stability and instability of periodic travelling wave solutions for the critical korteweg de vries and nonlinear schrodinger equations abstract in this paper we establish new results about the existence stability and instability of periodic travelling wave solutions related to the critical korteweg de vries equation u t 5 u 4 u x u x x x 0 and the critical nonlinear schrodinger equation i v t v x x v 4 v 0 the periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario the stability approach is based on the theory developed by angulo natali in j angulo f natali positivity properties of the fourier transform and the stability of periodic travelling wave solutions siam j math anal 40 2008 1123 1151 for positive periodic travelling wave solutions associated to dispersive evolution equations of korteweg de vries type the instability approach is based on an extension to the periodic setting of arguments found in bona souganidis strauss j l bona p e souganidis w a strauss stability and instability of solitary waves of korteweg de vries type proc roy soc london ser a 411 1987 395 412 regarding the critical schrodinger equation stability instability theories similar to the critical korteweg de vries equation are obtained by using the classical grillakis shatah strauss theory in m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry ii j funct anal 94 1990 308 348 m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry i j funct anal 74 1987 160 197 the arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations

optimal control of an eddy current problem with a dipole source abstract this paper is concerned with the analysis of a class of optimal control problems governed by a time harmonic eddy current system with a dipole source which is taken as the control variable a mathematical model is set up for the state equation where the dipole source takes the form of a dirac mass located in the interior of the conducting domain a non standard approach featuring the fundamental solution of a curl curl id operator is proposed to address the well posedness of the state problem leading to a split structure of the state field as the sum of a singular part and a regular part the aim of the control is the best approximation of desired electric and magnetic fields via a suitable l 2 quadratic tracking cost functional here special attention is devoted to establishing an adjoint calculus which is consistent with the form of the state variable and in this way first order optimality conditions are eventually derived

invariance of white noise for kdv on the line we consider the korteweg de vries equation with white noise initial data posed on the whole real line and prove the almost sure existence of solutions moreover we show that the solutions obey the group property and follow a white noise law at all times past or future as an offshoot of our methods we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting

an alternative daugavet property abstract we introduce a strictly weaker version of the daugavet property as follows a banach space x has this alternative daugavet property adp in short if the norm identity ade max 1 id t 1 t holds for all rank one operators t x x in such a case all weakly compact operators on x also satisfy ade we give some geometric characterizations of the alternative daugavet property in terms of the space and its successive duals we prove that the adp is stable for c0 l1 and l sums and characterize when some vector valued function spaces have the property finally we show that a c algebra or the predual of a von neumann algebra has the adp if and only if its atomic projection respectively the atomic projection of the algebra are central we also establish some geometric properties of jb triples and characterize jb triples possessing the adp and the daugavet property

maximal parabolic regularity for divergence operators including mixed boundary conditions abstract we show that elliptic second order operators a of divergence type fulfill maximal parabolic regularity on distribution spaces even if the underlying domain is highly non smooth the coefficients of a are discontinuous and a is complemented with mixed boundary conditions applications to quasilinear parabolic equations with non smooth data are presented

the daugavet equation for polynomials on c algebras abstract in this paper we study the daugavet equation and the alternative daugavet equation for polynomials on c algebras we study separately the cases in which the c algebras are commutative and non commutative in these cases we obtain necessary and sufficient conditions on such c algebras in order that certain polynomial classes satisfy the daugavet equation and the alternative daugavet equation

a priori error estimates for three dimensional parabolic optimal control problems with pointwise control in this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise dirac type control in space on three dimensional domains the two dimensional case was treated in d leykekhman and b vexler siam j numer anal 51 2013 pp 2797 2821 however the three dimensional case is technically much more involved to approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we establish mathcal o sqrt k h order of convergence rate for the control in the l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou siam j control optim 52 2014 pp 97 119 and in addition does not require any relationship between the time step k and the mesh size h the main technical tools are discrete maximal parabolic regularity results and the best approximation type estim

second order and stability analysis for state constrained elliptic optimal control problems with sparse controls an optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state the main novelty of the paper is the presence of the l 1 norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions second order sufficient optimality conditions are analyzed they are applied to show the convergence of optimal solutions for vanishing l 2 regularization parameter for the control the associated convergence rate is estimated

a convex penalty for switching control of partial differential equations abstract a convex penalty for promoting switching controls for partial differential equations is introduced such controls consist of an arbitrary number of components of which at most one should be simultaneously active using a moreau yosida approximation a family of approximating problems is obtained that is amenable to solution by a semismooth newton method the efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

