nonlinear spectral instability of steady state flow of a viscous liquid past a rotating obstacle we show that a steady state solution to the system of equations of a navier stokes flow past a rotating body is nonlinearly unstable if the associated linear operator mathcal l has a part of the spectrum in the half plane lambda in mathbb c mathrm re lambda 0 our result does not follow from known methods mainly because the basic nonlinear operator is not bounded in the same space in which the instability is studied as an auxiliary result of independent interest we also show that the uniform growth bound of the c 0 semigroup mathrm e mathcal l t is equal to the spectral bound of operator mathcal l

the steady motion of a navier stokes liquid around a rigid body let mathcal r be a body moving by prescribed rigid motion in a navier stokes liquid mathcal l that fills the whole space and is subject to given boundary conditions and body force under the assumptions that with respect to a frame mathcal f attached to mathcal r these data are time independent and that their magnitude is not too large we show the existence of one and only one corresponding steady motion of mathcal l with respect to mathcal f such that the velocity field at the generic point x in space decays like x 1 these solutions are physically reasonable in the sense of finn 10 in particular they are unique and satisfy the energy equation among other things this result is relevant in engineering applications involving orientation of particles in viscous liquid 14

first hit analysis of algorithms for computing quadratic irregularity the author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields it has been conjectured that the bernoulli numbers or alternatively the values of the riemann zeta function at odd negative integers are uniformly distributed modulo p for every p this is the basis of a well known heuristic given by siegel estimating the frequency of irregular primes so far analyses have shown that if q d is a real quadratic field then the values of the zeta function d 1 2m q d 1 2m at negative odd integers are also distributed as expected modulo p for any p we use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude we also discuss alternative ways of collecting large amounts of data to test the heuristic

steady state navier stokes flows past a rotating body leray solutions are physically reasonable a rigid body fancyscript b moves in a navier stokes liquid fancyscript l filling the whole space outside fancyscript b we assume that when referred to a frame attached to fancyscript b the nonzero velocity of the center of mass and the angular velocity of fancyscript b are constant and that the flow of fancyscript l is steady our main theorem implies that every weak steady state solution in the sense of leray is in fact physically reasonable in the sense of finn for data of arbitrary size such a theorem improves and generalizes an analogous famous result of babenko math ussr sb 20 1 25 1973 obtained in the case 0

how far do chemotaxis driven forces influence regularity in the navier stokes system the chemotaxis navier stokes system nt u n n n c c ct u c c nf c ut u u u p n u 0 0 1 is considered under boundary conditions of homogeneous neumann type for n and c and dirichlet type for u in a bounded convex domain r with smooth boundary where w and and f are sufficiently smooth given functions generalizing the prototypes const and f s s for s 0 it is known that for all suitably regular initial data n0 c0 and u0 satisfying 0 6 n0 0 c0 0 and u0 0 a corresponding initial boundary value problem admits at least one global weak solution which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems the present paper shows that after some relaxation time this solution enjoys further regularity properties and thereby complies with the concept of eventual energy solutions which is newly introduced here and which inter alia requires that two quasi dissipative inequalities are ultimately satisfied moreover it is shown that actually for any such eventual energy solution n c u there exists a waiting time t0 0 with the property that n c u is smooth in t0 and that n x t n0 c x t 0 and u x t 0 hold as t uniformly with respect to x this resembles a classical result on the three dimensional navier stokes system asserting eventual smoothness of arbitrary weak solutions thereof which additionally fulfill the associated natural energy inequality in consequence our results inter alia indicate that under the considered boundary conditions the possibly destabilizing action of chemotactic cross diffusion in does not substantially affect the regularity properties of the fluid flow at least on large time scales

global solution of a diffusive predator prey model with prey taxis abstract we consider the prey taxis system u t d 1 u u v u a u b u f v x t 0 v t d 2 v v c v u f v x t 0 in a smoothly bounded domain r n with zero flux boundary condition where a d 1 d 2 b c are positive constants and is a non negative constant we first investigate the global existence and local boundedness of solution for the case 0 moreover when 0 we show that the solution exists globally and is uniformly bounded provided is large enough

comparison of algorithms to calculate quadratic irregularity of prime numbers in previous work the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k0 using the values of the zeta function k0 at negative integers as our higher bernoulli numbers in the case where k0 is a real quadratic field siegel presented two formulas for calculating these zeta values one using entirely elementary methods and one which is derived from the theory of modular forms the author would like to thank henri cohen for suggesting an analysis of the second formula we briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of k0 irregularity more generally quadratic irregularity of a prime number

l p theory of the navier stokes flow in the exterior of a moving or rotating obstacle in this paper we describe two recent approaches for the lp theory of the navier stokes flow in the exterior of a moving or rotating obstacle

an lq approach with generalized anisotropic weights of the weak solution of the oseen flow around a rotating body in the whole space abstract in the present paper we study the stationary oseen type system for flow around a rotating body we study the system in the whole three dimensional space we establish the existence and uniqueness of a weak solution in weighted l q spaces we also investigate the regularity of the solutions

