DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

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ANALYTIC SOLUTIONS OF UNSTEADY CRYSTALS AND RAYLEIGH-TAYLOR BUBBLES XUMING XIE We study the initial value problem for 2-dimensional dendritic crystal growth with zero surface tension and classical Rayleigh-Taylor problems. If the initial data is analytic, it is proved that unique analytic solution exists locally in time. The analysis is based on a Nirenberg theorem on abstract Cauchy-Kovalevsky problem in properly chosen Banach spaces. Copyright © 2006 Xuming Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The phenomenon of dendritic crystal growth is one of the earliest scientific problems, tackled first by Kepler [4] in 1661 in his work on six-sided snowflake crystals. It has long been a subject of continued interest to physicists, metallurgists as well as mathematicians. Dendrite constitutes a good example of pattern selection and stability in nonequilibrium systems. From mathematical point of view, dendrite formation is a free boundary problem like the Stefan problem. Many review papers on this subject have appeared in the literature, for example, Langer [8], Kessler et al. [5], Pelce [13], Levine [9]. For unsteady dendritical crystal growth, Kunka et al. [6, 7] studied the linear theory of localized disturbances and a class exact zero-surface-tension solutions if the initial conditions include only poles. They also studied the singular behavior of unsteady dendritical crystal with surface tension. In those situations, a zero of the conformal map that describes the crystal gives birth to a daughter singularity that moves away from the zero and approaches the interface. The motion of the interface of a heavy fluid resting above a lighter fluid in the presence of gravity (Rayleigh-Taylor flow) is very basic but important problem. When the fluids are immiscible, the sharp interface deforms into a pattern containing rising bubbles of lighter fluid and falling spikes of heavier fluid. Model equations for the location of the interface have been derived (see Baker et al. [2], Moore [10], Sharp [16], and references therein). These studies are numerical and asymptotic, but important to furthering physical understanding of the flow dynamics. Numerical calculation ran into the traditional Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1149–1157
TWO-WEIGHT INEQUALITIES FOR SOME OPERATORS APPLIED TO HARMONIC FORMS YUMING XING We obtain two-weight integral inequalities for the composition T ◦ G, whereT is the homotopy operator and G is Green’s operator applied to A-harmonic forms. Copyright © 2006 Yuming Xing. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and statement of results The aim of this paper is to prove different versions of two-weight embedding inequalities for the composition of homotopy operator T and Green’s operator G applied to the A-harmonic forms on manifolds and establish two-weight norm estimates for the composition T ◦ G. Our main results are presented in the following Theorems 1.1 and 1.2, respectively. Theorem 1.1. Let u ∈ L p loc (Λl M,w α ), l = 1,...,n, 1
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Proceedings of the Conference on Di
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Proceedings of the Conference on Di
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CONTENTS Preface...................
Page 8 and 9:
Inequalities for positive solutions
Page 10 and 11:
