nonlinear equalization of hammerstein ofdm systems a practical orthogonal frequency division multiplexing ofdm system can generally be modelled by the hammerstein system that includes the nonlinear distortion effects of the high power amplifier hpa at transmitter in this contribution we advocate a novel nonlinear equalization scheme for ofdm hammerstein systems we model the nonlinear hpa which represents the static nonlinearity of the ofdm hammerstein channel by a b spline neural network and we develop a highly effective alternating least squares algorithm for estimating the parameters of the ofdm hammerstein channel including channel impulse response coefficients and the parameters of the b spline model moreover we also use another b spline neural network to model the inversion of the hpa s nonlinearity and the parameters of this inverting b spline model can easily be estimated using the standard least squares algorithm based on the pseudo training data obtained as a byproduct of the hammerstein channel identification equalization of the ofdm hammerstein channel can then be accomplished by the usual one tap linear equalization as well as the inverse b spline neural network model obtained the effectiveness of our nonlinear equalization scheme for ofdm hammerstein channels is demonstrated by simulation results

on the soliton resolution for equivariant wave maps to the sphere we consider finite energy corotational wave maps with target manifold s2 we prove that for a sequence of times they decompose as a sum of decoupled harmonic maps in the light cone and a smooth wave map in the blowup case or a linear scattering term in the global case up to an error which tends to 0 in the energy space 2015 wiley periodicals inc

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

on the multivariate horner scheme we present an efficient version of the horner scheme for the evaluation of multivariate polynomials and study its stability properties in particular we show that the backward error of evaluation is bounded by a quantity that is linear in the total degree of the polynomial which itself is usually significantly smaller than the number of operations involved in the evaluation process moreover we show that the same type of estimates holds also for the forward error provided that the polynomial basis is chosen suitably

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

the cauchy problem and a generalization of the hille yosida theorem on demontre une generalisation du theoreme de hille yosida en caracterisant les generateurs d une certaine classe de semi groupes d operateurs non bornes sur un espace de banach on resout le probleme de cauchy abstrait

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 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

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

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

construction of type ii blowup solutions for the 1 corotational energy supercritical wave maps abstract we consider the energy supercritical wave maps from r d into the d sphere s d with d 7 under an additional assumption of 1 corotational symmetry the problem reduces to the one dimensional semilinear wave equation t 2 u r 2 u d 1 r r u d 1 2 r 2 sin 2 u we construct for this equation a family of c solutions which blow up in finite time via concentration of the universal profile u r t q r t where q is the stationary solution of the equation and the speed is given by the quantized rates t c u t t l l n l d 1 2 the construction relies on two arguments the reduction of the problem to a finite dimensional one thanks to a robust universal energy method and modulation techniques developed by merle raphael and rodnianski 49 for the energy supercritical nonlinear schrodinger equation then we proceed by contradiction to solve the finite dimensional problem and conclude using the brouwer fixed point theorem

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

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

finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii we prove the existence of weak solutions to the harmonic map heat flow and wave maps into spheres of nonconstant radii weak solutions are constructed as proper limits of iterates from a fully practical scheme based on lowest order conforming finite elements where discrete lagrange multipliers are employed to exactly meet the sphere constraint at mesh points computational studies are included to motivate interesting dynamics in two and three spatial dimensions

evaluation of the derivative of a polynomial in bernstein form in geometric modelling and computer aided design a polynomial is usually represented in bernstein form forward error analysis for the computation of the derivative of a polynomial in bernstein form is performed the corresponding running error is also analyzed we have compared our results with those corresponding to a polynomial represented in power basis and using legendre and chebyshev forms since the conversions between the bernstein and the power bases and between bernstein and chebyshev polynomials were already studied in the literature for the sake of completeness the conversion between bernstein and chebyshev polynomials of the first kind is analyzed in an appendix

