on the l p solvability of higher order parabolic and elliptic systems with bmo coefficients we prove the solvability in sobolev spaces for both divergence and non divergence form higher order parabolic and elliptic systems in the whole space on a half space and on a bounded domain the leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders for the proof we develop a set of new techniques to produce mean oscillation estimates for systems on a half space

the heat equation with rough boundary conditions and holomorphic functional calculus abstract in this paper we consider the laplace operator with dirichlet boundary conditions on a smooth domain we prove that it has a bounded h calculus on weighted l p spaces for power weights which fall outside the classical class of a p weights furthermore we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity in particular we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

existence and uniqueness of weak solutions to ginzburg landau equation with external noise and stochastic perturbation abstract in this paper the ginzburg landau and fractional ginzburg landau equations with external noise and stochastic perturbations are studied and we obtain the global existence and uniqueness of weak l 1 solutions our proof is supported by the generalized ito formula and a careful analysis of the nonlinear transport equation after founding the well posedness for the nonlinear transport problem in the class of l p 1 p we also gain the well posedness of the ginzburg landau and fractional ginzburg landau equations in the same spaces correspondingly

global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

boundedness of riesz transforms for elliptic operators on abstract wiener spaces let e h be an abstract wiener space and let dv vd where d denotes the malliavin derivative and v is a closed and densely defined operator from h into another hilbert space h given a bounded operator b on h coercive on the range r v we consider the operators a v bv in h and a vv b in h as well as the realisations of the operators l dv bdv and l dvdv b in lp e and lp e h respectively where 1 p our main result asserts that the following four assertions are equivalent 1 d l d dv with lf p dvf p for f d l 2 l admits a bounded h functional calculus on r dv 3 d a d v with ah vh for h d a 4 a admits a bounded h functional calculus on r v moreover if these conditions are satisfied then d l d dv2 d da the equivalence 1 4 is a non symmetric generalisation of the classical meyer inequalities of malliavin calculus where h h v i b 12i a one sided version of 1 4 giving lp boundedness of the riesz transform dv l in terms of a square function estimate is also obtained as an application let a generate an analytic c0 contraction semigroup on a hilbert space h and let l be the lp realisation of the generator of its second quantisation our results imply that two sided bounds for the riesz transform of l are equivalent with the kato square root property for a the boundedness of the riesz transform is used to obtain an lp domain characterisation for the operator l

a non local regularization of first order hamilton jacobi equations abstract in this paper we investigate the regularizing effect of a non local operator on first order hamilton jacobi equations we prove that there exists a unique solution that is c 2 in space and c 1 in time in order to do so we combine viscosity solution techniques and green s function techniques viscosity solution theory provides the existence of a w 1 solution as well as uniqueness and stability results a duhamel s integral representation of the equation involving the green s function permits to prove further regularity we also state the existence of c solutions in space and time under suitable assumptions on the hamiltonian we finally give an error estimate in l norm between the viscosity solution of the pure hamilton jacobi equation and the solution of the integro differential equation with a vanishing non local part

hilbert space frames containing a riesz basis and banach spaces which have no subspace isomorphic toc0 abstract we prove that a hilbert space frame fi i icontains a riesz basis if every subfamily fi i j j i is a frame for its closed span secondly we give a new characterization of banach spaces which do not have any subspace isomorphic toc0 this result immediately leads to an improvement of a recent theorem of holub concerning frames consisting of a riesz basis plus finitely many elements

maximum robustness and surgery of frames in finite dimensions abstract we consider frames in a finite dimensional hilbert space h n where frames are exactly the spanning sets of the vector space we present a method to determine the maximum robustness of a frame we present results on tight subframes and surgery of frames we also answer the question of when length surgery resulting in a tight frame set for h n is possible

robustness and surgery of frames abstract we characterize frames in r n that are robust to k erasures the characterization is given in terms of the support of the null space of the synthesis operator of the frame a necessary and sufficient condition is given for when an r k surgery on unit norm tight frames in r 2 are possible also a generalization of a known characterization of the existence of tight frames with prescribed norms is given

local l estimates weak harnack inequality and stochastic continuity of solutions of spdes we consider stochastic partial differential equations under minimal assumptions the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition in particular the diffusion term is allowed to be scaling critical we derive local supremum estimates with a stochastic adaptation of de giorgi s iteration and establish a weak harnack inequality for the solutions the latter is then used to obtain pointwise almost sure continuity

