350 research outputs found
An elementary proof of uniqueness of the particle trajectories for solutions of a class of shear-thinning non-Newtonian 2D fluids
We prove some regularity results for a class of two dimensional non-Newtonian
fluids. By applying results from [Dashti and Robinson, Nonlinearity, 22 (2009),
735-746] we can then show uniqueness of particle trajectories
A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions
Stents are medical devices designed to modify blood flow in aneurysm sacs, in
order to prevent their rupture. Some of them can be considered as a locally
periodic rough boundary. In order to approximate blood flow in arteries and
vessels of the cardio-vascular system containing stents, we use multi-scale
techniques to construct boundary layers and wall laws. Simplifying the flow we
turn to consider a 2-dimensional Poisson problem that conserves essential
features related to the rough boundary. Then, we investigate convergence of
boundary layer approximations and the corresponding wall laws in the case of
Neumann type boundary conditions at the inlet and outlet parts of the domain.
The difficulty comes from the fact that correctors, for the boundary layers
near the rough surface, may introduce error terms on the other portions of the
boundary. In order to correct these spurious oscillations, we introduce a
vertical boundary layer. Trough a careful study of its behavior, we prove
rigorously decay estimates. We then construct complete boundary layers that
respect the macroscopic boundary conditions. We also derive error estimates in
terms of the roughness size epsilon either for the full boundary layer
approximation and for the corresponding averaged wall law.Comment: Dedicated to Professor Giovanni Paolo Galdi 60' Birthda
Quantum graphs with singular two-particle interactions
We construct quantum models of two particles on a compact metric graph with
singular two-particle interactions. The Hamiltonians are self-adjoint
realisations of Laplacians acting on functions defined on pairs of edges in
such a way that the interaction is provided by boundary conditions. In order to
find such Hamiltonians closed and semi-bounded quadratic forms are constructed,
from which the associated self-adjoint operators are extracted. We provide a
general characterisation of such operators and, furthermore, produce certain
classes of examples. We then consider identical particles and project to the
bosonic and fermionic subspaces. Finally, we show that the operators possess
purely discrete spectra and that the eigenvalues are distributed following an
appropriate Weyl asymptotic law
Adaptive FE-BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction
We analyze an adaptive finite element/boundary element procedure for scalar
elastoplastic interface problems involving friction, where a nonlinear
uniformly monotone operator such as the p-Laplacian is coupled to the linear
Laplace equation on the exterior domain. The problem is reduced to a
boundary/domain variational inequality, a discretized saddle point formulation
of which is then solved using the Uzawa algorithm and adaptive mesh refinements
based on a gradient recovery scheme. The Galerkin approximations are shown to
converge to the unique solution of the variational problem in a suitable
product of L^p- and L^2-Sobolev spaces.Comment: 27 pages, 3 figure
Nonexistence of self-similar singularities for the 3D incompressible Euler equations
We prove that there exists no self-similar finite time blowing up solution to
the 3D incompressible Euler equations. By similar method we also show
nonexistence of self-similar blowing up solutions to the divergence-free
transport equation in . This result has direct applications to the
density dependent Euler equations, the Boussinesq system, and the
quasi-geostrophic equations, for which we also show nonexistence of
self-similar blowing up solutions.Comment: This version refines the previous one by relaxing the condition of
compact support for the vorticit
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Determination of tip transfer function for quantitative MFM using frequency domain filtering and least squares method
Magnetic force microscopy has unsurpassed capabilities in analysis of nanoscale and microscale magnetic samples and devices. Similar to other Scanning Probe Microscopy techniques, quantitative analysis remains a challenge. Despite large theoretical and practical progress in this area, present methods are seldom used due to their complexity and lack of systematic understanding of related uncertainties and recommended best practice. Use of the Tip Transfer Function (TTF) is a key concept in making Magnetic Force Microscopy measurements quantitative. We present a numerical study of several aspects of TTF reconstruction using multilayer samples with perpendicular magnetisation. We address the choice of numerical approach, impact of non-periodicity and windowing, suitable conventions for data normalisation and units, criteria for choice of regularisation parameter and experimental effects observed in real measurements. We present a simple regularisation parameter selection method based on TTF width and verify this approach via numerical experiments. Examples of TTF estimation are shown on both 2D and 3D experimental datasets. We give recommendations on best practices for robust TTF estimation, including the choice of windowing function, measurement strategy and dealing with experimental error sources. A method for synthetic MFM data generation, suitable for large scale numerical experiments is also presented
Spectral Duality for Planar Billiards
For a bounded open domain with connected complement in
and piecewise smooth boundary, we consider the Dirichlet Laplacian
on and the S-matrix on the complement . We
show that the on-shell S-matrices have eigenvalues converging to 1
as exactly when has an eigenvalue at energy
. This includes multiplicities, and proves a weak form of
``transparency'' at . We also show that stronger forms of transparency,
such as having an eigenvalue 1 are not expected to hold in
general.Comment: 33 pages, Postscript, A
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