Patterns in a Smoluchowski Equation

We analyze the dynamics of concentrated polymer solutions modeled by a 2D Smoluchowski equation. We describe the long time behavior of the polymer suspensions in a fluid. \par When the flow influence is neglected the equation has a gradient structure. The presence of a simple flow introduces significant structural changes in the dynamics. We study the case of an externally imposed flow with homogeneous gradient. We show that the equation is still dissipative but new phenomena appear. The dynamics depend on both the concentration intensity and the structure of the flow. In certain limit cases the equation has a gradient structure, in an appropriate reference frame, and the solutions evolve to either a steady state or a tumbling wave. For small perturbations of the gradient structure we show that some features of the gradient dynamics survive: for small concentrations the solutions evolve in the long time limit to a steady state and for high concentrations there is a tumbling wave.Comment: Minor typos fixed. References adde

Equivalence of weak formulations of the steady water waves equations

We prove the equivalence of three weak formulations of the steady water waves equations, namely the velocity formulation, the stream function formulation, and the Dubreil-Jacotin formulation, under weak Holder regularity assumptions on their solutions

Refined approximation for a class of Landau-de Gennes energy minimizers

We study a class of Landau-de Gennes energy functionals in the asymptotic regime of small elastic constant $L>0$. We revisit and sharpen the results in [18] on the convergence to the limit Oseen-Frank functional. We examine how the Landau-de Gennes global minimizers are approximated by the Oseen-Frank ones by determining the first order term in their asymptotic expansion as $L\to 0$. We identify the appropriate functional setting in which the asymptotic expansion holds, the sharp rate of convergence to the limit and determine the equation for the first order term. We find that the equation has a ``normal component'' given by an algebraic relation and a ``tangential component'' given by a linear system

Partial regularity and smooth topology-preserving approximations of rough domains

For a bounded domain $\Omega\subset\mathbb{R}^m, m\geq 2,$ of class $C^0$, the properties are studied of fields of `good directions', that is the directions with respect to which $\partial\Omega$ can be locally represented as the graph of a continuous function. For any such domain there is a canonical smooth field of good directions defined in a suitable neighbourhood of $\partial\Omega$, in terms of which a corresponding flow can be defined. Using this flow it is shown that $\Omega$ can be approximated from the inside and the outside by diffeomorphic domains of class $C^\infty$. Whether or not the image of a general continuous field of good directions (pseudonormals) defined on $\partial\Omega$ is the whole of $\mathbb{S}^{m-1}$ is shown to depend on the topology of $\Omega$. These considerations are used to prove that if $m=2,3$, or if $\Omega$ has nonzero Euler characteristic, there is a point $P\in\partial\Omega$ in the neighbourhood of which $\partial\Omega$ is Lipschitz. The results provide new information even for more regular domains, with Lipschitz or smooth boundaries.Comment: Final version appeared in Calc. Var PDE 56, Issue 1, 201

Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

Mathematical studies of nematic liquid crystals address in general two rather different perspectives: That of fluid mechanics and that of calculus of variations. The former focuses on dynamical problems while the latter focuses on stationary ones. The two are usually studied with different mathematical tools and address different questions. The aim of this brief review is to give the practitioners in each area an introduction to some of the results and problems in the other area. Also, aiming to bridge the gap between the two communities, we will present a couple of research topics that generate natural connections between the two areas. This article is part of the theme issue 'Topics in mathematical design of complex materials'