the justification of approximate bounded solutions of mixed sturm liouville problems the approximate solution of a homogeneous singular integral equation or equivalently a homogeneous infinite algebraic system is fully justified and the error resulting from the truncation is estimated this equation algebraic system is the standard form to which many even mixed sturm liouville problems are reducible

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

the spectrum of unbounded operator matrices with non diagonal domain abstract systems of linear evolution equations involving non diagonal boundary conditions yield operator matrices having non diagonal domains in products of banach spaces we show how to compute the spectrum of these matrices by using virtual matrix elements and characteristic operator functions

the spectrum of an infinite graph abstract the spectrum of a locally finite countable graph is defined some theorems known from the theory of spectra of finite graphs are extended to infinite graphs

analysis and computation of the pressure field of preatherosclerotic transport in the artery wall prerequisite to atherogenesis is low density lipoprotein ldl cholesterol transport from the blood into the artery wall across its endothelial cell ec lining whereas most ec junctions are tight enough to preclude ldl passage some approx 1 2000 6000 rare ecs have temporarily widened junctions typically approx 10 5 of the problem s radial scale that allow passage of large molecular aggregates like ldl in a purely diffusive transport model laplace s equation describes the steady tracer profile throughout the two major subendothelial wall layers connected by matching conditions in a diffusive and transmural pressure driven advective transport model laplace s equation describes the steady pressure distribution in the wall whose gradient is proportional to the advecting velocity both problems yield the same boundary value problem that has mixed dirichlet robin boundary conditions we construct a linear system of integral equations that approximates the series expansion coefficients o

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

the eigenvalue problem for networks of beams abstract in this paper we consider the spectral analysis of different models of networks of euler bernoulli beams we first give the characteristic equation for the spectrum secondly in some particular situations we show that the spectrum depends only on the structure of the graph thirdly we investigate the asymptotic behaviour of the eigenvalues by proving the so called weyl s formula

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

statistical equilibrium measures and coherent states in two dimensional turbulence the equilibrium statistics of the euler equations in two dimensions are studied and a new continuum model of coherent or organized states is proposed this model is defined by a maximum entropy principle similar to that governing the miller robert model except that the family of global vorticity invariants is relaxed to a family of inequalities on all convex enstrophy integrals this relaxation is justified by constructing the continuum model from a sequence of lattice models defined by gibbs measures whose invariants are derived from the exact vorticity dynamics not a spectral truncation or spatial discretization of it the key idea is that the enstrophy integrals can be partially lost to vorticity fluctuations on a range of scales smaller than the lattice microscale while energy is retained in the larger scales a consequence of this relaxation is that many of the convex enstrophy constraints can be inactive in equilibrium leading to a simplification of the mean field equation for the coherent state specific examples of these simplified theories are established for vortex patch dynamics in particular a universal relation between mean vorticity and stream function is obtained in the dilute limit of the vortex patch theory which is different from the sinh relation predicted by the montgomery joyce theory of point vortices 1999 john wiley sons inc

the method of integral equation formulation and the unbounded solutions of elastic contact problems abstract it is shown that a modification of the integral equation formulation 1 can be used to find an expression of the unbounded contact stress of problems in the theory of elasticity the modifications consist in reducing the problem to a hilbert type singular integral equation rather than that of the cauchy s kernel one the reduction is carried out in analogous procedures to that followed in 1 but here the unknown function is the contact stress in contrast to the previous formulation in which the unknown function was a necessarily continuous displacement the hilbert equation is inverted to define the contact stress and further reduced to an infinite algebraic system its solution completes the definition with the aid of the physical conditions the truncation of the algebraic system is justified and the error is estimated

the integral equation formulation of mixed finite time dependent elastic problems abstract this paper establishes through a concrete example that the integral equation formulation of time dependent mixed boundary value problems can be extended for problems in the theory of elasticity to this end the method applied to the resulting integral equation is the one begun by cherski 1 and evolved by eckhardt and el sheikh 2 still further to solve intial mixed boundary value problems new relationships between the fundamental coefficients characterizing the technique are obtained these simplify the procedures and reveal more of the nature of the method

