new formulations for interface problems in polar coordinates in this paper numerical methods are proposed for some interface problems in polar or cartesian coordinates the new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution into a problem with a smooth solution the new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution in conjunction with the fast immersed interface method a fast solver has been developed for the interface problems with a piecewise constant but a discontinuous coefficient using the new formulation in a polar coordinate system

instability of the cauchy kovalevskaya solution for a class of nonlinear systems we prove that in any c infty neighborhood of an analytic cauchy datum there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first order nonlinear systems we use a method initiated by g m etivier for elliptic systems based on the representation of solutions and on the fbi transform in our case the system can be hyperbolic at initial time but the characteristic roots leave the real line at positive times

fast solvers for 3d poisson equations involving interfaces in a finite or the infinite domain in this paper numerical methods are proposed for poisson equations defined in a finite or infinite domain in three dimensions in the domain there can exists an interface across which the source term the flux and therefore the solution may be discontinuous the existence and uniqueness of the solution are also discussed to deal with the discontinuity in the source term and in the flux the original problem is transformed to a new one with a smooth solution such a transformation can be carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three dimensional function an auxiliary sphere is used to separate the infinite region into an interior and exterior domain the kelvin s inversion is used to map the exterior domain into an interior domain the two poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously by choosing the mesh size carefully and exploiting the fast fourier transform the resulting finite difference equations can be solved efficiently the approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain numerical results demonstrate second order accuracy of our algorithms

survival probability for the stadium billiard abstract we consider the open stadium billiard consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides due to the hyperbolic nature of the stadium billiard the initial decay of trajectories due to loss through the hole appears exponential however some trajectories bouncing ball orbits persist and survive for long times and therefore form the main contribution to the survival probability function at long times using both numerical and analytical methods we concur with previous studies that the long time survival probability for a reasonably small hole drops like c o n s t a n t t i m e 1 here we obtain an explicit expression for the constant

statistical properties for flows with unbounded roof function including the lorenz attractor for geometric lorenz attractors including the classical lorenz attractor we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors we also obtain the functional central limit theorem and moment estimates as well as iterated versions of these results a consequence is deterministic homogenisation convergence to a stochastic differential equation for fast slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two

invariant helical subspaces for the navier stokes equations three dimensional solutions with helical symmetry are shown to form an invariant subspace for the navier stokes equations uniqueness of weak helical solutions in the sense of leray is proved and these weak solutions are shown to be regular strong solutions existing for arbitrary time t the global universal attractor for the infinite dimensional dynamical system generated by the corresponding semi group of helical flows is shown to be compact and finite dimensional the hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class with the most detailed results obtained for rotating hagen poiseuille pipe flow in this case the dimension either hausdorff or fractal up to an absolute constant is bounded from above by frac operatorname re sqrt alpha 2 n 2 where is the axial wavenumber n is the azimuthal wavenumber and re is the reynolds number based on the radius of the pipe these upper bounds are independent of the rotation rate

on the time fractional navier stokes equations abstract this paper is concerned with the navier stokes equations with time fractional derivative of order 0 1 this type of equations can be used to simulate anomalous diffusion in fractal media we establish the existence and uniqueness of local and global mild solutions in h q meanwhile we also give local mild solutions in j q moreover we prove the existence and regularity of classical solutions for such equations in j q

zero viscosity limit of the navier stokes equations in the analytic setting in this paper we consider the zero viscosity limit of the navier stokes equations in a half space with non slip boundary condition based on the vorticity formulation and the use of conormal derivative we develop an energy method to justify the zero viscosity limit for the analytic data

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

rates of mixing for non markov infinite measure semiflows we develop an abstract framework for obtaining optimal rates of mixing and higher order asymptotics for infinite measure semiflows previously such results were restricted to the situation where there is a first return poincare map that is uniformly expanding and markov as illustrations of the method we consider semiflows over non markov pomeau manneville intermittent maps with infinite measure and we also obtain mixing rates for semiflows over collet eckmann maps with nonintegrable roof function

accurate numerical methods for micromagnetics simulations with general geometries in current fft based algorithms for micromagnetics simulations the boundary is typically replaced by a staircase approximation along the grid lines either eliminating the incomplete cells or replacing them by complete cells sometimes the magnetizations at the boundary cells are weighted by the volume of the sample in the corresponding cell we show that this leads to large errors in the computed exchange and stray fields one consequence of this is that the predicted switching mechanism depends sensitively on the orientation of the numerical grid we present a boundary corrected algorithm to efficiently and accurately handle the incomplete cells at the boundary we show that this boundary corrected algorithm greatly improves the accuracy in micromagnetics simulations we demonstrate by using a arrott s example of a hexagonal element that the switching mechanism is predicted independently of the grid orientation

