Tobias Black

Sublinear signal production in a two-dimensional Keller–Segel–Stokes system

Abstract We study the chemotaxis–fluid system { n t = Δ n − ∇ ⋅ ( n ∇ c ) − u ⋅ ∇ n , x ∈ Ω , t > 0 , c t = Δ c − c + f ( n ) − u ⋅ ∇ c , x ∈ Ω , t > 0 , u t = Δ u + ∇ P + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , where Ω ⊂ R 2 is a bounded and convex domain with smooth boundary, ϕ ∈ W 2 , ∞ ( Ω ) and f ∈ C 1 ( [ 0 , ∞ ) ) satisfies 0 ≤ f ( s ) ≤ K 0 s α for all s ∈ [ 0 , ∞ ) , with K 0 > 0 and α ∈ ( 0 , 1 ] . This system models the chemotactic movement of actively communicating cells in slow moving liquid. We will show that in the two-dimensional setting for any α ∈ ( 0 , 1 ) the classical solution to this Keller–Segel–Stokes-system is global and remains bounded for all times.

Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D

Abstract We study the chemotaxis–fluid system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n c ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, where Ω ⊂ R 2 is a bounded domain with smooth boundary and ϕ ∈ C 2 ( Ω ¯ ) . 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 ∫ Ω 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 ( Ω ) and in L log ⁡ L ( Ω ) , u 0 in L 4 ( Ω ) , and of ∇ c 0 in L 2 ( Ω ) .

Boundedness in a Keller–Segel system with external signal production

Abstract We study the Neumann initial-boundary problem for the chemotaxis system { u t = Δ u − ∇ ⋅ ( u ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u + f ( x , t ) , x ∈ Ω , t > 0 , ∂ u ∂ ν = ∂ v ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω in a smooth, bounded domain Ω ⊂ R n with n ≥ 2 and f ∈ L ∞ ( [ 0 , ∞ ) ; L n 2 + δ 0 ( Ω ) ) ∩ C α ( Ω × ( 0 , ∞ ) ) with some α > 0 and δ 0 ∈ ( 0 , 1 ) . First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of n = 2 and f being constant in time, requiring the nonnegative initial data u 0 to fulfill the property ∫ Ω u 0 d x 4 π ensures that the solution is global and remains bounded uniformly in time. Thereby we extend the well known critical mass result by Nagai, Senba and Yoshida for the classical Keller–Segel model (coinciding with f ≡ 0 in the system above) to the case f ≢ 0 . Under certain smallness conditions imposed on the initial data and f we furthermore show that for more general space dimension n ≥ 2 and f not necessarily constant in time, the solutions are also global and remain bounded uniformly in time. Accordingly we extend a known result given by Winkler for the classical Keller–Segel system to the present situation.

Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion

We consider the chemotaxis-fluid system given by

Singular sensitivity in a Keller–Segel-fluid system

n_{t}+u\cdot\!\nabla n=\Delta n^m-\nabla\!\cdot(n\nabla c)

Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant

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On the weakly competitive case in a two-species chemotaxis model

c_{t}+u\cdot\!\nabla c=\Delta c-c+n

Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals

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Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions.

u_{t}+(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi

Stabilization in the Keller--Segel system with signal-dependent sensitivity.

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