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

rough solutions for the wave maps equation we consider the wave maps equation with values into a riemannian manifold which is isometrically embedded in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i our main result asserts that the cauchy problem is globally well posed for initial data which is small in the critical sobolev spaces this extends and completes recent work of tao and other authors

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

asymptotic stability of large energy harmonic maps under the wave map from 2d hyperbolic spaces to 2d hyperbolic spaces abstract in this paper we prove that the large energy harmonic maps from h 2 to h 2 are asymptotically stable under the wave map equation this work is the last paper in a series of works 46 45 44 which together prove the asymptotic stability for all admissible harmonic maps of arbitrary size without any symmetry assumptions

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

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

global well posedness for the three dimensional simplified inertial ericksen leslie systems near equilibrium abstract we study a simplified inertial ericksen leslie system for the nematic liquid crystal flow which can be viewed as a system coupling navier stokes equations and wave map equations we prove the global existence of classical solution with initial data near equilibrium

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

quantization of time like energy for wave maps into spheres in this article we consider large energy wave maps in dimension 2 1 as in the resolution of the threshold conjecture by sterbenz and tataru commun math phys 298 1 139 230 2010 commun math phys 298 1 231 264 2010 but more specifically into the unit euclidean sphere mathbb s n 1 subset mathbb r n with n geq2 and study further the dynamics of the sequence of wave maps that are obtained in sterbenz and tataru commun math phys 298 1 231 264 2010 at the final rescaling for a first finite or infinite time singularity we prove that on a suitably chosen sequence of time slices at this scaling there is a decomposition of the map up to an error with asymptotically vanishing energy into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm concentrating on the null boundary and converging to a constant locally in the interior of the cone in the energy space similar and stronger results have been recently obtained in the equivariant setting by several authors cote commun pure appl math 68 11 1946 2004 2015 cote commun pure appl math 69 4 609 612 2016 cote am j math 137 1 139 207 2015 cote et al am j math 137 1 209 250 2015 krieger commun math phys 250 3 507 580 2004 where better control on the dispersive term concentrating on the null boundary of the cone is provided and in some cases the asymptotic decomposition is shown to hold for all time here however we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to lin and riviere ann math 149 2 785 829 1999 duke math j 111 177 193 2002 which we make work here in the hyperbolic context of sterbenz and tataru commun math phys 298 1 231 264 2010

complex network synchronizability analysis and control in this paper the investigation is first motivated by showing two examples of simple regular symmetrical graphs which have the same structural parameters such as average distance degree distribution and node betweenness centrality but have very different synchronizabilities for a given network with identical node dynamics it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability in searching for conditions under which the network synchronizability may be increased by adding edges it is found that for networks with disconnected complementary graphs adding edges never decreases their synchronizability moreover it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region therefore to effectively enhance the network synchronizability a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region for the case where the synchronous state is an equilibrium point of the network

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

global schrodinger maps in dimensions d 2 small data in the critical sobolev spaces we consider the schrodinger map initial value problem t on r d r 0 0 where r d r s 2 r 3 is a smooth function in all dimensions d 2 we prove that the schrodinger map initial value problem admits a unique global smooth solution 2 c r h 1 q q2 s 2 provided that the data 02 h 1 q is smooth and satises the smallness condition

three structural properties reflecting the synchronizability of complex networks during the process of adding links we find that the synchronizability of the classical barabasi albert ba scale free or watts strogatz ws small world networks can be statistically quantified by three essentially structural quantities of these networks i e the eccentricity variance of the degree distribution and clustering coefficients the results indicate that both the eccentricity and clustering coefficient are positively linearly correlated with synchronizability while the variance is negatively linearly correlated moreover the efficiency of some particular strategies of adding links to change the synchronizability is also investigated this information can be used to guide us to design corresponding strategies of structure evolving processes to manipulate the synchronizability of a given network

