On the existence of bounded solutions for a nonlinear elliptic system

This work deals with the system $(-\Delta)^m u= a(x) v^p$, $(-\Delta)^m v=b(x) u^q$ with Dirichlet boundary condition in a domain \Omega\subset\RR^n, where $\Omega$ is a ball if $n\ge 3$ or a smooth perturbation of a ball when $n=2$. We prove that, under appropriate conditions on the parameters ($a,b,p,q,m,n$), any non-negative solution $(u,v)$ of the system is bounded by a constant independent of $(u,v)$. Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case $m=1$ was considered by Souplet in \cite{PS}. Our paper generalize to $m\ge 1$ the results of that paper

DSMC-LBM mapping scheme for rarefied and non-rarefied gas flows

We present the formulation of a kinetic mapping scheme between the Direct Simulation Monte Carlo (DSMC) and the Lattice Boltzmann Method (LBM) which is at the basis of the hybrid model used to couple the two methods in view of efficiently and accurately simulate isothermal flows characterized by variable rarefaction effects. Owing to the kinetic nature of the LBM, the procedure we propose ensures to accurately couple DSMC and LBM at a larger Kn number than usually done in traditional hybrid DSMC-Navier-Stokes equation models. We show the main steps of the mapping algorithm and illustrate details of the implementation. Good agreement is found between the moments of the single particle distribution function as obtained from the mapping scheme and from independent LBM or DSMC simulations at the grid nodes where the coupling is imposed. We also show results on the application of the hybrid scheme based on a simpler mapping scheme for plane Poiseuille flow at finite Kn number. Potential gains in the computational efficiency assured by the application of the coupling scheme are estimated for the same flow.Comment: Submitted to Journal of Computational Scienc