the global well posedness and spatial decay of solutions for the derivative complex ginzburg landau equation in h1 abstract the global well posedness for the cauchy problem of the derivative complex ginzburg landau equation is shown in the h1 critical and h1 subcritical cases a spatial decaying estimate of solutions in h 1 is also obtained and the decaying rate is independent of time t 0

on self similar singular solutions of the complex ginzburg landau equation we address the open problem of existence of singularities for the complex ginzburg landau equation using a combination of rigorous results and numerical computations we describe a countable family of self similar singularities our analysis includes the supercritical nonlinear schrodinger equation as a special case we also consider the problem of stability of these singularities 2001 john wiley sons inc

bounds for the solutions of the complex ginzburg landau equation in terms of the dispersion parameters abstract the diameter in the l norm of the global attractor of the complex ginzburg landau equation ut 1 i u ru 1 i u2 u is estimated by using weighted energy estimates for the solutions on the whole space r d for all parameters d and for which global existence is known we obtain the bound c d r 1 2 upper estimates for c are given which are polynomial in for d

limit behavior of saturated approximations of nonlinear schrodinger equation we consider the solutionu t of the saturated nonlinear schrodinger equation i partial u partial t delta u left u right 4 n u varepsilon left u right q 1 uandu 0 varphi 1 where n geqslant 2 varepsilon 0 1 4 n 0 such that mathop lim limits t to t left u t right h 1 infty whereu t is solution of i partial u partial t delta u left u right 4 n uandu 0 varphi 1 for 0 fixed u t is defined for all time we are interested in the limit behavior as 0 ofu t fort t in the case where there is no loss of mass inu at infinity in a sense to be made precise we describe the behavior ofu as goes to zero and we derive an existence result for a solution of 1 after the blow up timet in a certain sense nonlinear schrodinger equation with supercritical exponents are also considered

complex ginzburg landau equations as perturbations of nonlinear schrodinger equations a melnikov approach abstract we study the persistence of quasiperiodic and homoclinic solutions of generalized nonlinear schrodinger equations under ginzburg landau perturbations in this paper the first of a series melnikov criteria for the persistence of quasiperiodic and homoclinic solutions are derived directly from the governing partial differential equations via an averaging technique for families of tori of quasiperiodic solutions such as rotating waves and traveling waves that arise within critical sets of linear combinations of conserved functionals we find that usually only isolated tori will satisfy these selection criteria moreover in some simple cases these criteria are sufficient to conclude that a torus persists we also demonstrate the nonpersistence of solutions that are homoclinic to rotating waves under a broad class of ginzburg landau perturbations which satisfy a convexity condition

large deviations for the stochastic derivative ginzburg landau equation with multiplicative noise abstract this paper first proves the existence of a unique mild solution to the stochastic derivative ginzburg landau equation the fixed point theorem for the corresponding truncated equation is used as the main tool since we restrict our study to the one dimensional case it is not necessary to introduce another banach space and thus the estimates of the stochastic convolutions in the banach space are avoided secondly we also consider large deviations for the stochastic derivative ginzburg landau equation perturbed by a small noise since the underlying space considered is polish using the weak convergence approach we establish a large deviations principle by proving a laplace principle

dirichlet inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following complex ginzburg landau equation with cubic nonlinearity on r n for t 0 u t a i u b i u 2 u a b 0 under initial and dirichlet boundary conditions u x 0 h x for x u x t q x t on where h q are given smooth functions under suitable conditions we prove the existence of a global solution in h1 further this solution approaches to the solution of the nls limit under identical initial and boundary data as a b 0

the limit behavior of solutions for the cauchy problem of the complex ginzburg landau equation this paper is devoted to the study of the limit behavior as e 0 or e 0 and a 0 for the solutions of the cauchy problem of the complex ginzburg landau equation ut e u i u a i u u 0 u 0 x u0 x 4 n 4 n 2 0 0 s 0 1 and s n 2 2 moreover the sharp convergence rate in such spaces is also given 2002 john wiley sons inc

