Abstract
The chemotaxis system is considered under the boundary conditions and v = v ⋆ on ∂Ω, where is a ball and v ⋆ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case n = 2, while global weak solutions are constructed when n ∈ {3, 4, 5}. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.
Highlights
Chemotaxis systems, if posed in bounded domains, are usually studied with homogeneous Neumann boundary conditions
Where the chemotactic agents partially direct their motion toward higher concentrations of a signal which they consume instead of produce, other boundary conditions may become relevant
Arising from a line of investigations concerned with pattern formation in colonies of B. subtilis in a fluid environment [29], such chemotaxis systems with signal consumption, coupled to a Stokes- or Navier-Stokes fluid have been studied extensively over the last decade
Summary
Chemotaxis systems, if posed in bounded domains, are usually studied with homogeneous Neumann boundary conditions. To make appropriate use of these symmetry assumptions, at a first stage of our analysis we shall rely on the essentially one-dimensional framework thereby generated in order to step by step turn the basic properties of mass conservation in the first component, and uniform L∞ boundedness in the second, into knowledge on higher order regularity features locally outside the spatial origin (see Section 3 and especially Corollary 3.7) This will enable us to appropriately control boundary integrals which due to the presence of possibly nonzero normal derivatives arise in a spatially global energy analysis related to that in (1.2) (Section 4). )e−λσ dσ for all t ∈ (0, Tmax,ε), which establishes (2.7)
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