A Nash stratification inequality and global regularity for a chemotaxis-fluid system on general 2D domains
Incompressible fluid advection has been shown to facilitate singularity suppression in various differential equations, often by mixing-enhanced diffusion or by dimension-reduction effects. To aid with the study of such scenarios, we prove a refinement of the classical Nash inequality, $$ \|f-f_M\|_{L^2}^2 \lesssim \|f-f_M\|_{L^1}^{8/7}\|\nabla f\|_{L^2}^{6/7} + \|\partial_1 f\|_{\dot{H}_0^{-1}}^{2\vartheta} \|f-f_M\|_{L^1}^{1-\vartheta}\|\nabla f\|_{L^2}^{1-\vartheta}, $$ for $f \in H^1$ with mean $f_M$ over a smooth bounded planar domain under the main constraint of having connected horizontal cross-sections. The first term on the right-hand side follows the classical Nash scaling for a formal dimension of $3/2$. The second term introduces a mixing norm that measures how far $f$ is from being stratified. The proof provides an explicit exponent $0 < \vartheta \ll 1$. As an application, we study the 2D parabolic-elliptic Patlak--Keller--Segel (PKS) chemotaxis model over the aforementioned large class of bounded domains. This aggregation-diffusion equation is well-known to produce finite-time singularity formation for large-mass data in finite domains. Using the above Nash stratification inequality, we prove that the 2D PKS equation becomes globally regular when actively coupled via buoyancy to a fluid obeying Darcy's law for incompressible porous media flows. This result holds for arbitrarily large $C^\infty$ initial data, far from perturbative regimes, and for arbitrarily weak coupling strength. Moreover, the spatial domain can have "bottle neck" regions and boundary segments of large curvature. The argument conceptualizes and generalizes recent work by Hu, Yao, and the first author on the analogous result for the periodic channel.