Local well-posedness of kinetic chemotaxis models (Q946274)

Using a semigroup approach, the local well-posedness is shown for a class of kinetic models of chemotaxis which describes the evolution of the distribution function \(f=f(t,x,v)\geq 0\) of cells at time \(t\geq 0\) and position \(x\in {\mathbb R}^N\) with velocity \(v\) in a compact subset \(V\) of \({\mathbb R}^N\) and the concentration of chemoattractant \(S=S(t,x)\geq 0\). The model couples a linear transport equation for \(f\) \[ \partial_{t}f + v \cdot \nabla_{x} f = {\mathcal T}[S] f \] with a parabolic equation for \(S\). The turning operator \({\mathcal T}[S]\) is an integral operator and describes how the velocities of the cells change, this process being biased by the chemoattractant concentration \(S\).