transport and instability for perfect fluids incompressible perfect fluids are described by the euler equations we provide a new simple proof for well posedness for velocities in c 1 alpha and linear and nonlinear instability results using transport techniques the results have an important consequence the topology of c 1 alpha is too fine for interesting questions about large time behavior

the unstable spectrum of oscillating shear flows a complete description is given for the unstable spectrum of the euler equation linearized about shear flows with sinusoidal profiles the spectral problem is treated analytically and numerically the spectral asymptotics are determined using the method of averaging for general rapidly oscillating profiles such flows are always unstable

fast growth of the vorticity gradient in symmetric smooth domains for 2d incompressible ideal flow abstract we construct initial data for the two dimensional euler equation in a bounded smooth symmetric domain such that the gradient of vorticity in l grows as a double exponential in time for all time our construction is based on the recent result by kiselev and sverak 9

mellin transform methods applied to integral evaluation taylor series and asymptotic approximations mellin transform methods have been used to obtain taylor series or asymptotic approximations to definite integrals whose integrands consist of the product of two generalized hypergeometric functions this approach has proved extremely useful in many applied physics problems including the analysis of electromagnetic propagation in a turbulent medium the main result in this article is a generalization of the method to integrands containing the product of an arbitrary number of hypergeometric functions providing a very powerful and general technique for integral evaluation with many applications in physics

on the asymptotic limit of the navier stokes system on domains with rough boundaries abstract we study the asymptotic behavior of solutions to the incompressible navier stokes system considered on a sequence of spatial domains whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non degenerate meaning not oriented in a single direction

the euler equations in planar domains with corners when the velocity field is not a priori known to be globally almost lipschitz global uniqueness of solutions to the two dimensional euler equations has been established only in some special cases and the solutions to which these results apply share the property that the diffuse part of the vorticity is constant near the points where the velocity is insufficiently regular assuming that the latter holds initially the challenge is then to propagate this property along the euler dynamic via an appropriate control of the lagrangian trajectories in domains with obtuse corners and sufficiently smooth elsewhere yudovich solutions fail to be almost lipschitz only near these corners and we investigate the necessary and sufficient conditions for the vorticity to remain constant there we show that if the vorticity is initially constant near the whole boundary then it remains so forever and global weak solutions are unique provided that no corner has angle greater than pi we also show that this fails in general for domains that do have such corners

small scale creation for solutions of the incompressible two dimensional euler equation we construct an initial data for the two dimensional euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times this estimate is known to be sharp the double exponential growth is the fastest possible growth rate

the two dimensional euler equations on singular exterior domains this paper is a follow up of gerard varet and lacave arch ration mech anal 209 1 131 170 2013 on the existence of global weak solutions to the two dimensional euler equations in singular domains in gerard varet and lacave arch ration mech anal 209 1 131 170 2013 we have established the existence of weak solutions for a large class of bounded domains with initial vorticity in lp p 1 for unbounded domains we have proved a similar result only when the initial vorticity is in l p c p 2 and when the domain is the exterior of a single obstacle the goal here is to retrieve these two restrictions we consider general initial vorticity in l 1 cap l p p 1 outside an arbitrary number of obstacles not reduced to points

hardy spaces and the two dimensional euler equations with nonnegative vorticity we prove that certain quadratic expressions involving the gradient of a weakly superharmonic function in r belong to a local hardy space as an application we provide a new proof of j m delort s convergence theorem for solutions of the two dimensional euler equations with vorticities of one sign department of mathematics university of california berkeley california 94720 sfb 256 institut fur angewandte mathematik beringstrasse 4 d5300 bonn 1 ger many this content downloaded from 207 46 13 101 on sat 08 oct 2016 05 44 35 utc all use subject to http about jstor org terms

on an integral related to biaxially anisotropic media an integral arising in certain studies of theoretical electromagnetics is evaluated and its properties are discussed in some detail the integral has three integer two real and one complex parameter the integrand involves a product of bessel functions of different argument and order several generalizations are discussed

