In biology, the behavior of a bacterial suspension in an incompressible fluid drop is modeled by the chemotaxis-Navier-Stokes equations.
In this lecture, we introduce an exchange of oxygen between the drop and its environment and an additionally logistic growth of the bacteria population.
A prototype system is given by
\begin{equation*}
\left\{\begin{aligned}
n_t+u\cdot\nabla n &= \Delta n-\nabla\cdot(n\nabla c)+n-n^2, & x\in\Omega, \ t>0,\\
c_t+u\cdot\nabla c &= \Delta c-nc, & x\in\Omega, \ t>0,\\
u_t &= \Delta u+u\cdot\nabla u+\nabla P-n \nabla\phi, & x\in\Omega, \ t>0,\\
\nabla\cdot u&=0, & x\in\Omega, \ t>0
\end{aligned}\right.
\end{equation*}
in conjunction with the initial data \((n,c,u)(\cdot,0) = (n_0,c_0,u_0)\) and the boundary conditions
\begin{equation*}
\left.\begin{aligned}
\frac{\partial c}{\partial \nu}&=1-c,\
\frac{\partial n}{\partial \nu}=n\frac{\partial}{\partial\nu}c, \ u= 0, &x\in\partial\Omega, \ t>0.
\end{aligned}\right.
\end{equation*}
Here, the fluid drop is described by \(\Omega\subset\mathbb{R}^N\) being a bounded convex domain with smooth boundary. Moreover, \(\phi\) is a given smooth gravitational potential.
Requiring sufficiently smooth initial data, the lecture gives an outline of the proofs of the existence of
- a global bounded classical solution for \(N=2\) and
- a global weak solution for \(N=3\).
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