on hartree equations with derivatives abstract we consider the cauchy problem of two types of hartree equations with exchange correlation correction terms i u t u v k u u in r 1 n k 1 2 u 0 in r n n 1 where v 1 u x 1 u 2 2 u 2 v 2 u x u 2 we establish the well posedness of cauchy problems and show the smoothing effect of solutions for each 0 n and 0 1

vanishing viscosity with short wave long wave interactions for multi d scalar conservation laws we consider a system coupling a multidimensional semilinear schrodinger equation and a multidimensional nonlinear scalar conservation law with viscosity which is motivated by a model of short wave long wave interaction introduced by benney 1977 we prove the global existence and uniqueness of the solution of the cauchy problem for this system we also prove the convergence of the whole sequence of solutions when the viscosity e and the interaction parameter approach zero so that o e1 2 we also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of schrodinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled

further results on global stabilization of discrete nonlinear systems in this paper we show how recent advances 11 12 in global stabilization of continuous time nonaffine nonlinear systems with stable free dynamics can be nontrivially generalized to their discrete time counterparts by means of passivity and bounded state feedback as a consequence global stabilization results of discrete time nonlinear systems with triangular structure are established

strichartz estimates for the wave and schrodinger equations with the inverse square potential abstract we prove spacetime weighted l 2 estimates for the schrodinger and wave equation with an inverse square potential we then deduce strichartz estimates for these equations

smooth output feedback stabilization of planar systems without controllable observable linearization this note considers the problem of global stabilization by output feedback for a family of planar systems whose jacobian linearization is neither controllable nor observable the problem cannot be dealt with by existing output feedback design methods most of them are based on the separation principle under appropriate growth conditions we propose an output feedback control scheme that does not rely on the separation principle and achieves global asymptotic stabilization the novelty of our control scheme lies in the explicit design of a dynamic output compensator which combines a nonlinear gain observer design and the technique of adding a power integrator as a consequence an interesting global stabilization result by output feedback can be obtained for feedback linearizable systems in a triangular form which turns out to be new even in the two dimensional case

gaussian heat kernel upper bounds via the phragmen lindelof theorem we prove that in presence of l gaussian estimates so called davies gaffney estimates on diagonal upper bounds imply precise off diagonal gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

factorization in lorentz spaces with an application to centralizers abstract we give an explicit formula to factorize functions in the lorentz spaces l p q some applications to centralizers twisted sums and interpolation theory are included

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

generalized strichartz estimates and scattering for 3d zakharov system we obtain scattering for the 3d zakharov system with non radial small data in the energy space with angular regularity of degree one the main ingredient is a generalized strichartz estimate for the schrodinger equation in the space of l2 angular integrability

strichartz estimates in similarity coordinates and stable blowup for the critical wave equation we establish strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a time dependent self similar potential as an application we consider the critical wave equation and prove the asymptotic stability of the ode blowup profile in the energy space

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

constructive stabilization for quadratic input nonlinear systems in this paper stabilization of nonlinear systems with quadratic multi input is considered with the help of control lyapunov function clf a constructive parameterization of controls that globally asymptotically stabilize the system is proposed two different cases are considered firstly under certain regularity assumptions the feasible control set is parameterized and continuous feedback stabilizing controls are designed then for the general case piecewise continuous stabilizing controls are proposed the design procedure can also be used to verify whether a candidate clf is indeed a clf several illustrative examples are presented as well

the radial defocusing energy supercritical nls in dimension four abstract we consider the radial defocusing nonlinear schrodinger equations i u t u u p u with supercritical exponent p 4 in four space dimensions and prove that any radial solution that remains bounded in the critical sobolev space must be global and scatter

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

stein weiss and caffarelli kohn nirenberg inequalities with angular integrability abstract we prove an extension of the stein weiss weighted estimates for fractional integrals in the context of l p spaces with different integrability properties in the radial and the angular direction in this way the classical estimates can be unified with their improved radial versions a number of consequences are obtained in particular we deduce precised versions of weighted sobolev embeddings caffarelli kohn nirenberg estimates and strichartz estimates for the wave equation which extend the radial improvements to the case of arbitrary functions

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

analyticity of solutions to a quadratic system of schrodinger equations without mass resonance condition abstract we consider the cauchy problem for a quadratic system of schrodinger equations in space dimensions n 4 without mass resonance condition analytic smoothing effect for the quadratic system of schrodinger equations under the mass resonance condition for data which satisfy exponentially decaying condition has been studied in 14 in this study we construct global solutions in the analytic hardy space via the phase modulation operator based on the scale critical sobolev space h n 2 2 with data like the test functions of fourier hyperfunctions under the condition m 1 m 2 1 4 n m 1 between two masses m 1 and m 2

