No abstract uploaded!

We prove the existence of the global weak entropy solution to the Doi–Saintillan–Shelley model for active and passive rod-like particle suspensions, which couples a Fokker–Planck equation with the incompressible Navier–Stokes or Stokes equation, under the no-flux boundary conditions, L2(Ω;L1(Sd611))L2(Ω;L1(Sd611)) initial data, and finite initial entropy for the particle distribution function in two and three dimensions. Furthermore, for the model with the Stokes equation, we obtain the global L2(Ω×Sd611)L2(Ω×Sd611) weak solution in two and three dimensions and the uniqueness in two dimension.

The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established.

In this paper we consider the well-posedness of the compressible Prandtl boundary layer for the Navier–Stokes–Poisson equations with nonslip boundary condition and obtain the linear stability of the approximate solution constructed by boundary layer expansion.

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their$L$_{2}Whitney averages belongs to$L$_{2}on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to$L$_{2}on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to$L$_{$p$}generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.