11,055 research outputs found
Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility
In this work, the sharp interface limit of the degenerate Cahn-Hilliard
equation (in two space dimensions) with a polynomial double well free energy
and a quadratic mobility is derived via a matched asymptotic analysis involving
exponentially large and small terms and multiple inner layers. In contrast to
some results found in the literature, our analysis reveals that the interface
motion is driven by a combination of surface diffusion flux proportional to the
surface Laplacian of the interface curvature and an additional contribution
from nonlinear, porous-medium type bulk diffusion, For higher degenerate
mobilities, bulk diffusion is subdominant. The sharp interface models are
corroborated by comparing relaxation rates of perturbations to a radially
symmetric stationary state with those obtained by the phase field model.Comment: 27 pages, 2 figure
Kinetic Solvers with Adaptive Mesh in Phase Space
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for
solving multi-dimensional kinetic equations by the discrete velocity method. A
Cartesian mesh for both configuration (r) and velocity (v) spaces is produced
using a tree of trees data structure. The mesh in r-space is automatically
generated around embedded boundaries and dynamically adapted to local solution
properties. The mesh in v-space is created on-the-fly for each cell in r-space.
Mappings between neighboring v-space trees implemented for the advection
operator in configuration space. We have developed new algorithms for solving
the full Boltzmann and linear Boltzmann equations with AMPS. Several recent
innovations were used to calculate the discrete Boltzmann collision integral
with dynamically adaptive mesh in velocity space: importance sampling,
multi-point projection method, and the variance reduction method. We have
developed an efficient algorithm for calculating the linear Boltzmann collision
integral for elastic and inelastic collisions in a Lorentz gas. New AMPS
technique has been demonstrated for simulations of hypersonic rarefied gas
flows, ion and electron kinetics in weakly ionized plasma, radiation and light
particle transport through thin films, and electron streaming in
semiconductors. We have shown that AMPS allows minimizing the number of cells
in phase space to reduce computational cost and memory usage for solving
challenging kinetic problems
Existence of solutions for a higher order non-local equation appearing in crack dynamics
In this paper, we prove the existence of non-negative solutions for a
non-local higher order degenerate parabolic equation arising in the modeling of
hydraulic fractures. The equation is similar to the well-known thin film
equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann
operator, corresponding to the square root of the Laplace operator on a bounded
domain with Neumann boundary conditions (which can also be defined using the
periodic Hilbert transform). In our study, we have to deal with the usual
difficulty associated to higher order equations (e.g. lack of maximum
principle). However, there are important differences with, for instance, the
thin film equation: First, our equation is nonlocal; Also the natural energy
estimate is not as good as in the case of the thin film equation, and does not
yields, for instance, boundedness and continuity of the solutions (our case is
critical in dimension in that respect)
Symmetry-broken dissipative exchange flows in thin-film ferromagnets with in-plane anisotropy
Planar ferromagnetic channels have been shown to theoretically support a
long-range ordered and coherently precessing state where the balance between
local spin injection at one edge and damping along the channel establishes a
dissipative exchange flow, sometimes referred to as a spin superfluid. However,
realistic materials exhibit in-plane anisotropy, which breaks the axial
symmetry assumed in current theoretical models. Here, we study dissipative
exchange flows in a ferromagnet with in-plane anisotropy from a dispersive
hydrodynamic perspective. Through the analysis of a boundary value problem for
a damped sine-Gordon equation, dissipative exchange flows in a ferromagnetic
channel can be excited above a spin current threshold that depends on material
parameters and the length of the channel. Symmetry-broken dissipative exchange
flows display harmonic overtones that redshift the fundamental precessional
frequency and lead to a reduced spin pumping efficiency when compared to their
symmetric counterpart. Micromagnetic simulations are used to verify that the
analytical results are qualitatively accurate, even in the presence of nonlocal
dipole fields. Simulations also confirm that dissipative exchange flows can be
driven by spin transfer torque in a finite-sized region. These results
delineate the important material parameters that must be optimized for the
excitation of dissipative exchange flows in realistic systems.Comment: 20 pages, 5 figure
Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches
This work has been partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258.Publicad
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