| Abstract: |
| This presentation investigates a two-dimensional Keller--Segel--Navier--Stokes system with a tensor-valued sensitivity $S(x,n,c)$. Under a signal-dependent power-decay condition $|S(x,n,c)|\le s_0(s_1+c)^{-\gamma}$, we establish the global boundedness of classical solutions for both fluid-coupled ($\gamma > 1/2$) and fluid-free ($\gamma > 0$) systems. In both cases, the result covers the logarithmic Keller-Segel system ($\gamma=1$). To overcome mathematical difficulties arising from signal production and fluid transport, our approach utilizes a sequence of localized energy estimates. Furthermore, under specific structural assumptions on the sensitivity tensor, we prove that solutions of the fluid-free system converge exponentially to the spatially homogeneous steady state. |
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