partial regularity of a generalized solution to the navier stokes equations in exterior domain in this note we prove two regularity theorems for solutions to the navier stokes equations of an i b v p in exterior domains namely we prove that the sets of the singular points of a solution if not empty has at most 1 hausdorff measureh1 s 0 moreover the sets is enclosed in a sphere of rayr for anyt 0 these results are obtained as corollaries to the partial regularity results furnished in 2

nonlinear stability of traveling waves to a hyperbolic parabolic system modeling chemotaxis we prove nonlinear stability of traveling waves of arbitrary amplitudes to a hyperbolic parabolic system modeling repulsive chemotaxis in contrast to the previous related results where various sm

asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis abstract in this paper we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed by a change of variable from the well known keller segel model describing cell bacteria movement toward the concentration gradient of the chemical that is consumed by the cells we prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates

global existence and decay rate of the boussinesq burgers system with large initial data abstract in this paper we study the global existence and the long time asymptotic behavior of classical solutions to the cauchy problem for the boussinesq burgers system with large initial data more precisely we first show that the classical solutions exist globally in time with large initial data by using the l p p 2 estimates other than the conventional l 2 estimate then we prove that the global classical solutions converge to constant equilibrium states with an algebraic decay rate as time approaches infinity

convergence to traveling waves of a singular pde ode hybrid chemotaxis system in the half space abstract this paper is concerned with the asymptotic stability of the initial boundary value problem of a singular pde ode hybrid chemotaxis system in the half space r 0 we show that when the non zero flux boundary condition at x 0 is prescribed and the initial data are suitably chosen the solution of the initial boundary value problem converges as time tend to infinity to a shifted traveling wavefront restricted in the half space 0 where the wave profile and speed are uniquely selected by the boundary flux data the results are proved by a cole hopf type transformation and weighted energy estimates along with the technique of taking the anti derivative

optimal decay rates for a chemotaxis model with logistic growth logarithmic sensitivity and density dependent production consumption rate abstract we consider a keller segel type chemotaxis model with logistic growth logarithmic sensitivity and density dependent production consumption rate it is a 2 2 reaction diffusion system describing the interaction of cells and a chemical signal we study cauchy problem for the original system and its transformed system which is one of hyperbolic parabolic balance laws our initial data are generic perturbations of a constant ground state i e the initial mass of perturbation is non zero in the case of non diffusive chemical we obtain optimal l 2 time decay rates for the solution with finite initial data in the case of diffusive chemical optimal l 2 rates are also obtained with additional assumption on the smallness of the initial amplitude but still allowing large oscillation

l3 solutions of the liquid crystals system abstract in this article we study the simplified system of the original ericksen leslie equations for the flow of liquid crystals it is shown that the l 3 solutions of the cauchy problem for the three dimensional incompressible liquid crystal system are smooth

monotonic decreasing and asymptotic behavior of the kinetic energy for weak solutions of the navier stokes equations in exterior domains existence of weak solutions of the three dimensional navier stokes problem was first proved by j leray in the case of the cauchy problem cf leray 1934 as is well known these solutions are important in that they are the only solutions which so far are known to exist for all times without restriction on the data unfortunately however the question of whether they are classical in an ordinary sense is still open even though partial conclusions regarding regularity are avilable leray 1934 scheffer 1976 1980 caffarelli kohn nirenberg 1982 subsequently e hopf 1951 using a different technique constructed weak solutions for a general initial boundary value problem however hopf s solution even for the cauchy problem has weaker properties than leray s solution among others we refer to the energy inequality for the velocity field u x t i e ntimits mega u 2 x t dx ntimits mega u 2 x s sx eqq 2ntimits s t ntimits mega abla u abla udxdau i for s 0 for almost all s 0 and for all t s

on energy inequality smoothness and large time behavior in l 2 for weak solutions of the navier stokes equations in exterior domains soit un domaine exterieur de r n n 3 4 a frontiere lisse sur le cylindre d espace temps 0 on considere le probleme de navier stokes non stationnaire u t u u u p f dans 0 u 0 dans 0 u 0 sur 0 u x 0 a x avec des donnees prescrites f et a