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

neumann inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following neumann inhomogeneous boundary value problem for the complex ginzburg landau equation on r n n 3 u t a i u b i u 2 u a b t 0 under initial condition u x 0 h x for x and neumann boundary condition u n k x t on where h k are given functions under suitable conditions we prove the existence of a global solution in h1

mass concentration phenomena for the l 2 critical nonlinear schrodinger equation in this paper we show that any solution of the nonlinear schr o dinger equation iu t delta u pm u frac 4 n u 0 which blows up in finite time satisfies a mass concentration phenomena near the blow up time our proof is essentially based on the bourgain s one cite mr99f 35184 which has established this result in the bidimensional spatial case and on a generalization of strichartz s inequality where the bidimensional spatial case was proved by moyua vargas and vega cite mr1671214 we also generalize to higher dimensions the results in keraani cite mr2216444 and merle and vega cite mr1628235

a dirichlet inhomogeneous boundary value problem for a generalized ginzburg landau equation abstract the ginzburg landau equation has been used as a simplified mathematical model for various pattern formation systems in mechanics physics and chemistry in this paper we study a generalized complex ginzburg landau equation in two spatial dimension with a fifth order nonlinear term and cubic terms containing spatial derivatives we prove a global existence and uniqueness theorem

existence and decay estimates of solutions to complex ginzburg landau type equations abstract this paper deals with the initial boundary value problem denoted by cgl for the complex ginzburg landau type equation u t i u i u q 1 u u 0 with initial data u 0 l p in the case 1 q 1 2 p n where is bounded or unbounded in r n 0 r there are a lot of studies on local and global existence of solutions to cgl including the physically relevant case q 3 and 0 this paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view the physically relevant case can be considered as a special case of the results moreover in the case 0 local and global existence of solutions with the decay estimate u t l p c 1 e c 2 t c 1 c 2 are positive constants is obtained under some conditions the key to the local existence is to construct a semigroup e t i and its l p l q estimate on the other hand the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with re v p 2 v i v

weakly nonlinear dissipative approximations of hyperbolic parabolic systems with entropy hyperbolic parabolic systems have spatially homogenous stationary states when the dissipation is weak one can derive weakly nonlinear dissipative approximations that govern perturbations of these constant states these approximations are quadratically nonlinear when the original system has an entropy the approximation is formally dissipative in a natural hilbert space we show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that hilbert space we also prove a weak strong uniqueness theorem for it in addition we give a kawashima type criterion for this approximation to be strictly dissipative we apply the theory to the compressible navier stokes system

superconvergence analysis of a two grid finite element method for ginzburg landau equation abstract in this paper a two grid method tgm is presented for the complex ginzburg landau equation in which the original nonlinear system is analyzed on the coarse grid based on the newton iteration method for the backward euler fully discrete scheme and then a simple linearized problem developed through taylor s expansion on the fine grid is solved by use of the character of the element and some special techniques the superclose estimation in the h1 norm is deduced for the tgm scheme furthermore the global superconvergent result is derived through interpolation postprocessing skill numerical results illustrate that the proposed tgm is very effective and its computing cost is much less than that of the traditional galerkin finite element method fem without loss of accuracy

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

a nonstiff euler discretization of the complex ginzburg landau equation in one space dimension a nonstiff discretization method is applied to the complex ginzburg landau cgl equation with periodic boundary conditions begin eqnarray partial t u 1 i nu partial xx 2 u ru 1 i mu u 2 u u t geq 0 end eqnarray this parabolic equation is discretized first in space by a truncated fourier series that is a finite fourier sum and then in time by an explicit euler method the exponential decay of fourier modes in both the original cgl equation and the resulting equation for the truncated fourier series independently from the number of fourier modes in the truncation is used in an essential way to prove the nonstiffness also dissipativity of the discretized equation is established and numerical results are discussed

finite dimensional behaviour for the derivative ginzburg landau equation in two spatial dimensions abstract in this paper we study the long time behaviour of the solutions for the generalized derivative ginzburg landau equation in two spatial dimensions we show the existence of the global attractor for this equation and establish the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractor

