remarks on the blow up rate for critical nonlinear schrodinger equation with harmonic potential abstract we consider the blow up solutions of the cauchy problem for the critical nonlinear schrodinger equation with a harmonic potential which models the attractive bose einstein condensate we establish the sharp lower and upper bounds of blow up rate as t t blow up time which improve the result of q liu y zhou j zhang upper and lower bound of the blow up rate for nonlinear schrodinger equation with a harmonic potential appl math comput 172 2006 1121 1132

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

existence and stability of a solution blowing up on a sphere for an l 2 supercritical nonlinear schrodinger equation we consider the quintic two dimensional focusing nonlinear schrodinger equation iut u u 4u which is l2 supercritical even though the existence of finite time blow up solutions in the energy space h1 is known very little is understood concerning the singularity formation numerics suggest the existence of a stable blow up dynamic corresponding to a self similar blowup at one point in space we prove the existence of a different type of dynamic and exhibit an open set among the h1 radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere this is the first result of an explicit description of a blow up dynamic in the l2 supercritical setting

control of rare intense events in spatiotemporally chaotic systems we address the problem of using feedback control for the purpose of suppressing rare intense events in spatially extended systems as an example we investigate the use of control to suppress turbulent spikes in the complex ginzburg landau equation in the limit of small dissipation we explore how information obtained by forecasting can be used to implement spatially and temporally localized control parameter changes and how control strength and cost are related to effectiveness in this framework the effects of model error and imperfect state measurement are also considered

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

on the blow up phenomenon of the critical nonlinear schrodinger equation abstract in this paper we consider the blow up phenomenon of critical nonlinear schrodinger equations in dimension 1d and 2d we define the minimal mass as the l 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism asymptotic compactness properties and l 2 concentration are proved the proof relies on linear and nonlinear profile decompositions

on an elliptic equation related to the blow up phenomenon in the nonlinear schrodinger equation this paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in n the equation is derived from the blow up problem in the non linear schrodinger equation

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

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

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

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

some remarks on the nonlinear schrodinger equation with fractional dissipation we consider the cauchy problem for the l2 critical focussing nonlinear schrodinger equation with a fractional dissipation according to the order of the fractional dissipation we prove the global existence or the existence of finite time blowup dynamics with the log log blow up speed for u t l2

on the role of quadratic oscillations in nonlinear schrodinger equations ii the l 2 critical case we consider a nonlinear semi classical schrodinger equation for which quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator the relevance of the nonlinearity was discussed by r carles c fermanian kammerer and i gallagher for l 2 supercritical power like nonlinearities and more general initial data the present results concern the l 2 critical case in space dimensions 1 and 2 we describe the set of non linearizable data which is larger due to the conformal invariance as an application we precise a result by f merle and l vega concerning finite time blow up for the critical schrodinger equation the proof relies on linear and nonlinear profile decompositions

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

rough blowup solutions to the l 2 critical nls we study the singularity formation for the cubic focusing l 2 critical nonlinear schrodinger equation on mathbb r 2 in a series of recent works merle and raphael have completely described the so called log log blowup regime and proven its stability in the energy space h 1 our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space l 2 by blending the merle raphael techniques with the quantitative i method developed by colliander keel staffilani takaoka and tao for the study of the cauchy problem for rough data we obtain the stability of the log log regime in h s for all s 0

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

determination of the blow up rate for a critical semilinear wave equation in this paper we determine the blow up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance

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 the stability of critical chemotactic aggregation we consider the two dimensional parabolic elliptic keller segel model of chemotactic aggregation for radially symmetric initial data we show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

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

on the exponential inequalities for negatively dependent random variables a number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors the proofs use the truncation technique together with the control of the bounded terms and unbounded terms in this paper we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables our results improve on the corresponding ones in the literature

geometric motives and the h topology the aim of this paper is to give a detailed proof of a comparison of voevodsky s categories of geometric motives with and without transfers respectively the latter category is defined by means of h topology introduced by voevodsky a topology essentially given by zariski coverings finite coverings and blowups