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

ergodicity of the 2d navier stokes equations with random forcing we consider the navier stokes equation on a two dimensional torus with a random force acting at discrete times and analytic in space for arbitrarily small viscosity coefficient we prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time

a family of balance relations for the two dimensional navier stokes equations with random forcing for the 2d navier stokes equation perturbed by a random force of a suitable kind we show that if g f is an arbitrary real continuous function with at most polynomial growth then the stationary in time vorticity field t x satisfies mathbb e g omega t x nabla omega t x 2 frac 1 2 m 1 mathbb e g omega t x where m 1 is a number independent of g which measures the strength of the random forcing another way of stating this result is that in the unique stationary measure of this system the random variables g t x and t x 2 are uncorrelated for each t and each x

an index theory for uniformly locally finite graphs abstract an index theory for uniformly locally finite ulf graphs is developed based on the adjacency operator a acting on the space of bounded sequences defined on the vertices it turns out that the characterization by upper and lower nonnegative eigenvectors is an appropriate tool to overcome the difficulties imposed by the l setting a distinctive property of the spectral radius r a in l is the identity r sup 0 x l x 0 a x x i whilethe l 2 spectral radius r 2 of the adjacency operator satisfies r 2 inf 0 x l x 0 a x x the index i as well as other order indices can serve in classifying ulf graphs and enables connections with various graph invariants e g the chromatic number can be estimated from above by 1 r moreover results on the index i in the periodic case the regular one and for graphs having only finitely many essential ramification nodes are presented

spectral properties and asymptotic periodicity of flows in networks we combine functional analytical and graph theoretical methods in order to study flows in networks we show that these flows can be described by a strongly continuous operator semigroup on a banach space using perron frobenius spectral theory we then prove that this semigroup behaves asymptotically periodic

control of a network of euler bernoulli beams abstract the aim is to study the boundary controllability of a system modeling the vibrations of a network of n euler bernoulli beams connected by n vibrating point masses using the classical hilbert uniqueness method the control problem is reduced to the obtention of an observability inequality the solution is then expressed in terms of fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value this property called spectral gap holds as soon as the roots of a function denoted by f and giving the asymptotic behaviour of the eigenvalues are all simple for a network of n 2 different beams this assumption on the multiplicity of the roots of f denoted by a is proved to be satisfied and controllability follows for higher values of n a numerical approach allows one to prove a in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open

remarks on the balance relations for the two dimensional navier stokes equation with random forcing we use the balance relations for the stationary in time solutions of the randomly forced 2d navier stokes equations found in 10 to study these solutions further we show that the vorticity t x of a stationary solution has a finite exponential moment and that for any a in mathbb r t geq 0 the expectation of the integral of nabla x xi over the level set x mid xi t x a up to a constant factor equals the expectation of the integral of nabla x xi 1 over the same set

martingale solutions for the three dimensional stochastic nonhomogeneous incompressible navier stokes equations driven by levy processes abstract in this paper the three dimensional stochastic nonhomogeneous incompressible navier stokes equations driven by levy processes consisting of the brownian motion the compensated poisson random measure and the poisson random measure are considered in a bounded domain we obtain the existence of martingale solutions the construction of the solution is based on the classical galerkin approximation method the stopping times the stochastic compactness method and the jakubowski skorokhod theorem

inviscid limit for damped and driven incompressible navier stokes equations in mathbb r 2 we consider the zero viscosity limit of long time averages of solutions of damped and driven navier stokes equations in mathbb r 2 we prove that the rate of dissipation of enstrophy vanishes stationary statistical solutions of the damped and driven navier stokes equations converge to renormalized stationary statistical solutions of the damped and driven euler equations these solutions obey the enstrophy balance

on divergence free drifts we investigate the validity and failure of liouville theorems and harnack inequalities for parabolic and elliptic operators with low regularity coefficients we are particularly interested in operators of the form t b resp b with a divergence free drift b we prove the liouville theorem and harnack inequality when b l bmo 1 resp b bmo 1 and provide a counterexample demonstrating sharpness of our conditions on the drift our results generalize to divergence form operators with an elliptic symmetric part and a bmo skew symmetric part we also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions depending on the non scale invariant norm b l1 in three dimensions on the other hand bounded solutions with l1 drifts may be discontinuous