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

inviscid limits of the complex ginzburg landau equation in the inviscid limit the generalized complex ginzburg landau equation reduces to the nonlinear schrodinger equation this limit is proved rigorously with h1 data in the whole space for the cauchy problem and in the torus with periodic boundary conditions the results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of l2 in the focusing case in any spatial dimension furthermore optimal convergence rates are proved the proofs are based on estimates of the schrodinger energy functional and on gagliardo nirenberg inequalities

global attractors for the complex ginzburg landau equation abstract this paper concerns the long time behavior of the following complex ginzburg landau equations u t i u i u p 2 u u 0 without any restriction on p 2 under the assumptions 1 4 we first prove the well posedness of strong solutions for the complex ginzburg landau equations and then the existence of absorbing sets in l 2 h 0 1 l p and h 2 l 2 p 1 respectively for the semigroup s t t 0 generated by 1 1 1 3 is established finally we prove the existence of global attractors in l 2 and h 0 1 for the semigroup s t t 0 generated by 1 1 1 3 by the sobolev compactness embedding theorem and prove the existence of global attractor in l p for the semigroup s t t 0 generated by 1 1 1 3 using interpolation inequality

momentum estimates and ergodicity for the 3d stochastic cubic ginzburg landau equation with degenerate noise abstract in this paper we consider the ergodicity of invariant measures for the stochastic ginzburg landau equation with degenerate random forcing first we show the existence and pathwise uniqueness of strong solutions with h 1 initial data and then the existence of an invariant measure for the feller semigroup by the krylov bogoliubov method then in the case of degenerate additive noise using the notion of asymptotically strong feller property we prove the uniqueness of invariant measures for the transition semigroup

global well posedness for the averaged euler equations in two dimensions abstract we prove global well posedness of the two dimensional averaged euler or euler equations for initial potential vorticity of class l2 this model generalizes the one dimensional fokas fuchssteiner camassa holm equation which describes the propagation of unidirectional waves on the surface of shallow water as such it can be realized as a geodesic equation for the h1 metric on the lie algebra of vector fields moreover in two dimensions the model obeys an advection equation for the so called potential vorticity in close analogy to the vorticity form of the euler equations we construct solutions to the weak form of the potential vorticity equation by taking the inviscid limit of solutions to a system regularized by artificial viscosity since the streamfunction vorticity relation is of order four we can show uniqueness even for potential vorticities in l2

exact blow up solutions to the cauchy problem for the davey stewartson systems we present exact blow up solutions to the cauchy problem for the davey stewartson systems it is shown that for any prescribed blow up time there is an exact solution whose mass density converges to the dirac measure as time goes to the blow up time and that the solution extends beyond the blow up time and behaves like the free solution as time tends to infinity

approximate high order dynamic theory of a fluid layer in between two thick solids a thin layer of heavy fluid with bound faces is considered first for the sake of simplicity the effects of vortex and viscosity are neglected two cases of compressible and incompressible fluids are treated assuming that a thickness over wavelength ratio is a small parameter for the internal state of medium the relations between the quantities on the layer surfaces are deduced in contrast to the standard approach which expands the propagator matrix into power series of a small wave number the asymptotic integration of 3d equations and boundary conditions of fluid dynamics is performed respective relations are represented in a recurrent form and permit one to obtain the high order components of the displacements and pressure rather simply when considering two thick solids with a fluid in between this result is used to derive the so called impedance boundary conditions ibc with a relative asymptotic error up to tenth order tests show their validity until the first quasiresonance frequency of a layer so the analysis is not just long wave but essentially low frequency thus these ibc are applicable to reduce the dimension in the analysis of challenged multicomponent system of fluid coupled solids in a reasonable frequency range