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

a counter example to strichartz estimates for the inhomogeneous schrodinger equation abstract we disprove strichartz estimates for the solution of the inhomogeneous schrodinger equation in a certain range of the lebesgue exponents values

stability improvement of nonlinear systems by feedback this paper is concerned with the property of stabilizing a nonlinear system to a specified equilibrium point arbitrarily fast by appropriate smooth feedback two closely related forms of this property are explored one refers to the asymptotic transfer to the critical point with exponential decay and it is shown that the systems possessing the above property are precisely those whose corresponding linearization is controllable the second stabilizability form defined here amounts to driving the system back to an arbitrarily small neighborhood of the given point arbitrarily fast by smooth feedback a global stabilization result is also included finding application in hamiltonian systems special emphasis is given to the bilinear case

sharp null form estimates for the wave equation null form estimates from dot h alpha 1 times dot h alpha 2 to l q t l r x for the wave equation in bbb r n 1 are studied for n ge4 we obtain the sharp null form estimates except for the endpoints for n 2 3 we obtain the estimates under the additional assumption 4 n q when 2 r

dispersive and strichartz estimates for the three dimensional wave equation with a scaling critical class of potentials we consider the three dimensional linear wave equation perturbed by a potential v belonging to a scaling critical class which is the closure of bounded compactly supported functions with respect to the global kato norm without any further smallness assumptions we first prove the dispersive estimate for the solution of the wave equation as a byproduct we prove strichartz estimates under a slightly stronger condition

some remarks on strichartz estimates for homogeneous wave equation abstract we give several remarks on strichartz estimates for homogeneous wave equation with special attention to the cases of l x estimates radial solutions and initial data from the inhomogeneous sobolev spaces in particular we give the failure of the endpoint estimate l t 4 l x for n 2

on the cauchy problem for the nonlinear schrodinger equations with time dependent linear loss gain abstract this paper is devoted to the cauchy problem for the nonlinear schrodinger equation with time dependent loss gain which reads i u t u u u i a t u 0 this equation appears in the recent studies of bose einstein condensates and optical systems we obtain some global existence and blow up results which depend on the size of the loss gain coefficient in particular we prove the global existence for the energy critical nonlinearity by scaling and compactness arguments we also discuss asymptotic profiles and concentration properties of blow up solutions

nonlinear generalized dunkl wave equations and applications abstract in this paper we introduce a class of nonlinear wave equations associated with the generalized dunkl laplace operator we study local and global well posedness next we establish the linearization of bounded energy solutions in the spirit of gerard 1996 9 the proof uses strichartz type inequalities and the energy estimate

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

a schrodinger equation with time oscillating critical nonlinearity abstract in this paper we study the time oscillating critical nonlinear schrodinger equation i u t u t u 4 n 2 u 0 in r n n 3 where is a periodic function we show that for a given initial condition u 0 in h 1 the solution u converges as to the solution u of the limiting equation i u t u i u 4 n 2 u 0 with the same initial condition where i is the average of we also show that if the solution u is global and has a certain decay property then u is also global if is sufficiently large similar results for the subcritical problem are given by cazenave and scialom 2010 9

parametrices and dispersive estimates for schrodinger operators with variable coefficients in this article we consider variable coefficient time dependent schr odinger evolutions in bbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

homogeneity with incremental degrees and global stabilisation of a class of high order upper triangular systems the problem of global asymptotic stabilisation is investigated for a class of upper triangular systems with uncontrollable linearisation a notion of generalised homogeneity is introduced in this article providing a new insight and a deeper perspective on how to categorise nonlinear terms into the higher order and equal order terms in a non traditional way the delicate analysis and new categorisation of the vector field make it possible to encompass more general nonlinearities a globally stabilising controller is recursively constructed by a bottom up step by step procedure containing the equal order terms as its indispensable components it is illustrated that the results obtained in this article incorporate and extend existing results for the global stabilisation of upper triangular systems

near sharp strichartz estimates with loss in the presence of degenerate hyperbolic trapping we consider an n dimensional spherically symmetric asymptotically euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic by separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond ehrenfest time we show that solutions to the linear schrodinger equation with initial conditions localized on a spherical harmonic satisfy strichartz estimates with a loss depending only on the dimension n and independent of the degeneracy the strichartz estimates are sharp up to an arbitrary 0 loss this is in contrast to christianson and wunsch amer j math 2013 where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set independent of the dimension