neumann problem for generalized n poisson equation using a hierarchy of integral operators having higher order neumann functions and their derivatives as kernels the neumann problem for a 2nth order linear partial complex differential equation is discussed the solvability of the problem is obtained

malliavin calculus for the stochastic 2d navier stokes equation we consider the incompressible two dimensional navier stokes equation with periodic boundary conditions under the effect of an additive white in time stochastic forcing under mild restrictions on the geometry of the scales forced we show that any finite dimensional projection of the solution possesses a smooth strictly positive density with respect to lebesgue measure in particular our conditions are viscosity independent we are mainly interested in forcing that excites a very small number of modes all of the results rely on proving the nondegeneracy of the infinite dimensional malliavin matrix c 2006 wiley periodicals inc

compressible fluids driven by stochastic forcing the relative energy inequality and applications we show the relative energy inequality for the compressible navier stokes system driven by a stochastic forcing as a corollary we prove the weak strong uniqueness property pathwise and in law and convergence of weak solutions in the inviscid incompressible limit in particular we establish a yamada watanabe type result in the context of the compressible navier stokes system that is pathwise weak strong uniqueness implies weak strong uniqueness in law

on landau damping going beyond the linearized study has been a longstanding problem in the theory of landau damping in this paper we establish exponential landau damping in analytic regularity the damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables rather than exchanges of energy phase mixing is the driving mechanism the analysis involves new families of analytic norms measuring regularity by comparison with solutions of the free transport equation new functional inequalities a control of non linear echoes sharp deflection estimates and a newton approximation scheme our results hold for any potential no more singular than coulomb or newton interaction the limit cases are included with specific technical effort as a side result the stability of homogeneous equilibria of the non linear vlasov equation is established under sharp assumptions we point out the strong analogy with the kam theory and discuss physical implications finally we extend these results to some gevrey non analytic distribution functions

local times for solutions of the complex ginzburg landau equation and the inviscid limit abstract we consider the behaviour of the distribution for stationary solutions of the complex ginzburg landau equation perturbed by a random force it was proved in s kuksin and a shirikyan 2004 4 that if the random force is proportional to the square root of the viscosity 0 then the family of stationary measures possesses an accumulation point as 0 we show that if is such a point then the distributions of the l 2 norm and of the energy possess a density with respect to the lebesgue measure the proofs are based on ito s formula and some properties of local time for semimartingales

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

existence and uniqueness for stochastic 2d euler flows with bounded vorticity the strong existence and the pathwise uniqueness of solutions with l infty vorticity of the 2d stochastic euler equations are proved the noise is multiplicative and it involves the first derivatives a lagrangian approach is implemented where a stochastic flow solving a nonlinear flow equation is constructed the stability under regularizations is also proved

eigenvalue bracketing for discrete and metric graphs abstract we develop eigenvalue estimates for the laplacians on discrete and metric graphs using various types of boundary conditions at the vertices of the metric graph via an explicit correspondence of the equilateral metric and discrete graph spectrum also in the exceptional values of the metric graph corresponding to the dirichlet spectrum we carry over these estimates from the metric graph laplacian to the discrete case we apply the results to covering graphs and present examples where the covering graph laplacians have spectral gaps

well posedness and symmetries of strongly coupled network equations we consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different possibly non adjacent edges this generalizes the setting that is common in the literature where the only considered interactions take place at the boundary i e in the nodes of the network we discuss well posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions finally we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space i e of symmetries of the associated physical system applications to a neurobiological model as well as to a system of linear schrodinger equations on a quantum graph are discussed

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

the spectrum of the hilbert space valued second derivative with general self adjoint boundary conditions abstract we consider a large class of self adjoint elliptic problems associated with the second derivative acting on a space of vector valued functions we present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems the first more general one allows to replace a secular equation which is well known in some special cases by an abstract rank condition the second one though available in general seems to apply particularly well to a specific boundary condition the sometimes dubbed anti kirchhoff condition in the literature that arises in the theory of differential operators on graphs it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities in particular some results on the symmetry of the spectrum in either case

a characteristic equation associated to an eigenvalue problem on c2 networks abstract a characteristic equation for the spectrum of an eigenvalue problem on c 2 networks is deduced and those special cases are investigated in which the spectrum depends mainly on the structure of the abstract graph belonging to the c 2 network

the finite mixed sturm liouville problems of the dirichlet newton type abstract the mixed sturm liouville problem for the laplace operator in the unit circle is considered the boundary conditions consist in the dirichlet condition prescribed partially at the circumference while the complementary part is subjected to the newton condition the problem is reduced to a homogeneous wing singular integro differential equation and this in turn to an infinite system of algebraic equations whose solution can conveniently be obtained through truncation the procedures of the reduction suggest that almost any problem of the type considered in this paper can be reduced to wing equations of the same class the calculations are carried out right to the numerical steps the first few eigenvalues and their corresponding eigenfunctions are obtained

