bilinear restriction estimates for surfaces with curvatures of different signs recently the sharp l 2 bilinear adjoint restriction estimates for the cone and the paraboloid were established by wolff and tao respectively their results rely on the fact that for the cone and the paraboloid the nonzero principal curvatures have the same sign we generalize those bilinear restriction estimates to surfaces with curvatures of different signs

extension of finite volume compressible flow solvers to multi dimensional variable density zero mach number flows when attempting to compute unsteady variable density flows at very small or zero mach number using a standard finite volume compressible flow solver one faces at least the following difficulties i spatial pressure variations vanish as the mach numberm

restriction theorems for a surface with negative curvature we prove a bilinear restriction theorem for a surface of negative curvature this is the analogue of the results of t wolff 19 and t tao 14 15 for cones and paraboloids as a consequence we obtain an almost sharp linear restriction theorem

a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

restriction estimates for some surfaces with vanishing curvatures abstract we obtain bilinear restriction estimates for surfaces with vanishing curvatures as application we also prove new linear restriction estimates for some class of conic surfaces

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

transference of local to global l2 maximal estimates for dispersive partial differential equations abstract in this paper we give an elementary proof for transference of local to global maximal estimates for dispersive pdes this is done by transferring local l 2 estimates for certain oscillatory integrals with rough phase functions to the corresponding global estimates the elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context and instead is based on scalings and classical oscillatory integral estimates

improved restriction estimate for hyperbolic surfaces in r3 abstract recently l guth improved the restriction estimate for the surfaces with strictly positive gaussian curvature in r 3 in this paper we extend his restriction estimate to the quadratic surfaces with strictly negative gaussian curvature in r 3

unconditionally stable gauge uzawa finite element schemes for incompressible natural convection problems with variable density abstract we construct in this paper two gauge uzawa schemes one in conserved form and the other in convective form for solving natural convection problems with variable density and prove that the first order versions of both schemes are unconditionally stable we also show that a full discretized version of the conserved scheme with finite elements is also unconditionally stable these schemes lead to a sequence of decoupled elliptic equations to solve at each step hence they are very efficient and easy to implement we present several numerical tests to validate the analysis and demonstrate the effectiveness of these schemes for simulating natural convection problems with large density differences

multi scale bilinear restriction estimates for general phases we prove adjoint bilinear restriction estimates for general phases at different scales in the full non endpoint mixed norm range and give bounds with a sharp and explicit dependence on the phases these estimates have applications to high low frequency interactions for solutions to partial differential equations as well as to the linear restriction problem for surfaces with degenerate curvature as a consequence we obtain new bilinear restriction estimates for elliptic phases and wave klein gordon interactions in the full bilinear range and give a refined strichartz inequality for the klein gordon equation in addition we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

refined and microlocal kakeya nikodym bounds of eigenfunctions in higher dimensions we prove a kakeya nikodym bound on eigenfunctions and quasimodes which sharpens a result of the authors blair and sogge in anal pde 8 747 764 2015 and extends it to higher dimensions as in the prior work the key intermediate step is to prove a microlocal version of these estimates which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic geodesic flow in a companion paper blair and sogge in j differ geom 2015 it will be seen that these sharpened estimates yield improved l q m bounds on eigenfunctions in the presence of nonpositive curvature when 2 q frac 2 d 1 d 1

an hybrid finite volume finite element method for variable density incompressible flows this paper is devoted to the numerical simulation of variable density incompressible flows modeled by the navier stokes system we introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint the breakthrough relies on the definition of a suitable footbridge between the two methods through the design of compatibility condition in turn the method is very flexible and allows to deal with unstructured meshes several numerical tests are performed to show the scheme capabilities in particular the viscous rayleigh taylor instability evolution is carefully investigated

fourier restriction to a hyperbolic cone abstract using a bilinear restriction theorem of lee and a bilinear to linear argument of stovall we obtain the conjectured range of fourier restriction estimates for a conical hypersurface in r 4 with hyperbolic cross sections

