Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems?

Abstract

Abstract We consider the spatially 2D version of the model $$\begin{equation*} \qquad\quad\left\{ \begin{array}{@{}rcll} n_t + u\cdot\nabla n &=& \Delta n - \nabla \cdot \big(nS(x,n,c) \cdot \nabla c \big), \qquad &\qquad x\in \Omega, \ t>0, \\ c_t + u\cdot \nabla c &=& \Delta c - n f(c), \qquad &\qquad x\in \Omega, \ t>0, \\ u_t &=& \Delta u + \nabla P + n\nabla\phi, \qquad \nabla\cdot u=0, \qquad &\qquad x\in \Omega, \ t>0, \end{array} \right. \qquad \qquad (\star) \end{equation*}$$for nutrient taxis processes, possibly interacting with liquid environments. Here the particular focus is on the situation when the chemotactic sensitivity $S$ is not a scalar function but rather attains general values in ${\mathbb{R}}^{2\times 2}$, thus accounting for rotational flux components in accordance with experimental findings and recent modeling approaches. Reflecting significant new challenges that mainly stem from apparent loss of energy-like structures, especially for initial data with large size, the knowledge on ($\star$) so far seems essentially restricted to results on global existence of certain generalized solutions with possibly quite poor boundedness and regularity properties; widely unaddressed seem aspects related to possible effects of such non-diagonal taxis mechanisms on the qualitative solution behavior, especially with regard to the fundamental question whether spatial structures may thereby be supported. The present work answers the latter in the negative in the following sense: under the assumptions that the initial data $(n_0,c_0,u_0)$ and the parameter functions $S$, $f$, and $\phi$ are sufficiently smooth, and that $S$ is bounded and $f$ is positive on $(0,\infty )$ with $f(0)=0$, it is shown that any nontrivial of these solutions eventually becomes smooth and satisfies $$\begin{equation*} n(\cdot,t)\to - \int_\Omega n_0, \quad c(\cdot,t)\to 0 \quad \text{and} \quad u(\cdot,t)\to 0 \qquad \text{as} \ t\to\infty, \end{equation*}$$uniformly with respect to $x\in \Omega$. By not requiring any smallness condition on the initial data, the latter seems new even in the corresponding fluid-free version obtained on letting $u\equiv 0$ in ($\star$).