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

a remark on the blowing up of solutions to the cauchy problem for nonlinear schrodinger equations on considere des solutions de iu t u u p 1 u u o u 0 ou x appartient a un domaine lisse cr n et on demontre que sous de bonnes conditions sur p n et u 0 h 2 h 0 1 u t l 2 eclate en un temps fini

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

l2 concentration of blow up solutions for the nonlinear schrodinger equation with critical power nonlinearity abstract we consider the solution of the nonlinear schrodinger equation i u t u f u and u 0 where u is defined on 0 t rn and f is a nonlinear complex valued function we consider the case of the critical power that is f z z 4 n z as z in a sense which will be made precise in addition we suppose that u t blows up in the norm h1 at time t we prove that u t has no limit in l2 as t t in the particular case where has a spherical symmetry we further show a phenomenon of l2 concentration at the origin

self focusing on bounded domains the critical nonlinear schrodinger equation nls on bounded domains models the propagation of cw laser beams in hollow core fibers unlike the nls on unbounded domains which models propagation in bulk media the ground state waveguide solutions are stable and the condition of critical power for singularity formation is generically sharp

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

on the initial value problem for the generalized two dimensional ginzburg landau equation abstract the ginzburg landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics physics and chemistry most work so far concentrates on ginzburg landau type equations with one spatial dimension 1d in this paper the authors study a complex generalized ginzburg landau equation with two spatial dimensions 2d sufficient conditions for the existence and uniqueness of global solutions for the initial value problem of the generalized 2d ginzburg landau equation are obtained this rigorously establishes the foundation for further investigation of this type of model

symmetry and chaos in the complex ginzburg landau equation ii translational symmetries abstract the complex ginzburg landau cgl equation on a one dimensional domain with periodic boundary conditions has a number of different symmetries and solutions of the equation may or may not be fixed by the action of these symmetries we investigate the stability of chaotic solutions that are spatially periodic with period l with respect to subharmonic perturbations that have a spatial period kl for some integer k 1 this is done by considering the isotypic decomposition of the space of solutions and finding the dominant lyapunov exponent associated with each isotypic component we find a region of parameter space in which there exist chaotic solutions with spatial period l and homogeneous neumann boundary conditions that are stable with respect to perturbations of period 2l on varying the parameters it is possible to arrange for this solution to become unstable to perturbations of period 2l while remaining chaotic leading to a supercritical subharmonic blowout bifurcation for a large number of parameter values checked chaotic solutions with spatial period l were found to be unstable with respect to perturbations of period 3l we conclude that while periodic boundary conditions are often convenient mathematically we would not expect to see chaotic spatially periodic solutions forming starting with an arbitrary non periodic initial condition

large orbits in coprime actions of solvable groups let g be a solvable group of automorphisms of a finite group k if g and k are coprime then there exists an orbit of g on k of size at least g it is also proved that in a solvable group the largest normal subgroup is the intersection of at most three hall subgroups

blow up profile for the complex ginzburg landau equation abstract we construct a solution to the complex ginzburg landau equation which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude two major difficulties arise in the proof the linearized operator around the profile is not self adjoint and it has a second neutral mode in the last section the interpretation of the parameters of the finite dimensional problem in terms of the blow up time and the blow up point gives the stability of the constructed solution with respect to perturbations in the initial data

existence of suitable weak solutions of complex ginzburg landau equations and properties of the set of singular points in this paper we consider the supercritical complex ginzburg landau equation we discuss the existence of suitable weak solution in where is a bounded domain in rn or the whole space we also discuss the properties of the set of the singular points of the suitable weak solution in rn which means that the possible singular points are located in a bounded ball for any given time and there is no singular point on the whole space after limited time