feedback stabilization of a class of nonsmooth systems in this paper we investigate global stabilization of nonsmooth systems the aim is to provide sufficient condition based on standard derivative for feedback stabilization of nonsmooth systems by means of a feedback of class cr r 1 the main result is a generalization of a result concerning smooth systems see for instance j control optimization 32 1994 358 to systems that are only continuous and locally lipschitz

strichartz estimates for long range perturbations we study local in time strichartz estimates for the schroedinger equation associated to long range perturbations of the flat laplacian on the euclidean space we prove that in such a geometric situation outside of a large ball centered at the origin the solutions of the schroedinger equation enjoy the same strichartz estimates as in the non perturbed situation the proof is based on the isozaki kitada parametrix construction if in addition the metric is non trapping we prove that the strichartz estimates hold in the whole space

uniform resolvent estimates for schrodinger operator with an inverse square potential abstract we study the uniform resolvent estimates for schrodinger operator with a hardy type singular potential let l v v x where is the usual laplacian on r n and v x v 0 r 2 where r x x x and v 0 c 1 s n 1 is a real function such that the operator v 0 n 2 2 4 is a strictly positive operator on l 2 s n 1 we prove some new uniform weighted resolvent estimates and also obtain some uniform sobolev estimates associated with the operator l v

on strictly singular nonlinear centralizers abstract we show that c 0 is the only banach space with unconditional basis that satisfies the equation ext x x 0 this partially improves an old result by kalton and peck we prove that the kalton peck maps are strictly singular on a number of sequence spaces including l p for 0 p tsirelson and schlumprecht spaces and their duals as well as certain super reflexive variations of these spaces in the last section we give estimates of the projection constants of certain finite dimensional twisted sums of kalton peck type

wave equation on riemannian symmetric spaces we consider the wave equation on riemannian symmetric spaces of the noncompact type we prove that the solutions of the homogeneous equation have dispersion properties and we deduce strichartz type estimates for these solutions we use these estimates to treat nonlinear equations with power type as a byproduct we prove the global well posedness for small energy

a centre stable manifold for the focussing cubic nls in mathbb r 1 3 consider the focussing cubic nonlinear schrodinger equation in mathbb r 3 i psi t delta psi psi 2 psi quad 0 1 it admits special solutions of the form eit where phi in mathcal s mathbb r 3 is a positive 0 solution of delta phi alpha phi phi 3 quad 0 2 the space of all such solutions together with those obtained from them by rescaling and applying phase and galilean coordinate changes called standing waves is the 8 dimensional manifold that consists of functions of the form e i v cdot gamma phi cdot y alpha we prove that any solution starting sufficiently close to a standing wave in the sigma w 1 2 mathbb r 3 cap x 1 l 2 mathbb r 3 norm and situated on a certain codimension one local lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves furthermore we show that mathcal n is invariant under the hamiltonian flow locally in time and is a centre stable manifold in the sense of bates jones batjon the proof is based on the modulation method introduced by soffer and weinstein for the l2 subcritical case and adapted by schlag to the l2 supercritical case an important part of the proof is the keel tao endpoint strichartz estimate in mathbb r 3 for the nonselfadjoint schrodinger operator obtained by linearizing 0 1 around a standing wave solution all results in this paper depend on the standard spectral assumption that the hamiltonian mathcal h left begin array cc delta 2 phi cdot alpha 2 alpha quad phi cdot alpha 2 phi cdot alpha 2 quad delta 2 phi cdot alpha 2 alpha end array right quad 0 3 has no embedded eigenvalues in the interior of its essential spectrum infty alpha cup alpha infty

global asymptotic stabilization of periodic nonlinear systems with stable free dynamics abstract we consider a nonlinear control system with periodic coefficients we study the problem of asymptotic stabilization of the equilibrium x 0 of the closed loop system by state feedback we assume that the free dynamic system possesses a periodic lyapunov function ensuring lyapunov stability of the equilibrium x 0 we have developed a method for constructing a damping control for affine systems with periodic coefficients based on a generalization of weak jurdjevic quinn conditions using an extension of the notion of the commutator to non stationary vector fields using this approach we obtain sufficient conditions for uniform local and global asymptotic stabilization of general nonlinear systems in particular affine control systems with periodic coefficients stabilization results generalize known results for time invariant systems to time varying periodic systems