the optimal trilinear restriction estimate for a class of hypersurfaces with curvature abstract in 4 nearly optimal l 1 trilinear restriction estimates in r n 1 are established under transversality assumptions only in this paper we show that the curvature improves the range of exponents by establishing l p estimates for any p 2 n 4 3 n 2 in the case of double conic surfaces the exponent 2 n 4 3 n 2 is shown to be the universal threshold for the trilinear estimate

continuum traffic model with the consideration of two delay time scales this paper presents a continuum traffic model the derivation of this model is based upon the assumption that the stream velocity u reaches the equilibrium velocity u e within the relaxation time t while the equilibrium velocity u e is adjusted to be attained through the driver s reaction time t r it is also assumed that the former delay time scale is greater than the latter a motion equation with nonconstant propagation velocity of a disturbance in traffic flow is derived that can reflect the anisotropy of disturbance propagation in real traffic unlike some other higher order continuum models it indicates that in our model the undesirable wrong way travel phenomenon and gas like behavior have been eliminated the formation and diffusion of traffic shock can be better simulated

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

endpoint bilinear restriction theorems for the cone and some sharp null form estimates recently wolff 25 obtained a nearly sharp l 2 bilinear restriction theorem for bounded subsets of the cone in general dimension we obtain the endpoint of wolff s estimate and generalize to the case when one of the subsets is large as a consequence we are able to deduce some nearly sharp l p null form estimates

bilinear identities involving the k plane transform and fourier extension operators we prove certain l2 n bilinear estimates for fourier extension operators associated to spheres and hyperboloids under the action of the k plane transform as the estimates are l2 based they follow from bilinear identities in particular these are the analogues of a known identity for paraboloids and may be seen as higher dimensional versions of the classical l2 2 bilinear identity for fourier extension operators associated to curves in 2

on the cone multiplier in r3 abstract we prove the sharp l 3 bounds for the cone multiplier in r 3 and the associated square function which is known as mockenhaupt s square function

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

a sharp l p l q fourier restriction theorem for a conical surface of finite type in this paper we study the fourier restriction problem for certain conical surfaces where the sections are curves of finite type we obtain the sharp l p l q range for this surfaces our methods rely on a variation of the affine arclength measure

an extended residual based variational multiscale method for two phase flow including surface tension abstract in this study an extended residual based variational multiscale method is proposed for two phase flow including surface tension the extended residual based variational multiscale method combines a residual based form of the variational multiscale method and the extended finite element method xfem by extending the solution spaces it is possible to reproduce discontinuities of the solution fields inside elements intersected by the interface in particular we propose a quasi static enrichment to reproduce time dependent discontinuities kink enrichments of both velocity and pressure as well as kink enrichments of velocity combined with jump enrichments of pressure are considered here to capture the interface between the phases on a fixed grid a level set approach is used a residual based variational multiscale method is employed for computing both flow and interface motion the presented method is tested for various two phase flow examples exhibiting small and large density and viscosity ratios with and without surface tension a two phase couette flow a rayleigh taylor instability a sloshing tank and a three dimensional rising bubble to the best of our knowledge these are the first simulation results for representative time dependent three dimensional two phase flow problems using an extended finite element method stable and accurate results are obtained for all test examples

restriction of the fourier transform to a quadratic surface in n we obtain estimates for the restriction of the fourier transform to a certain k dimensional quadratic submanifold of open image in new window n

gauge uzawa methods for incompressible flows with variable density two new gauge uzawa schemes are constructed for incompressible flows with variable density one is in the conserved form while the other is in the convective form it is shown that the first order versions of both schemes in their semi discretized form are unconditionally stable numerical experiments indicate that the first order resp second order versions of the two schemes lead to first order resp second order convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

nonlinear profile decomposition for the h 12 h 12 rd energy subcritical wave equation abstract we consider the nonlinear wave equation u t t u u u 4 d 1 0 in dimensions d 2 this equation is critical in h 1 2 h 1 2 r d and subcritical with respect to the energy we prove the nonlinear profile decomposition the proof must address the lack of compactness due to the lorentz symmetry

