two bubble dynamics for threshold solutions to the wave maps equation we consider the energy critical wave maps equation mathbb r 1 2 rightarrow mathbb s 2 in the equivariant case with equivariance degree k ge 2 it is known that initial data of energy 8 pi k and topological degree zero leads to global solutions that scatter in both time directions we consider the threshold case of energy 8 pi k we prove that the solution is defined for all time and either scatters in both time directions or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction in the latter case we describe the asymptotic behavior of the scales of the two harmonic maps the proof combines the classical concentration compactness techniques of kenig merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation

radially symmetric wave maps from 1 2 dimensional minkowski space to the sphere by a blow up analysis as in 8 for a related problem we rule out concentration of energy for radially symmetric wave maps from the 1 2 dimensional minkowski space to the sphere when combined with the local existence and regularity results of christodoulou and tahvildar zadeh for this problem our result implies global existence of smooth solutions to the cauchy problem for radially symmetric wave maps for smooth radially symmetric data

global regularity of wave maps from r3 1 to surfaces we consider wave maps with smooth compactly supported initial data of small norm from r3 1 to certain 2 dimensional riemannian manifolds and show that they stay smooth globally in time our methods are based on the introduction of a global coulomb gauge as in 17 followed by a dynamic separation as in 8 we then rely on an adaptation of t tao s methods used in his recent breakthrough result 24

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

characterization of large energy solutions of the equivariant wave map problem ii we consider 1 equivariant wave maps from bbb r 1 2 to bbb s 2 of finite energy we establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map q in particular for each global energy solution of topological degree one we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term together with a companion article part i where we consider the case of finite time blow up this gives a characterization of all 1 equivariant degree 1 wave maps in the energy regime e q 3e q

scattering for wave maps exterior to a ball abstract we consider 1 equivariant wave maps from r t r x 3 b s 3 where b is a ball centered at 0 and b gets mapped to a fixed point on s 3 we show that 1 equivariant maps of degree zero scatter to zero irrespective of their energy for positive degrees we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree

uniqueness of weak solutions of 1 1 dimensional wave maps in this paper we prove that finite energy distributional solutions to the cauchy problem of wave maps from 1 1 dimensional minkowsky space to any complete riemannian manifold are unique

a proof of price s law for the collapse of a self gravitating scalar field a well known open problem in general relativity dating back to 1972 has been to prove price s law for an appropriate model of gravitational collapse this law postulates inverse power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates it is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon in this paper we prove a well defined upper bound formulation of price s law for the collapse of a self gravitating scalar field with spherically symmetric initial data we also allow the presence of an additional gravitationally coupled maxwell field our results are obtained by a new mathematical technique for understanding the long time behavior of large data solutions to the resulting coupled non linear hyperbolic system of p d e s in 2 independent variables the technique is based on the interaction of the conformal geometry the celebrated red shift effect and local energy conservation we feel it may be relevant for the problem of non linear stability of the kerr solution when combined with previous work of the first author concerning the internal structure of charged black holes which had assumed the validity of price s law our results can be applied to the strong cosmic censorship conjecture for the einstein maxwell real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data under christodoulou s c0 formulation the conjecture is proven to be false

low regularity solutions for the wave map equation into the 2 d sphere a class of weak wave map solutions with initial data in sobolev space of order s 1 is studied a non uniqueness result is proved for the case when the target manifold is a two dimensional sphere using an equivariant wave map ansatz a family of self similar solutions is constructed this construction enables one to show ill posedness of the inhomogeneous cauchy problem for wave maps

on collapse of wave maps abstract we derive the universal collapse law of degree 1 equivariant wave maps solutions of the sigma model from the 2 1 minkowski space time to the 2 sphere to this end we introduce a nonlinear transformation from original variables to blowup ones our formal derivations are confirmed by numerical simulations

on the dynamics of gowdy space times we study the behavior near the singularity t 0 of gowdy metrics we prove existence of an open dense set of boundary points near which the solution is smoothly asymptotically velocity term dominated avtd we show that the set of avtd solutions satisfying a uniformity condition is open in the set of all solutions we analyze in detail the asymptotic behavior of power law solutions at the hitherto unchartered points at which the asymptotic velocity equals zero or one several other related results are established 2004 wiley periodicals inc

global existence and uniqueness of schrodinger maps in dimensions d 4 abstract in dimensions d 4 we prove that the schrodinger map initial value problem t s s s on r d r s 0 s 0 admits a unique solution s r d r s 2 r 3 s c r h q provided that s 0 h q and s 0 q h d 2 1 where q s 2

concentration compactness and universal profiles for the non radial energy critical wave equation abstract in this paper we give an overview of the authors work on applications of the method of concentration compactness to global well posedness scattering blow up and universal profiles for the energy critical wave equation in the non radial setting new results and proofs are also given

weak convergence of wave maps from 1 2 dimensional minkowski space to riemannian manifolds the weak limit of a sequence of wave maps with uniformly bounded energy is again a wave map essential ingredients in the proof are hodge structures related to harmonic maps a concise description of the concentration set of a sequence of wave maps and the use of 1 bmo duality combined with an elliptic gauge condition in the hyperbolic context

asymptotic decomposition for semilinear wave and equivariant wave map equations in this paper we give a unified proof to the soliton resolution conjecture along a sequence of times for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations including the four dimensional radial yang mills equation without using outer energy type inequalities such inequalities have played a crucial role in previous works with similar results roughly speaking we prove that along a sequence of times t n to t the maximal time of existence the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space plus a free radiation term in the global case or a regular part in the finite time blow up case the main difficulty is that in general for instance for the radial four dimensional yang mills case and the radial six dimensional semilinear wave case we do not have a favorable outer energy inequality for the associated linear wave equations our main new input is the simultaneous use of two virial identities

numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces this article studies the numerical approximation of harmonic maps mto surfaces i e critical pomts for the dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold an iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised and its global convergence to a discrete harmonic map is proved under general conditions weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations numerical simulations illustrate the performance of the algorithm for curved domains

free versus constrained evolution of the 2 1 equivariant wave map we compare the numerical solutions of the 2 1 equivariant wave map problem computed with the symplectic constraint respecting rattle algorithm and the well known fourth order runge kutta method we show the advantages of the rattle algorithm for the constrained system compared to the free evolution with the runge kutta method we also present an expression which represents the energy loss due to constraint violation taking this expression into account we can achieve energy conservation for the runge kutta scheme which is better than with the rattle method using the symplectic scheme with constraint enforcement we can reproduce previous calculations of the equivariant case without imposing the symmetry explicitly thereby confirming that the critical behaviour is stable

strong solutions to stochastic wave equations with values in riemannian manifolds let m be a d dimensional compact riemannian manifold we prove existence of a unique global strong solution of the stochastic wave equation dt tu dx xu yu tu xu w where y is a c1 smooth transformation and w is a spatially homogeneous wiener process on r whose spectral measure has finite moments up to order 2

on the stability of stationary wave maps equivariant wave maps from into have smooth stationary solutions which are critical points of the energy subject to constant charge these solutions are globally stable under equivariant perturbations consequently there exists a large set of initial data with no degree or energy restrictions for which the cauchy problem is globally well posed

updated semi discretization method for periodic delay differential equations with discrete delay an updated version of the semi discretization method is presented for periodic systems with a single discrete time delay the delayed term is approximated as a weighted sum of two neighbouring discrete delayed state values and the transition matrix over a single period is determined stability charts are constructed for the damped and delayed mathieu equation for different time period time delay ratios the convergence of the method is investigated by examples stability charts are constructed for 1 and 2 degree of freedom milling models the codes of the algorithm are also attached in the appendix copyright 2004 john wiley sons ltd

on the continuity of the solution operator to the wave map system we investigate the continuity properties of the solution operator to the wave map system from n to a general nonflat target of arbitrary dimension and we prove by an explicit class of counterexamples that this map is not uniformly continuous in the critical norms on any neighborhood of 0 2003 wiley periodicals inc

local discontinuous galerkin schemes for a nonlinear variational wave equation modelling liquid crystals abstract we consider a nonlinear variational wave equation that models the dynamics of nematic liquid crystals discontinuous galerkin schemes that either conserve or dissipate a discrete version of the energy associated with these equations are designed numerical experiments illustrating the stability and efficiency of the schemes are presented an interesting feature of these schemes is their ability to approximate two distinct weak solutions of the underlying system

singularity formation in 2 1 wave maps we present numerical evidence that singularities form in finite time during the evolution of 2 1 wave maps from spherically equivariant initial data of sufficient energy

energy conservative solutions to a one dimensional full variational wave system we establish the existence of a conservative weak solution to the initial value problem for a complete system of variational wave equations modeling liquid crystals in one space dimension in which the director has two degrees of freedom the solutions exist globally in time and singularities may develop in finite time but the energy of the solutions is conserved across singular times the method for existence also yields continuous dependence of solutions on the initial data 2011 wiley periodicals inc

existence of constant mean curvature foliations in spacetimes with two dimensional local symmetry it is shown that in a class of maximal globally hyperbolic spacetimes ad mitting two local killing vectors the past defined with respect to an appropriate time orientation of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces moreover the mean cur vature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity the simplest examples occur when the spatial topology is that of a torus with the standard global killing vectors but more exotic topologies are also covered in the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds the symmetry singles out a compact surface passing through any given point of spacetime and the hawking mass of any such surface is non negative if the hawking mass of any one of these surfaces is zero then the entire spacetime is flat

global regularity of wave maps from r 2 1 to h 2 small energy we demonstrate that wave maps with smooth initial data and small energy from r2 1 to the lobatchevsky plane stay smooth globally in time our method is similar to the one employed in 18 however the multilinear estimates required are considerably more involved and present novel technical challenges in particular we shall have to work with a modification of the functional analytic framework used in 30 33 18

conservative solutions to a system of variational wave equations of nematic liquid crystals in this paper we solve a system of hyperbolic equations by modelling nematic liquid crystals

the wave maps equation the wave maps equation has become a very popular topic in recent years the aim of these expository notes is to present a non technical survey of the ideas and methods which have proved useful in the study of wave maps leading up to the latest results the remaining open problems are also stated and explained

no blow up to a variational wave equation in liquid crystals we establish the global existence of smooth solutions to the cauchy problem for a system of variational wave equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and bend coefficients

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc

transference of strong type maximal inequalities by separation preserving representations let x lp ii where 1 p r is a uniformly bounded strongly continuous representation of g in x by separation preserving operators for instance by positivity pre serving operators we show that r transfers to x strong type maximal estimates for sequences of convolution operators defined on lp g if r also has a uniformly bounded version in l2 t then r will transfer to lp t strong type bounds for maximal operators defined by sequences of multipliers which are continuous on the dual group g these theo rems which cease to be valid if the separation preserving hypothesis is removed provide strong type maximal operator counterparts for the corresponding single operator theorems of coifman weiss transference theory one consequence of the second theorem described above is a maximal theorem counterpart of the homomorphism theorem for sin gle multiplier transforms

on stable self similar blowup for equivariant wave maps reference epfl article 166023doi 10 1002 cpa 20366 url http arxiv org abs 1003 0707 record created on 2011 05 23 modified on 2017 05 12