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

boundedness of weak solutions of a chemotaxis stokes system with slow p laplacian diffusion abstract in this paper a chemotaxis stokes system with slow p laplacian diffusion n t u n n p 2 n n c x t 0 c t u c c n c x t 0 u t u p n x t 0 u 0 x t 0 is considered under homogeneous boundary conditions of neumann type for n and c and of dirichlet type for u where is a smooth bounded domain in r 3 and w 2 is a given function it is proved that global bounded weak solutions exist whenever p 23 11

constrained convex bodies with extremal affine surface areas abstract given a convex body k r n and p r we introduce and study the extremal inner and outer affine surface areas i s p k sup k k as p k and o s p k inf k k as p k where as p k denotes the l p affine surface area of k and the supremum is taken over all convex subsets of k and the infimum over all convex compact subsets containing k the convex body that realizes i s 1 k in dimension 2 was determined in 3 where it was also shown that this body is the limit shape of lattice polytopes in k in higher dimensions no results are known about the extremal bodies we use a thin shell estimate of 23 and the lowner ellipsoid to give asymptotic estimates on the size of i s p k and o s p k surprisingly it turns out that both quantities are proportional to a power of volume

relation between pole assignment and lq regulator in this paper we demonstrate a connection between the pole assignment technique and the linear optimal regulator technique namely there exist common lyapunov functions between two closed loop systems synthesized by each method from this property we derive a useful theorem for pole assignment

boundedness in a chemotaxis model with oxygen consumption by bacteria abstract this paper deals with the keller segel model u t u u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 with nonnegative initial data u 0 v 0 w 1 r 2 for some r n u 0 l 1 0 and v 0 l 0 this model describes bacteria movement toward the concentration gradient of the oxygen that is consumed by the bacteria it is proved that if 0 1 6 n 1 v 0 l then the corresponding initial boundary value problem possesses a unique global solution that is uniformly bounded

comments on on optimal pole assignment in a specified region in a recent paper juang and lee 1984 presented a method of optimal pole placement for linear multivariable systems the method is based on theorem 1 and corollary 1 of juang and lee 1984 unfortunately neither the theorem nor the corollary can be correct furthermore the application of corollary 1 in proposition 2 and in cases 2 3 and 4 is valid only in a special case where the number of inputs m is at most equal to the number of shifted poles one or two poles in each step

asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity abstract this paper establishes the nonlinear stability of traveling wave solutions to a chemotaxis model with singular or logarithmic sensitivity and its transformed parabolic hyperbolic system depending on the parameter signs we discuss the linear instability of traveling wave solutions using the spectral analysis and nonlinear asymptotic stability of traveling wave solutions with zero end state by the weighted energy estimates where the latter result solves the open question left in a previous work li and wang 2009 7

the small convection limit in a two dimensional chemotaxis navier stokes system this paper deals with an initial boundary value problem for the chemotaxis navier stokes system begin aligned left begin array lcll n t u cdot nabla n delta n nabla cdot n nabla c qquad quad x in omega t 0 c t u cdot nabla c delta c nc qquad quad x in omega t 0 u t kappa u cdot nabla u delta u nabla p n nabla phi qquad quad x in omega t 0 nabla cdot u 0 qquad quad x in omega t 0 end array right end aligned in a bounded convex domain omega subset mathbb r 2 with smooth boundary with kappa in mathbb r and a given smooth potential phi omega rightarrow mathbb r it is known that for each kappa in mathbb r and all sufficiently smooth initial data this problem possesses a unique global classical solution n kappa c kappa u kappa the present work asserts that these solutions stabilize to n 0 c 0 u 0 uniformly with respect to the time variable more precisely it is shown that there exist mu 0 and c 0 such that whenever kappa in 1 1 begin aligned big vert n kappa cdot t n 0 cdot t big vert l infty omega big vert c kappa cdot t c 0 cdot t big vert l infty omega quad big vert u kappa cdot t u 0 cdot t big vert l infty omega le c kappa e mu t end aligned for all t 0 this result thereby provides an example for a rigorous quantification of stability properties in the stokes limit process as frequently considered in the literature on chemotaxis fluid systems in application contexts involving low reynolds numbers