difference norms for vector valued bessel potential spaces with an application to pointwise multipliers abstract in this paper we prove a randomized difference norm characterization for bessel potential spaces with values in umd banach spaces the main ingredients are r boundedness results for fourier multiplier operators which are of independent interest as an application we characterize the pointwise multiplier property of the indicator function of the half space on these spaces all results are proved in the setting of weighted spaces

heat kernels and analyticity of non symmetric jump diffusion semigroups let d geqslant 1 and alpha in 0 2 consider the following non local and non symmetric levy type operator on mathbb r d begin aligned fancyscript l kappa alpha f x hbox p v int mathbb r d f x z f x frac kappa x z z d alpha mathord mathrm d z end aligned where 0 kappa 0 leqslant kappa x z leqslant kappa 1 kappa x z kappa x z and kappa x z kappa y z leqslant kappa 2 x y beta for some beta in 0 1 using levi s method we construct the fundamental solution also called heat kernel p kappa alpha t x y of fancyscript l kappa alpha and establish its sharp two sided estimates as well as its fractional derivative and gradient estimates we also show that p kappa alpha t x y is jointly holder continuous in t x the lower bound heat kernel estimate is obtained by using a probabilistic argument the fundamental solution of fancyscript l kappa alpha gives rise a feller process x mathbb p x x in mathbb r d on mathbb r d we determine the levy system of x and show that mathbb p x solves the martingale problem for fancyscript l kappa alpha c 2 b mathbb r d furthermore we show that the c 0 semigroup associated with fancyscript l kappa alpha is analytic in l p mathbb r d for every p in 1 infty a maximum principle for solutions of the parabolic equation partial t u fancyscript l kappa alpha u is also established as an application of the main result of this paper sharp two sided estimates for the transition density of the solution of mathord mathrm d x t a x t mathord mathrm d y t is derived where y is a rotationally symmetric stable process on mathbb r d and a x is a holder continuous d times d matrix valued function on mathbb r d that is uniformly elliptic and bounded

stochastic evolution equations in umd banach spaces we discuss existence uniqueness and space time holder regular ity for solutions of the parabolic stochastic evolution equation du t au t f t u t dt b t u t dwh t t 2 0 t0 u 0 u0 where a generates an analytic c0 semigroup on a umd banach space e and wh is a cylindrical brownian motion with values in a hilbert space h we prove that if the mappings f 0 t e e and b 0 t e l h e satisfy suitable lipschitz conditions and u0 is f0 measurable and bounded then this problem has a unique mild solution which has trajectories in c 0 t d a provided 0 and 0 satisfy 1 2 vari ous extensions of this result are given and the results are applied to parabolic stochastic partial dierential equations

frames graphs and erasures two uniform frames and their use for the coding of vectors are the main subject of this paper these frames are known to be optimal for handling up to two erasures in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero here we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost we derive general error bounds for two uniform frames when more than two erasures occur and apply these to concrete examples we show that among the 227 known equivalence classes of two uniform 36 15 frames arising from hadamard matrices there are 5 that give smallest error bounds for up to 8 erasures

an lq lp theory for the time fractional evolution equations with variable coefficients abstract we introduce an l q l p theory for the semilinear fractional equations of the type 0 1 t u t x a i j t x u x i x j t x f t x u t 0 x r d here 0 2 p q 1 and t is the caupto fractional derivative of order uniqueness existence and l q l p estimates of solutions are obtained the leading coefficients a i j t x are assumed to be piecewise continuous in t and uniformly continuous in x in particular a i j t x are allowed to be discontinuous with respect to the time variable our approach is based on classical tools in pde theories such as the marcinkiewicz interpolation theorem the calderon zygmund theorem and perturbation arguments

trajectory statistical solutions and liouville type equations for evolution equations abstract results and applications abstract in this article we first prove from the viewpoint of infinite dynamical system sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a liouville type equation moreover we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the liouville type equation finally we study the equations of three dimensional incompressible magneto micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations

on the ito wentzell formula for distribution valued processes and related topics we prove the ito wentzell formula for processes with values in the space of generalized functions by using the stochastic fubini theorem and the ito wentzell formula for real valued processes appropriate versions of which are also proved

