Long-term behaviour in a chemotaxis-fluid system with logistic source Johannes LankeitWe consider the coupled chemotaxis Navier-Stokes model with logistic source termsin a bounded, smooth domain Ω ⊂ R 3 under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u and with given functions f ∈ L ∞ (Ω × (0, ∞)) satisfying certain decay conditions and Φ ∈ C 1+β (Ω) for some β ∈ (0, 1). We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state ( κ µ , 0, 0).

We investigate a family of isotropic volumetric-isochoric decoupled strain energiesbased on the Hencky-logarithmic (true, natural) strain tensor log U , where µ > 0 is the infinitesimal shear modulus, κ = 2µ+3λ 3 > 0 is the infinitesimal bulk modulus with λ the first Lamé constant, k, k are dimensionless parameters, F = ∇ϕ is the gradient of deformation, U = √ F T F is the right stretch tensor and devn log U = log U − 1 n tr(log U ) • 1 1 is the deviatoric part of the strain tensor log U . For small elastic strains, W eH approximates the classical quadratic Hencky strain energywhich is not everywhere rank-one convex. In plane elastostatics, i.e. n = 2, we prove the everywhere rankone convexity of the proposed family W eH , for k ≥ 1 4 and k ≥ 1 8 . Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for n = 2, 3 and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family W eH is not preserved in dimension n = 3 and that the energiesare not rank-one convex.

We prove existence of global weak solutions to the chemotaxis systemunder homogeneous Neumann boundary conditions in a smooth bounded convex domain Ω ⊂ R n , for arbitrarily small values of μ > 0.Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that κ is not too large. In this case, we also consider their large-time behaviour: We prove decay if κ ≤ 0 and the existence of an absorbing set if κ > 0 is sufficiently small.

The coupled chemotaxis fluid system

The chemotaxis systemis considered in a bounded domain Ω ⊂ R n with smooth boundary, where χ > 0.An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter χ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis thatfor all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that u ∈ L 1 loc (Ω × [0, ∞)).

We define and (for q > n) prove uniqueness and an extensibility property of W 1,q -solutions toin balls in R n , which we then use to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system -thereby extending recent results of Winkler [25] to the higher dimensional (radially symmetric) case.

A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity Johannes Lankeit * † Communicated by M. EfendievWe consider the parabolic chemotaxis modelin a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for 2 .0, 0 / for some 0 > 1, thereby proving that the value D 1 is not critical in this regard. Our main tool is consideration of the energy functionalfor a > 0, b 0, where using nonzero values of b appears to be new in this context.

This article deals with an initial-boundary value problem for the coupled chemotaxishaptotaxis system with nonlinear diffusionunder homogeneous Neumann boundary conditions in a bounded smooth domain Ω ⊂ R n , n = 2, 3, 4, where χ, ξ and µ are given nonnegative parameters. The diffusivity D(u) is assumed to satisfy D(u) ≥ δu m−1 for all u > 0 with some δ > 0. It is proved that for sufficiently regular initial data global bounded solutions exist whenever m > 2 − 2 n . For the case of non-degenerate diffusion (i.e. D(0) > 0) the solutions are classical; for the case of possibly degenerate diffusion (D(0) ≥ 0), the existence of bounded weak solutions is shown.Math Subject Classification (2010): 35K55 (primary), 35B40, 92C17