LINEAR EQUATION FOR (1) THE EQUATION I?T? = ??? HAS THE FUNDAMENTAL SOLUTION EIT?(X, Y) = 4T APPLYING (7) WE THEN OBTAIN THE FUNDAMENTAL SOLUTION TO I?TU = HA,V U: E?ITHA,V (X, Y) = ? 2?I SIN(2?T) N EI ? 2 (|X|2+|Y|2) COT(2?T)E?I? (ETM X) Y SIN(2?T) (12) THIS EXPRESSION IS SIGNI?CANTLY SIMPLER THAN THE ONE IN [10] PER MEHLER’S FORMULA TO OUR BEST KNOWLEDGE, (12) MIGHT BE THE ?RST SIMPLY UNI?ED EXPLICIT FORMULA COMPARED WITH [2,14–16] NOW WE ARE READY TO PROVE THEOREM 1 1 PROOF OF THEOREM 1 1 LET U ?C([0, T), H 1) BE THE BLOWUP SOLUTION TO THE PROBLEM (1), WHERE [0, T) IS THE MAXIMAL INTERVAL OF EXISTENCE THEN BY (8), THERE IS A ?(T, X) ?C([0, TAN(2?T ) 2? ), H 1) THAT SOLVES (2), WHERE [0, TAN(2?T ) 2? ) IS THE MAXIMAL INTERVAL OF EXISTENCE NOTE THAT U0 = ?0, AND ACCORDING TO THEOREM 3 1, THE CONDITION (4) SUGGESTS THAT ? IS A BLOWUP SOLUTION RECALL FROM (7), FOR T ?(0, ? 4? ] U(T, X) = COS N 2 (2?T)E?I ? 2 |X|2 TAN(2?T)? TAN(2?T) 2? , ETMX COS(2?T) NON LINEAR WAVES IN DISPERSIVE MEDIA THEN FOR ALL T ?[0, T), WE HAVE ?XU(T, X) = ?I?X SIN(2?T) COS N 2 +1(2?T)E?I ? 2 |X|2 TAN(2?T)? TAN(2?T) 2? , ETMX COS(2?T) + COS N 2 +1(2?T)E?I ? 2 |X|2 TAN(2?T)(ETM)T ?? TAN(2?T) 2? , ETMX COS(2?T) := I1 + I2 (13) FOR I1, A CHANGE OF VARIABLE GIVES |I1|2 = ? SIN(2?T) X? TAN(2?T) 2? , X 2 THEN J(T) = J(0) + J?(0)T + T H 1, J??(T) = 8E(?0) THUS X? TAN(2?T) 2? , X = J TAN(2?T) 2? ?|J(0)| + |J?(0)|TAN(2?T) 2? + 4E(?0) TAN(2?T) 2? H 1 TAN(2?T) 2? + 4E(?0) TAN(2?T) 2? , AND SO |I1|2 ?C(?0, T) (14) FOR I2, IN VIEW OF LEMMA 4 2, ETMT ?SO(N), A CHANGE OF VARIABLE GIVES FOR T ?[0, T), (T ??/4?) |I2|2 = COS(2?T) ?? TAN(2?T) 2? , X AS T ?T, TAN(2?T) 2? ?TAN(2?T ) 2? , SO BY THEOREM 3 1, LIM T?T ?? TAN(2?T) 2? , X TAN(2?T ) 2? ?TAN(2?T) 2? LOG LOG TAN(2?T ) 2? ?TAN(2?T) 2? = AND LOG SIN(2?(T ?T)) LOG(T ?T) ?1, NON LINEAR WAVES IN DISPERSIVE MEDIA SO THE ABOVE BLOWUP RATE CAN BE SIMPLI?ED AS LIM T?T ?? TAN(2?T) 2? , X ? 2? ? 4 1 BLOWUP RATE AT THE GROUND STATE Q WE CONCLUDE WITH SOME DISCUS- SIONS ON THE WAVE COLLAPSE RATES FOR (1) WHEN THE INITIAL DATA IS NEAR Q, WHICH COULD BE A SUBTLE ISSUE NOTICE THAT THIS Q IS NOT THE GROUND STATE FOR (1), INSTEAD, IT IS THE ONE FOR (2) IF |U0|2 = |Q|2, THEN THE WAVE COLLAPSE FOR (1) IS DI?ERENT THAN IN THE CASE |U0|2 > |Q|2 APPLYING THE TRANSFORM (7) TO THE SOLITARY WAVE ? = EITQ WE CAN CONSTRUCT A BLOWUP SOLUTION WITH BLOWUP RATE (T ?T)?1: U(T, X) = COS N 2 (2?T)E?I ? 