global regularity for some classes of large solutions to the navier stokes equations in previous works by the first two authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the main feature of the initial data considered in one of those studies is that it varies slowly in one direction though in some sense it is well prepared its norm is large but does not depend on the slow parameter the aim of this article is to generalize that setting to an ill prepared situation the norm blows up as the small parameter goes to zero as in those works the proof uses the special structure of the nonlinear term of the equation

compressible navier stokes system large solutions and incompressible limit abstract here we prove the existence of global in time regular solutions to the two dimensional compressible navier stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density 0 for large volume bulk viscosity the result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible navier stokes equations supplemented with the divergence free projection of v 0 is global the systems are examined in r d with d 2 in the critical b 2 1 s besov spaces framework

dispersing billiards with cusps slow decay of correlations dispersing billiards introduced by sinai are uniformly hyperbolic and have strong statistical properties exponential decay of correlations and various limit theorems however if the billiard table has cusps corner points with zero interior angles then its hyperbolicity is nonuniform and statistical properties deteriorate until now only heuristic and experimental results existed predicting the decay of correlations as mathcal o 1 n we present a first rigorous analysis of correlations for dispersing billiards with cusps

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

long time well posedness of prandtl system with small and analytic initial data abstract in this paper we investigate the long time existence and uniqueness of small solution to d for d 2 3 dimensional prandtl system with small initial data which is analytic in the horizontal variables in particular we prove that d dimensional prandtl system has a unique solution with the life span of which is greater than e 4 3 if the initial data is of size e and the value on the boundary of the tangential velocity of the outflow are of size e 5 3 we mention that the tool developed in 4 5 to make the analytical type estimates and the special structure of the nonlinear terms to this system play an essential role in the proof of this result

limit theorems for dispersing billiards with cusps dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos exhibiting intermittent behavior that alternates between regular and chaotic patterns their statistical properties are therefore weak and delicate they are characterized by a slow power law decay of correlations and as a result the classical central limit theorem fails we prove that a non classical central limit theorem holds with a scaling factor of sqrt n log n replacing the standard sqrt n we also derive the respective weak invariance principle and we identify the class of observables for which the classical clt still holds

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

large global solutions to the navier stokes equations slowly varying in one direction in to previous papers by the authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions in the whole space contrary to the previous examples the initial data has no particular oscillatory properties but varies slowly in one direction the proof uses the special structure of the nonlinear term of the equation

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

global strong solutions for equations related to the incompressible viscoelastic fluids with a class of large initial data abstract we investigate the global strong solutions for a system related to the incompressible viscoelastic fluids of oldroyd b type with a class of large initial data e 0 e 0 i j n n v 0 v 01 v 0 n we obtain the global existence and uniqueness of a solution provided that e 0 h v 0 h are sufficiently small in b 2 1 n 2 1 b 2 1 n 2 b 2 1 n 2 1 while e 0 n and v 0 n are large where e 0 h e 0 i j n 1 n v 0 h v 01 v 0 n 1 and e 0 n e 0 n 1 e 0 n n

central limit theorem and stable laws for intermittent maps in the setting of abstract markov maps we prove results concerning the convergence of renormalized birkhoff sums to normal laws or stable laws they apply to one dimensional maps with a neutral fixed point at 0 of the form x x1 for 0 1 in particular for 1 2 we show that the birkhoff sums of a holder observable f converge to a normal law or a stable law depending on whether f 0 0 or f 0 0 the proof uses spectral techniques introduced by sarig and wiener s lemma in non commutative banach algebras

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

non stationary almost sure invariance principle for hyperbolic systems with singularities we investigate a wide class of two dimensional hyperbolic systems with singularities and prove the almost sure invariance principle asip for the random process generated by sequences of dynamically holder observables the observables could be unbounded and the process may be non stationary and need not have linearly growing variances our results apply to anosov diffeomorphisms sinai dispersing billiards and their perturbations the random processes under consideration are related to the fluctuation of lyapunov exponents the shrinking target problem etc