schauder estimates for stochastic transport diffusion equations with levy processes abstract in this study we consider a transport diffusion equation with levy noise and holder continuous coefficients by using the heat kernel estimates we derive the schauder estimates for the mild solutions moreover when the transport term vanishes and p 2 we show that the holder index in the space variable is optimal

global existence for the magnetohydrodynamic system in critical spaces in this article we show that the magnetohydrodynamic system in mathbb r n with variable density variable viscosity and variable conductivity has a local weak solution in the besov space dot b n p 1 p 1 1 mathbb r n times dot b n p 2 1 p 2 1 mathbb r n times dot b n p 2 1 p 2 1 mathbb r n for all 1 and some 1 if the initial density approaches a positive constant moreover this solution is unique if we impose the restrictive condition 1 we also prove that the constructed solution exists globally in time if the initial data are small in particular this allows us to work in the framework of besov space with negative regularity indices and this fact is particularly important when the initial data are strongly oscillating

a bmo estimate for stochastic singular integral operators and its application to spdes abstract in this paper we investigate kernel conditions on k t s x so that the stochastic singular integral operator 0 t k t s g s x d w s has a bounded mean oscillation as an application we prove that for the solution u of the stochastic heat equation 0 1 d u t x a i j t u x i x j d t g t k x d w t k u 0 0 t t the q th order bmo quasi norm of the derivatives of u is controlled by g l

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

unique solvability for the density dependent navier stokes equations abstract in this paper we consider the incompressible navier stokes equations with a density dependent viscosity in a bounded domain of r n n 2 3 we prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition this condition is also necessary for a very general initial data moreover we provide a blow up criterion for the regularity of the strong solution for these results the initial density need not be strictly positive it may vanish in an open subset of

bmo and morrey campanato estimates for stochastic convolutions and schauder estimates for stochastic parabolic equations abstract in this paper we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t x 2 u t x d t g t x d t u 0 0 t 0 t x g for a random field u t x 0 t g u t x u t x r where 2 2 0 2 is the fractional laplacian t 0 is arbitrarily fixed g r d is a bounded domain g 0 t g r is a joint measurable coefficient and t t 0 is either a brownian motion or a levy process on a given filtered probability space f p f t t 0 t to this end we derive the bmo estimates and morrey campanato estimates respectively for stochastic singular integral operators arising from the equations concerned then by utilizing the embedding theory between the campanato space and the holder space we establish the controllability of the norm of the space c 2 d where 0 d 0 t g with all these in hand we are able to show that the q th order bmo quasi norm of the q 0 order derivative of the solution u is controlled by the norm of g under the condition that t is a levy process finally we derive the schauder estimate for the p moments of the solution of the above stochastic fractional heat equations driven by levy noise

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

a non markovian liquidation problem and backward spdes with singular terminal conditions we establish the existence uniqueness and regularity of solution results for a class of backward stochastic partial differential equations with singular terminal condition the equation describes the value function of non markovian stochastic optimal control problem in which the terminal state of the controlled process is prespecified the analysis of such control problems is motivated by models of optimal portfolio liquidation

parabolic bmo estimates for pseudo differential operators of arbitrary order abstract in this article we prove the bmo l estimate 2 u bmo r d 1 n t u a t u l r d 1 u c c r d 1 for a wide class of pseudo differential operators a t of order 0 the coefficients of a t are assumed to be merely measurable in time variable as an application to the equation t u a t u f t r we prove that for any u c c r d 1 u t l p r d 1 2 u l p r d 1 n u t a t u l p r d 1 where p 1 and the constant n is independent of u

pathwise holder convergence of the implicit linear euler scheme for semi linear spdes with multiplicative noise in this article we prove pathwise holder convergence with optimal rates of the implicit euler scheme for the abstract stochastic cauchy problem begin aligned left begin aligned du t au t dt f t u t dt g t u t dw h t quad t in 0 t u 0 x 0 end aligned right end aligned here a is the generator of an analytic c 0 semigroup on a umd banach space x w h is a cylindrical brownian motion in a hilbert space h and the functions f 0 t times x rightarrow x theta f and g 0 t times x rightarrow fancyscript l h x theta g satisfy appropriate local lipschitz conditions the results are applied to a class of second order parabolic spdes driven by multiplicative space time white noise