Modelling the effect of surgical st
Page 12:
Maximum principles for elliptic equ
Page 16 and 17:
A UNIFIED APPROACH TO MONOTONE METH
Page 18 and 19:
IMPULSIVE CONTROL OF THE POPULATION
Page 20 and 21:
BOUNDARY ESTIMATES FOR BLOW-UP SOLU
Page 22 and 23:
SECOND-ORDER DIFFERENTIAL GRADIENT
Page 24 and 25:
ON STRUCTURE OF FRACTIONAL SPACES G
Page 26 and 27:
ON THE STABILITY OF THE DIFFERENCE
Page 28 and 29:
TWO-WEIGHTED POINCARÉ-TYPE INTEGRA
Page 30 and 31:
PRODUCT DIFFERENCE EQUATIONS APPROX
Page 32 and 33:
QUASIDIFFUSION MODEL OF POPULATION
Page 34 and 35:
FIXED-SIGN EIGENFUNCTIONS OF TWO-PO
Page 36 and 37:
MULTIPLE POSITIVE SOLUTIONS OF SUPE
Page 38 and 39:
SPECTRAL STABILITY OF ELLIPTIC SELF
Page 40 and 41:
HO CHARACTERISTIC EQUATIONS SURPASS
Page 42 and 43:
ASYMPTOTIC STABILITY IN DISCRETE MO
Page 44 and 45:
CONTINUOUS AND DISCRETE NONLINEAR I
Page 46 and 47:
ON PARABOLIC SYSTEMS WITH DISCONTIN
Page 48 and 49:
INEQUALITIES FOR POSITIVE SOLUTIONS
Page 50 and 51:
NONOSCILLATION OF ONE OF THE COMPON
Page 52 and 53:
ANNULAR JET AND COAXIAL JET FLOW JO
Page 54 and 55:
JUSTIFICATION OF QUADRATURE-DIFFERE
Page 56 and 57:
VLASOV-ENSKOG EQUATION WITH THERMAL
Page 58 and 59:
SUBDIFFERENTIAL OPERATOR APPROACH T
Page 60 and 61:
A DISCRETE EIGENFUNCTIONS METHOD FO
Page 62 and 63:
ON NONLINEAR SCHRÖDINGER EQUATIONS
Page 64 and 65:
NUMERICAL SOLUTION OF A TRANSMISSIO
Page 66 and 67:
THE SEMILINEAR EVOLUTION EQUATION F
Page 68 and 69:
ON DOUBLY PERIODIC SOLUTIONS OF QUA
Page 70 and 71:
DYNAMICS OF A DISCONTINUOUS PIECEWI
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PROBABILISTIC SOLUTIONS OF THE DIRI
Page 74 and 75:
MONOTONICITY RESULTS AND INEQUALITI
Page 76 and 77:
WELL-POSEDNESS AND BLOW-UP OF SOLUT
Page 78 and 79:
REARRANGEMENT ON THE UNIT BALL FOR
Page 80 and 81: CONVERGENCE OF SERIES OF TRANSLATIO
Page 82 and 83: ON MINIMAL (MAXIMAL) COMMON FIXED P
Page 84 and 85: BOUNDEDNESS OF SOLUTIONS OF FUNCTIO
Page 86 and 87: MODELLING THE EFFECT OF SURGICAL ST
Page 88 and 89: LIMIT CYCLES OF LIÉNARD SYSTEMS M.
Page 90 and 91: MAXIMUM PRINCIPLES AND DECAY ESTIMA
Page 92 and 93: LIPSCHITZ REGULARITY OF VISCOSITY S
Page 94 and 95: ON AN ELASTIC BEAM FULLY EQUATION W
Page 96 and 97: BOUNDARY VALUE PROBLEMS ON THE HALF
Page 98 and 99: EXISTENCE AND UNIQUENESS OF AN INTE
Page 100 and 101: STABILITY OF SOME MODELS OF CIRCULA
Page 102 and 103: LIPSCHITZ CLASSES OF A-HARMONIC FUN
Page 104 and 105: A TWO-DEGREES-OF-FREEDOM HAMILTONIA
Page 106 and 107: QUANTIZATION OF LIGHT FIELD IN PERI
Page 108 and 109: CONSERVATION LAWS IN QUANTUM SUPER
Page 110 and 111: ON SETS OF ZERO HAUSDORFF s-DIMENSI
Page 112 and 113: ON THE GLOBAL ATTRACTOR IN A CLASS
Page 114 and 115: SANDWICH PAIRS MARTIN SCHECHTER Sin
Page 116 and 117: EXISTENCE RESULTS FOR EVOLUTION INC
Page 118 and 119: AVERAGING DOMAINS: FROM EUCLIDEAN S
Page 120 and 121: CONCENTRATION-COMPACTNESS PRINCIPLE
Page 122 and 123: NONLINEAR VARIATIONAL INCLUSION PRO
Page 124 and 125: MAXIMUM PRINCIPLES FOR ELLIPTIC EQU
Page 126 and 127: GLOBAL CONVERGENT ALGORITHM FOR PAR
Page 128 and 129: SECOND-ORDER NONLINEAR OSCILLATIONS
Page 132 and 133: UNBOUNDEDNESS OF SOLUTIONS OF PLANA
Page 134 and 135: QUENCHING OF SOLUTIONS OF NONLINEAR
Page 136 and 137: VARIATION-OF-PARAMETERS FORMULAE AN
Page 138: DIFFERENCE EQUATIONS AND THE ELABOR
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DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

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