The stability analysis of a 2D Keller-Segel-Navier-Stokes system in fast signal diffusion (Q6622940)

The authors investigate the stability of the Keller-Segel-Navier-Stokes system, focusing on the convergence of the parabolic-parabolic-fluid system to the parabolic-elliptic-fluid system as the relaxation time scale \(\epsilon\) (the ratio between the diffusivity of the cells and that of the chemoattractant) approaches \(0\).\N\NUnder appropriate regularity and volume-filling hypothesis on the chemotactic sensitivity and a fairly general class of initial data, the authors showed that on a bounded smooth domain of \(\mathbb{R}^2\)\N\N\begin{itemize}\N\item The global classical solutions (not just a subsequence) of the full PP-fluid system will converge to the solution of the corresponding PE-fluid system, with exponential growth in time convergence rate.\N\N\item As a by-product, the global well-posedness of the PE-fluid system for general initial data has been established.\N\N\item Established exponential time decay estimates of PP-Fluid solutions uniformly in \(\epsilon\) for small initial data, which in particular ensure an improvement of convergence rate to at most \(O(t^{\frac{1}{2}})\)-growth.\N\N\item Finally, the convergence behavior on \(\epsilon\) and \(t\) through numerical experiments of three different types of solution have been explored: the nontrivial and the constant equilibriums and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.\N\N\end{itemize}\N\NOne key step in their argument is obtaining uniform \( L^4\)-regularity estimates on the cell densities of the PP-fluid system. After acquiring this essential bound, the damping effect of the chemical concentration provides the required uniform bounds, which could ensure the convergence of some subsequences.\N\NThe next essential step is obtaining a linear (in time) growth estimate for the mixed interaction of the chemical concentrations of the PP-fluid and PE-fluid systems. This estimate will provide the convergence rate for general initial data.\N\NThen the authors show that the solution of the PP-fluid system exponentially decays to the constant steady state uniformly in \(\epsilon\) for appropriate small initial data, which ensures that one can improve the growth to \(t^{ \ \frac {1}{2}}\), by investigating the time evolution of a mixed functional.