high order approximate low frequency theory of elastic anisotropic lining and coating a problem of the dynamic behavior of an elastic layer coupled to one or two thick elastic solids is considered all the materials may possess a general anisotropy and the layer is assumed to be thin enough with respect to the characteristic wavelength introducing the asymptotic power series with respect to the thickness over wavelength ratio for the main quantities and using the asymptotic integration method the displacements and stresses on the layer surfaces are related thus the so called impedance boundary conditions ibc are deduced for three cases for a coated substrate with given displacements or with given stresses on the surface and for two substrates with a layer in between in contrast to previous papers these ibc are obtained for the most general situation with the asymptotic accuracy up to the sixth order uniform with respect to the representation of the displacements and stresses presented theory can be used for studying the surface and interface phenomena as well as for calculating fie

theory of singular vortex solutions of the nonlinear schrodinger equation we present a systematic study of singular vortex solutions of the critical and supercritical two dimensional nonlinear schrodinger equation in particular we study the critical power for collapse and the asymptotic blowup profile of singular vortices

singular limits of the klein gordon equation we establish the singular limits including semiclassical nonrelativistic and nonrelativistic semiclassical limits of the cauchy problem for the modulated defocusing nonlinear klein gordon equation for the semiclassical limit hbar to 0 we show that the limit wave function of the modulated defocusing cubic nonlinear klein gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation the nonrelativistic limit c of the modulated defocusing nonlinear klein gordon equation is the defocusing nonlinear schrodinger equation the nonrelativistic semiclassical limit hbar to 0 c hbar alpha to infty for some 0 of the modulated defocusing cubic nonlinear klein gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function

large varvec p prime orbits for varvec p p nilpotent linear groups let g be a p nilpotent linear group on a finite vector space v of characteristic p suppose that g v is odd let p be a sylow p subgroup of g we show that there exist vectors v 1 and v 2 in v such that c g v 1 cap c g v 2 p a striking conjecture of malle and navarro offers a simple global criterion for the nilpotence in the sense of broue and puig of a p block of a finite group our result implies that this conjecture holds for groups of odd order

exponential attractor for the 3d ginzburg landau type equation abstract in this paper we consider a complex ginzburg landau type equation with periodic initial value condition in three spatial dimensions sufficient conditions for existence and uniqueness of global solutions are obtained by uniform a priori estimates of solutions furthermore the existence of a global attractor with finite hausdorff and fractal dimensions is proved finally the existence of the exponential attractor is proved

asymptotic evaluation of the pulse train radiated by an angled beam and fluid coupled rectangular ultrasonic transducer the near field underneath the ultrasonic probe fluid coupled to an isotropic solid is approximated in the frequency domain by a closed form asymptotic solution the approximation is based on the problem decomposition and uses the stationary phase method evaluating the response to an equivalent surface source this results in a sum of contributions each dominating in a specific geometrical region the main beam or a side lobe the transitional zones are also described the pulse trains are computed using the harmonic synthesis and compared with the results obtained by direct calculation of fourier integrals it is shown that the asymptotic approach permits us to elucidate the physics of problem and leads to a numerical algorithm which is about 104 times faster than the direct computations the near field underneath the ultrasonic probe fluid coupled to an isotropic solid is approximated in the frequency domain by a closed form asymptotic solution the approximation is based on the problem decomposition and uses the stationary phase method evaluating the response to an equivalent surface source this results in a sum of contributions each dominating in a specific geometrical region the main beam or a side lobe the transitional zones are also described the pulse trains are computed using the harmonic synthesis and compared with the results obtained by direct calculation of fourier integrals it is shown that the asymptotic approach permits us to elucidate the physics of problem and leads to a numerical algorithm which is about 104 times faster than the direct computations

well posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials abstract the current paper is devoted to the well posedness and dynamics of the stochastic coupled fractional ginzburg landau equation which describes a class of nonlinear optical fiber materials with active and passive coupled cores by the commutation estimates and fourier galerkin approximation the global existence of weak solutions and the uniqueness criterion are established moreover the existence of a global attractor is shown finally we consider the long time behavior of the stochastic coupled fractional ginzburg landau equation scfgl with multiplicative noise and prove the existence of a random attractor for the random dynamical system generated by the scfgl equation