statistical mechanics of nonlinear wave equations 4 cubic schrodinger the cubic schrodinger equation is considered on the circle both in the de focussing and the focussing case the existence of the flow is proved together with the invariance of the appropriate gibbsian measure namely the petit canonical measure in the defocussing case and the micro canonical measure in the focussing case

on the cr structure of compact group orbits associated with bounded symmetric domains let e be a complex vector space of finite dimension and let k gl e be a compact connected subgroup then for fixed a e the orbit k k a is a real analytic submanifold of e that inherits various structures from e for instance choosing a k invariant positive definite inner product x y on e makes k a riemannian manifold on which k acts transitively by isometries on the other hand k inherits from e a cauchy riemann structure cr structure that is given by the distribution of the maximal complex subspaces hx k tx k itx k of the real tangent spaces tx k e x k together with the complex structure on every hx k multiplication by i the subspace hx k is called the holomorphic tangent space to k at x see 9 and 4 as general references for cr manifolds of interest for the geometry of the orbit k k a with respect to its cr structure is the study of the cr functions or more generally cr mappings on k i e of smooth functions f k c that satisfy the tangential cauchy riemann differential equations in the sense that the restriction of the differential df to every holomorphic tangent space is complex linear for instance all holomorphic functions defined in an open neighbourhood of k e give by restriction real analytic cr functions on k actually we deal with the more general continuous cr functions on k which satisfy by definition the tangential cr equations in the distribution sense or equivalently which are locally uniform limits of sequences of smooth cr functions due to the approximation theorem of baouenditreves 6 in this context it is of interest to determine the space of all

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

concentration of the invariant measures for the periodic zakharov kdv nls and gross piatevskii equations in 1d and 2d abstract this paper concerns gibbs measures for some nonlinear pde over the d torus t d the hamiltonian h t d u 2 t d u p has canonical equations with solutions in n u l 2 t d u 2 n this n is a parameter in quantum field theory analogous to the number of particles in a classical system for d 1 and 2 p 6 n supports the gibbs measure d u z 1 e h u x t d u x which is normalized and formally invariant under the flow generated by the pde the paper proves that n l 2 is a metric probability space of finite diameter that satisfies the logarithmic sobolev inequalities for the periodic kdv the focussing cubic nonlinear schrodinger equation and the periodic zakharov system for suitable subset of n a logarithmic sobolev inequality also holds in the critical case p 6 for d 2 the gross piatevskii equation has h t 2 u 2 t 2 v u 2 u 2 for a suitable bounded interaction potential v and the gibbs measure lies on a metric probability space h s which satisfies lsi in the above cases d is the limit in l 2 transportation distance of finite dimensional n n given by fourier sums

invariant measures for nls in infinite volume we study gibbs measures and invariant dynamics for 1d defocusing nls iut uxx u u u p 2 0 when the period l for all p weak solutions are typically holder continuous on bounded space time regions if p 4 uniqueness and continuous dependence on the initial data may be shown

optimal control of pdes in a complex space setting application to the schrodinger equation in this paper we discuss optimality conditions for abstract optimization problems over complex spaces we then apply these results to optimal control problems with a semigroup structure as an application we detail the case when the state equation is the schrodinger one with pointwise constraints on the bilinear control we derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly

a complexity approach to the soliton resolution conjecture the soliton resolution conjecture is one of the most interesting open problems in the theory of nonlinear dispersive equations roughly speaking it asserts that a solution with generic initial condition converges to a finite number of solitons plus a radiative term in this paper we use the complexity of a finite object a notion introduced in algorithmic information theory to show that the soliton resolution conjecture is equivalent to the analogous of the second law of thermodynamics for the complexity of a solution of a dispersive equation

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

directional sparsity in optimal control of partial differential equations we study optimal control problems in which controls with certain sparsity patterns are preferred for time dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval the approach uses a nondifferentiable cost functional to implement the sparsity requirements additionally bound constraints for the optimal controls can be included we study the resulting problem in appropriate function spaces and present two solution methods of newton type based on different formulations of the optimality system using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

an additional gibbs state for the cubic schrodinger equation on the circle an invariant gibbs state for the nonlinear schrodinger equation on the circle was constructed by bourgain 2 3 and mckean 6 out of the basic hamiltonian using a trigonometric cutoff the cubic nonlinear schrodinger equation is a completely integrable system having an infinite number of additional integrals of motion in this paper we construct the second invariant gibbs state from one of these additional integrals for the cubic nls on the circle this additional gibbs state is singular with respect to the gibbs state previously constructed from the basic hamiltonian our approach employs the ablowitz ladik system a completely integrable discretization of the cubic schrodinger equation 2001 john wiley sons inc