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

blowup dynamics for smooth data equivariant solutions to the critical schrodinger map problem we consider the energy critical schrodinger map problem with the 2 sphere target for equivariant initial data of homotopy index k 1 we show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm which generates finite time blowup solutions we give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

construction of type ii blow up solutions for the energy critical wave equation in dimension 5 abstract we consider the semilinear wave equation with focusing energy critical nonlinearity in space dimension n 5 t t u u u 4 3 u with radial data it is known 7 that a solution u t u which blows up at t 0 in a neighborhood in the energy norm of the family of solitons w decomposes in the energy space as u t t u t w t u 0 u 1 o 1 where lim t 0 t t 0 and u 0 u 1 h 1 l 2 we construct a blow up solution of this type such that the asymptotic profile u 0 u 1 is any pair of sufficiently regular functions with u 0 0 0 for these solutions the concentration rate is t t 4 we also provide examples of solutions with concentration rate t t 1 for 8 related to the behavior of the asymptotic profile near the origin

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

renormalization and blow up for the critical yang mills problem we consider the yangs mills equations in 4 1 dimensions this is the energy critical case and we show that it admits a family of solutions which blow up in finite time they are obtained by the spherically symmetric ansatz in the so 4 gauge group and result by rescaling of the instanton solution the rescaling is done via a prescribed rate which in this case is a modification of the self similar rate by a power of vertical bar log t vertical bar the powers themselves take any value exceeding 3 2 and thus form a continuum of distinct rates leading to blow up the methods are related to the authors previous work on wave maps and the energy critical semi linear equation however in contrast to these equations the linearized yang mills operator around an instanton exhibits a zero energy eigenvalue rather than a resonance this turns out to have far reaching consequences amongst which are a completely different family of rates leading to blow up logarithmic rather than polynomial corrections to the self similar rate c 2009 elsevier inc all rights reserved

on blowup of co rotational wave maps in odd space dimensions abstract we consider co rotational wave maps from the 1 d dimensional minkowski space into the d sphere for d 3 odd this is an energy supercritical model which is known to exhibit finite time blowup via self similar solutions based on a method developed by the second author and schorkhuber we prove the asymptotic nonlinear stability of the ground state self similar solution

gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space in this paper we study k equivariant wave maps from the hyperbolic plane into the 2 sphere as well as the energy critical equivariant su 2 s u 2 yang mills problem on 4 dimensional hyperbolic space the latter problem bears many similarities to a 2 equivariant wave map into a surface of revolution as in the case of 1 equivariant wave maps considered in 9 both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target then it is asymptotically stable in the energy space however for a stationary solution that covers a large enough portion of the target we prove that the schrodinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap as there is no a priori nonlinear obstruction to asymptotic stability this gives evidence for the existence of metastable states i e solutions with anomalously slow decay rates in these simple geometric models

convergent discretization of heat and wave map flows to spheres using approximate discrete lagrange multipliers we propose fully discrete schemes to approximate the harmonic map heat flow and wave maps into spheres the finite element based schemes preserve a unit length constraint at the nodes by means of approximate discrete lagrange multipliers satisfy a discrete energy law and iterates are shown to converge to weak solutions of the continuous problem comparative computational studies are included to motivate finite time blow up behavior in both cases

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

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

integrated semigroups and integrated solutions to abstract cauchy problems abstract concepts which have been developed in the theory of weakly continuous semigroups on dual banach spaces are extended to integrated semigroups the relation to integrated solutions of homogeneous and inhomogeneous abstract cauchy problems is established the results are illustrated for the wave equation in l2 rm

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

least supported bases and local linear independence we introduce the concept of least supported basis which is very useful for numerical purposes we prove that this concept is equivalent to the local linear independence of the basis for any given locally linearly independent basis we characterize all the bases of the space sharing the same property several examples for spline spaces are given

on collapsing ring blow up solutions to the mass supercritical nls we consider the nonlinear schr odinger equation i pa tu delta u u u p 1 0 in dimension n 2 and in the mass super critical and energy subcritical range 1 4n p min n 2n 2 5 for initial data u0 h1 with radial symmetry we prove a universal upper bound on the blow up speed we then prove that this bound is sharp and attained on a family of collapsing ring blow up solutions first formally predicted by gavish fibich and wang