dynamics of the 3 d fractional complex ginzburg landau equation abstract we study the initial boundary value problem of the fractional complex ginzburg landau equation in three spatial dimensions with the dissipative effect given by a fractional laplacian a priori estimates are derived when the nonlinearity satisfies certain growth conditions using galerkin s method the existence of a global smooth solution is established uniqueness is also proved furthermore the existence of a global attractor is proved and estimates of the hausdorff and fractal dimensions for the global attractor are obtained

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

linearized adi schemes for two dimensional space fractional nonlinear ginzburg landau equation abstract space and time approximations for two dimensional space fractional complex ginzburg landau equation are examined the schemes under consideration are discreted by the second order backward differential formula bdf2 in time and two classes of the fractional centered finite difference methods in space a linearized technique is employed by the extrapolation we prove the unique solvability and stability for both numerical methods the convergence of both numerical methods is analyzed at length utilizing the energy argument and the convergence orders under the optimal step size ratio are o 2 h 2 and o 2 h 4 in the sense of the discrete l 2 norm where is the time step size h max h x h y and h x h y are spatial grid sizes in the x direction and y direction respectively in addition we construct a multistep alternating direction implicit adi scheme and a multistep compact adi scheme based on bdf2 for the efficiently numerical implementation finally numerical examples are carried out to verify our theoretical results

time space fractional stochastic ginzburg landau equation driven by fractional brownian motion abstract the current paper is devoted to the time space fractional ginzburg landau equation driven by fractional brownian motion the spatial temporal regularity of the nonlocal stochastic convolution is firstly established and then the existence and uniqueness of the global mild solution are obtained by the banach fixed point theorem and mittag leffler functions finally the numerical simulations for the time fractional stochastic ginzburg landau equation are provided to verify the mathematical analysis results

asymptotic behavior for non autonomous fractional stochastic ginzburg landau equations on unbounded domains in this paper we first prove the existence and uniqueness of tempered pullback random attractors for a non autonomous stochastic fractional ginzburg landau equation driven by multiplicative noise with 0 1 in l2r3 then we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero due to the lack of the compactness of sobolev embeddings on unbounded domains we establish the pullback asymptotic compactness of solutions in l2 r3 by the tail estimates of solutions

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

stochastic nonlinear schrodinger equations abstract this paper is devoted to the well posedness of stochastic nonlinear schrodinger equations in the energy space h 1 r d and is a continuation of our recent work barbu et al 2014 we allow general complex coefficients in the noise thus covering both the conservative and non conservative case the latter being important to include quantum measurement effects in the underlying physical model we consider both focusing and defocusing nonlinearities and prove global well posedness in h 1 r d including also the pathwise continuous dependence on initial conditions with exponents of the nonlinearity in exactly the same sharp range as in the deterministic case in particular in the special conservative case this work improves earlier results in de bouard and debussche 2003 where the critical exponents could not be reached for d 6 in the defocusing and for d 4 in the focusing case respectively moreover the local existence uniqueness and blowup alternative are also established for the energy critical case the approach presented here is mainly based on the rescaling approach already used in barbu et al 2014 to study the l 2 case and also on the strichartz estimates established in marzuola et al 2008 for large perturbations of the laplacian

limit of the solution of a nonlinear schrodinger equation at blow up time we consider the solution of the equation i v t and 0 where v is defined on 0 t rn and is a nonlinear complex valued function let us denote the function x v t x by v t we suppose that v blows up in the norm h1 at time t under suitable assumptions on and we prove the existence of a limit for v t when t t in appropriate spaces furthermore we specify the behavior of v near blow up time