strichartz estimates and local smoothing estimates for asymptotically flat schrodinger equations abstract in this article we study global in time strichartz estimates for the schrodinger evolution corresponding to long range perturbations of the euclidean laplacian this is a natural continuation of a recent article d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 of the third author where it is proved that local smoothing estimates imply strichartz estimates by d tataru parametrices and dispersive estimates for schrodinger operators with variable coefficients amer j math 130 2008 571 634 the local smoothing estimates are known to hold for small perturbations of the laplacian here we consider the case of large perturbations in three increasingly favorable scenarios i without non trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term ii with non trapping assumptions we prove global estimates modulo a lower order spatially localized error term and iii for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

a sharp condition for scattering of the radial 3d cubic nonlinear schrodinger equation we consider the problem of identifying sharp criteria under which radial h 1 finite energy solutions to the focusing 3d cubic nonlinear schrodinger equation nls i t u u u 2 u 0 scatter i e approach the solution to a linear schrodinger equation as t the criteria is expressed in terms of the scale invariant quantities u 0 l 2 nabla u 0 l 2 and m u e u where u 0 denotes the initial data and m u and e u denote the conserved in time mass and energy of the corresponding solution u t the focusing nls possesses a soliton solution e it q x where q is the ground state solution to a nonlinear elliptic equation and we prove that if m u e u q l 2 nabla q l 2 then the solution blows up in finite time the technique employed is parallel to that employed by kenig merle 17 in their study of the energy critical nls

passivity and absolute stabilization of a class of discrete time nonlinear systems abstract by using the theory of discrete time passive systems and the concept of feedback equivalence we present an approach toward deriving sufficient conditions for the global stabilization of discrete time nonlinear systems in the form x k 1 tf x k g x k u k the central feature of the approach is that a unifying set of stability criteria can be obtained and the stability results recover the previous stabilization schemes proposed in the literature

nonlinear scattering theory for a class of wave equations in hs abstract for the semi linear higher order wave equation and the nonlinear higher order schrodinger equation we show that the scattering operators map a band in h s into h s if the nonlinearities have sub critical powers in h s the smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing nls equation are also shown provided that the nonlinearities have subcritical growth in h 1 moreover the spatial decaying behavior of solutions in energy space for the defocusing nls equation are obtained

losslessness feedback equivalence and the global stabilization of discrete time nonlinear systems in this paper a necessary and sufficient condition for a nonlinear system of the form spl sigma given by x k 1 f x k g x k u k y k h x k j x k u k to be lossless is given and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero state observable then we investigate conditions under which spl sigma can be rendered lossless via smooth state feedback in particular we show that this is possible if and only if the system in question has relative degree spl lcub 0 0 spl rcub and has lossless zero dynamics under suitable controllability like rank conditions we prove that nonlinear systems having relative degree spl lcub 0 0 spl rcub and lossless zero dynamics can be globally stabilized by smooth state feedback as a consequence we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable the global stabilization problem of the nonlinear system spl sigma without output is also investigated in this paper by means of feedback equivalence some of the results are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

strichartz estimates for the schrodinger equation on products of compact groups and odd dimensional spheres abstract we generalize the framework developed in anal pde 13 4 2020 1173 1219 to prove certain strichartz estimates which are scale invariant up to an e loss on any product of compact groups and odd dimensional spheres some partial results toward such strichartz estimates on a general compact globally symmetric space are also given including a kernel estimate sharp up to an e loss near rational times and near corners of a maximal torus

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

output feedback control of a class of nonlinear systems a nonseparation principle paradigm considers the problem of global stabilization by output feedback for a family of nonlinear systems that are dominated by a triangular system satisfying a linear growth condition the problem has remained unsolved due to the violation of the commonly assumed conditions in the literature using a feedback domination design method which is not based on the separation principle we explicitly construct a linear output compensator making the closed loop system globally exponentially stable

stabilizability of the angular velocity of a rigid body revisited abstract we prove that the jurdjevic quinn techniques for smooth stabilization can be applied to the equation of the angular velocity of a rigid body an explicit polynomial control is given for the symmetric case

secular growth estimates for hyperbolic systems abstract we consider a model problem for the secular growth which covers all the cases likely to happen in multi scales bkw expansions for nonlinear hyperbolic systems this model problem consists in studying the growth in time of the solution of a nonhomogeneous hyperbolic system whose source term is a product of various functions which solve homogeneous hyperbolic systems the secular growth is due to resonances that we try to control when this is not possible other tools such as decay properties or strichartz estimates must be used

linearized control systems and applications to smooth stabilization for a control system dot x f x u the author proves that for generic feedback laws u such that f x u x does not vanish the linearized control systems around the trajectories of dot x f x u x have the same strong accessibility algebra as f applications are given to the smooth stabilization problem