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

rational approximation schemes for bi continuous semigroups abstract this paper extends the hille phillips functional calculus and rational approximations results due to r hersh t kato p brenner and v thomee to generators of bi continuous semigroups the method yields error estimates for rational time discretization schemes for such semigroups in particular for dual semigroups feller semigroups such as the ornstein uhlenbeck semigroup the heat semigroup semigroups induced by nonlinear flows implemented semigroups and evolution semigroups furthermore the results provide error estimates for a new class of inversion formulas for the laplace transform

on the focusing critical semi linear wave equation the wave equation partial derivative tt psi delta psi psi 5 0 in r 3 is known to exhibit finite time blowup for data of negative energy furthermore it admits the special static solutions phi x a 3a 1 4 1 a x 2 1 2 for all a 0 which are linearly unstable we view these functions as a curve in the energy space h over dot 1 x l 2 we prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well defined long time asymptotic behavior as the sum of a bulk term plus a scattering term moreover this family forms a co dimension one manifold m of small diameter in a suitable topology loosely speaking m acts as a center stable manifold with the curve phi a as an attractor in m

distribution semigroups and abstract cauchy problems we present a new definition of distribution semigroups covering in particular non densely defined generators we show that for a closed operator a in a banach space e the following assertions are equivalent a a generates a distribution semigroup b the convolution operator i a has a fundamental solution in d l e d where d denotes the domain of a supplied with the graph norm and i denotes the inclusion d e c a generates a local integrated semigroup we also show that every generator of a distribution semigroup generates a regularized semigroup introduction distribution semigroups in banach spaces and their generators were introduced by j l lions in 16 as a generalization of c0 semigroups and their generators like the generators of c0 semigroups the generators of the distribution semigroups in 16 are densely defined it was then shown in 16 that the densely defined closed linear operators a that generate a distribution semigroup are exactly those for which the convolution operator i a where i denotes the inclusion d a e has a fundamental solution treating cauchy problems as convolution equations has turned out to be very useful see 6 12 13 and 14 but in doing so the assumption that d a is dense in e is unnecessary in most of the results on the other hand in the last years there has been an interest in dealing with operators that are not densely defined starting perhaps with 7 reasons for this interest may be that it might not be so easy to decide whether a given operator is densely defined and the difficulties that arise when passing from e to e or l e so the question how the theory of distribution semigroups introduced in 16 can be extended to cover non densely defined generators in such a way that the above characterization via fundamental solutions holds seems natural it turns out that this indeed is possible in a way that even simplifies the original definitions given in 16 moreover the local strongly continuous representations see section 4 of the distribution semigroups thus obtained are exactly the local integrated semigroups for the distribution semigroups in the sense of 16 a relation to local integrated semigroups has been obtained in 23 theorem 5 6 see also 15 theorem 7 6 or 2 theorem 7 2 so the theory presented here received by the editors october 17 1995 and in revised form february 6 1997 1991 mathematics subject classification primary 47d03 34g10 47a10 46f10 c 1999 american mathematical society

the instability of bourgain wang solutions for the l2 critical nls we consider the two dimensional l 2 critical nonlinear schr odinger equation i partial tu delta u uu 2 0 in their pioneering 1997 work bourgain and wang have constructed smooth solutions which blow up in finite time t nabla u t l 2 sim 1 over t t and which display some decoupling between the regular and the singular part of the solution at blow up time we prove that this dynamic is unstable more precisely we show that any such solution with small super critical l 2 mass lies on the boundary of both h 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log log regime exhibited in work by f merle and p rapha e l we moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in f merle s 1992 it comm pure appl math paper in the critical mass case

