337 research outputs found
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 . 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 . 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 of class ,
the properties are studied of fields of `good directions', that is the
directions with respect to which 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
, in terms of which a corresponding flow can be defined. Using
this flow it is shown that can be approximated from the inside and the
outside by diffeomorphic domains of class . Whether or not the image
of a general continuous field of good directions (pseudonormals) defined on
is the whole of is shown to depend on the
topology of . These considerations are used to prove that if ,
or if has nonzero Euler characteristic, there is a point
in the neighbourhood of which 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'
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