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

eigenvalue estimates for the laplacian with anti kirchhoff conditions on a metric tree abstract this paper deals with the eigenvalue estimates for laplace operators with anti kirchhoff conditions on compact trees the results include lower estimates for k involving the edge lengths or diameter of when is a star and upper estimates for k involving the average edge length or the diameter of when is a general tree we get the upper eigenvalue estimates by proving a monotonicity principle for the eigenvalues with respect to the domain graph two geometric versions of the ambartsumian theorem for are also proven

tight wavelet frames on multislice graphs we present a framework for the design of wavelet transforms tailored to data defined on multislice graphs i e multiplex or dynamic graphs graphs with multiple types of interactions are ubiquitous in real life motivating the extension of wavelets to these complex domains our framework generalizes the recently proposed spectral graph wavelet transform sgwt d hammond p vandergheynst and r gribonval wavelets on graphs via spectral graph theory appl comput harmon anal vol 30 pp 129 150 mar 2011 which is designed in the spectral frequency domain of an arbitrary finite weighted graph we extend the sgwt to form a tight frame which conserves energy in the wavelet domain and define the relationship between conventional and spectral graph wavelets we then propose a design for multislice graphs that is based on the higher order singular value decomposition hosvd a powerful tool from multilinear algebra in particular the multiple adjacency matrices are stacked to form a tensor and the hosvd decomposition provides information about its third order structure analogous to that provided by matrix factorizations we obtain a set of eigennetworks and from these graph wavelets which exploit the variability across the graphs we demonstrate the feasibility of our method 1 by capturing different orientations of a gray scale image and 2 by decomposing brain signals from functional magnetic resonance imaging we show its effectiveness to identify variability across graph edges and provide meaningful decompositions

mathematical aspects of vacuum energy on quantum graphs we use quantum graphs as a model to study various mathematical aspects of the vacuum energy such as convergence of periodic path expansions consistency among different methods trace formulae versus method of images and the possible connection with the underlying classical dynamics in our study we derive an expansion for the vacuum energy in terms of periodic paths on the graph and prove its convergence and smooth dependence on the bond lengths of the graph for an important special case of graphs with equal bond lengths we derive a simpler explicit formula with minor changes this formula also applies to graphs with rational up to a common factor bond lengths the main results are derived using the trace formula we also discuss an alternative approach using the method of images and prove that the results are consistent this may have important consequences for other systems since the method of images unlike the trace formula includes a sum over special bounce paths we succeed in showing that in our model bounce paths do not contribute to the vacuum energy finally we discuss the proposed possible link between the magnitude of the vacuum energy and the type chaotic versus integrable of the underlying classical dynamics within a random matrix model we calculate the variance of the vacuum energy over several ensembles and find evidence that the level repulsion leads to suppression of the vacuum energy

harmonic solutions of a mixed boundary problem arising in the modeling of macromolecular transport into vessel walls the earliest events leading to atherosclerosis involve the transport of lpw density lipooprotein ldl cholesterol from the blood across endothelial cells that line the artery wall laplace s equation describes the steady state diffusion profile of a tracer through the vessel wall this gives rise to a boundary value problem with mixed dirichlet and robin conditions we construct a linear system of integral equations that approximate the coefficients of the series expansion of the solution we prove the existence of the solution to this problem analytically by using gershgorin s theorem on the location of the eigenvalues of the corresponding matrix we give a uniqueness proof using miranda s theorem c miranda p d e of elliptic type springer verlag berlin 1970 the analytical construction method forms the basis for a numerical calculation algorithm we apply our results to the transport problem above and use them to interpret experimental observations of the growth of localized tracer leakage spots with tracer circulation time