stochastic de giorgi iteration and regularity of stochastic partial differential equations under general conditions we devise a stochastic version of de giorgi iteration scheme for semilinear stochastic parabolic partial differential equation of the form tu div a u f t x u gi t x u w ti with progressively measurable diffusion coefficients we use the scheme to show that the solution of the equation is almost surely holder continuous in both space and time variables

bmo estimates for the p laplacian we prove bmo estimates of the inhomogeneous p laplace system given by div jruj p 2 ru divf we show that f 2 bmo impliesjruj p 2 ru2 bmo which is the limiting case of the nonlinear calder on zygmund theory this extends the work of dibenedetto and manfredi dm93 which was restricted to the super quadratic case p 2 to the full case 1 p 1 and even more general growth moreover we prove that a ru inherits the campanato and vmo regularity of f

traces and embeddings of anisotropic function spaces in this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities to a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in sobolev and mixed derivative embeddings with fixed integrability we apply the general results to prove maximal l p l q regularity for the linearized fully inhomogeneous two phase stefan problem with gibbs thomson correction

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

stochastic maximal lp regularity in this article we prove a maximal lp regularity result for stochastic convolutions which extends krylov s basic mixed lp lq inequality for the laplace operator on d to large classes of elliptic operators both on d and on bounded domains in d with various boundary conditions our method of proof is based on mcintosh s h functional calculus r boundedness techniques and sharp lp lq square function estimates for stochastic integrals in lq spaces under an additional invertibility assumption on a a maximal space time lp regularity result is obtained as well

lp theory of semi linear spdes on general measure spaces and applications abstract in general settings applying evolutional semigroup arguments we prove the existence and uniqueness of l p solutions to semi linear spdes of the type d u t x l u t x f t x u t d t k g k t x u t d w t k u 0 x u 0 x x e where l is an unbounded linear negative operator on l p e b w t k t 0 k 1 2 is a sequence of independent brownian motions and e b is a general measure space we also discuss the regularities of solutions in sobolev spaces moreover a time discretized approximation for above equation is proved to convergence in holder spaces as applications we study several classes of solutions for different types spdes on abstract wiener space and riemannian manifold

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

regular dependence on initial data for stochastic evolution equations with multiplicative poisson noise abstract we prove existence uniqueness and lipschitz dependence on the initial datum for mild solutions of stochastic partial differential equations with lipschitz coefficients driven by wiener and poisson noise under additional assumptions we prove g teaux and frechet differentiability of solutions with respect to the initial datum as an application we obtain gradient estimates for the resolvent associated to the mild solution finally we prove the strong feller property of the associated semigroup

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

well posedness for the fractional fokker planck equations in this paper we study the fractional fokker planck equation and obtain the existence and uniqueness of weak lp solutions 1 p under the assumptions that the coefficients are only in sobolev spaces moreover to l solutions we gain the well posedness for bv coefficients besides the non negative weak lp solutions and renormalized solutions are derived after then we achieve the stability for stationary solutions

on the regularity of the stochastic heat equation on polygonal domains in r2 abstract we establish existence uniqueness and higher order weighted l p sobolev regularity for the stochastic heat equation with zero dirichlet boundary condition on angular domains and on polygonal domains in r 2 we use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable in this way we can capture the influence of both main sources for singularities the incompatibility between noise and boundary condition on the one hand and the singularities of the boundary on the other hand the range of admissible powers of the distance to the vertexes is described in terms of the maximal interior angle and is sharp

conical stochastic maximal l p regularity for 1 leqslant p infty let a mathrm div a cdot nabla be a second order divergence form elliptic operator on mathbb r n with bounded measurable real valued coefficients and let w be a cylindrical brownian motion in a hilbert space h our main result implies that the stochastic convolution process begin aligned u t int 0 t e t s a g s dw s quad t geqslant 0 end aligned satisfies for all 1 leqslant p infty a conical maximal l p regularity estimate begin aligned mathbb e vert nabla u vert t 2 p 2 mathbb r times mathbb r n p leqslant c p p mathbb e vert g vert t 2 p 2 mathbb r times mathbb r n h p end aligned here t 2 p 2 mathbb r times mathbb r n and t 2 p 2 mathbb r times mathbb r n h are the parabolic tent spaces of real valued and h valued functions respectively this contrasts with krylov s maximal l p regularity estimate begin aligned mathbb e vert nabla u vert l p mathbb r l 2 mathbb r n mathbb r n p leqslant c p mathbb e vert g vert l p mathbb r l 2 mathbb r n h p end aligned which is known to hold only for 2 leqslant p infty even when a delta and h mathbb r the proof is based on an l 2 estimate and extrapolation arguments which use the fact that a satisfies suitable off diagonal bounds our results are applied to obtain conical stochastic maximal l p regularity for a class of nonlinear spdes with rough initial data