on the foundation of bases of spline spaces the concept of foundation is a sharper concept than support which allows us to recognize spline bases suitable for design and computational purposes we present new results on least founded bases of univariate spaces in the case of tensor product spaces we find under very general conditions bases of minimally founded functions although we show that in contrast to the univariate case they are no least founded bases we also analyze related questions in a space of splines defined on angular regions of the plane

on fourier transforms of distribution semigroups abstract a one to one correspondence is established between fourier transforms of ultradistribution semigroups in the sense of beurling and some class of pseudoresolvents characterized by conditions concerning their domains of existence and growth

stability of stationary equivariant wave maps from the hyperbolic plane in this paper we initiate the study of equivariant wave maps from 2d hyperbolic space bbb h 2 into rotationally symmetric surfaces this problem exhibits markedly different phenomena than its euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain in particular when the target is bbb s 2 we find a family of equivariant harmonic maps bbb h 2 to bbb s 2 indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere these maps have energies taking all values between zero and the energy of the unique co rotational euclidean harmonic map q rm euc from bbb r 2 to bbb s 2 given by stereographic projection we prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere indeed we prove strichartz estimates for the operator obtained by linearizing around such a harmonic map however for harmonic maps with energies approaching the euclidean energy of q rm euc asymptotic stability via a perturbative argument based on strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator when the target is bbb h 2 we find a continuous family of asymptotically stable equivariant harmonic maps bbb h 2 to bbb h 2 with arbitrarily small and arbitrarily large energies this stands in sharp contrast to the corresponding problem on euclidean space where all finite energy solutions scatter to zero as time tends to infinity

comparative performance of complex valued b spline and polynomial models applied to iterative frequency domain decision feedback equalization of hammerstein channels complex valued cv b spline neural network approach offers a highly effective means for identifying and inverting practical hammerstein systems compared with its conventional cv polynomial based counterpart a cv b spline neural network has superior performance in identifying and inverting cv hammerstein systems while imposing a similar complexity this paper reviews the optimality of the cv b spline neural network approach advantages of b spline neural network approach as compared with the polynomial based modeling approach are extensively discussed and the effectiveness of the cv neural network based approach is demonstrated in a real world application more specifically we evaluate the comparative performance of the cv b spline and polynomial based approaches for the nonlinear iterative frequency domain decision feedback equalization nifddfe of single carrier hammerstein channels our results confirm the superior performance of the cv b spline based nifddfe over its cv polynomial based counterpart

semi implicit approximation of wave maps into smooth or convex surfaces a semi implicit scheme using linear finite elements for the approximation of wave maps into smooth or convex surfaces is devised and its stability is analyzed convergence is established for the case of the unit sphere as a target manifold which is unconditional in the case of 2 1 minkowski space numerical experiments illustrate the theoretical results

blow up for the critical generalized korteweg de vries equation i dynamics near the soliton we consider the quintic generalized korteweg de vries equation gkdv u t u xx u 5 x 0 which is a canonical mass critical problem for initial data in h1 close to the soliton in earlier works on this problem finite or infinite time blow up was proved for non positive energy solutions and the solitary wave was shown to be the universal blow up profile see 16 26 and 20 for well localized initial data finite time blow up with an upper bound on blow up rate was obtained in 18

equivariant wave maps in two space dimensions singularities of corotational wave maps from 1 2 dimensional minkowski space into a surface n of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into in consequence for noncompact target surfaces of revolution the cauchy problem for wave maps is globally well posed 2003 wiley periodicals inc