on the boussinesq burgers equations driven by dynamic boundary conditions abstract we study the qualitative behavior of the boussinesq burgers equations on a finite interval subject to the dirichlet type dynamic boundary conditions assuming h 1 h 2 initial data which are compatible with boundary conditions and utilizing energy methods we show that under appropriate conditions on the dynamic boundary data there exist unique global in time solutions to the initial boundary value problem and the solutions converge to the boundary data as time goes to infinity regardless of the magnitude of the initial data

approximate polytope membership queries in the polytope membership problem a convex polytope k in mathbb r d is given and the objective is to preprocess k into a data structure so that given any query point q in mathbb r d it is possible to determine efficiently whether q in k we consider this problem in an approximate setting given an approximation parameter varepsilon the query can be answered either way if the distance from q to k s boundary is at most varepsilon times k s diameter we assume that the dimension d is fixed and k is presented as the intersection of n halfspaces previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by dudley approx theory 10 1974 pp 227 236 and bentley faust and preparata commun acm 25 1982 pp 64 68 the former is optimal in the worst case with respect to space and the latter is optimal with respect to query time we present four main results first we show how

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis abstract the asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper by a cole hopf type transformation the singular chemotaxis is converted into a non singular hyperbolic system then we study the transformed system and establish the global existence asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates the main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established

nonlinear stability of strong traveling waves for the singular keller segel system with large perturbations abstract this paper is concerned with the nonlinear stability of traveling wave solutions for a conserved system of parabolic equations derived from a singular chemotaxis model describing the initiation of tumor angiogenesis when the initial datum is a continuous small perturbation with zero integral from the spatially shifted traveling wave the asymptotic stability of the large amplitude strong traveling waves has been established in a series of works 29 34 35 by the second author with his collaborators in this paper we shall show that similar stability results indeed hold true for large and discontinuous initial data i e the initial perturbation from the traveling wave could be discontinuous and has large oscillations such as riemann data with large jumps to the best of our knowledge this paper provides a first result on the asymptotic stability of large amplitude traveling waves with large initial perturbation for a system of conservation laws although similar results have been available for the scalar equations cf 8 42 we also extend existing results to the initial data with lower regularity

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

self similar measures and intersections of cantor sets it is natural to expect that the arithmetic sum of two cantor sets should have positive lebesgue measure if the sum of their dimensions exceeds 1 but there are many known counterexamples e g when both sets are the middle ca cantor set and ca e 1 1 we show that for any compact set k and for a e ag e 0 1 the arithmetic sum of k and the middle ae cantor set does indeed have positive lebesgue measure when the sum of their hausdorff dimensions exceeds 1 in this case we also determine the essential su premum as the translation parameter t varies of the dimension of the intersection of k t with the middle ca cantor set we also establish a new property of the infinite bernoutlli convolutions lj the distributions of random series e 0 a where the signs are chosen independently with probabilities p 1 p let 1 ql q2 2 for p 2 near 2 and for a e a in some nonempty interval iv is absolutely continuous 2 1 and its density is in lil but not in lq2 we also answer a question of kahane concerning the fourier transform of vl 2

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

global existence and asymptotic behavior of the boussinesq burgers system abstract this paper is concerned with the boussinesq burgers system which models the propagation of bores by combing the dissipation dispersion and nonlinearity we establish the global existence and asymptotical behavior of classical solutions of the initial value boundary problem of the boussinesq burgers system with the help of a lyapunov functional and the technique of moser iteration particularly we show that the solution converges to the unique constant stationary solution exponentially as time tends to infinity

boundary layer problem on a hyperbolic system arising from chemotaxis abstract this paper is concerned with the boundary layer problem for a hyperbolic system transformed via a cole hopf type transformation from a repulsive chemotaxis model with logarithmic sensitivity proposed in 23 34 modeling the biological movement of reinforced random walkers which deposit a non diffusible or slowly moving signal that modifies the local environment for succeeding passages by prescribing the dirichlet boundary conditions to the transformed hyperbolic system in an interval 0 1 we show that the system has the boundary layer solutions as the chemical diffusion coefficient e 0 and further use the formal asymptotic analysis to show that the boundary layer thickness is e 1 2 our work justifies the boundary layer phenomenon that was numerically found in the recent work 25 however we find that the original chemotaxis system does not possess boundary layer solutions when the results are reverted to the pre transformed system