2 |X|2 TAN(2?T)EI TAN(2?T) 2? Q ETMX COS(2?T) (16) ONE EASILY CHECKS THAT U BLOWS UP AT T = ? 4? SATISFYING |?U|2 ? AS T ?T = ? 4? NOTE THAT SOLUTIONS OF THE FORM (16) WITH SUCH BLOWUP TIME AND SINGULARITY CAN ALSO BE OBTAINED WITH OTHER NONPOSITIVE, NON-RADIAL BOUND STATES QB AS THE PRO?LE IN PLACE OF Q, WHERE |QB|2 > |Q|2 AND N ?2 SUPPOSE THAT U IS A BLOWUP SOLUTION TO (1) WITH BLOWUP TIME T ?< ? 4? AND |U0|2 = |Q|2 THEN BY MEANS OF (8) WE MAY DE?NE A BLOWUP SOLUTION TO (2) WITH THE SAME INITIAL DATA WHICH BLOWS UP AT T0 = TAN(2?T ) 2? MERLE [19] SHOWED THAT UP TO THE SCALING AND PHASE INVARIANCES OF (2), THE ONLY MINIMAL MASS BLOWUP SOLUTIONS ARE OF THE FORM ?(T, X) = FOR SOME X0 ?RN BY THE R-TRANSFORM (7) AND THE UNIQUENESS OF (1) (PROPOSITION 2 1), WE THEN ESTABLISH A CHARACTERIZATION FOR ALL MINIMAL MASS BLOWUP SOLUTIONS OF (1) PROPOSITION 4 5 LET |U0|2 = |Q|2 LET U BE A BLOWUP SOLUTION OF (1) ON [0, T) WITH T := T ?< ? 4? THEN U MUST ASSUME THE FOLLOWING FORM (UP TO SCALING AND NON LINEAR WAVES IN DISPERSIVE MEDIA PHASE INVARIANCE): THERE EXISTS X1 ?RN SUCH THAT (17) U(T, X) = 2? COS(2?T) SIN(2?(T ?T)) N |Q|2+ N , IT IS EASY TO VERIFY U0 ?B??AND CONDITION (4) REMARK 4 7 FOR LARGE INITIAL DATA ONE CAN ALSO DERIVE A GENERAL LOWER BOUND FOR THE COLLAPSE RATE IF THE SOLUTION OF (1) SATIS?ES LIMT?T ?|?U|2 = ?, THEN THERE EXISTS C = CP,N > 0 SUCH THAT |?U(T)|2 ?C(T ??T)?( ) THIS FOLLOWS FROM QUITE STANDARD ARGUMENT AS IN [9] THAT IS USED TO SHOW THE L W P AND BLOWUP ALTERNATIVE FOR THE CAUCHY PROBLEM (1) ON [0, T ?) IN THE L2-CRITICAL CASE P = 1 + [6] N BASHARAT, H HAJAIEJ, Y HU, S ZHENG, THRESHOLD FOR BLOWUP AND STABILITY FOR NONLINEAR SCHR¨ODINGER EQUATION WITH ROTATION PREPRINT [7] R CARLES, CRITICAL NONLINEAR SCHR¨ODINGER EQUATIONS WITH AND WITHOUT HARMONIC PO- TENTIAL, MATH MODELS METHODS APPL SCI NON LINEAR WAVES IN DISPERSIVE MEDIA [9] T CAZENAVE AND F B WEISSLER, THE CAUCHY PROBLEM FOR THE CRITICAL NONLINEAR SCHR¨ODINGER EQUATION IN HS, NONLINEAR ANAL [23] G PERELMAN, ON THE FORMATION OF