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

convoluted c cosine functions and semigroups relations with ultradistribution and hyperfunction sines abstract convoluted c cosine functions and semigroups in a banach space setting extending the classes of fractionally integrated c cosine functions and semigroups are systematically analyzed structural properties of such operator families are obtained relations between convoluted c cosine functions and analytic convoluted c semigroups introduced and investigated in this paper are given through the convoluted version of the abstract weierstrass formula which is also proved in the paper ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups several examples of operators generating convoluted cosine functions analytic convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors as an application it is proved that the polyharmonic operator 2 n n n acting on l 2 0 with appropriate boundary conditions generates an exponentially bounded k n convoluted cosine function and consequently an exponentially bounded analytic k n 1 convoluted semigroup of angle 2 for suitable exponentially bounded kernels k n and k n 1

global existence of smooth solutions for wave maps in de sitter spacetime abstract in this paper we study the problem of a harmonic map from de sitter spacetime into a geodesic on an n dimensional complete riemannian manifold we show that the global smooth solution exists for such wave map when the solution is a small perturbation along the geodesic solution by the method of weighted energy estimates

self similar solutions for the schrodinger map equation we study in this article the equivariant schrodinger map equation in dimension 2 from the euclidean plane to the sphere a family of self similar solutions is constructed this provides an example of regularity breakdown for the schrodinger map these solutions do not have finite energy and hence do not fit into the usual framework for solutions for data of infinite energy but small in some norm we build up associated global solutions

stable manifolds for all monic supercritical focusing nonlinear schrodinger equations in one dimension keywords critical schrodinger equation stable manifolds modulation theory spectral theory klein gordon equations non linear schrodinger blow up rate standing waves nonintegrable equations invariant manifolds stability theory solitary waves ground states instability reference epfl article 155183doi 10 1090 s0894 0347 06 00524 8 record created on 2010 11 19 modified on 2017 05 12

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

standing ring blow up solutions to the n dimensional quintic nonlinear schrodinger equation we consider the quintic nonlinear schr odinger equation in dimension n 3 this problem is energy critical in dimension n 3 and energy super critical for n 4 we prove the existence of a radially symmetric blow up mechanism with l2 concentration along the unit sphere of rn this singularity formation is moreover stable by smooth and radially symmetric perturbation of the initial data this result extends the result obtained for n 2 in 29 and is the first result of description of a singularity formation in the energy supercritical class for nls type problems our main tool is the proof of the propagation of regularity outside the blow up sphere in the presence a so called log log type singularity

b splines and optimal stability it is proved that among all nonnegative bases of its space the b spline basis is optimally stable for evaluating spline functions

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

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

on spectra of linearized operators for keller segel models of chemotaxis abstract we consider the phenomenon of collapse in the critical keller segel equation ks which models chemotactic aggregation of micro organisms underlying many social activities e g fruiting body development and biofilm formation also ks describes the collapse of a gas of self gravitating brownian particles we find the fluctuation spectrum around the collapsing family of steady states for these equations which is instrumental in the derivation of the critical collapse law to this end we develop a rigorous version of the method of matched asymptotics for the spectral analysis of a class of second order differential operators containing the linearized keller segel operators and as we argue linearized operators appearing in nonlinear evolution problems we explain how the results we obtain are used to derive the critical collapse law as well as for proving its stability

c semigroups and the abstract cauchy problem where by a solution u x we mean that u x e c 0 co x u t x e d u for t 2 0 and acp u x is satisfied it is well known that the acp is closely related to semigroup theory and covers a number of problems of interest to the applications see s 6 9 141 recently davies and pang 2 introduced exponentially bounded c semigroups and studied their connection with the acp other relating results are in tanaka ll later delaubenfels 3 investigated c semigroups that may not be exponentially bounded and their connection with the acp roughly speaking the results in 2 3 111 are as follows

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

slow blow up solutions for the h 1 mathbb r 3 critical focusing semilinear wave equation given 1 2 and 0 arbitrary we prove the existence of energy solutions of ttu u u5 0 0 1 in r3 1 which blow up exactly at r t 0 as t 0 these solutions are radial and of the form u t 1 2w t r r t inside the cone r t where t t 1 w r 1 r2 3 1 2 is the stationary solution of 0 1 and is a radiation term with r t x t 2 t x t 2 x t 6 dx 0 t 0 outside of the light cone there is the energy bound r t u x t 2 ut x t 2 u x t 6 dx 0 the regularity of u increases with as in our accompanying article on wave maps 10 the argument is based on a renormalization method for the soliton profile w r