poincare perron s method for the dirichlet problem on stratified sets abstract we extend standard poincare perron s method to the dirichlet problem on a class of multistructures this method is based on the spherical mean theorem the construction of fundamental solutions and on harnack s inequality on such domains that we first establish

spectral asymptotics for quantum graphs with equal edge lengths if q is smooth on the edges of a finite metric graph with all edges of length one the eigenvalues of the schrodinger operator d2 q with standard vertex conditions have generalized asymptotic expansions to all orders

on distribution of energy and vorticity for solutions of 2d navier stokes equation with small viscosity we study distributions of some functionals of space periodic solutions for the randomly perturbed 2d navier stokes equation and of their limits when the viscosity goes to zero the results obtained give explicit information on distribution of the velocity field of space periodic turbulent 2d flows

on the loss of continuity for super critical drift diffusion equations we show that there exist solutions of drift diffusion equations in two dimensions with divergence free super critical drifts that become discontinuous in finite time we consider classical as well as fractional diffusion however in the case of classical diffusion and time independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local l 1 norm of the drift which is a super critical quantity

on the mixed initial dirichlet neumann problem for the wave equation in the circle the method of integral equation formulation proposed by eckhardt and el sheikh is extended to solve an even initial mixed problem for the first time the truncation of the integral equation or equivalently the infinite homogeneous algebraic system to which the problem is converted is justified in addition having carried out the progression of the procedures right to the numerical results at several cases a clarified and comprehensive idea about the usefulness of the method is revealed

global l2 solutions of stochastic navier stokes equations this paper concerns the cauchy problem in r d for the stochastic navier stokes equation 1 u u u u p f u u p g u o w u 0 u 0 div u 0 driven by white noise w under minimal assumptions on regularity of the coefficients and random forces the existence of a global weak martingale solution of the stochastic navier stokes equation is proved in the two dimensional case the existence and pathwise uniqueness of a global strong solution is shown a wiener chaos based criterion for the existence and uniqueness of a strong global solution of the navier stokes equations is established

local martingale and pathwise solutions for an abstract fluids model we establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system the primary application of this abstract framework is to infer the local existence of strong pathwise solutions to the 3d primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise instead of developing our results specifically for the 3d primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations atmosphere oceans coupled atmosphere ocean on the sphere on the beta plane approximation etc and the incompressible navier stokes equations in applications all of the details are given for the beta plane approximation of the oceans equations

spectral analysis of the schrodinger operator on binary tree shaped networks and applications abstract in this paper we analyse the spectrum of the dissipative schrodinger operator on binary tree shaped networks as applications we study the stability of the schrodinger system using a riesz basis as well as the transfer function associated to the system

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

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

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

on inviscid limits for the stochastic navier stokes equations and related models we study inviscid limits of invariant measures for the 2d stochastic navier stokes equations as shown by kuksin j stat phys 115 1 2 469 492 2004 the noise scaling sqrt nu is the only one which leads to non trivial limiting measures which are invariant for the 2d euler equations using a moser type iteration for stochastic drift diffusion equations we show that any limiting measure mu 0 is in fact supported on bounded vorticities relationships of mu 0 to the long term dynamics of 2d euler in l infty with the weak topology are discussed we also obtain a drift independent modulus of continuity for a stationary deterministic model problem which leads us to conjecture that in fact mu 0 is supported on c 0 moreover in view of the batchelor krainchnan 2d turbulence theory we consider inviscid limits for a weakly damped stochastic navier stokes equation in this setting we show that only an order zero noise scaling with respect to leads to a nontrivial limiting measure in the inviscid limit

surgery principles for the spectral analysis of quantum graphs we present a systematic collection of spectral surgery principles for the laplacian on a compact metric graph with any of the usual vertex conditions natural dirichlet or delta type which show

the justification of the truncation applied to homogeneous integral equations with cauchy s kernel abstract this paper is devoted to studying the influence of the truncation applied to a homogeneous cauchy type singular integral equation on its eigenvalues and eigenfunctions this equation represents the class to which mixed sturm liouville problems of the dirichlet neuman type are reducible the study is illustrated through considering a concrete problem it has been shown that an eigenvalue of the truncated equation tends to the exact corresponding one on increasing the order of the truncation set on that equation as for a truncated eigenfunction it leads to the designation of the exact one in the limit where the corresponding eigenvalue is sufficiently precise an illustrative example is given