global well posedness for the lake equations abstract we prove global well posedness for the lake equations these equations arise to leading order in a low aspect ratio low froude number i e low wave speed and very small wave amplitude expansion of the three dimensional incompressible euler equations in a basin with a free upper surface and a spatially varying bottom topography well posedness means that there exists a solution that it is unique and that it depends continuously on the data i e the initial condition and the bottom topography our approach follows the works of yudovitch and bardos in constructing the solutions as the inviscid limit of solutions to a system with artificial viscosity which is the analog of the navier stokes with respect to the euler equations one of the main assumptions in the present work is a nondegenerate bottom topography

asymptotics for the generalized two dimensional ginzburg landau equation abstract in this paper the authors have studied a generalized ginzburg landau equation in two spatial dimensions 2d they have shown that this equation under periodic boundary conditions has the maximal attractor with finite hausdorff dimension this rigorously establishes the foundation for further investigation of this type of model

existence of the weak solution of coupled time dependent ginzburg landau equations in this paper we investigate the existence of weak solutions of the coupled time dependent ginzburg landau equations and establish the global existence of weak solutions to the equations by galerkin method and compactness theory

global existence of solutions to the derivative 2d ginzburg landau equation abstract in this paper we study a complex derivative ginzburg landau equation with two spatial variables 2d we obtain sufficient conditions for the existence and uniqueness of global solutions for the initial boundary value problem of the derivative 2d ginzburg landau equation and improve the known results

the time periodic solution to a 2d generalized ginzburg landau equation abstract in this paper we study a 2d generalized ginzburg landau equation with a periodic boundary condition the existence and uniqueness of a time periodic solution to this equation is proved

long time behaviour for generalized complex ginzburg landau equation in this paper the two dimensional generalized complex ginzburg landau equation cgl ut u u 1 i u 2u 1 i u 2 u 1 u 2u 2 u 2 is studied the existence of global attractor for this equation with periodic boundary condition is established and upper bounds of hausdorff and fractal dimensions of attractor are obtained

length scales in solutions of the complex ginzburg landau equation abstract we generalise and in certain cases improve on previous a priori estimates of sobolev norms of solutions to the generalised complex ginzburg landau equation a set of dynamic length scales based on ratios of these norms is defined we are able to derive lower bounds for time averages and long time limits of these length scales the bounds scale like the inverses of our l bounds

standing waves of the complex ginzburg landau equation abstract we prove the existence of nontrivial standing wave solutions of the complex ginzburg landau equation t e i e i k on a bounded domain with dirichlet boundary conditions our result requires that 0 be sufficiently small

blow up for the 1d nonlinear schrodinger equation with point nonlinearity i basic theory abstract we consider the 1d nonlinear schrodinger equation nls with focusing point nonlinearity 0 1 i t x 2 p 1 0 where x is the delta function supported at the origin in this work we show that 0 1 shares many properties in common with those previously established for the focusing autonomous translationally invariant nls 0 2 i t p 1 0 the critical sobolev space h c for 0 1 is c 1 2 1 p 1 whereas for 0 2 it is c d 2 2 p 1 in particular the l 2 critical case for 0 1 is p 3 we prove several results pertaining to blow up for 0 1 that correspond to key classical results for 0 2 specifically we 1 obtain a sharp gagliardo nirenberg inequality analogous to weinstein 44 2 apply the sharp gagliardo nirenberg inequality and a local virial identity to obtain a sharp global existence blow up threshold analogous to weinstein 44 glassey 17 in the case c 0 and duyckaerts holmer roudenko 12 guevara 18 and fang xie cazenave 13 for 0 c 1 3 prove a sharp mass concentration result in the l 2 critical case analogous to tsutsumi 43 merle tsutsumi 36 and 4 show that minimal mass blow up solutions in the l 2 critical case are pseudoconformal transformations of the ground state analogous to merle 28