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

on the global wellposedness to the 3 d incompressible anisotropic navier stokes equations corresponding to the wellposedness result 2 for the classical 3 d navier stokes equations ns with initial data in the scaling invariant besov space mathcal b 1 frac3p p infty here we consider a similar problem for the 3 d anisotropic navier stokes equations ans where the vertical viscosity is zero in order to do so we first introduce the besov sobolev type spaces mathcal b frac12 frac12 4 and mathcal b frac12 frac12 4 t then with initial data in the scaling invariant space mathcal b frac12 frac12 4 we prove the global wellposedness for ans provided the norm of initial data is small enough compared to the horizontal viscosity in particular this result implies the global wellposedness of ans with high oscillatory initial data 1 2

the second iterate for the navier stokes equation we consider the iterative resolution scheme for the navier stokes equation and focus on the second iterate more precisely on the map from the initial data to the second iterate at a given time t we investigate boundedness properties of this bilinear operator this new approach yields very interesting results a new perspective on koch tataru solutions a first step towards weak strong uniqueness for koch tataru solutions and finally an instability result in b q 1 for q 2

global well posedness of compressible navier stokes equations for some classes of large initial data we prove the global well posedness of three dimensional compressible navier stokes equations for some classes of large initial data which may have large oscillation for the density and large energy for the velocity the proof uses the special structure of the system especially the effective viscous flux

a class of solutions of the navier stokes equations with large data we are concerned with global regularity of solutions in a periodic domain q 0 1 3 we prove that in the class of solutions oscillating in the vertical direction the solutions are smooth under natural conditions on the horizontal derivatives of the horizontal components of the velocity the derivative in the vertical direction and the vertical average of the initial data the obtained conditions admit data whose bmo 1 norm has algebraic dependence on 1 h where h is the period of oscillation and generate global solutions also the results allow non zero force and large data which do not decay in time

structure of helicity and global solutions of incompressible navier stokes equation in this paper we derive a new energy identity for the three dimensional incompressible navier stokes equations by a special structure of helicity the new energy functional is critical with respect to the natural scalings of the navier stokes equations moreover it is conditionally coercive as an application we construct a family of finite energy smooth solutions to the navier stokes equations whose critical norms can be arbitrarily large

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

global large solutions to 3 d inhomogeneous navier stokes system with one slow variable under a nonlinear smallness condition on the isotropic critical besov norm to the fluctuation of the initial density and the critical anisotropic besov norm of the horizontal components of the initial velocity which have to be exponentially small compared with the critical anisotropic besov norm to the third component of the initial velocity we investigate the global wellposedness of 3 d inhomogeneous incompressible navier stokes equations 1 2 in the critical besov spaces the novelty of this results is that the isotropic space structure to the inhomogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field in the second part we apply the same idea to prove the global wellposedness of 1 2 with some large data which are slowly varying in one direction

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

almost global existence for the prandtl boundary layer equations we consider the prandtl boundary layer equations on the half plane with initial datum that lies in a weighted h1 space with respect to the normal variable and is real analytic with respect to the tangential variable the boundary trace of the horizontal euler flow is taken to be a constant we prove that if the prandtl datum lies within varepsilon of a stable profile then the unique solution of the cauchy problem can be extended at least up to time t varepsilon geqq rm exp varepsilon 1 rm log varepsilon 1

limit theorems in the stadium billiard we prove that the birkhoff sums for almost every relevant observable in the stadium billiard obey a non standard limit law more precisely the usual central limit theorem holds for an observable if and only if its integral along a one codimensional invariant set vanishes otherwise a open image in new window normalization is needed as one of the two key steps in the argument we obtain a limit theorem that holds in young towers with exponential return time statistics in general an abstract result that seems to be applicable to many other situations

between homogeneous and inhomogeneous navier stokes systems the issue of stability abstract we construct large velocity vector solutions to the three dimensional inhomogeneous navier stokes system the result is proved via the stability of two dimensional solutions with constant density under the assumption that initial density is point wisely close to a constant key elements of our approach are estimates in the maximal regularity regime and the lagrangian coordinates considerations are done in the whole r 3

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

on almost sure versions of classical limit theorems for dynamical systems the purpose of this article is to support the idea that whenever we can prove a limit theorem in the classical sense for a dynamical system we can prove a suitable almost sure version based on an empirical measure with log average we follow three different approaches martingale methods spectral methods and induction arguments our results apply among others to axiom a maps or flows to systems inducing a gibbs markov map and to the stadium billiard

