absolutely γ summing multilinear operators abstract in this paper we introduce an abstract approach to the notion of absolutely summing multilinear operators we show that several previous results on different contexts absolutely summability almost summability cohen summability are particular cases of our general results

multilinear variants of pietsch s composition theorem various concepts of multilinear summing operators were introduced in the last years by extending the well known one from the linear case in this paper we prove that as in the linear case there is a splitting theorem for dominated operators as a consequence of this result we prove various multilinear variants of pietsch s composition theorem

hyper ideals of multilinear operators abstract we introduce and develop the notion of hyper ideals of multilinear operators between banach spaces while the well studied notion of ideals of multilinear operators multi ideals relies on the composition with linear operators the notion we propose by considering the composition with multilinear operators explores more deeply the nonlinear feature of the subject the results we prove show that although more restrictive a priori hyper ideals enjoy nice general properties and their theory is rich enough to provide distinguished examples in every situation where the corresponding multi ideal fails to be a hyper ideal

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

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 existence and boundedness in a keller segel stokes system involving a tensor valued sensitivity with saturation the 3d case abstract in this paper we continue to deal with the initial boundary value problem for the coupled keller segel stokes system n t u n n n s x n c c x t 0 t c t u c c c n x t 0 t u t p u n x t 0 t u 0 x t 0 t where r d is a bounded domain with smooth boundary and the chemotactic sensitivity s is not a scalar function but rather attains values in r d d and satisfies s x n c c s 1 n with some c s 0 and 0 when d 2 our previous work j differential equations 2015 has established the existence of global bounded classical solutions under the subcritical assumption 0 which is consistent with the corresponding results of the fluid free system but the method seems to be invalid in the three dimensional setting in this paper for the case d 3 we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption 1 2 which is slightly stronger than the corresponding subcritical assumption 1 3 on the fluid free system where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions the key idea here is to establish the general l p regularity of u from a rather low l p regularity of n which will be obtained by a new combinational functional

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

some techniques on nonlinear analysis and applications in this paper we present two different results in the context of nonlinear analysis the first one is essentially a nonlinear technique that in view of its strong generality may be useful in different practical problems the second result more technical but also connected to the first one is an extension of the well known pietsch domination theorem the last decade witnessed the birth of different families of pietsch domination type results and some attempts of unification our result that we call full general pietsch domination theorem is potentially a definitive pietsch domination theorem which unifies the previous versions and delimits what can be proved in this line the connections to the recent notion of weighted summability are traced

finite time blowup and global in time unbounded solutions to a parabolic parabolic quasilinear keller segel system in higher dimensions abstract in this paper we consider quasilinear keller segel type systems of two kinds in higher dimensions in the case of a nonlinear diffusion system we prove an optimal with respect to possible nonlinear diffusions generating explosion in finite time of solutions finite time blowup result in the case of a cross diffusion system we give results which are optimal provided one assumes some proper non decay of a nonlinear chemical sensitivity moreover we show that once we do not assume the above mentioned non decay our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross diffusion terms to this end we provide an example interesting by itself of global in time unbounded solutions to the nonlinear cross diffusion keller segel system with chemical sensitivity decaying fast enough in a range of parameters in which there is a finite time blowup result in a corresponding case without nonlinear cross diffusion

decay estimates of the coupled chemotaxis fluid equations in r 3 abstract in this paper we are concerned with a chemotaxis navier stokes model arising from biology which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations with transport and external force the optimal convergence rates of classical solutions to the chemotaxis navier stokes system for small initial perturbation around constant states are obtained by pure energy method under the assumption the initial data belong to h s h n n 3 0 s 3 2 the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution compared to the result in 5 we obtain the optimal decay rates of the higher order spatial derivatives of the solutions

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

existence and uniqueness theorem on mild solutions to the keller segel system coupled with the navier stokes fluid abstract we consider the keller segel system coupled with the navier stokes fluid in the whole space and prove the existence of global mild solutions with the small initial data in the scaling invariant space our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data as a byproduct we show the asymptotic stability of solutions as the time goes to infinity since we may deal with the initial data in the weak l p spaces the existence of self similar solutions provided the initial data are small homogeneous functions

multiscale models of taxis driven patterning in bacterial populations spatially distributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants in the past decade a great deal has been learned about signal transduction metabolism and movement in e coli and other bacteria but translating the individual level behavior into population level dynamics is still a challenging problem however this is a necessary step because it is computationally impractical to use a strictly cell based model to understand patterning in growing populations since the total number of cells may reach 10 12 10 14 in some experiments in the past phenomenological equations such as the patlak keller segel equations have been used in modeling the cell movement that is involved in the formation of such patterns but the question remains as to how the microscopic behavior can be correctly described by a macroscopic equation significant progress has been made for

an abstract result on cohen strongly summing operators abstract we present an abstract result that characterizes the coincidence of certain classes of linear operators with the class of cohen strongly summing linear operators our argument is extended to multilinear operators and as a consequence we establish some alternative characterizations for the class of cohen strongly summing multilinear operators