impermeability through a perforated domain for the incompressible two dimensional euler equations we study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain the porous medium is composed of inclusions of size varepsilon separated by distances d varepsilon and the fluid fills the exterior if the inclusions are distributed on the unit square the asymptotic behavior depends on the limit of frac d varepsilon varepsilon when varepsilon goes to zero if frac d varepsilon varepsilon to infty then the limit motion is not perturbed by the porous medium namely we recover the euler solution in the whole space if on the contrary frac d varepsilon varepsilon to 0 then the fluid cannot penetrate the porous region namely the limit velocity verifies the euler equations in the exterior of an impermeable square if the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion it is determined by the limit of frac d varepsilon varepsilon 2 frac1 gamma where gamma in 0 infty is related to the geometry of the lateral boundaries of the obstacles if frac d varepsilon varepsilon 2 frac1 gamma to infty then the presence of holes is not felt at the limit whereas an impermeable wall appears if this limit is zero therefore for a distribution in one direction the critical distance depends on the shape of the inclusions in particular it is equal to varepsilon 3 for balls

bounds for ratios of the membrane eigenvalues abstract for a bounded planar region in r 2 we obtain the ratios of lower order eigenvalues of laplace operator combining our results with the recursive formula in cheng and yang 2007 11 we can obtain better upper bound of the k 1 th k 3 membrane eigenvalues

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

two dimensional incompressible viscous flow around a small obstacle abstract in this article we study the asymptotic behavior of incompressible ideal time dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small our main purpose is to identify the equation satisfied by the limit flow we will see that the asymptotic behavior depends on the circulation around the obstacle for smooth flow around a single obstacle is a conserved quantity which is determined by the initial data we will show that if 0 the limit flow satisfies the standard incompressible euler equations in the full plane but if 0 the limit equation acquires an additional forcing term we treat this problem by first constructing a sequence of approximate solutions to the incompressible 2d euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation we use an explicit treatment of the green s

boundary layer analysis of the navier stokes equations with generalized navier boundary conditions we study the weak boundary layer phenomenon of the navier stokes equations with generalized navier friction boundary conditions u n 0 s u n tan au 0 in a bounded domain in r3 when the viscosity e 0 is small here s u is the symmetric gradient of the velocity u and a is a type 1 1 tensor on the boundary when a i we obtain navier boundary conditions and when a is the shape operator we obtain the conditions u n curlu n 0 by constructing an explicit corrector we prove the convergence as e tends to zero of the navier stokes solutions to the euler solution both in the natural energy norm and uniformly in time and space

smoothness of the trajectories of ideal fluid particles with yudovich vorticities in a planar bounded domain abstract we consider the incompressible euler equations in a possibly multiply connected bounded domain of r 2 for flows with bounded vorticity for which yudovich 1963 proved in 29 global existence and uniqueness of the solution we prove that if the boundary of the domain is c respectively gevrey of order m 1 then the trajectories of the fluid particles are c respectively gevrey of order m 2 our results also cover the case of slightly unbounded vorticities for which yudovich 1995 extended his analysis in 30 moreover if in addition the initial vorticity is holder continuous on a part of then this holder regularity propagates smoothly along the flow lines finally we observe that if the vorticity is constant in a neighborhood of the boundary the smoothness of the boundary is not necessary for these results to hold

on the vanishing viscosity limit in a disk we say that a solution of the navier stokes equations converges in the vanishing viscosity limit to a solution of the euler equations if their velocities converge in the energy l2 norm uniformly in time as the viscosity vanishes we show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space time energy density of the solution to the navier stokes equations in a boundary layer of width proportional to vanish with and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1

exponential growth of the vorticity gradient for the euler equation on the torus abstract we prove that there are solutions to the euler equation on the torus with c 1 vorticity and smooth except at one point such that the vorticity gradient grows in l at least exponentially as t the same result is shown to hold for the vorticity hessian and smooth solutions our proofs use a version of a recent result by kiselev and sverak 6

a sufficient condition for instability in the limit of vanishing dissipation a general theorem is presented concerning an evolution pde whose leading order term is of viscous dissipative type it is proved that under certain conditions instability in the underlying nondissipative equation is a sufficient condition for viscous instability in the limit of vanishing viscosity this theorem is applied to give a sufficient condition for instability of a rotating body with a cavity filled by a viscous fluid

blow up for the 2d euler equation on some bounded domains abstract we find a smooth solution of the 2d euler equation on a bounded domain which exists and is unique in a natural class locally in time but blows up in finite time in the sense of its vorticity losing continuity the domain s boundary is smooth except at two points which are interior cusps