nonreal eigenvalues for second order differential operators on networks with circuits it is shown that on every finite network with at least one circuit there exist second order differential operators having an infinite number of nonreal eigenvalues the presence of nonreal eigenvalues implies that these operators cannot be selfadjoint with respect to any metric these eigenvalues reveal also the existence of oscillatory solutions for the corresponding time dependent partial differential equations

a fourier method for initial value problems with mixed boundary conditions abstract works concerning solutions of initial problems with mixed boundary conditions are very rare in the literature and essentially depend on the geometry of some individual problems k b ranger for example considered the cooling of a finite needle and strip with prescribed cooling temperature in a medium bounded by two parallel planes perpendicular to the axis and maintained at zero temperature 1 because of the axial symmetries of these problems ranger could employ integral operators which map 2 d harmonics into the solution of the reduced wave equation and into that of the heat equation thereby in this work a systematic method for solving initial plane problems with mixed boundary conditions is proposed it can be considered as a natural generalization of the fourier method that deals with the uniform initial problems on the one hand and of that started by ju i cersi 2 dealing with steady mixed boundary problems on the other by gradual modifications the latter method has become of wide application to solutions of problems in several branches of mathematical physics this has recently been expounded 3 we shall start with presenting the method by means of a concrete example that geometrically as well as physically is rather simplified the heat conductivity in the unit circle with dirichlet neumann boundary conditions the method is applicable to problems with different configurations further it can be extended to deal with more complicated physical situations of mixed character such as dynamical problems of the theory of elasticity with the typical stress strain boundary conditions

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

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

the l2 strong maximum principle on arbitrary countable networks abstract we study the strong maximum principle for the heat equation associated with the dirichlet form on countable networks we start by analysing the boundedness properties of the incidence operators on a countable network subsequently we prove that the strong maximum principle is equivalent to the underlying graph being connected after deletion of the nodes with infinite degree using this result we prove that the number of connected components of the graph with respect to the heat flow equals the number of maximal invariant ideals of the adjacency matrix

stochastic equations in hilbert space with application to navier stokes equations in any dimension abstract we give an existence theorem for an abstract nonlinear stochastic evolution equation in a hilbert space the result is applicable to the stochastic navier stokes equation in any dimension with a nonlinear noise term

ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes we consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general nonlinear dissipative parabolic pdes results on invariant measures and their exponential mixing properties are proved and applications to 2d navier stokes systems are included

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

the estimation of the error resulting due to the truncation applied to homogeneous integral equations with cauchy s kernel abstract many mixed sturm liouville problems can be formulated in a standard form of a discrete riemann problem when it is concerned with the holder continuous entity at the points where the boundary condition changes the discrete problem can be transformed to a homogeneous integral equation with cauchy s kernel this equation can in general be approximately solved and namely through truncation this gives rise to several questions about the justification of the truncation applied to a homogeneous operator as well as the influence of this truncation on the eigenfunctions and eigenvalues in this paper it has been shown that provided an eigenvalue of this integral equation is precisely given the corresponding solution of the truncated integral equations tends to the unique solution of the exact equation on increasing the truncations order indefinitely and the relative error is estimated as for the influence of the truncation on the eigenvalues of that equation it may constitute the subject of another study

ito s formula for the lp norm of stochastic w 1 _ p valued processes we prove ito s formula for the lp norm of a stochastic w 1 p valued processes appearing in the theory of spdes in divergence form

concrete method for recovering the euler characteristic of quantum graphs trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs the basis of our work is the result by kurasov which links the euler characteristic of metric graphs to the spectrum of their standard laplacian these ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured in the present work we analyse sufficient hypotheses which guarantee the successful recovery of we also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required finally we compare our findings with numerical examples surprisingly just a few dozens of eigenvalues can be enough

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

multiplicity of solutions of second order linear differential equations on networks abstract an upper optimal bound is obtained for the multiplicity of solutions of second order linear differential equations on networks with general boundary and node conditions this bound depends only on the abstract topological associated graph in particular the result holds for the multiplicity of eigenvalues of the corresponding operators