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

conjecture and improved extension theorems for paraboloids in the finite field setting we study the extension estimates for paraboloids in d dimensional vector spaces over finite fields mathbb f q with q elements we use the connection between l 2 based restriction estimates and l p rightarrow l r extension estimates for paraboloids as a consequence we improve the l 2 rightarrow l r extension results obtained by lewko and lewko proc am math soc 140 2013 2028 2012 in even dimensions d ge 6 and odd dimensions d 4 ell 3 for ell in mathbb n our results extend the consequences for 3 d paraboloids due to lewko adv math 270 1 457 479 2015 to higher dimensions we also clarifies conjectures on finite field extension problems for paraboloids

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a new fractional time stepping method for variable density incompressible flows this paper describes a new method for solving variable density incompressible viscous flows we have dealt with the momentum equation and the divergence free constraint in a new manner by rewriting the original equations the originality of the proposed approach is that we have used different numerical methods to evaluate the evolution of the velocity and pressure compared with some established methods the proposed approach is parameter free more flexible and can avoid the difficulties caused by the original equations the stability analysis of the method is performed to show that our method is stable finally numerical experiments are given to show the accuracy efficiency and validness of this method for variable density incompressible flows

linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces abstract we obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces as applications we improve the known estimates for oscillatory integrals

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

weighted restriction estimates using polynomial partitioning we use the polynomial partitioning method of guth j amer math soc 29 2016 371 413 to prove weighted fourier restriction estimates in r3 with exponents p that range between 3 and 3 25 depending on the weight as a corollary to our main theorem we obtain new non weighted local and global restriction estimates for compact c surfaces s r3 with strictly positive second fundamental form for example we establish the global restriction estimate ef lp r3 f lq s in the full conjectured range of exponents for p 3 25 up to the sharp line and the global restriction estimate ef lp f l2 s for p 3 and certain sets r3 of infinite lebesgue measure as a corollary to our main theorem we also obtain new results on the decay of spherical means of fourier transforms of positive compactly supported measures on r3 with finite dimensional energies

regular article extension of finite volume compressible flow solvers to multi dimensional variable density zero mach number flows when attempting to compute unsteady variable density flows at very small or zero mach number using a standard finite volume compressible flow solver one faces at least the following difficulties i spatial pressure variations vanish as the mach number m 0 but they do affect the velocity field at leading order ii the resulting spatial homogeneity of the leading order pressure implies an elliptic divergence constraint for the energy flux iii violations of this constraint crucially affect the transport of mass preventing a code to properly advect even a constant density distribution we overcome these difficulties through a new algorithm for constructing numerical fluxes in the context of multi dimensional finite volume methods in conservation form the construction of numerical fluxes involves 1 an explicit upwind step yielding predictions for the nonlinear convective flux components 2 a first correction step that introduces pressure gradients which guarantee compliance of the convective fluxes with a divergence constraint this step requires the solution of a first poisson type equation 3 a second projection step which provides the yet unknown non convective pressure contribution to the total flux of momentum this second projection requires the solution of another poisson type equation and yields the cell centered velocity field at the new time this velocity field exactly satisfies a divergence constraint consistent with the asymptotic limit step 1 can be done by any standard finite volume compressible flow solver the input to steps 2 and 3 involves solely the fluxes from step 1 and is independent of how these were obtained thus our approach allows any such solver to be extended to compute variable density incompressible flows

improved bound for the bilinear bochner riesz operator we study l p times l q rightarrow l r bounds for the bilinear bochner riesz operator mathcal b alpha alpha 0 in mathbb r d d ge 2 which is defined by open image in new window we make use of a decomposition which relates the estimates for mathcal b alpha to the square function estimates for the classical bochner riesz operators in consequence we significantly improve the previously known bounds

weak version of restriction estimates for spheres and paraboloids in finite fields abstract we study l p l r restriction estimates for algebraic varieties in d dimensional vector spaces over finite fields unlike the euclidean case if the dimension d is even then it is conjectured that the l 2 d 2 d 3 l 2 stein tomas restriction result can be improved to the l 2 d 4 d 4 l 2 estimate for both spheres and paraboloids in finite fields in this paper we show that the conjectured l p l 2 restriction estimate holds in the specific case when test functions under consideration are restricted to d coordinate functions or homogeneous functions of degree zero to deduce our result we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in d 1 dimensions