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

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

graded poisson lie structures on classical complex lie groups the external algebra over holomorphic first order differential forms on a complex lie groupg is endowed with the structure of a graded poisson lie algebra this structure is introduced via graded bicovariant brackets that are shown to be in one to one correspondence withg invariant tensors of special symmetry complete classification of graded poisson lie structures defined by homogeneous brackets is obtained for the case of classical complex lie groups

existence and decay of global smooth solutions to the coupled chemotaxis fluid model abstract in this paper we consider a coupled chemotaxis fluid model in r 3 which describes the so called chemotaxis boycott effect arising from the interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions it is shown that the cauchy problem has a unique global in time solution n c u x t on r 3 0 provided the invariant initial norm u 0 c 0 l 3 n 0 l 3 2 is suitably small or the diffusion coefficients of cells substrate and fluid i e are large enough we also show that the invariant norm u c t l 3 3 n t l 3 2 3 2 is monotone decreasing in t for all t 0

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

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

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

summing multilinear operators by blocks the isotropic and anisotropic cases abstract unifying several directions of the development of the study of summing multilinear operators between banach spaces we construct a general framework that studies under one single definition multilinear operators that are summing with respect to sums taken over any number of indices iterated and non iterated sums the isotropic and the anisotropic cases sums over arbitrary blocks and over several different sequence norms a large number of special classes of multilinear operators and of methods of generating classes of multilinear operators are recovered as particular instances ideal properties and coincidence theorems for the general classes are proved

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

global existence and boundedness in a keller segel navier stokes system with signal dependent sensitivity abstract in this paper we consider the following keller segel navier stokes system n t u n n n c c x t 0 c t u c c c n x t 0 u t u u u p n x t 0 u 0 x t 0 where r n n 2 3 is a bounded domain with smooth boundary r and c is assumed to generalize the prototype c 0 1 c 2 c 0 it is proved that i for 0 and n 2 or 0 and n 2 3 the corresponding initial boundary problem admits a unique global classical solution which is bounded ii for 0 and n 3 the corresponding initial boundary problem possesses at least one global weak solution

new critical exponents in a fully parabolic quasilinear keller segel system and applications to volume filling models abstract we carry on our studies related to the fully parabolic quasilinear keller segel system started in 6 and continued in 7 in the above mentioned papers we proved finite time blowup of some radially symmetric solutions to the quasilinear keller segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds on the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time the present paper is devoted to looking for critical exponents distinguishing between those two behaviors moreover we apply our results to the so called volume filling models with a power type jump probability function the most important consequence of our investigations of the latter is a critical mass phenomenon found in dimension 2 namely we find a value m such that when the solution to the two dimensional volume filling keller segel system starts with mass smaller than m then it is bounded while for some initial data with mass exceeding m solutions are unbounded though being defined for any time t 0

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

boundedness in quasilinear keller segel systems of parabolic parabolic type on non convex bounded domains abstract this paper deals with the quasilinear fully parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a bounded domain r n with smooth boundary n n the diffusivity d u is assumed to satisfy some further technical conditions such as algebraic growth and d 0 0 which says that the diffusion is allowed to be not only non degenerate but also degenerate the global in time existence and uniform in time boundedness of solutions are established under the subcritical condition that s u d u k u e for u 0 with 2 n k 0 and e 0 when d 0 0 this paper represents an improvement of tao and winkler 17 because the domain does not necessarily need to be convex in this paper in the case r n and d 0 0 uniform in time boundedness is an open problem left in a previous paper 7 this paper also gives an answer to it in bounded domains

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

global weak solutions for the three dimensional chemotaxis navier stokes system with nonlinear diffusion we consider an initial boundary value problem for the incompressible chemotaxis navier stokes equations generalizing the porous medium type diffusion model nt u n nm n c c x t 0 ct u c c nf c x t 0 ut u u u p n x t 0 u 0 x t 0 in a bounded convex domain r3 here r w1 0 c2 0 and 0 f c1 0 with f 0 0 it is proved that under appropriate structural assumptions on f and for any choice of m 23 and all sufficiently smooth initial data n0 c0 u0 the model possesses at least one global weak solution

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

preventing blow up in a chemotaxis model abstract this paper deals with a chemotactic model of keller segel type the main feature of the keller segel model is the possibility of blow up of solutions in a finite time to eliminate the possibility of blow up a modified version of keller segel model is introduced the blow up control relies on the presence of a pressure function which increases faster than a logarithm for high enough cells densities for such a pressure function the solutions cannot blow up in a finite time some other conditions are introduced to ensure the global boundedness