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

strichartz estimates and wave equation in a conic singular space consider the metric cone x c y 0 infty r times y with metric g dr 2 r 2h where the cross section y is a compact n 1 dimensional riemannian manifold y h let delta g be the positive friedrichs extension laplacian on x and let delta h be the positive laplacian on y and consider the operator mathcal l v delta g v 0 r 2 where v 0 in mathcal c infty y such that delta h v 0 n 2 2 4 is a strictly positive operator on l 2 y in this paper we prove global in time strichartz estimates without loss regularity for the wave equation associated with the operator mathcal l v it verifies a conjecture in wang remark 2 4 in ann inst fourier 56 1903 1945 2006 for wave equation the range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of delta h v 0 n 2 2 4 to prove the result we show a sobolev inequality and a boundedness of a generalized riesz transform in this setting in addition as an application we study the well posed theory and scattering theory for energy critical wave equation with small data on this setting of dimension n ge 3

fractal strichartz estimate for the wave equation abstract we consider strichartz estimates for the wave equation with respect to general measures which satisfy certain growth conditions in r 3 1 we obtain the sharp estimate and in higher dimensions improve the previous results

feedback stabilization of general nonlinear control systems a passive system approach abstract this paper develops sufficient conditions for a general nonlinear control system 1 x f x u to be locally resp globally asymptotically stabilizable via smooth state feedback in particular it is shown that as in the case of affine systems this is possible if the unforced dynamic system of 1 is lyapunov stable and appropriate controllability like rank conditions are satisfied our results incorporate a series of well known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems

on persistently exciting observers and a non linear separation principle application to the stabilization of a generator we provide a separation principle for a class of non linear time varying systems the only assumption we make on the state feedback control law is that it does not vanish as the states grow unboundedly for the observer we propose a design method based on the assumption that the system is transformable into a form affine in the unmeasured variable and a property of persistency of excitation this assumption covers but is not restricted to the common observability condition that the vector field multiplying the unmeasured variable is separated from zero for all state values our separation principle relies on stability results for cascades systems

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

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

bounded smooth state feedback and a global separation principle for non affine nonlinear systems abstract in this paper we show that a single input nonlinear system z 0 x g 1 x u g l x u l is globally asymptotically stabilizable by arbitrarily small smooth state feedback if unforced dynamics of are lyapunov stable and satisfies appropriate controllability like rank conditions characterized by the lie bracket of vector fields f0 x and g1 x an explicit smooth state feedback control law is constructed this in turn leads to an analytic resp smooth state feedback control law which solves the problem of global stabilization for analytic resp smooth systems x f x u based on these results we then establish a global separation principle for the system with an output finally we briefly discuss how the results can be extended to a multi input system

analyticity of the scattering operator for semilinear dispersive equations we present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic and to compute the coefficients of its taylor series at any point we illustrate this method in the case of the schrodinger equation with power like nonlinearity or with hartree type nonlinearity and in the case of the wave and klein gordon equations with power nonlinearity finally we discuss the link of this approach with inverse scattering and with complete integrability

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

time varying feedback control of nonaffine nonlinear systems without drift sufficient conditions are presented under which a general nonlinear system without drift is globally asymptotically stabilizable by time varying state feedback a novel approach is developed for the design of a time varying smooth state feedback controller the controller is explicitly constructed by using the bounded state feedback strategy lin 1995 1996 combined with lyapunov technique as well as lossless systems theory this work incorporates earlier global stabilization results coron 1992 pomet 1992 for controllable affine systems without drift which are known not to be smoothly stabilizable via any time invariant state feedback

contribution to the classification of minimal extensions abstract we show that certain quasi banach spaces arising from quasi linear functions on l 1 are not isomorphic we also show that if is a large set then every quasi linear function on l 1 is near to a true linear map on some infinite dimensional subspace

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

hs global well posedness for semilinear wave equations abstract we consider the cauchy problem for semilinear wave equations in r n with n 3 making use of bourgain s method in conjunction with the endpoint strichartz estimates of keel and tao we establish the hs global well posedness with s