a refinement of the strichartz inequality on the saddle and applications abstract we consider the nonlinear nonelliptic schrodinger equation defined by i t u x 2 y 2 u u 2 u 0 with initial datum in l 2 r 2 we show that if the solution blows up in finite time then there is a mass concentration phenomenon near the blow up time the key ingredient is a refinement of the strichartz inequality on the saddle

restriction and kakeya phenomena for finite fields the restriction and kakeya problems in euclidean space have received much attention in the last few decades and they are related to many problems in harmonic analysis partial differential equations pdes and number theory in this paper we initiate the study of these problems on finite fields in many cases the euclidean arguments carry over easily to the finite setting and are in fact somewhat cleaner but there are some new phenomena in the finite case which deserve closer study

essence of conservation forms in the traveling wave solutions of higher order traffic flow models this paper shows the essence of conservation forms when applying the weak solution theory to solve the traveling wave solution of a wide cluster in the payne whitham pw model the consideration of the conservation form for the acceleration equation is an important ingredient in the development of higher order traffic flow models but it is largely ignored in the research community to fix the idea we define two conservation forms for the same pw model and consequently derive two solutions with different sets of characteristic parameters of the wide cluster the analytical results are in good agreement with those that are obtained from numerical simulations moreover these two solutions are also shown to be asymptotic to those of the well known kuhne and kerner konhauser models with a viscosity term more importantly the careful treatment of the conservation form for the acceleration equation closes the important gap in the literature without the conservation form the solution obtained depends very much on the design of numerical schemes and can be quite arbitrary and may not adequately conform to the physically relevant properties

the schrodinger equation along curves and the quantum harmonic oscillator abstract we consider the schrodinger equation associated to the harmonic oscillator i t u h u where h x 2 with initial data in the sobolev space h s r d with d 2 and s 3 8 we prove almost everywhere convergence of the solution to its initial data as time tends to zero which improves a result of yajima 1990 30 to this end we consider almost everywhere convergence for the free schrodinger along curves as it turns out these problems are more or less equivalent to that of the free schodinger equation along vertical lines our results are obtained by showing the equivalence of related maximal estimates

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

uniform estimates for the x ray transform restricted to polynomial curves abstract we establish near optimal mixed norm estimates for the x ray transform restricted to polynomial curves with a weight that is a power of the affine arclength the bounds that we establish depend only on the spatial dimension and the degree of the polynomial some of our results are new even in the well curved case

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

linear adjoint restriction estimates for paraboloid we prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices this result generalizes the paraboloid restriction estimate in radial case from shao rev mat iberoam 25 2009 1127 1168 as well as the result from miao et al proc ams 140 2012 2091 2102 as an application we show a local smoothing estimate for a solution of the linear schrodinger equation under the assumption that the initial datum has additional angular regularity

interactions of waves in the speed gradient traffic flow model abstract in this paper we use the speed gradient sg model proposed by jiang et al transportation research part b 36 2002 405 419 to study the initial value problem of traffic flow the evolution of multi traffic waves is examined through setting piecewise continuous initial density numerical results show that distinct shocks and smooth transition layers appear instantaneously for some initial densities and the shock front is smoothed by the smooth transition layer ahead of or behind it if in the initial density the length of shock section or the length of lower density section before the shock is very short we thus verify that the sg model can satisfyingly reproduce the evolution of multi traffic waves with various initial densities

an extended continuum model considering optimal velocity change with memory and numerical tests in this paper an extended continuum model of traffic flow is proposed with the consideration of optimal velocity changes with memory the new model s stability condition and kdv burgers equation considering the optimal velocities change with memory are deduced through linear stability theory and nonlinear analysis respectively numerical simulation is carried out to study the extended continuum model which explores how optimal velocity changes with memory affected velocity density and energy consumption numerical results show that when considering the effects of optimal velocity changes with memory the traffic jams can be suppressed efficiently both the memory step and sensitivity parameters of optimal velocity changes with memory will enhance the stability of traffic flow efficiently furthermore numerical results demonstrates that the effect of optimal velocity changes with memory can avoid the disadvantage of historical information which increases the stability of traffic flow on road and so it improve the traffic flow stability and minimize cars energy consumptions

