We study the global existence of weak solutions and asymptotic flocking behavior for the Cucker-Smale-Navier-Stokes (CS-NS) system in an infinitely long cylindrical domain under no-slip and specular reflection boundary conditions. This coupled kinetic-fluid system provides the modeling for the collective dynamics of flocking particles immersed in an incompressible viscous fluid, where the specular boundary condition allows elastic particle-wall interactions. We extend our previous results on Stokes-type coupled systems to the Navier-Stokes regime and establish global weak solutions. In particular, we verify asymptotic convergence toward flocking states. To the best of our knowledge, this is new for kinetic-fluid models with boundary walls. Furthermore, we perform numerical simulations under inflow-outflow boundary conditions to validate analytic results and compare the long-time dynamics of CS-NS with its Stokes counterpart. These simulations illustrate that fluid-particle coupling and inter-particle communication enhance apparent shear-thinning, non-Newtonian flow behaviors, and we compare these findings with shear-thinning phenomena measured in real suspensions of orientable particles, such as blood and swimming bacteria.
We study a class of nonlocal pseudo-parabolic equations with Sonine kernels, where the nonlinearity involves time-varying delays. Our main goal is to address some qualitative questions including the solvability, stability and the Ho & uml;lder regularity for mild solutions. In addition, we show a result on the convergence of solutions to ones of corresponding subdiffusion equations. The main ingredients for our analysis are regularity estimates of resolvents, a nonlo cal Halanay inequality, the fixed point argument and the embedding of fractional Sobolev spaces.
The low-magnetic-Reynolds-number approximation has been widely used in mathematical modeling of conducting fluids under the influence of external magnetic fields. The aim of this paper is to justify this approximation using a rigorous analysis in the appropriate functional setting. We derive absolute and relative error estimates for both three-dimensional magnetohydrodynamic duct flow and two-dimensional shallow-channel flow. The derived relative error estimate vanishes as the magnetic Reynolds number tends to zero, thereby validating the approximation and also providing its order of accuracy and the range of applicability.