SINGULARITIES IN SOLUTIONS OF THE CRITICAL NON- LINEAR SCHR¨ODINGER EQUATION, ANN HENRI POINCAR´E DEPARTMENT OF POPULATION HEALTH SCIENCES, AUGUSTA UNIVERSITY, AUGUSTA, GEORGIA EMAIL ADDRESS: NYLABASHARAT12@GMAIL COM DEPARTMENT OF MATHEMATICAL SCIENCES, GEORGIA SOUTHERN UNIVERSITY, STATESBORO, GEORGIA EMAIL ADDRESS: YIHU@GEORGIASOUTHERN EDU DEPARTMENT OF MATHEMATICAL SCIENCES, GEORGIA SOUTHERN UNIVERSITY, STATESBORO, GEORGIA EMAIL ADDRESS: SZHENG@GEORGIASOUTHERN EDU NON LINEAR WAVES IN DISPERSIVE MEDIA GRADIENT ESTIMATES FOR WEAK SOLUTIONS OF LINEAR ELLIPTIC SYSTEMS WITH SINGULAR-DEGENERATE COE?CIENTS ABSTRACT THIS PAPER ESTABLISHES CALDER´ON-ZYGMUND TYPE REGULARITY ESTI- MATES FOR SOLUTIONS OF THE CONORMAL DERIVATIVE PROBLEM FOR A CLASS OF LINEAR ELLIPTIC SYSTEMS IN DIVERGENCE-FORM WITH SINGULAR, DEGENERATE COE?CIENTS IN BOUNDED DOMAINS IN OUR CLASS OF EQUATIONS, THE PRINCIPAL TERMS ARE FOURTH ORDER TENSORS OF MEASURABLE FUNCTIONS THAT BEHAVE AS SOME WEIGHT FUNCTION IN THE MUCKENHOUPT CLASS OF A2-WEIGHTS REGULARITY ESTIMATES FOR GRADIENT OF WEAK SOLUTIONS IN WEIGHTED LEBESGUE SPACES ARE ESTABLISHED UNDER SOME NAT- URAL SMALLNESS CONDITIONS ON THE MEAN OSCILLATION OF COE?CIENTS THE RESULTS OBTAINED RECOVER KNOWN RESULTS WHEN THE COE?CIENTS ARE UNIFORMLY ELLIPTIC THESE RESULTS CAN BE CONSIDERED AS THE SOBOLEV COUNTERPARTS OF THE CLASSICAL H¨OLDER’S REGULARITY ESTIMATES ESTABLISHED BY B FABES, C E KENIG, AND R P SERAPIONI 1 INTRODUCTION THIS WORK STUDIES REGULARITY ESTIMATES IN WEIGHTED SOBOLEV SPACES FOR WEAK SOLUTIONS OF A CLASS OF LINEAR SYSTEMS OF ELLIPTIC EQUATION WITH PRESCRIBED DEGENERATE, SINGULAR COE?CIENTS IN BOUNDED DOMAINS WITH CONORMAL BOUNDARY CONDITIONS THE SYSTEM WE ARE STUDYING IS GIVEN BY (1 1) DIV[A(X)?U] = DIV[F] IN ?, ?A(X)?U ?F, N? = A?? IJ ?J ??I ?, ?? = (?I ?) ?RNN, A E X ??, AND (1 3) A?? IJ (X) ???1?(X) A E X ??, AND ALL ?, ? = 1, , N; I, J = 1, N ?(B?(X) ??) ˆ B?(X)?? |F(Y) ??F?B?(X)??|DY, AND ?F?B?(X)?? = |B?(X) ??| ˆ B?(X)?? F(Y)DY NON LINEAR WAVES IN DISPERSIVE MEDIA OBSERVE THAT THE CLASSICAL JOHN-NIRENBERG BMO IN ? CORRESPONDS TO ? = 1 AND R0 = DIAM(?) THE CLASS OF MUCKENHOUPT AP- WEIGHTS WILL BE RECALLED IN DE?NITION 2 2 OUR ?RST MAIN RESULT OF THIS PAPER IS THE FOLLOWING THEOREM THEOREM 1 2 LET ? > 0, M0 ?1, AND P ?2 BE GIVEN THERE EXISTS A SUF- ?CIENTLY SMALL CONSTANT ? = ?