58 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
Polydispersity and surface energy strength in nematic colloids
We consider a Landau-de Gennes model for a polydisperse, inhomogeneous
suspension of colloidal inclusions in a nematic host, in the dilute regime. We
study the homogenised limit and compute the effective free energy of the
composite material. By suitably choosing the shape of the inclusions and
imposing a quadratic, Rapini-Papoular type surface anchoring energy density, we
obtain an effective free energy functional with an additional linear term,
which may be interpreted as an "effective field" induced by the inclusions.
Moreover, we compute the effective free energy in a regime of "very strong
anchoring", that is, when the surface energy effects dominate over the volume
free energy.Comment: 24 pages, 1 figur
On the motion of a large number of small rigid bodies in a viscous incompressible fluid
We consider the motion of rigid bodies -- compact sets
-- immersed in a viscous incompressible fluid
contained in a domain in the Euclidean space , .
We show the fluid flow is not influenced by the presence of the infinitely
many bodies in the asymptotic limit and
as soon as
{\rm diam}[\mathcal{S}^i_\varepsilon ] \to 0 \ \mbox{as}\ \varepsilon \to 0
,\ i=1,\cdots, N(\varepsilon).
The result depends solely on the geometry of the bodies and is independent of
their mass densities. Collisions are allowed and the initial data are arbitrary
with finite energy
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