Zhaoyin Xiang

In this paper, we first apply the method of hypocoercivity with a time-dependent weight, introduced by Wei and Zhang (Sci. China Math. 62 (2019): 1219-1232), to establish a semigroup estimate for the passive scalar equation advected by a Poiseuille flow $ (Ay^2, 0)^\top $ in $ \mathbb{T} \times \mathbb{R} $. Our result shows that the $ x $-dependent modes of solutions to this linear problem decay at a rate proportional to $ \mathcal{O}(A^{1/2}) $, which is faster than the heat dissipation time scale $ \mathcal{O}(A) $, a phenomenon known as enhanced dissipation. This result is then applied to the classical parabolic-elliptic Patlak-Keller-Segel (PKS) system, transported by the passive Poiseuille flow $ (Ay^2, 0)^\top $, to show that the solution to the PKS system remains global in time for any large initial mass, provided the Poiseuille flow is sufficiently strong, i.e., the amplitude $ A $ is suitably large. The key decay estimates for the $ x $-dependent modes of the PKS system are derived by adapting the approach of Li, Wei, and Zhang (Comm. Pure Appl. Math. 73 (2020): 465-557).

In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid:$ \begin{equation*} \left\{ \begin{split} \partial_t n + {\bf{u}}\cdot\nabla n & = \Delta n - \nabla\cdot\left(\chi(c)n\nabla c\right) + \nabla\cdot(n\nabla\phi), \\ \partial_t c + {\bf{u}}\cdot\nabla c & = \Delta c - nf(c), \\ \partial_t{\bf{u}} + \kappa({\bf{u}}\cdot\nabla){\bf{u}} + \nabla P & = \Delta{\bf{u}} - n\nabla\phi + \chi(c)n\nabla c, \\ \nabla\cdot{\bf{u}} & = 0 \end{split} \right. \end{equation*} $in $ \mathbb R^d\times(0,T)\, (d = 2,3) $. One of the novelties and difficulties here is that the coupling in this model is stronger and more nonlinear than the most-studied chemotaxis-fluid model due to the additional term $ \chi(c)n\nabla c $ in the third equation. We will first establish several extensibility criteria of classical solutions, which ensure us to extend the local solutions to global ones in the three dimensional chemotaxis-Stokes case and in the two dimensional chemotaxis-Navier-Stokes version under suitable smallness assumption on $ \|c_0\|_{L^{\infty}} $ with the help of a new entropy functional inequality. Some further decay estimates are also obtained under some suitable growth restriction on the potential $ \nabla \phi $ at infinity. As a byproduct of the entropy functional inequality, we also establish the global-in-time existence of weak solutions to the three dimensional chemotaxis-Navier-Stokes system. To the best of our knowledge, this seems to be the first work addressing the global well-posedness and decay property of solutions to the Cauchy problem of self-consistent chemotaxis-fluid system.