(?, M0, N, P) > 0 SUCH THAT THE FOLLOWING STATEMENT HOLDS SUPPOSE ALSO THAT ? ?A2 WITH [?]A2 ?M0, ? IS A C1-DOMAIN, (1 2)-(1 3) HOLD ON ?, AND [A]BMOR0(?,?) ?? FOR SOME R0 > 0 THEN EVERY WEAK SOLUTION U ?W 1,2(?, ?, RN) OF (1 1) CORRESPONDING TO |F|/? ?LP(?, ?, RN) SATIS?ES THE ESTIMATE ??U?LP(?,?) ?C F ? LP(?,?) , WHERE C IS A CONSTANT DEPENDING ONLY ON N, ?, P, M0, R0 AND ? LOCAL REGULARITY ESTIMATES FOR WEAK SOLUTIONS OF (1 1) ARE NOT ONLY OF GREAT INTEREST BY THEMSELVES, BUT ALSO IMPORTANT IN MANY APPLICATIONS SINCE THEY ONLY REQUIRE LOCAL INFORMATION ON THE DATA THIS PAPER ALSO ESTABLISHES INTERIOR REGULARITY ESTIMATES, AND LOCAL BOUNDARY ESTIMATES TO STATE THE LOCAL RESULTS WE USE THE NOTATION BR TO DENOTE FOR A BALL OF RADIUS R CENTERED AT THE ORIGIN, B+ R ITS UPPER PART AND TR THE ?AT PART OF THE BOUNDARY OF B+ R THE NEXT RESULT PRESENTS THE INTERIOR LOCAL GRADIENT ESTIMATE THEOREM 1 3 LET ? > 0, M0 ?1, AND P ?2 BE GIVEN THERE EXISTS A SUF- ?CIENTLY SMALL CONSTANT ? = ?(?, M0, N, P) > 0 SUCH THAT THE FOLLOWING STATEMENT HOLDS SUPPOSE THAT ? ?A2 WITH [?]A2 ?M0, AND (1 2)-(1 3) HOLD FOR A GIVEN TEN- SOR MATRIX A ON B2 MOREOVER, ASSUME THAT [A]BMO1(B1,?) ?? THEN, FOR EVERY F SUCH THAT F/? ?LP(B2, ?, RN), IF U ?W 1,2(B2, ?, RN) IS A WEAK SOLUTION TO (1 4) DIV[A?U] = DIV(F) IN B2, THEN ?U ?LP(B1, ?) AND ??U?LP(B1,?) ?C (?(B1)) 2 ??U?L2(B2,?) + ?F/??LP(B2,?) WHERE C IS A CONSTANT DEPENDING ONLY ON ?, P, N, M0 OUR LAST RESULT IS A LOCAL BOUNDARY REGULARITY ONE THEOREM 1 4 LET ? > 0, M0 ?1, AND P ?2 BE GIVEN THERE EXISTS A SUF- ?CIENTLY SMALL CONSTANT ? = ?(?, M0, N, P) > 0 SUCH THAT THE FOLLOWING STATEMENT HOLDS SUPPOSE THAT ? ?A2 WITH [?]A2 ?M0, AND (1 2)-(1 3) HOLD FOR THE GIVEN TENSOR MATRIX A ON B+ 2 MOREOVER, ASSUME THAT [A]BMO1(B+ 1 ,?) ?? THEN, FOR EVERY F SUCH THAT F/? ?LP(B+ 2 , ?), IF U ?W 1,2(B+ 2 , ?, RN) IS A WEAK SOLUTION TO (1 5) DIV[A?U] = DIV(F) IN B+ 2 ?A?U ?F, N?= ON T2, THEN ?U ?LP(B+ 1 , ?) AND ??U?LP(B+ 1 ,?) ?C (?(B1)) 2 ??U?L2(B+ 2 ,?) + ?F/??LP(B+ 2 ,?) , WHERE C IS A CONSTANT DEPENDING ONLY ON ?, P, N, M0 NON LINEAR WAVES IN DISPERSIVE MEDIA