ABSTRACT We investigate a kinetic model for the sedimentation of dilute suspensions of rod-like particles under gravity, deduced by Helzel, Otto, and Tzavaras (2011), which couples the impressible (Navier-)Stokes equation with the Fokker-Planck equation. With a no-flux boundary condition for the distribution function, we establish the existence and uniqueness of a global weak solution to the two dimensional model involving the Stokes equation.

We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class $\gamma^{s_{0}}$ and the Cauchy data belong to $\gamma^{s_{1}}$ , then the Cauchy problem has a solution in $\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R}))$ for some$T$^{*}>0, provided 1≤$s$_{1}≤2−1/$s$_{0}. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤$s$_{1}≤$s$_{0}.

In this paper, we prove two existence results of solutions to mean field equations Δu+e^u=ρδ_0 and Δu=λe^u(e^u−1)+4π\sum_{j=1}^M δ_{p_j} on an arbitrary connected finite graph, where ρ>0 and λ>0 are constants, M is a positive integer, and p_1,…,p_M are arbitrarily chosen distinct vertices on the graph.

In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.

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