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

asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with gibbs thomson relation abstract in this paper we study a free boundary problem modeling the growth of solid tumor spheroid it consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumor respectively the new feature is that nutrient concentration on the boundary is less than external supply due to a gibbs thomson relation and the problem has two radial stationary solutions which differs from widely studied tumor spheroid model with surface tension effect we first establish local well posedness by using a functional approach based on fourier multiplier method and analytic semigroup theory then we investigate stability of each radial stationary solution by employing a generalized principle of linearized stability we prove that the radial stationary solution with a smaller radius is always unstable and there exists a positive threshold value of cell to cell adhesiveness such that the radial stationary solution with a larger radius is asymptotically stable for and unstable for 0

smoothing estimates for the schrodinger equation with unbounded potentials abstract we prove a local in time smoothing estimate for a magnetic schrodinger equation with coefficients growing polynomially at spatial infinity the assumptions on the magnetic field are gauge invariant and involve only the first two derivatives the proof is based on the multiplier method and no pseudodifferential techniques are required

on convergence of solutions to equilibria for quasilinear parabolic problems abstract we show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non discrete but forms a finite dimensional c 1 manifold which is normally hyperbolic our results do not depend on the presence of an appropriate lyapunov functional as in the ojasiewicz simon approach but are of local nature

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

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

regularity of solutions to nonlinear dispersive equations on considere le probleme de cauchy pour les equations dispersives non lineaires de la forme t u ip d u f u x r n t u x 0 u 0 x

bifurcation from stability to instability for a free boundary problem modeling tumor growth by stokes equation we consider a free boundary problem modeling tumor growth in fluid like tissue the model equations include a diffusion equation for the nutrient concentration and the stokes equation with a source which represents the proliferation of tumor cells the proliferation rate and the cell to cell adhesiveness which keeps the tumor intact are two parameters which characterize the aggressiveness of the tumor for any positive radius r there exists a unique radially symmetric stationary solution with radius r r for a sequence mn r there exist symmetry breaking bifurcation branches of solutions with free boundary r r eyn 0 o e 2 n even 2 for small e w hereyn 0 is the spherical harmonic of mode n 0 furthermore the smallest mn r s aymn r is such that n n r as r in this paper we prove that the radially symmetric stationary solution with r rs is linearly stable if n rs w heren rs mn rs and we prove that strict inequality holds if is small or if is large the biological implications of these results are discussed at the end of the paper

uniqueness properties of solutions of schrodinger equations under suitable assumptions on the potentials v and a we prove that if u c 0 1 h1 is a solution of the linear schrodinger equation i t x u vu a xu on rd 0 1 and if u 0 in x r 0 1 for some r 0 then u 0 in rd 0 1 as a consequence we obtain uniqueness properties of solutions of nonlinear schrodinger equations of the form i t x u g x t u u xu xu on rd 0 1 where g is a suitable nonlinear term the main ingredient in our proof is a carleman inequality of the form e x1 v lx2lt2 e x1 xv bx 2lt2 c e x1 i t x v bx1 2lt2 for any v c r h1 with v t 0 for t 0 1 in this inequality bx 2 and bx1 2 are banach spaces of functions on rd and e x1 is a suitable weight

the general quasilinear ultrahyperbolic schrodinger equation abstract we consider the general quasilinear schrodinger equation whose second order coefficients are given by a real symmetric non degenerate matrix we deduce conditions which guarantee that the associated initial value problem is locally well posed

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

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

smoothing effects and dispersion of singularities for the schrodinger evolution group we describe smoothing effects and dispersion of singularities for the schrodinger evolution group in the weighted sobolev spaces under a fairly general assumption on the potential it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay we measure the regularity gained by the wavefunction by the decay property of the initial state no assumptions on the regularity of the initial state are imposed throughout the paper

maximal function estimates of solutions to general dispersive partial differential equations let u x t sqf x t be the solution of the general dispersive initial value problem ot uiq d u o u x o f x x t ee rn x r and s f the global maximal operator of sqf sharp weighted lp esimates for sq f with f e h rtn are given for general phase functions q

quasi static motion of a capillary drop ii the three dimensional case abstract a theory is presented to analyze the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia the theory is developed here on the three dimensional version of the relevant free boundary model for stokes equations within this context we show that if the free boundary initiates close to a sphere r 1 e 0 e small then there exists a global in time solution with free boundary r 1 t e 1 n 1 n t e n which approaches a sphere exponentially fast as t moreover we prove that if 0 is analytic resp c in then the velocity u x t e the pressure p x t e and the free boundary are all jointly analytic resp c in x e in an earlier paper we considered the analogous problem for a two dimensional drop although the three dimensional problem proceeds along similar lines the analysis is more complicated due to the fact that we work here with spherical harmonics and vector spherical harmonics

smoothing effects for schrodinger equations with electro magnetic potentials and applications to the maxwell schrodinger equations abstract we consider schrodinger equations in r 1 2 with electro magnetic potentials the potentials belong to h 1 and typically they are time independent or determined as solutions to inhomogeneous wave equations we prove kato type smoothing estimates for solutions we also apply this result to the maxwell schrodinger equations in the lorentz gauge and prove unique solvability of this system in the energy space