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

existence of weak and regular solutions for keller segel system with degradation coupled to fluid equations abstract we establish the global well posedness for the following chemotaxis fluid system t n u n n n c n q t c u c c c n t u u u p u n u 0 in r d d 2 3 where 0 q 2 1 d and 0 1 for either q 2 d 1 2 or q 2 d 0 3 we prove the global existence of regular solutions in case that q 2 1 d and 0 very weak solutions are constructed as well

on heat convection equations in a half space with non decaying data and stokes semi group on besov spaces based on l abstract the local existence theorem for the boussinesq equations in the half space r n 1 x x x n r n 1 x r n x n 1 0 with non decaying data is shown to this end the stokes operator is shown to generate an analytic semi group on the besov spaces of negative order based on l

initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis abstract this paper is devoted to the analytical study of initial boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis model with logarithmic sensitivity in the first part of the paper we show that subject to the dirichlet boundary conditions classical solutions exist globally in time for large initial data asymptotically in time the solutions are shown to converge to their boundary data at an exponential rate as time goes to infinity numerical simulations are supplied to corroborate the analytical results the analytic approach developed herein can be utilized to handle a family of initial and initial boundary value problems of the model and related models with similar mathematical structure as a demonstration of the effectiveness of our approach in the second part of the paper we show that subject to the neumann dirichlet boundary conditions classical solutions exist and converge to constant equilibrium states for large initial data and for arbitrary values of the chemical diffusion coefficient this improves a previous result obtained in 30 where the smallness of the chemical diffusion coefficient was required

the spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

initial and boundary values for lαq lp solution of the navier stokes equations in the half space abstract in this paper we study the initial and boundary value problem of the navier stokes equations in the half space we prove the existence of weak solution u l q 0 l p r n 1 2 1 n p 2 q 0 n p with u l loc q 2 0 l loc p 2 r n for the solenoidal initial data h b p q 1 n p r n and the boundary data g l q 0 b p p 1 p r n 1 when h b p q 1 n p r n g l q 0 b p p 1 p r n 1 is small enough moreover the solution is unique in the class l q 0 t l p r n for any t if 0 and for some t if 0

nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion abstract we consider a simplified chemotaxis model of tumor angiogenesis described by a keller segel system on the two dimensional infinite cylindrical domain x y r s where s is the circle of perimeter 0 the domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located the system is known to allow planar traveling wave solutions of an invading type in this paper we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if is sufficiently small the same result for the corresponding system in one dimension was obtained by li li wang 2014 17 our result solves the problem remained open in chae choi kang lee 2018 3 at which only linear stability of the planar traveling waves was obtained under certain artificial assumption

regularity criteria for suitable weak solutions of the navier stokes equations near the boundary abstract we present some new regularity criteria for suitable weak solutions of the navier stokes equations near the boundary in dimension three we prove that suitable weak solutions are holder continuous up to the boundary provided that the scaled mixed norm l x t p q with 3 p 2 q 2 2 q p q 3 2 is small near the boundary our methods yield new results in the interior case as well partial regularity of weak solutions is also analyzed under some additional integral conditions

oscillatory traveling wave solutions to an attractive chemotaxis system abstract this paper investigates oscillatory traveling wave solutions to an attractive chemotaxis system the convective part of this system changes its type when crossing a parabola in the phase space the oscillatory nature of the traveling wave comes from the fact that one far field state is in the elliptic region and another in the hyperbolic region such traveling wave solutions are shown to be linearly unstable detailed construction of some traveling wave solutions is presented

asymptotic nonlinear stability of traveling waves to a system of coupled burgers equations abstract this paper is concerned with the existence and asymptotic nonlinear stability of traveling wave solutions to a system of coupled burgers equations which arises in a number of various physical contexts we establish the existence of traveling fronts based on the phase plane analysis method the asymptotic nonlinear stability of traveling wave solutions is proved by the method of energy estimates in contrast to the previous related results our results do not require any smallness assumption on the wave strengths or on the coefficients