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

continuous phase transitions for dynamical systems we study the asymptotic expansion of the topological pressure of one parameter families of potentials at a point of non analyticity the singularity is related qualitatively and quantitatively to non gaussian limit laws and to slow decay of correlations with respect to the equilibrium measure

ill posedness of the navier stokes equations in a critical space in 3d abstract we prove that the cauchy problem for the three dimensional navier stokes equations is ill posed in b 1 in the sense that a norm inflation happens in finite time more precisely we show that initial data in the schwartz class s that are arbitrarily small in b 1 can produce solutions arbitrarily large in b 1 after an arbitrarily short time such a result implies that the solution map itself is discontinuous in b 1 at the origin

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

limit laws and recurrence for the planar lorentz process with infinite horizon as bleher j stat phys 66 1 315 373 1992 observed the free flight vector of the planar infinite horizon periodic lorentz process s n n 0 1 2 belongs to the non standard domain of attraction of the gaussian law actually with the sqrt n log n scaling our first aim is to establish his conjecture that indeed frac s n sqrt n log n converges in distribution to the gaussian law a global limit theorem here the recent method of balint and gouezel commun math phys 263 461 512 2006 helped us to essentially simplify the ideas of our earlier sketchy proof szasz d varju t in modern dynamical systems and applications pp 433 445 2004 moreover we can also derive a the local version of the global limit theorem b the recurrence of the planar infinite horizon periodic lorentz process and finally c the ergodicity of its infinite invariant measure

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

on the spectra of randomly perturbed expanding maps we consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing we do this by proving the robustness of the spectra of their perron frobenius operators

decay of correlations and invariance principles for dispersing billiards with cusps and related planar billiard flows following recent work of chernov markarian and zhang it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1 n since the collisions inside a cusp occur in quick succession it is reasonable to expect a much faster decay rate in continuous time in this paper we prove that the flow is rapid mixing correlations decay faster than any polynomial rate a consequence is that the flow admits strong statistical properties such as the almost sure invariance principle even though the billiard map does not the techniques in this paper yield new results for other standard examples in planar billiards including bunimovich flowers and stadia

global well posedness of the incompressible fractional navier stokes equations in fourier besov spaces with variable exponents abstract we study the cauchy problem of the fractional navier stokes equations in critical variable exponent fourier besov spaces f b p q 4 2 3 p we discuss some properties of variable exponent fourier besov spaces and prove a general global well posedness result which covers some recent works about classical navier stokes equations

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

discontinuous bubble scheme for elliptic problems with jumps in the solution abstract we propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface by considering a suitable function which has the same jumps as the solution we transform the problem into one without jumps then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in 1 2 3 some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested our method can also handle the case when the interface passes through grid points we believe this paper presents the first resolution of such cases numerical experiments for various problems show second order convergence in l2 and first order in h1 norms moreover the convergence order is very robust for all problems tested

spectral method for matching exterior and interior elliptic problems a spectral method is described for solving coupled elliptic problems on an interior and an exterior domain the method is formulated and tested on the two dimensional interior poisson and exterior laplace problems whose solutions and their normal derivatives are required to be continuous across the interface a complete basis of homogeneous solutions for the interior and exterior regions corresponding to all possible dirichlet boundary values at the interface are calculated in a preprocessing step this basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem chebyshev approximations are used to represent both the interior solutions and the boundary values a standard chebyshev spectral method is used to calculate the interior solutions the exterior harmonic solutions are calculated as the convolution of the free space green s function with a surface density this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a chebyshev expansion properties of chebyshev approximations insure that the basis of exterior harmonic functions represents the external near boundary solutions uniformly the method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle the resulting influence matrix is well conditioned and solutions converge exponentially as the resolution is increased the generalization of this approach to three dimensional problems is discussed in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum

global well posedness for compressible navier stokes equations with highly oscillating initial velocity in this paper we prove global well posedness for compressible navier stokes equations in the critical functional framework with the initial data close to a stable equilibrium this result allows us to construct global solutions for the highly oscillating initial velocity the proof relies on a new estimate for the hyperbolic parabolic system with convection terms 2010 wiley periodicals inc

random walks with internal degrees of freedom between two absorbing barriers consider a random walk with a finite number of internal degrees of freedom and with zero drift by using a functional analytic approach based on the spectral theory of matrix polynomials the asymptotics of the first hitting probabilities is obtained when the distance of the barriers tends to infinity

on the hydrostatic approximation of the navier stokes equations in a thin strip abstract in this paper we first prove the global well posedness of a scaled anisotropic navier stokes system and the hydrostatic navier stokes system in a 2 d striped domain with small analytic data in the tangential variable then we justify the limit from the anisotropic navier stokes system to the hydrostatic navier stokes system with analytic data