state spaces of jb triples an atomic decomposition is proved for banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process this is then applied to yield under appropriate assumptions geometric characterizations up to isometry of the unit ball of the dual space of a jb triple and up to complete isometry of one sided ideals in c algebras

adaptive finite element method for parabolic equations with dirac measure abstract in this paper we study the adaptive finite element method for parabolic equations with dirac measure two kinds of problems with separate measure data in time and measure data in space are considered it is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the dirac measure and thus fit to adaptive fem for space discretization and variable time steps for time discretization for both cases we use piecewise linear and continuous finite elements for the space discretization and backward euler scheme or equivalently piecewise constant discontinuous galerkin method for the time discretization the a posteriori error estimates based on energy and l 2 norms for the fully discrete problems are then derived to guide the adaptive procedure numerical results are provided at the end of the paper to support our theoretical findings

optimal control of semilinear elliptic equations in measure spaces optimal control problems in measure spaces governed by semilinear elliptic equations are considered first order optimality conditions are derived and structural properties of their solutions in particular sparsity are discussed necessary and sufficient second order optimality conditions are obtained as well on the basis of the sufficient conditions stability of the solutions is analyzed highly nonlinear terms can be incorporated by utilizing an l infty omega regularity result for solutions of the first order necessary optimality conditions

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

on maximal parabolic regularity for non autonomous parabolic operators abstract we consider linear inhomogeneous non autonomous parabolic problems associated to sesquilinear forms with discontinuous dependence of time we show that for these problems the property of maximal parabolic regularity can be extrapolated to time integrability exponents r 2 this allows us to prove maximal parabolic l r regularity for discontinuous non autonomous second order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations

optimal control of semilinear parabolic equations by bv functions optimal control problems for semilinear parabolic equations with control costs involving the total bounded variation seminorm are analyzed this choice of control cost favors optimal controls which are piecewise constant and it penalizes the number of jumps it is an appropriate choice if a simple structure of the optimal controls is desired which however is still sufficiently flexible so that good tracking properties can be maintained existence of optimal controls necessary and sufficient optimality conditions and sparsity properties of the derivatives are obtained convergence of a finite element approximation is analyzed and numerical examples illustrating structural properties of the optimal controls are provided

optimal a priori error estimates for an elliptic problem with dirac right hand side it is well known that finite element solutions for elliptic pdes with dirac measures as source terms converge due to the fact that the solution is not in h 1 suboptimal in classical norms a standard remedy is to use graded meshes then quasioptimality i e optimal up to a log factor for low order finite elements can be recovered e g in the l 2 norm here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi uniform meshes in an l 2 seminorm the seminorm is defined as an l 2 norm on a fixed subdomain which excludes the locations of the delta source terms our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error e g in dimension reduced settings or is not the quantity of interest e g in optimal control problems the quasi optimal and optimal order a priori bounds are obtained recursively by using aub

spike controls for elliptic and parabolic pdes abstract we analyze the use of measures of the minimal norm to control elliptic and parabolic equations we prove the sparsity of the optimal control in the parabolic case we prove that the solution of the optimization problem is a borel measure supported in a set of lebesgue measure zero in both cases the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations we also analyze the corresponding dual problem

stabilization by sparse controls for a class of semilinear parabolic equations stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional this formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state such a property is not present for commonly chosen control mechanisms to establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons which is otherwise not available

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

continuity of the flow of kdv with regard to the wasserstein metrics and application to an invariant measure abstract in this paper we prove the continuity of the flow of kdv on spaces of probability measures with respect to a combination of wasserstein distances on h s s 0 and l 2 we are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined

statistical mechanics of the periodic benjamin ono equation the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations

weakly nonlinear schrodinger equation with random initial data it is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data while for the general case a proof of the kinetic limit remains open we report on first progress as wave equation we consider the nonlinear schrodinger equation discretized on a hypercubic lattice since this is a hamiltonian system a natural choice of random initial data is distributing them according to the corresponding gibbs measure with a chemical potential chosen so that the gibbs field has exponential mixing the solution t x of the nonlinear schrodinger equation yields then a stochastic process stationary in x d and t if denotes the strength of the nonlinearity we prove that the space time covariance of t x has a limit as 0 for t 2 with fixed and sufficiently small the limit agrees with the prediction from kinetic theory

the galerkin finite element method for a multi term time fractional diffusion equation we consider the initial boundary value problem for a diffusion equation involving multiple time fractional derivatives on a bounded convex polyhedral domain we analyze a space semidiscrete scheme based on the standard galerkin finite element method using continuous piecewise linear functions nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived which cover both smooth and nonsmooth data further we develop a fully discrete scheme based on a finite difference discretization of the time fractional derivatives and discuss its stability and error estimate extensive numerical experiments for one and two dimensional problems confirm the theoretical convergence rates