polyhedral approximation of smooth convex bodies in 3 mcclure and vitale have obtained a series of precise asymptotic results on the approximation of smooth convex curves in the plane by inscribed or circumscribed polygons with a given number of vertices in the present note we generalize one of their results to higher dimensions let s be the set of convex bodies compact convex subsets with interior points in d dimensional euclidean space w with scalar product and norm i i i for c e x we denote by 9 c the set of convex polytopes containing c with at most n facets let

design of linear quadratic regulators with assigned eigenstructure a recursive design procedure is presented which assigns the closed loop eigenstructure in linear quadratic regulators at each stage the required solution for the steady state riccati matrix which shifts a pole or pole pair to specified values is obtained for pole pair placement a free parameter in the solution permits selection of closed loop eigenvectors the interactive design procedure is summarized and illustrated by an example

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

derivative estimates for the navier stokes equations in a three dimensional region on etablit des estimations fractionnaires pour les solutions de l equation de navier stokes dans le cas d un domaine avec une limite sur laquelle les composantes des solutions s annulent la condition de limite fixee appropriee pour des ecoulements visqueux

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

non blowup of a generalized boussinesq burgers system with nonlinear dispersion relation and large data abstract we study the qualitative behavior of classical solutions to the cauchy problem of a generalized boussinesq burgers system in one space dimension assuming initial data belong to h 2 r and utilizing energy methods we show that there exist unique global in time classical solutions to the cauchy problem of the model and the solutions converge to constant equilibrium states as time goes to infinity regardless of the magnitude of the initial data moreover it is shown that the viscous and inviscid models are consistent in the process of vanishing viscosity limit

time global existence and finite time blow up criterion for solutions to the keller segel system coupled with the navier stokes fluid abstract we will deal with the chemotaxis model under the effect of the navier stokes fluid i e the incompressible viscous fluid we shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n 0 l 1 r 2 and u 0 l 2 r 2 our method is based on the perturbation of linearization together with the l p l q estimates of the heat semigroup as a by product of our method we shall prove the smoothing effect and uniqueness of our mild solution in addition we shall show a blow up criterion which almost covers the well known threshold number 8 of the size n 0 l 1 r 2 under the rest state of the fluid motion furthermore the blow up rate will be also discussed

global dynamics of a hyperbolic parabolic model arising from chemotaxis we prove global existence and qualitative behavior of classical solutions for a hyperbolic parabolic system describing chemotaxis on bounded domains it is shown that classical solutions to the initial boundary value problem of the one dimensional model exist globally in time for large initial data and the solutions converge to constant equilibrium states exponentially in time which rigorously demonstrates the collapsing of cell populations in chemotaxis moreover similar results are established for the multidimensional model when the initial data are small

stability of planar traveling waves in a keller segel equation on an infinite strip domain abstract we consider a simplified model of tumor angiogenesis described by a keller segel equation on the two dimensional domain x y r s where s is the circle of perimeter it is known that the system allows planar traveling wave solutions of an invading type in case that is sufficiently small we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted sobolev space when chemical diffusion is present it can be shown that the system is linearly stable lastly we prove that any solution with our front condition eventually becomes planar under certain regularity conditions

well posedness of a 3d parabolic hyperbolic keller segel system in the sobolev space framework abstract we investigate global strong solution to a 3 dimensional parabolic hyperbolic system arising from the keller segel model we establish the global well posedness and asymptotic behavior in the energy functional setting precisely speaking if the initial difference between cell density and its mean is small in l 2 and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in h 1 then they remain to be small in l 2 h 1 for all time moreover if the mean value of the initial cell density is smaller than some constant then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity the proof relies on an application of fourier analysis to a linearized parabolic hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration

the convex hull of the integer points in a large ball where fk p denotes the number of k dimensional faces of the polytope p the limit as r of the average of r 2 3f0 pr on an interval r r h is determined by balog and deshoullier bd and turns out to be 3 453 h must be large our main result extends 1 1 to any d 2 and to any fk pr with k 0 d 1 theorem 1 for every d 2 there are constants c1 d and c2 d such that for all k 0 d 1

asymptotic profile of a parabolic hyperbolic system with boundary effect arising from tumor angiogenesis this paper concerns a parabolic hyperbolic system on the half space r with boundary effect the system is derived from a singular chemotaxis model describing the initiation of tumor angiogenesis we show that the solution of the system subject to appropriate boundary conditions converges to a traveling wave profile as time tends to infinity if the initial data is a small perturbation around the wave which is shifted far away from the boundary but its amplitude can be arbitrarily large