localized estimates and cauchy problem for the logarithmic complex ginzburg landau equation we prove that the localization of the l2 and similar estimates for the complex ginzburg landau equation 1 known to hold in the power case g also holds in the logarithmic case g log for large with 2 as a consequence the theory of the cauchy problem in local spaces given in a previous paper also extends to the logarithmic case

blow up of solutions to the nonlinear schrodinger equations on manifolds in this paper we partially settle down the long standing open problem of the finite time blow up property about the nonlinear schrodinger equations on some riemannian manifolds such as the standard 2 sphere s2 and the hyperbolic 2 space h2 1 using the similar idea we establish such blow up results on higher dimensional standard sphere and hyperbolic n space extensions to n dimensional riemannian warped product manifolds with n 2 are also given

existence of standing waves for the complex ginzburg landau equation abstract we study the existence of standing wave solutions of the complex ginzburg landau equation gl t e i i e i 0 in r n where 0 n 2 4 0 and r we show that for any 2 2 there exists e 0 such that gl has a non trivial standing wave solution if e analogous result is obtained in a ball r n for 1 where 1 is the first eigenvalue of the laplace operator with dirichlet boundary conditions

existence and upper semicontinuity of random attractors for non autonomous fractional stochastic ginzburg landau equations this paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional ginzburg landau equations driven by multiplicative noise with 0 1 we first transform the stochastic equation into a random equation the solutions of which generate a random dynamical system consequently we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain finally we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

on the blow up solutions for the nonlinear fractional schrodinger equation abstract this paper is dedicated to the blow up solutions for the nonlinear fractional schrodinger equation arising from pseudorelativistic boson stars first we compute the best constant of a gg n inequality by the profile decomposition theory and variational arguments then we find the sharp threshold mass of the existence of finite time blow up solutions finally we study the dynamical properties of finite time blow up solutions around the sharp threshold mass by giving a refined compactness lemma

concentration properties of blow up solutions and instability results for zakharov equation in dimension two part ii we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right

the cauchy problem in local spaces for the complex ginzburg landau equation ii contraction methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

the orbital stability of the ground states and the singularity formation for the gravitational vlasov poisson system we study the gravitational vlasov poisson system f t v cdot nabla x f e cdot nabla v f 0 0 where e x nabla x phi x delta x phi rho x rho x int mathbb r n f x v rm d x rm d v in dimension n 3 4 in dimension n 3 where the problem is subcritical we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space this implies in particular the orbital stability in the energy space of the spherically symmetric polytropes and improves the nonlinear stability results obtained for this class in 11 16 19 in dimension n 4 where the problem is l1 critical we obtain the polytropic steady states as best constant minimizers of a suitable sobolev type inequality relating the kinetic and the potential energy we then derive using an explicit pseudo conformal symmetry the existence of critical mass finite time blow up solutions and prove more generally a mass concentration phenomenon for finite time blow up solutions this is the first result of description of a singularity formation in a vlasov setting the global structure of the problem is reminiscent of the one for the focusing nonlinear schrodinger equation iu t u u p 1u in the energy space h 1 mathbb r n

global well posedness and scattering for the defocusing mass critical nonlinear schrodinger equation for radial data in high dimensions we establish global well posedness and scattering for solutions to the defocusing mass critical pseudoconformal nonlinear schrodinger equation iut u u 4 nu for large spherically symmetric lx2 rn initial data in dimensions n 3 after using the concentration compactness reductions in 32 to reduce to eliminating blow up solutions that are almost periodic modulo scaling we obtain a frequency localized morawetz estimate and exclude a mass evacuation scenario somewhat analogously to 10 23 36 in order to conclude the argument

energy concentration of the focusing energy critical fnls abstract we consider the fractional nonlinear schrodinger equation fnls with non local dispersion and focusing energy critical hartree type nonlinearity x 2 u 2 u we consider the energy concentration phenomena of radial blowup solutions near the maximal existence time we use the concentration compactness approach of 19 for confined case and kinetic energy trapping approach of 21 for unconfined case