on singularity as a function of time of a conditional distribution of an exit time we establish the singularity with respect to lebesgue measure as a function of time of the conditional probability distribution that the sum of two one dimensional brownian motions will exit from the unit interval before time t given the trajectory of the second brownian motion up to the same time on the way of doing so we show that if one solves the one dimensional heat equation with zero condition on a trajectory of a one dimensional brownian motion which is the lateral boundary then for each moment of time with probability one the normal derivative of the solution is zero provided that the diffusion of the brownian motion is sufficiently large

global strong solutions to the 3d incompressible mhd equations with density dependent viscosity abstract this paper is concerned with the 3d incompressible mhd equations with density dependent viscosity in smooth bounded domains through time weighted a priori estimates the global existence of strong solutions is established under the assumption that the initial energy is suitably small this generalizes previous results for the 3d navier stokes equations in huang and wang 2015 and zhang 2015 which need u 0 l 2 to be small the initial vacuum is allowed

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

on the quasi linear reflected backward stochastic partial differential equations abstract this paper is concerned with the quasi linear reflected backward stochastic partial differential equation rbspde for short basing on the theory of backward stochastic partial differential equation and the parabolic capacity and potential we first associate rbspde to a variational problem and via the penalization method we prove the existence and uniqueness of the solution for linear rbspde with laplacian leading coefficients with the continuity approach we further obtain the well posedness of general quasi linear rbspdes related results including ito formulas for backward stochastic partial differential equations with stochastic regular measures the comparison principle for solutions of rbspdes and the connections with reflected backward stochastic differential equations and optimal stopping problems are addressed as well

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

parabolic littlewood paley inequality for a class of time dependent pseudo differential operators of arbitrary order and applications to high order stochastic pde in this paper we prove a parabolic version of the littlewood paley inequality for a class of time dependent local and non local operators of arbitrary order and as an application we show that this inequality gives a fundamental estimate for an lp theory of high order stochastic partial differential equations

well posedness and optimal regularity of stochastic evolution equations with multiplicative noises abstract in this paper we establish the well posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized lipschitz type coefficients driven by general multiplicative noises to ensure the well posedness of the problem the linear operator of the equations is only need to be a generator of a c 0 semigroup and the proposed noises are quite general which include space time white noise and rougher noises when the linear operator generates an analytic c 0 semigroup we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method

numerically erasure robust frames abstract given a channel with additive noise and adversarial erasures the task is to design a frame that allows for stable signal reconstruction from transmitted frame coefficients to meet these specifications we introduce numerically erasure robust frames we first consider a variety of constructions including random frames equiangular tight frames and group frames later we show that arbitrarily large erasure rates necessarily induce numerical instability in signal reconstruction we conclude with a few observations including some implications for maximal equiangular tight frames and sparse frames

degenerate parabolic stochastic partial differential equations quasilinear case we study the cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical wiener process in particular we adapt the notion of kinetic formulation and kinetic solution and develop a well posedness theory that includes also an l 1 contraction property in comparison to the first order case debussche and vovelle 2010 and to the semilinear degenerate parabolic case hofmanova 2013 the present result contains two new ingredients a generalized ito formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem

global strong solution to the 2d nonhomogeneous incompressible mhd system abstract in this paper we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible mhd system as long as the initial data satisfies some compatibility condition as a corollary the global existence of strong solution with vacuum to the 2d nonhomogeneous incompressible navier stokes equations is also established our main result improves all the previous results where the initial density need to be strictly positive the key idea is to use some critical sobolev inequality of logarithmic type which is originally due to brezis and wainger 1980 7

l p estimates for the uniform norm of solutions of quasilinear spde s in this paper we prove l p estimates p 2 for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations spde of parabolic type our method is based on a version of moser s iteration scheme developed by aronson and serrin in the context of non linear parabolic pde