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

blow up criterion for a hyperbolic parabolic system arising from chemotaxis abstract in this paper we investigate a hyperbolic parabolic system arising from chemotaxis a blow up criterion of local strong solutions to the model in r d is established based on the scaling invariance of the system as a by product we prove a global in time well posedness result when d 1 for large initial data

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

optimization and pole placement for a single input controllable system given the system xdot ax bu and the cost function j dt relations are to be determined among the open loop characteristic polynomial the closed loop characteristic polynomial and the matrices a and q those relations take a simple form if the system is in the standard controllable form in this case the optimal control law can be found easily without solving the matrix riccati equation while the minimum value of the cost function if it is required can be determined by solving a matrix equation of the form c t x xc d

global existence and stabilization in a degenerate chemotaxis stokes system with mildly strong diffusion enhancement abstract a class of chemotaxis stokes systems generalizing the prototype n t u n n m 1 n n c c t u c c n c u t p u n u 0 is considered in bounded convex three dimensional domains where w 2 is given the paper develops an analytical approach which consists in a combination of energy based arguments and maximal sobolev regularity theory and which allows for the construction of global bounded weak solutions to an associated initial boundary value problem under the assumption that 0 1 m 9 8 moreover the obtained solutions are shown to approach the spatially homogeneous steady state 1 n 0 0 0 in the large time limit this extends previous results which either relied on different and apparently less significant energy type structures or on completely alternative approaches and thereby exclusively achieved comparable results under hypotheses stronger than 0 1

a necessary condition for optimization in the frequency domain abstract a necessary condition in the frequeney domain is derived for the optimality of a linear multivariable feedback control system with respect to a quadratic performance index of infinite settling time

volume approximation of convex bodies by inscribed polytopes a convex body c in d dimensional euclidean space e is a compact convex subset of le a with non empty interior we say that c is of differentiability class k if its boundary considered as a manifold is of class k the symmetric difference metric 6 s also called nikodym metric on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference given a convex body c let c n d l d 2 denote the family of convex polytopes p having at most n vertices and being inscribed into c that is all vertices are on the boundary bdc of c it is well known that for each n there is a polytope p such that s c p s c inf s c p p

on the existence of local strong solutions for the navier stokes equations in completely general domains abstract there are only very few results on the existence of unique local in time strong solutions of the navier stokes equations for completely general domains r 3 although domains with edges and corners bounded or unbounded are very important in applications the reason is that the l q theory for the stokes operator a is available in general only in the hilbert space setting i e with q 2 our main result for a general domain is optimal in a certain sense consider an initial value u 0 l 2 and a zero external force then the condition 0 e t a u 0 4 8 d t is sufficient and necessary for the existence of a unique local strong solution u l 8 0 t l 4 in some interval 0 t 0 t with u 0 u 0 satisfying serrin s condition 2 8 3 4 1 note that fujita kato s sufficient condition u 0 d a 1 4 is strictly stronger and therefore not optimal

eventual smoothness of generalized solutions to a singular chemotaxis stokes system in 2d abstract we study the chemotaxis fluid system n t u n n n c c x t 0 c t u c c n c x t 0 u t p u n x t 0 u 0 x t 0 under homogeneous neumann boundary conditions for n and c and homogeneous dirichlet boundary conditions for u where r 2 is a bounded domain with smooth boundary and c 2 from recent results it is known that for suitable regular initial data the corresponding initial boundary value problem possesses a global generalized solution we will show that for small initial mass n 0 these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties moreover from the analysis of certain energy type inequalities arising during the investigation of the eventual regularity we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of n 0 in l 1 and in l log l u 0 in l 4 and of c 0 in l 2