Johannes Lankeit

Long-term behaviour in a chemotaxis-fluid system with logistic source

We consider the coupled chemotaxis Navier–Stokes model with logistic source terms: nt + u ⋅∇n = Δn − χ∇⋅ (n∇c) + κn − μn2, ct + u ⋅∇c = Δc − nc, ut + (u ⋅∇)u = Δu + ∇P + n∇Φ + f,∇⋅ u = 0 in a bounded, smooth domain Ω ⊂ ℝ3 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 ∈ C1+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).

A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity

We consider the parabolic chemotaxis model [ u_t=Delta u - chi nablacdot(frac uv nabla v), qquadqquad v_t=Delta v - v + u] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for

The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers

chiin(0,chi_0)

Boundedness in a chemotaxis–haptotaxis model with nonlinear diffusion

for some

Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm

chi_0>1

Uniqueness of integrable solutions to \({\nabla \zeta=G \zeta, \zeta|_\Gamma = 0}\) for integrable tensor coefficients G and applications to elasticity

, thereby proving that the value

On a Degenerate Nonlocal Parabolic Problem Describing Infinite Dimensional Replicator Dynamics

chi=1

On the convexity of the function C ↦ f(det C) on positive-definite matrices

is not critical in this regard. Our main tool is consideration of the energy functional [ mathcal{F}_{a,b}(u,v)=int_Omega uln u - a int_Omega uln v + b int_Omega |nabla sqrt{v}|^2 ] for

Singular sensitivity in a Keller–Segel-fluid system

a>0

Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

,