global existence and blow up for harmonic map heat flow abstract for degree one equivariant maps on bounded domains the question of finite time blow up vs global existence of solutions to the harmonic map heat flow has been well studied in this paper we study the cauchy problem for degree m equivariant harmonic map heat flow from 2 1 dimensional space time into the 2 sphere with initial energy close to the energy of harmonic maps it is proved that solutions are globally smooth for m 4 whereas for m 1 we show that finite time singularities can form for this class of data

blow up dynamics for equivariant critical schrodinger maps for the schrodinger map equation u t u times triangle u rm in mathbb r 2 1 with values in s2 we prove for any nu 1 the existence of equivariant finite time blow up solutions of the form u x t phi lambda t x zeta x t where phi is a lowest energy steady state lambda t t 1 2 nu and zeta t is arbitrary small in dot h 1 cap dot h 2

on energy critical half wave maps into mathbb s 2 s 2 we consider the energy critical half wave maps equation begin aligned partial t mathbf u mathbf u wedge nabla mathbf u 0 end aligned for mathbf u 0 t times mathbb r rightarrow mathbb s 2 we give a complete classification of all traveling solitary waves with finite energy the proof is based on a geometric characterization of these solutions as minimal surfaces with not necessarily free boundary on mathbb s 2 in particular we discover an explicit lorentz boost symmetry which is implemented by the conformal mobius group on the target mathbb s 2 applied to half harmonic maps from mathbb r to mathbb s 2 complementing our classification result we carry out a detailed analysis of the linearized operator l around half harmonic maps mathbf q with arbitrary degree m geqslant 1 and consisting of m identical blaschke factors here we explicitly determine the nullspace including the zero energy resonances in particular we prove the nondegeneracy of mathbf q moreover we give a full description of the spectrum of l by finding all its l 2 eigenvalues and proving their simplicity furthermore we prove a coercivity estimate for l and we rule out embedded eigenvalues inside the essential spectrum our spectral analysis is based on a reformulation in terms of certain jacobi operators tridiagonal infinite matrices obtained from a conformal transformation of the spectral problem posed on mathbb r to the unit circle mathbb s finally we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half harmonic maps our spectral results also have potential applications to the half harmonic map heat flow which is the parabolic counterpart of the half wave maps equation

type ii blow up for the four dimensional energy critical semi linear heat equation abstract we consider the energy critical four dimensional semi linear heat equation t u u u 3 0 we show the existence of type ii finite time blow up solutions and give a sharp description of the corresponding singularity formation these solutions concentrate a universal bubble of energy in the critical topology u t r 1 q r t u in h 1 where the blow up profile is given by the talenti aubin soliton q r 1 1 r 2 8 and with speed t t t log t t 2 as t t our approach uses a robust energy method approach developed for the study of geometrical dispersive problems raphael and rodnianski 2012 18 merle et al 2011 15 and lies in the continuation of the study of the energy critical harmonic heat flow raphael and schweyer 2011 19 and the energy critical four dimensional wave equation hillaret and raphael 2010 5

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

the adiabatic limit of wave map flow on a two torus the s2 valued wave map flow on a lorentzian domain r where is any flat two torus is studied the cauchy problem with initial data tangent to the moduli space of holomorphic maps s2 is onsidered in the limit of small initial velocity it is proved that wave maps in this limit converge in a precise sense to geodesics in the moduli space of holomorphic maps with respect to the l2 metric this establishes in a rigorous setting a long standing informal conjecture of ward

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

richardson method and totally nonnegative linear systems we show that modified richardson method converges for any nonsingular totally nonnegative stochastic matrix for any choice of the parameter between 0 and 2 we present a variant of the modified richardson method that is convergent for any nonsingular totally nonnegative matrix we obtain the optimal parameter value for this method and give a procedure for estimating it numerical experiments are presented