lp lq estimate of the stokes operator and navier stokes flows in the exterior of a rotating obstacle we consider the motion of a viscous fluid filling the whole three dimensional space exterior to a rotating obstacle with constant angular velocity we develop the l p l q estimates and the similar estimates in the lorentz spaces of the stokes semigroup with rotation effect we next apply them to the navier stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in l3 weak l3 space

global smooth solvability of a parabolic elliptic nutrient taxis system in domains of arbitrary dimension abstract this paper deals with the nutrient taxis system u t u u v 0 v u v v r x t in a bounded domain r n n 1 with smooth boundary where 0 is a parameter and r c 1 0 is a given nonnegative function it is shown that for any prescribed initial data u 0 w 1 with u 0 0 in the corresponding neumann initial boundary problem admits a global classical solution with regard to qualitative aspects it is moreover inter alia seen that if r additionally satisfies t t 1 r 2 0 as t then in the large time limit the solution component u stabilizes toward the constant 1 u 0 with respect to the norm in l 1 and that if furthermore sup t 0 r t l q for some q 1 fulfilling q n 2 then u is uniformly bounded

global solutions for chemotaxis navier stokes system with robin boundary conditions abstract we consider a chemotaxis navier stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account this phenomenon results in an inhomogeneous robin type boundary condition moreover the system is studied without the logistic growth of the bacteria population we prove that in one or two dimensions the system has a unique global classical solution while the existence of a global weak solution is shown in three dimensions in the latter case we show that the energy is bounded uniformly in time a key idea is to introduce and utilise a boundary energy to derive suitable a priori estimates moreover we are able to remove the convexity assumption on the domain

spatial pattern formation in a chemotaxis diffusion growth model abstract mimura and one of the authors 1996 proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis for this model tello and winkler 2007 22 obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution while kurata et al 2008 numerically showed several spatio temporal patterns in a rectangle motivated by their work we consider some qualitative behaviors of stationary solutions from global and local bifurcation viewpoints in the present paper first we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle for this case the directions of the branches near the bifurcation points are also obtained finally we exhibit several numerical results for the stationary and oscillating patterns

global stability of prey taxis systems abstract in this paper we prove the global boundedness and stability of the predator prey system with prey taxis in a two dimensional bounded domain with neumann boundary conditions by deriving an entropy like equality and a boundedness criterion we show that the intrinsic interaction between predators and preys is sufficient to prevent the population overcrowding even the prey taxis is included and strong furthermore by constructing appropriate lyapunov functionals we show that prey only steady state is globally asymptotically stable if the predation is weak and the co existence steady state is globally asymptotically stable under some conditions like the prey taxis is weak or the prey diffuses fast if the predation is strong the convergence rates of solutions to the steady states are derived in the paper

strong solutions to the navier stokes equations around a rotating obstacle we study the existence of strong solutions to the three dimensional navier stokes initial boundary value problem in the domain exterior to a rigid body that rotates with constant angular velocity we show that when the initial data u0 are prescribed in an appropriate functional class a strong solution exists at least in some finite time interval moreover the solution exists for all times provided u0 in suitable norm and the magnitude of do not exceed a certain constant depending only on the kinematic viscosity and on the regularity of in this latter case we also show that the velocity field converges to the velocity field of the corresponding steady state solution

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity we consider a stationary viscous incompressible flow around a translating and rotating body optimal rates of decay are derived for the velocity and its gradient on the basis of a representation formula involving a fundamental solution constructed by r b guenther and e a thomann the fundamental solution of the linearized navier stokes equations for spinning bodies in three spatial dimensions time dependent case j math fluid mech 8 2006 77 98 for a linearized system

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant we consider chemotaxis system with consumption of chemoattractant vt v uv ut u u v under homogeneous neumann boundary conditions it is proved that if either n 2 or 0 16 n 1 v x 0 l n 3 the global classical solution u v of this problem converges to u0 0 exponentially as t where u0 1 u x 0 dx

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

asymptotic homogenization in a three dimensional nutrient taxis system involving food supported proliferation abstract the taxis system u t u u v u v u v t d v u v v 1 v is considered in bounded convex domains r n n 1 where d and are positive but and are merely assumed to be nonnegative this particularly includes the special case 0 of a nutrient taxis system with food dependent cell proliferation but beyond that the system for general parameter choices is also used as a model for lotka volterra type interaction involving prey taxis it is firstly shown that if n 5 then for all suitably regular initial data an associated no flux initial boundary value problem admits a global weak solution to the best of our knowledge this inter alia provides the first result on global existence in a system of the form in a spatially three dimensional setting when arbitrarily large initial data and parameters are involved secondly under the additional hypotheses that n 3 0 and 16 d 2 it is seen that each of these solutions becomes eventually smooth and stabilizes toward a spatially homogeneous equilibrium in the sense that 0 1 u t u in l and v t 0 in l as t where u is a constant satisfying u 1 u 0 this on the one hand shows that in comparison to the well understood simple nutrient taxis system obtained on letting 0 and removing the proliferation term u v in the introduction of the latter does not substantially affect the tendency of the system to support relaxation into spatial homogeneity apart from that 0 1 complements previous findings on convergence to equilibria in prey taxis systems of type which seem to exclusively concentrate on cases in which unlike in 0 1 the constant limit functions are a priori known and in which thereby the corresponding asymptotic analysis is apparently simplified to a considerable extent

convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source abstract we study the global attractors to the chemotaxis system with logistic source u t u u v a u b u 2 v t v v u in r subject to the homogeneous neumann boundary conditions where smooth bounded domain r n with b 0 a r and 0 1 for the parabolic elliptic case with 0 and n 3 we obtain that the positive constant equilibrium a b a b is a global attractor if a 0 and b max n 2 n a 4 under the assumption n 3 it is proved that for either the parabolic elliptic case with 0 a 0 b max 3 a 4 or the parabolic parabolic case with 1 a 0 b a 4 large enough the system admits the positive constant equilibrium a b a b as a global attractor while the trivial equilibrium 0 0 is a global attractor if a 0 and b 0 it is pointed out that here the convergence rates are established for all of them the results of the paper mainly rely on parabolic regularity theory and lyapunov functionals carefully constructed

global existence of solutions and uniform persistence of a diffusive predator prey model with prey taxis this paper proves the global existence and boundedness of solutions to a general reaction diffusion predator prey system with prey taxis defined on a smooth bounded domain with no flux boundary condition the result holds for domains in arbitrary spatial dimension and small prey taxis sensitivity coefficient this paper also proves the existence of a global attractor and the uniform persistence of the system under some additional conditions applications to models from ecology and chemotaxis are discussed

irregular primes to one million using fast algorithms for power series inversion based on the fast fourier transform and multisectioning of power series we have calculated all irregular primes up to one million including their indices of irregularity and associated irregular pairs using this data we verified that fermat s last theorem and vandiver s conjecture are true for these primes two primes with index of irregularity five were already known we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six a pair of integers p k is said to be an irregular pair if p is a prime k is an even integer satisfying 2 p are known see 8 our calculations show that vandiver s conjecture is true for all p 106 the table of irregular pairs could also be used to calculate iwasawa invariants for the corresponding primes we did not do this as part of our calculations see 9 or 4 for a discussion of this problem previous computations of irregular pairs have used algorithms that take 0 p2 arithmetic operations for each prime p the bernoulli numbers are defined in received by the editor april 30 1991 and in revised form september 10 1991 1991 mathematics subject classification primary 11b68 secondary 1 1d41 65t20 1 1y99 the first author was partially supported by national science foundation grant dms 9012989 d 1992 american mathematical society 0025 5718 92 1 00 25 per page

eventual smoothness and stabilization of large data solutions in a three dimensional chemotaxis system with consumption of chemoattractant abstract this paper deals with positive solutions of u t u u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in bounded convex domains r 3 with smooth boundary it is shown that for arbitrarily large initial data this problem admits at least one global weak solution for which there exists t 0 such that u v is bounded and smooth in t moreover it is asserted that such solutions approach spatially constant equilibria in the large time limit

an existence theorem for the navier stokes flow in the exterior of a rotating obstacle we consider the three dimensional navier stokes initial value problem in the exterior of a rotating obstacle it is proved that a unique solution exists locally in time if the initial velocity possesses the regularity l 1 2 this regularity assumption is the same as that in the famous paper of fujita kato an essential step for the proof is the deduction of a certain smoothing property together with estimates near t 0 of the semigroup which is not an analytic one generated by the operator in the space l 2 where stands for the angular velocity of the rotating obstacle and p denotes the projection associated with the helmholtz decomposition

stationary stokes oseen and navier stokes equations with singular data the concept of very weak solution introduced by giga math z 178 287 329 1981 for the stokes equations has hardly been studied in recent years for either the navier stokes equations or the navier stokes type equations we treat the stationary stokes oseen and navier stokes systems in the case of a bounded open set connected of class mathcal c 1 1 of mathbb r 3 taking up once again the duality method introduced by lions and magenes problemes aus limites non homogenes et applications vols 1 2 dunod paris 1968 and giga math z 178 287 329 1981 for open sets of class mathcal c infty see also chapter 4 of necas les methodes directes en theorie des equations elliptiques french masson et cie ed paris academia editeurs prague 1967 which considers the hilbertian case p 2 for general elliptic operators we give a simpler proof of the existence of a very weak solution for stationary oseen and navier stokes equations when data are not regular enough based on density arguments and a functional framework adequate for defining more rigourously the traces of non regular vector fields in the stationary navier stokes case the results will be valid for external forces not necessarily small which lets us extend the uniqueness class of solutions for these equations considering more regular data regularity results in fractional sobolev spaces will also be discussed for the three systems all these results can be extended to other dimensions