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

on the existence of global solutions of the one dimensional cubic nls for initial data in the modulation space mp q r abstract we prove global existence for the one dimensional cubic nonlinear schrodinger equation in modulation spaces m p p for p sufficiently close to 2 in contrast to known results 9 and 14 our result requires no smallness condition on initial data the proof adapts a splitting method inspired by work of vargas vega hyakuna tsutsumi and grunrock to the modulation space setting and exploits polynomial growth of the free schrodinger group on modulation spaces

finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold this paper is devoted to the study of finite element approximations to parabolic optimal control problems with controls acting on a lower dimensional manifold the manifold can be a point a curve or a surface which may be independent of time or evolve in the time horizon and is assumed to be strictly contained in the space domain at first we obtain the first order optimality conditions for the control problems and the corresponding regularity results then for the control problems we consider the fully discrete finite element approximations based on the piecewise constant discontinuous galerkin scheme for time discretization and piecewise linear finite elements for space discretization and variational discretization to the control variable a priori error estimates are finally obtained for the fully discretized control problems and supported by numerical examples

statistical mechanics of the 2 dimensional focusing nonlinear schrodinger equation we study a natural construction of an invariant measure for the 2 dimensional periodic focusing nonlinear schrodinger equation with the critical cubic nonlinearity we find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased but that the natural construction successful for the 1 dimensional case and for the 2 dimensional defocusing case cannot produce an invariant measure our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of berlin and kac

local error analysis for the stokes equations with a punctual source term the solution of the stokes problem with a punctual force in source term is not h 1 times mathbb l 2 and therefore the approximation by a finite element method is suboptimal in the case of poisson problem with a dirac mass in the right hand side an optimal convergence for the lagrange finite elements has been shown on a subdomain which excludes the singularity in mathbb l 2 norm by koppl and wohlmuth here we show a quasi optimal local convergence in h 1 times mathbb l 2 norm for a mathbb p k mathbb p k 1 finite element method k geqslant 2 and for the mathbb p 1 mathrm b mathbb p 1 the error is still analysed on a subdomain which does not contain the singularity the proof is based on local arnold and liu error estimates a weak version of aubin nitsche duality lemma applied to the stokes problem and discrete inf sup conditions these theoretical results are generalized to a wide class of finite element methods before being illustrated by numerical simulations

optimal control of the two dimensional stationary navier stokes equations with measure valued controls the first author was supported by the spanish ministerio de economia industria y competitividad under research projects mtm2014 57531 p and mtm2017 83185 p the second author was supported by the erc advanced grant 668998 ocloc under the eu s h2020 research program

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

almost periodicity in time of solutions of the kdv equation we study the cauchy problem for the kdv equation tu 6u xu xu 0 with almost periodic initial data u x 0 v x we consider initial data v for which the associated schrodinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify and show the existence uniqueness and almost periodicity in time of solutions this establishes a conjecture of percy deift for this class of initial data the result is shown to apply to all small analytic quasiperiodic initial data with diophantine frequency vector

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

approximation of elliptic control problems in measure spaces with sparse solutions optimal control problems in measure spaces governed by elliptic equations are considered for distributed and neumann boundary control which are known to promote sparse solutions optimality conditions are derived and some of the structural properties of their solutions in particular sparsity are discussed a framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

measure valued directional sparsity for parabolic optimal control problems a directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems it allows for controls which are localized in space where the spatial support is independent of time well posedness of the optimal control problems is established and the optimality system is derived it is used to establish structural properties of the minimizer an a priori error analysis for finite element discretization is obtained and numerical results illustrate the effects of the proposed cost functional and the convergence results

statistical mechanics of the nonlinear schrodinger equation ii mean field approximation we investigate a mean field approximation to the statistical mechanics of complex fields with dynamics governed by the nonlinear schrodinger equation such fields whose hamiltonian is unbounded below may model plasmas lasers and other physical systems restricting ourselves to one dimensional systems with periodic boundary conditions we find in the mean field approximation a phase transition from a uniform regime to a regime in which the system is dominated by solitons we compute explicitly as a function of temperature and density l2 norm the transition point at which the uniform configuration becomes unstable to local perturbations static and dynamic mean field approximations yield the same result