decoupling process for better synchronizability on scale free networks we propose a decoupling process performed in scale free networks to enhance the synchronizability of the network together with preserving the scale free structure simulation results show that the decoupling process can effectively promote the network synchronizability which is measured in terms of eigenratio of the coupling matrix moreover we investigate the correlation between some important structural properties and the collective synchronization and find that the maximum vertex betweenness seems to be the most strongly correlated with the synchronizability among the major structural features considered we explain the effect of the decoupling process from a viewpoint of coupling information transmission our work provides some evidence that the dynamics of synchronization is related to that of information or vehicle traffic because of the low cost in modifying the coupling network the decoupling process may have potential applications

ill posedness for one dimensional wave maps at the critical regularity we show that the wave map equation in r 1 1 is in general ill posed in the critical space 1 2 and the besov space 1 2 1 2 the problem is attributed to the bad behavior of the one dimensional bilinear expression d 1 fdg in these spaces

absence of self similar blow up and local well posedness for the constant mean curvature wave equation abstract in this note we consider the constant mean curvature wave equation in 1 2 dimensions we show that it does not admit any self similar blow up we also remark that the equation is locally well posed for initial data in h 3 2

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

synchronizability of dynamical scale free networks subject to random errors in this paper the robustness of network synchronizability against random deletion of nodes i e errors in dynamical scale free networks was studied to this end two measures of network synchronizability namely the eigenratio of the laplacian and the order parameter quantifying the degree of phase synchrony were adopted and the synchronizability robustness on preferential attachment scale free graphs was investigated the findings revealed that as the network size decreases the robustness of its synchronizability against random removal of nodes declines i e the more the number of randomly removed nodes from the network the worse its synchronizability we also showed that this dependence of the synchronizability on the network size is different with that in the growing scale free networks the profile of a number of network properties such as clustering coefficient efficiency assortativity and eccentricity as a function of the network size was investigated in these two cases growing scale free networks and those with randomly removed nodes the results showed that these processes are also different in terms of these metrics

factors that predict better synchronizability on complex networks while shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network the suppressing tendency of the heterogeneity of the degree distribution even for shorter characteristic path length has also been reported to see this we investigate the effects of various factors such as the degree characteristic path length heterogeneity and betweenness centrality on synchronization and find a consistent trend between the synchronization and the betweenness centrality the betweenness centrality is thus proposed as a good indicator for synchronizability

relations between average distance heterogeneity and network synchronizability abstract by using the random interchanging algorithm we investigate the relations between average distance standard deviation of degree distribution and synchronizability of complex networks we find that both increasing the average distance and magnifying the degree deviation will make the network synchronize harder only the combination of short average distance and small standard deviation of degree distribution ensures strong synchronizability some previous studies assert that the maximal betweenness is the right quantity to estimate network synchronizability the larger the maximal betweenness the poorer the network synchronizability here we address an interesting case which strongly suggests that the single quantity maximal betweenness may not give a comprehensive description of network synchronizability

does simple random sampling provide adequate balance une etude empirique montre que sous un echantillonnage aleatoire simple les intervalles de confiance du total peuvent avoir des probabilites de recouvrement conditionnelles qui sont imposantes de facon persistante lorsque la taille de l echantillon croit

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

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

renormalization and blow up for wave maps from s 2 times mathbb r to s 2 we construct a one parameter family of finite time blow ups to the co rotational wave maps problem from s 2 x r to s 2 parameterized by nu is an element of 1 2 1 the longitudinal function u t alpha which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from r 2 to s 2 the domain of this harmonic map is identified with a neighborhood of the north pole in the domain s 2 via the exponential coordinates alpha theta in these coordinates u t alpha q lambda t alpha r t alpha where q r 2arctan r is the standard co rotational harmonic map to the sphere lambda t t 1 nu and r t alpha is the error with local energy going to zero as t 0 blow up will occur at t alpha 0 0 due to energy concentration and up to this point the solution will have regularity h1 nu

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

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