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

dimension of divergence sets for the pointwise convergence of the schrodinger equation abstract in this note we consider the pointwise convergence along curves for the schrodinger equation and obtain estimates for the capacitary dimension of divergence sets which extend our previous result in 6

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

quintic nls in the exterior of a strictly convex obstacle we consider the defocusing energy critical nonlinear schr odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions for the initial value problem with dirichlet boundary condition we prove global well posedness and scattering for all initial data in the energy space

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

local smoothing for the quantum liouville equation abstract we analyze in this work the regularity properties of the density operator solution to the quantum liouville equation as was recently done for the strichartz inequalities we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the schrodinger equation we show in particular that the local density associated to the solution to the free linear quantum liouville equation admits locally fractional derivatives of given order provided the data belong to some schatten spaces

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

on the interpolation with the potential bound for global solutions of the defocusing cubic wave equation on t2 abstract consider the solutions of the defocusing cubic wave equation t t u u u 3 with real data in h s t 2 h s 1 t 2 s 2 5 we prove that the solutions exist globally in time by contradiction assuming that one of the maximal times of existence is finite we prove that the sobolev norm of each of these solutions is bounded in an open neighborhood of it by using the i method see e g 4 it consists of introducing a mollified energy and estimating its growth in this neighborhood the growth is first estimated on small subintervals by means of dispersive bounds then it is estimated on the whole interval by an iteration process in order to enlarge the size of subintervals on which the dispersive bounds hold we interpolate with the potential bound for the low frequency part our results complement those obtained in 1

well posedness and analyticity of solutions to a water wave problem with viscosity abstract the water wave problem models the free surface evolution of an ideal fluid under the influence of gravity and surface tension the governing equations are a central model in the study of open ocean wave propagation but they possess a surprisingly difficult and subtle well posedness theory in this paper we establish the existence and uniqueness of spatially periodic solutions to the water wave equations augmented with physically inspired viscosity suggested in the recent work of dias et al 2008 16 as we show this viscosity which can be arbitrarily weak not only delivers an enormously simplified well posedness theory for the governing equations but also justifies a greatly stabilized numerical scheme for use in studying solutions of the water wave problem

strichartz estimates via the schrodinger maximal operator we consider the schrodinger operator e it delta acting on initial data f in dot h s we show that an affirmative answer to a question of carleson concerning the sharp range of s for which lim t to 0 e it delta f x f x a e x in mathbb r n would imply an affirmative answer to a question of planchon concerning the sharp range of q and r for which e it delta is bounded in l x q mathbb r n l r t mathbb r when n 2 we unconditionally improve the range for which the mixed norm estimates hold

magnetic virial identities weak dispersion and strichartz inequalities we show a family of virial type identities for the schrodinger and wave equations with electromagnetic potentials as a consequence some weak dispersive inequalities in space dimension n 3 involving morawetz and smoothing estimates are proved finally we apply them to prove strichartz inequalities for the wave equation with a non trapping electromagnetic potential with almost coulomb decay

extremizability of fourier restriction to the paraboloid abstract in this article we prove that nearly all valid scale invariant fourier restriction inequalities for the paraboloid in r 1 d have extremizers and that l p normalized extremizing sequences are precompact modulo symmetries this result had previously been established for the case q 2 in the range where the boundedness of the restriction operator is still an open question our result is conditional on improvements toward the restriction conjecture

global existence and scattering for rough solutions of a nonlinear schr odinger equation on r 3 we prove global existence and scattering for the defocusing cubic nonlinear schrodinger equation in h s r 3 for s 4 the main new estimate in the argument is a morawetz type inequality for the solution this estimate bounds k x t k l 4 r3 r whereas the well known morawetz type estimate of lin strauss controls r 1 0 r r3 x t 4 x dxdt

dispersive smoothing effects for kdv type equations in this paper we study the smoothness properties of solutions of some nonlinear equations of korteweg de vries kdv type which are of the form tu a x t u3 f u2 u1 u x t 1 wherex r uj jxu andkandjare nonnegative integers our principal condition is thata x t be positive and bounded so that the dispersion is dominant it is shown under certain additional conditions onaandfthatc solutionsu x t are obtained fort 0 if the initial datau x 0 decays faster than it does polynomially onr and has certain initial sobolev regularity a quantitative relationship between the rate of decay and the amount of gain of smoothness is given lets0be the sobolev index if ru2 x 0 1 x m dx 2 for an integerm 0 and the solution obeys u hs0 for an existence time 0 t t thenu x t hmloc r for all 0 t t andu x t l1 0 t h m 1 loc r our method can also be extended to address the fully nonlinear dispersive equations related to 1

