We consider a system coupling the incompressible Navier–Stokes equations to the Vlasov–Fokker–Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by the non-stiff convection operator, with an implicit treatment of the stiff drag term and the Fokker–Planck operator. Yet, consistent to a standard asymptotic-preserving Fokker–Planck solver or an incompressible Navier–Stokes solver, only the conjugate–gradient method and fast Poisson and Helmholtz solvers are needed. Numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme, with several interesting applications.

We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for a one-dimensional (1D) bipolar Vlasov--Poisson--Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type of decomposition around the local Maxwellian related to the bipolar VPB system in the previous work [H.-L. Li, Y. Wang, T. Yang, and M.-Y. Zhong, <i>Arch. Ration. Mech. Anal.</i>, 228 (2018), pp. 39--127], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to the 1D bipolar VPB system under some smallness conditions on the initial perturbations and wave strength, which implies that this typical superposition wave pattern is nonlinearly stable under the combined effects of the binary collisions, the electrostatic potential

We prove the existence of a resonance-free region in scattering by a strictly convex obstacle $\mathcal{O}$ with the Robin boundary condition $\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0$ . More precisely, we show that the scattering resonances lie below a cubic curve ℑ$ζ$=−$S$|$ζ$|^{1/3}+$C$. The constant$S$is the same as in the case of the Neumann boundary condition$γ$=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.

Motivated by a model proposed by Peng et al. [Advances in Coll. and Interf. Sci. 206 (2014)] for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDE for the film thickness and salt concentration subject to nonconservative effects representing evaporation. We analytically prove the global existence of solutions to this model with mobility exponents in several different ranges and the results are then validated against PDE simulations. We also numerically capture other interesting dynamics of the model, including finite-time rupture-shock phenomenon due to the instabilities caused by locally elevated evaporation rates, convergence to equilibrium and infinite-time thinning.

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