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

convergence of numerical approximations of the incompressible navier stokes equations with variable density and viscosity we consider numerical approximations of incompressible newtonian fluids having variable possibly discontinuous density and viscosity since solutions of the equations with variable density and viscosity may not be unique numerical schemes may not converge if the solution is unique then approximate solutions computed using the discontinuous galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution if the solution is not unique a subsequence of these approximate solutions will converge to a solution

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

logarithmic improvements in l p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature we consider the problem of proving l p bounds for eigenfunctions of the laplacian in the high frequency limit in the presence of nonpositive curvature and more generally manifolds without conjugate points in particular we prove estimates at the critical exponent p c frac 2 d 1 d 1 where a spectrum of scenarios for phase space concentration must be ruled out our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical l p bounds of the second author

high resolution numerical approximation of traffic flow problems with variable lanes and free flow velocities this paper develops macroscopic traffic flow models for a highway section with variable lanes and free flow velocities that involve spatially varying flux functions to address this complex physical property we develop a riemann solver that derives the exact flux values at the interface of the riemann problem based on this solver we formulate godunov type numerical schemes to solve the traffic flow models numerical examples that simulate the traffic flow around a bottleneck that arises from a drop in traffic capacity on the highway section are given to illustrate the efficiency of these schemes

mixed norm strichartz type estimates for hypersurfaces in three dimensions in their work ikromov and muller fourier restriction for hypersurfaces in three dimensions and newton polyhedra princeton university press princeton 2016 proved the full range l p l 2 fourier restriction estimates for a very general class of hypersurfaces in mathbb r 3 which includes the class of real analytic hypersurfaces in this article we partly extend their results to the mixed norm case where the coordinates are split in two directions one tangential and the other normal to the surface at a fixed given point in particular we resolve completely the adapted case and partly the non adapted case in the non adapted case the case when the linear height h text lin phi is below two is settled completely

stability of the car following model on two lanes in the case of two lane traffic vehicle drivers always worry about the lane changing actions from neighbor lane this paper studies the stability of a car following model on two lanes which incorporates the lateral effects in traffic the stability condition of the model is obtained by using the linear stability theory the modified korteweg de vries equation is constructed and solved and three types of traffic flows in the headway sensitivity space stable metastable and unstable are classified both analytical and simulation results show that the anxiousness about lane changing from neighbor lane indeed has influence upon people s driving behavior and the consideration of lateral effects could stabilize the traffic flows on both lanes

estimates for the x ray transform restricted to line complexes of kirillov type let there be given a piecewise continuous rectifiable curve r r n let g 1 n m 1 n be the usual grassmannian bundle in r n define an n dimensional submanifold m r n of m 1 n as the set of all copies of g 1 n along the curve following krillov we know that a nice function f x can be recovered from its x ray transform r 1 n f on m r n if and only if the curve intersects almost every affine hyperplane

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

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

on the restriction of the fourier transform to a conical surface let r be the su rface of a circular cone in r3 we show that if 1 p 4 3 1 q 3 1 1 p and f e lp r3 then the fourier transform of f belongs to lq r da for a certain natural measure a on r following p tomas we also establish bounds for restrictions of fourier transforms to conic annuli at the endpoint p 4 3 with logarithmic growth of the bound as the thickness of the annulus tends to zero

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

full velocity difference model for a car following theory in this paper we present a full velocity difference model for a car following theory based on the previous models in the literature to our knowledge the model is an improvement over the previous ones theoretically because it considers more aspects in car following process than others this point is verified by numerical simulation then we investigate the property of the model using both analytic and numerical methods and find that the model can describe the phase transition of traffic flow and estimate the evolution of traffic congestion

radial multipliers and restriction to surfaces of the fourier transform in mixed norm spaces in this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces l rad p l ang 2 left mathbb r n right we produce sharp bounds for the restriction of the fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier we also present some results for the discrete restriction conjecture and state an intriguing open problem

a sobolev estimate for the adjoint restriction operator in this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption we obtain the optimal h s l q estimates and their mixed norm generalizations as applications we prove some weighted strichartz estimates for the propagator varphi rightarrow e it delta alpha 2 varphi alpha 0