morphological stability of an elastic tumor host interface abstract the ability of tumors to metastasize is preceded by morphological instabilities such as chains or fingers that invade the host environment therefore parameters that control the morphology of the tumor may also contribute to its invasive ability previous analyses on morphological changes were performed using surface energy of the tumor host interface in an effort to understand the role the interface stiffness plays on tumor evolution here we model the tumor host interface as an elastic membrane governed by the helfrich bending energy using an energy variation approach we derive a modified laplace young condition for the stress jump across the interface in the stokes equation we then perform a linear stability analysis and investigate how physical parameters such as viscosity bending rigidity and apoptosis affect the morphological instability results show that increased bending rigidity versus mitosis rate contributes to a more stable morphological tumor behavior on the other hand increasing tumor viscosity or apoptosis may lead to invasive fingering morphologies comparison with experimental data on glioblastoma spheroids shows good agreement especially for tumors with high adhesion and low proliferation

thin film approximations of the two phase stokes problem abstract passing to the limit of small layer thickness in the two phase stokes problem we obtain when including only gravity resp surface tension effects a strongly coupled parabolic system of second resp fourth order in the non degenerate case we prove that the corresponding evolution problems are locally well posed for the gravity driven flow though we have to assume that the less dense fluid lies on top of the less dense layer moreover we show that the solutions converge exponentially fast towards a flat steady state which is uniquely determined by the volume of the two fluids provided they are initially close to this rest state

local well posedness for the modified kdv equation in almost critical widehat h r_s spaces we study the cauchy problem for the modified kdv equation u t u xxx u 3 x 0 u 0 u 0 for data u 0 in the space h r s defined by the norm u 0 h r s s u 0 lr local well posedness of this problem is established in the parameter range 2 r 1 s 1 2 1 2r so the case s r 0 1 which is critical in view of scaling considerations is almost reached to show this result we use an appropriate variant of the fourier restriction norm method as well as bi and trilinear estimates for solutions of the airy equation

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

local smoothing and convergence properties of schrodinger type equations abstract the initial value problem for the schrodinger type equation u t ip d u u 0 x u 0 x x r n is considered here p is a principal type polynomial of order m i e p c 1 m 1 r or a certain type of other real symbol such as p m m 1 the following results are proved a if u 0 l 2 r n then for any fixed r 0 write u 0 u 1 u 2 where u 1 and u 2 are square integrable and the fourier transform of u 1 vanishes for r and that of u 2 vanishes for r if u 1 t x e itp d u 1 x u 2 t x e itp d u 2 x then u 1 is real analytic in r r n and satisfies for every multi index sup t x d u 1 t x c u 0 l 2 while 1 x 2 1 2 u 2 t x l 2 r t h m 1 2 r x n b if u 0 l 2 s r n s 0 then for every t 0 u t h loc m 1 s r n c if u 0 h s r n s 1 2 then for a e x r n u t x u 0 x as t 0

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

on a class of solutions to the generalized derivative schrodinger equations ii abstract in this note we shall continue our study on the initial value problem associated for the generalized derivative schrodinger gdnls equation t u i x 2 u u x u x t r 0 1 and 1 inspiring by cazenave naumkin s works we shall establish the local well posedness for a class of data of arbitrary size in an appropriate weighted sobolev space thus removing the size restriction on the data required in our previous work the main new tool in the proof is the homogeneous and inhomogeneous versions of the kato smoothing effect for the linear schrodinger equation with lower order variable coefficients established by kenig ponce vega

best constants in some operator smoothness estimates abstract we provide a short proof of the inequality cf ben artzi and klainerman regularity and decay of evolution equations preprint and kato and yajima rev math phys 1 1989 481 496 1 x 2 1 2 1 1 4 e it u 2 dt c u 2 with explicit essentially exact values for c

on the dimension of divergence sets of dispersive equations we refine results of carleson sjogren and sjolin regarding the pointwise convergence to the initial data of solutions to the schrodinger equation we bound the hausdorff dimension of the sets on which convergence fails for example with initial data in h 1 mathbb r 3 the sets of divergence have dimension at most one

an endline bilinear cone restriction estimate for mixed norms we prove an l 2 times l 2 rightarrow l t ql x p bilinear fourier extension estimate for the cone when p q are on the critical line 1 q frac n 1 2 1 1 p this extends previous results by wolff tao and lee vargas

a free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids abstract this paper deals with a mathematical model describing the cell cycle dynamics and chemotactic driven cell movement in a multicellular tumor spheroid tumor cells consist of two types of cells proliferating cells and quiescent cells which have different chemotactic responses to an extracellular nutrient supply the model is a free boundary problem for a nonlinear system of reaction diffusion advection equations where the free boundary is the outer boundary of the spheroid the free boundary condition is quite novel due to different velocity of two types of cells the global existence and uniqueness of solutions to the model is proved the proof is based on a fixed point argument together with the l p theory for parabolic equations with the third boundary condition

a justification for the thin film approximation of stokes flow with surface tension in the free boundary problem of stokes flow driven by surface tension we pass to the limit of small layer thickness it is rigorously shown that in this limit the evolution is given by the well known thin film equation the main techniques are appropriate scaling and uniform energy estimates in sobolev spaces of sufficiently high order based on parabolicity