global small solutions of 2 d incompressible mhd system abstract in this paper we consider the global wellposedness of 2 d incompressible magneto hydrodynamical system with smooth initial data which is close to some non trivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system to achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation we first reformulate our system 1 1 in the lagrangian coordinates 2 19 then we employ anisotropic littlewood paley analysis to establish the key a priori l 1 r lip r 2 estimate for the lagrangian velocity field y t with this estimate we can prove the global wellposedness of 2 19 with smooth and small initial data by using the energy method we emphasize that the algebraic structure of 2 19 is crucial for the proofs to work the global wellposedness of the original system 1 1 then follows by a suitable change of variables

global regularity to the 3d incompressible mhd equations abstract in this paper we prove the global regularity to the 3d nonhomogeneous incompressible magnetohydrodynamic equations let 0 m 0 and h 0 be the initial density momentum and magnetic field respectively we establish a unique strong solution on r 3 0 t for any 0 t under the assumption that the viscosity coefficient 0 is sufficiently large or m 0 2 0 l 1 2 h 0 l 2 2 or u 0 l 2 2 h 0 h 1 2 is small enough moreover if the given data are more regular and satisfy an additional compatibility condition for the existence of strong solution then we show that the strong solution is indeed a classical one moreover the weak strong uniqueness of solutions is also obtained which generalizes the result in 13 to the case of vacuum

stability properties of stochastic maximal l p regularity in this paper we consider lp regularity estimates for solutions to stochastic evolution equations which is called stochastic maximal lp regularity our aim is to find a theory which is analogously to dore s theory for deterministic evolution equations he has shown that maximal lp regularity is independent of the length of the time interval implies analyticity and exponential stability of the semigroup is stable under perturbation and many more properties we show that the stochastic versions of these results hold

holder continuous of the solutions to stochastic nonlocal heat equations abstract in this paper we aim to obtain the holder continuous of solutions to stochastic nonlocal heat equations by using campanato estimates and sobolev embedding theorem we first prove the holder continuous of the mild solution of stochastic nonlocal diffusion equations locally in r d in the sense that the solution u belongs to the space c d t l p then by using tail estimates we obtain the estimates of the mild solution in l p c d t lastly we prove the holder continuous of the mild solution on bounded domain

on quasi orthogonal signatures for cdma systems sum capacity optimal signatures in synchronous code division multiple access cdma systems are functions of the codebook length as well as the number of active users a new signature set must be assigned every time the number of active users changes this correspondence considers signature sets that are less sensitive to changes in the number of active users equiangular signature sequences are proven to solve a certain max min signal to interference plus noise problem which results from their interference invariance unions of orthonormal bases have subsets that come close to satisfying the welch bound bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided connections are made between these signature design problems grassmannian line packing frame theory and algebraic geometry

global regularity of 3d nonhomogeneous incompressible magneto micropolar system with the density dependent viscosity abstract this paper is concerned with the 3d nonhomogeneous incompressible magneto micropolar equations with density dependent viscosity and vacuum in smooth bounded domains we prove that the system has a unique global smooth solution under the assumption that the initial energy is suitably small

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions

maximal regularity for stochastic convolutions driven by levy processes we generalize the maximal regularity result from da prato and lunardi atti accad naz lincei cl sci fis mat natur rend lincei 9 mat appl 9 1 25 29 1998 to stochastic convolutions driven by time homogenous poisson random measures and cylindrical infinite dimensional wiener processes

global well posedness for the incompressible navier stokes equations with density dependent viscosity coefficient abstract this paper is concerned with an initial boundary value problem of the incompressible navier stokes equations with density dependent viscosity in a smooth bounded domain r 3 the global well posedness of strong solutions with large oscillations is established in both non vacuum and vacuum cases provided the initial velocity is suitably small in certain sense it is worth pointing out that there isn t any smallness condition on the density and its gradient

global strong solution to the three dimensional density dependent incompressible magnetohydrodynamic flows abstract an initial boundary value problem is considered for the density dependent incompressible viscous magnetohydrodynamic flow in a three dimensional bounded domain the homogeneous dirichlet boundary condition is prescribed on the velocity and the perfectly conducting wall condition is prescribed on the magnetic field for the initial density away from vacuum the existence and uniqueness are established for the local strong solution with large initial data as well as for the global strong solution with small initial data furthermore the weak strong uniqueness of solutions is also proved which shows that the weak solution is equal to the strong solution with certain initial data