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

observations on the vanishing viscosity limit whether in the presence of a boundary solutions of the navier stokes equations converge to a solution to the euler equations in the vanishing viscosity limit is unknown in a seminal 1983 paper tosio kato showed that the vanishing viscosity limit is equivalent to having sufficient control of the gradient of the navier stokes velocity in a boundary layer of width proportional to the viscosity in a 2008 paper the present author showed that the vanishing viscosity limit is equivalent to the formation of a vortex sheet on the boundary we present here several observations that follow on from these two papers compiled on sunday 20 march 2016 1 definitions and past results 3 2 a 3d version of vorticity accumulation on the boundary 6 3 l norms of the vorticity blow up for p 1 7 4 improved convergence when vorticity bounded in l 8 5 some kind of convergence always happens 9 6 width of the boundary layer 2d 10 7 optimal convergence rate 2d 12 8 a condition on the boundary equivalent to v v 2d 14 9 examples where condition on the boundary holds 2d 17 10 on a result of bardos and titi 2d 22 appendix a a trace lemma 23 acknowledgements 25 references 26 the navier stokes equations for a viscous incompressible fluid in a domain rd d 2 with no slip boundary conditions can be written ns tu u u p u f in div u 0 in u 0 on date compiled on sunday 20 march 2016 2010 mathematics subject classification primary 76d05 76b99 76d10

weak solutions with decreasing energy of incompressible euler equations weak solution of the euler equations is an l2 vector field u x t satisfying certain integral relations which express incompressibility and the momentum balance our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations as both viscosity and compressibility tend to zero thus we believe that weak solutions describe turbulent flows with very high reynolds numbers

the two dimensional euler equations on singular domains we establish the existence of global weak solutions of the two dimensional incompressible euler equations for a large class of non smooth open sets loosely these open sets are the complements in a simply connected domain of a finite number of obstacles with positive sobolev capacity existence of weak solutions with lp vorticity is deduced from a property of domain continuity for the euler equations that relates to the so called convergence of open sets our results complete those obtained for convex domains in taylor progress in nonlinear differential equations and their applications vol 42 2000 or for domains with asymptotically small holes iftimie et al in commun partial differ equ 28 1 2 349 379 2003 lopes filho in siam j math anal 39 2 422 436 2007

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

electromagnetic fields in air of traveling wave currents above the earth the problem of the electromagnetic field created by a thin straight conductor of infinite length carrying a forward traveling wave current with a complex propagation constant above a homogeneous and isotropic planar earth of wave number k1 is formulated in terms of contour integrals in the limit where becomes equal to the free space wave number k0 the component of the magnetic field in air normal to both conductor and interface is evaluated in closed form in terms of known special functions while the remaining components of the field in air are expressed as series expansions in k02 k14 k04 1 2 via the application of a contour integration technique the new analytical formulas involve familiar transcendental functions and are valid at any distance from the source the analysis sheds light on the intricate nature of low frequency electromagnetic fields generated by transmission lines in the presence of a conducting or a dielectric half space

the euler equations in planar nonsmooth convex domains abstract as a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere we consider the euler system on a bounded convex planar domain endowed with non penetrating boundary conditions for 4 3 p 2 and initial and forcing data with l p vorticity we show the existence of a weak solution enriching and extending the results of taylor 2000 32 in the physical case of a rectangular domain 0 l 1 0 l 2 a similar result holds for all 2 p as well moreover by means of a new bmo type regularity estimate for the dirichlet problem on a planar domain with corners we prove uniqueness of solutions with bounded initial vorticity

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

some stability and instability criteria for ideal plane flows we investigate stability and instability of steady ideal plane flows for an arbitrary bounded domain first we obtain some general criteria for linear and nonlinear stability second we find a sufficient condition for the existence of a growing mode to the linearized equation third we construct a steady flow which is nonlinearly and linearly stable in the l2 norm of vorticity but linearly unstable in the l2 norm of velocity

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

cocycles and mane sequences with an application to ideal fluids abstract exponential dichotomy of a strongly continuous cocycle is proved to be equivalent to existence of a mane sequence either for or for its adjoint as a consequence we extend some of the classical results to general banach bundles the dynamical spectrum of a product of two cocycles one of which is scalar is investigated and applied to describe the essential spectrum of the euler equation in an arbitrary spacial dimension