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

a priori estimates and weak solutions for the derivative nonlinear schrodinger equation on torus below h 1 2 abstract we propose a priori estimates for a weak solution to the derivative nonlinear schrodinger equation dnls on torus with small l 2 norm datum in low regularity sobolev spaces these estimates allow us to show the existence of solutions in h s t with some s 1 2 in a relatively weak sense furthermore we make some remarks on the error estimates arising from the finite dimensional approximation solutions of dnls using the fourier lesbesgue type as auxiliary spaces despite the fact that nahmod oh rey bellet and staffilani 12 have already seen them

exponential relaxation to equilibrium for a one dimensional focusing non linear schrodinger equation with noise we construct generalized grand canonical and canonical gibbs measures for a hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear schrodinger equation with a focusing nonlinearity of order p 6 key properties of these gibbs measures in particular absence of phase transitions and regularity properties of field samples are established we then study a time evolution of this system given by the hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature the noise is of ornstein uhlenbeck type for the fourier modes of the field with the strength of the noise decaying to zero as the frequency of the mode tends to we prove exponential approach of the state of the system to a grand canonical gibbs measure at a temperature and chemical potential determined by the stochastic noise term

a statistical approach to the asymptotic behavior of a class of generalized nonlinear schrodinger equations a statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear schrodinger type which govern non integrable non singular dynamics in a bounded domain the solutions of these equations have been shown numerically to tend in the long time limit toward a gibbsian statistical equilibrium state consisting of a ground state solitary wave on the large scales and gaussian fluctuations on the small scales the main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit the large deviation principle pertains to a process governed by a gibbs ensemble that is canonical in energy and microcanonical in particle number some supporting monte carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation especially the high frequency growth of the dispersion relation the large deviation principle for the process governed by the gibbs ensemble is based on a large deviation principle for gaussian processes for which two independent proofs are given

sparse source identification of linear diffusion advection equations by adjoint methods abstract we present an algorithm for the time inversion of diffusion advection equations based on the adjoint methodology given a final state distribution our main aim is to recover sparse initial conditions constituted by a finite combination of kronecker deltas identifying their location and mass we discuss the strengths of the adjoint machinery and the difficulties that are to be faced in particular when the diffusivity coefficient or the time horizon is large

optimal a priori error estimates of parabolic optimal control problems with pointwise control in this paper we consider a parabolic optimal control problem with a pointwise dirac type control in space but variable in time in two space dimensions to approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we show almost optimal h 2 k convergence rate for the control in l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control tech report 2011 07 hamburger beitrage zur angewandten mathematik hamburg germany 2011

invariant measures for the periodic derivative nonlinear schrodinger equation we construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear schrodinger equation dnls for small data in l 2 and we show these measures to be absolutely continuous with respect to the gaussian measure the key ingredient of the proof is the analysis of the gauge group of transformations associated to dnls as an intermediate step for our main result we prove quasi invariance with respect to the gauge maps of the gaussian measure on l 2 with covariance mathbb i delta k 1 for any k geqslant 2

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

brownian motion with restoring drift the petit and micro canonical ensembles letf q be odd and positive near then the non linear wave equation 2q t2 2q x2 f q 0 considered on the circle 0 x l can be written in hamiltonian formq h p p h q with p q cdot and h tfrac 1 2 mathop smallint limits 0 l q 2 mathop smallint limits 0 l f q tfrac 1 2 mathop smallint limits 0 l p 2 the corresponding flow preserves the suitably interpreted petit ensemble e h d qd p and forl q settles down to the stationary diffusion with infinitesimal operator 1 2 2 q2 m q q m being the logarithmic derivative of the ground state of d2 dq2 f q this diffusion is the brownian motion with restoring drift see mckean vaninsky 1993 1 for reasons suggested by the paper of lebowitz rose speer 1988 on nls it is interesting to study the micro canonical ensemble obtained by restricting to the sphere int limits 0 l q 2 n and makingl with fixedd n l now forf q q2 the same type of diffusion appears but with drift arising from the modified potentialf q cq2 c being chosen so that the mean ofq2 is the assigned numberd the proof employs doblin s method of loops 1937 and steepest descent the same is true forf q m2q2 only now the proof is elementary the outcome is also the same iff q q2 0 providedd is smaller than the petit canonical mean ofq2 ford larger than this mean the matter is more subtle and the outcome is unknown

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals abstractfast first order proximal methods that solve linear and bilinear parabolic optimal control problems with a sparsity cost functional are discussed weak convergence of these methods is proved and for benchmarking purposes the proposed inexact proximal schemes are compared to an inexact semi smooth newton method results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes and to validate the theoretical estimates