404 research outputs found
On Singularity Formation of a Nonlinear Nonlocal System
We investigate the singularity formation of a nonlinear nonlocal system. This
nonlocal system is a simplified one-dimensional system of the 3D model that was
recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible
Navier-Stokes equations with swirl. The main difference between the 3D model of
Hou and Lei and the reformulated 3D Navier-Stokes equations is that the
convection term is neglected in the 3D model. In the nonlocal system we
consider in this paper, we replace the Riesz operator in the 3D model by the
Hilbert transform. One of the main results of this paper is that we prove
rigorously the finite time singularity formation of the nonlocal system for a
large class of smooth initial data with finite energy. We also prove the global
regularity for a class of smooth initial data. Numerical results will be
presented to demonstrate the asymptotically self-similar blow-up of the
solution. The blowup rate of the self-similar singularity of the nonlocal
system is similar to that of the 3D model.Comment: 28 pages, 9 figure
Absence of squirt singularities for the multi-phase Muskat problem
In this paper we study the evolution of multiple fluids with different
constant densities in porous media. This physical scenario is known as the
Muskat and the (multi-phase) Hele-Shaw problems. In this context we prove that
the fluids do not develop squirt singularities.Comment: 16 page
On the global well-posedness of a class of Boussinesq- Navier-Stokes systems
In this paper we consider the following 2D Boussinesq-Navier-Stokes systems
\partial_{t}u+u\cdot\nabla u+\nabla p+ |D|^{\alpha}u &= \theta e_{2}
\partial_{t}\theta+u\cdot\nabla \theta+ |D|^{\beta}\theta &=0 \quad with
and . When , , where is an explicit function
as a technical bound, we prove global well-posedness results for rough initial
data.Comment: 23page
New Algorithm for Parallel Laplacian Growth by Iterated Conformal Maps
We report a new algorithm to generate Laplacian Growth Patterns using
iterated conformal maps. The difficulty of growing a complete layer with local
width proportional to the gradient of the Laplacian field is overcome. The
resulting growth patterns are compared to those obtained by the best algorithms
of direct numerical solutions. The fractal dimension of the patterns is
discussed.Comment: Sumitted to Phys. Rev. Lett. Further details at
http://www.pik-potsdam.de/~ander
A phase-field model of Hele-Shaw flows in the high viscosity contrast regime
A one-sided phase-field model is proposed to study the dynamics of unstable
interfaces of Hele-Shaw flows in the high viscosity contrast regime. The
corresponding macroscopic equations are obtained by means of an asymptotic
expansion from the phase-field model. Numerical integrations of the phase-field
model in a rectangular Hele-Shaw cell reproduce finger competition with the
final evolution to a steady state finger the width of which goes to one half of
the channel width as the velocity increases
A maximum principle for the Muskat problem for fluids with different densities
We consider the fluid interface problem given by two incompressible fluids
with different densities evolving by Darcy's law. This scenario is known as the
Muskat problem for fluids with the same viscosities, being in two dimensions
mathematically analogous to the two-phase Hele-Shaw cell. We prove in the
stable case (the denser fluid is below) a maximum principle for the
norm of the free boundary.Comment: 16 page
Dynamical Systems approach to Saffman-Taylor fingering. A Dynamical Solvability Scenario
A dynamical systems approach to competition of Saffman-Taylor fingers in a
channel is developed. This is based on the global study of the phase space
structure of the low-dimensional ODE's defined by the classes of exact
solutions of the problem without surface tension. Some simple examples are
studied in detail, and general proofs concerning properties of fixed points and
existence of finite-time singularities for broad classes of solutions are
given. The existence of a continuum of multifinger fixed points and its
dynamical implications are discussed. The main conclusion is that exact
zero-surface tension solutions taken in a global sense as families of
trajectories in phase space spanning a sufficiently large set of initial
conditions, are unphysical because the multifinger fixed points are
nonhyperbolic, and an unfolding of them does not exist within the same class of
solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed
points is argued to be essential to the physically correct qualitative
description of finger competition. The restoring of hyperbolicity by surface
tension is discussed as the key point for a generic Dynamical Solvability
Scenario which is proposed for a general context of interfacial pattern
selection.Comment: 3 figures added, major rewriting of some sections, submitted to Phys.
Rev.
A CsI(Tl) Scintillating Crystal Detector for the Studies of Low Energy Neutrino Interactions
Scintillating crystal detector may offer some potential advantages in the
low-energy, low-background experiments. A 500 kg CsI(Tl) detector to be placed
near the core of Nuclear Power Station II in Taiwan is being constructed for
the studies of electron-neutrino scatterings and other keV-MeV range neutrino
interactions. The motivations of this detector approach, the physics to be
addressed, the basic experimental design, and the characteristic performance of
prototype modules are described. The expected background channels and their
experimental handles are discussed.Comment: 34 pages, 11 figures, submitted to Nucl. Instrum. Method
Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima Model
The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental
models that describe plasma turbulence. The model also appears as a simplified
reduced Rayleigh-B\'enard convection model. The mathematical analysis the
Hasegawa-Mima equation is challenging due to the absence of any smoothing
viscous terms, as well as to the presence of an analogue of the vortex
stretching terms. In this paper, we introduce and study a model which is
inspired by the inviscid Hasegawa-Mima model, which we call a
pseudo-Hasegawa-Mima model. The introduced model is easier to investigate
analytically than the original inviscid Hasegawa-Mima model, as it has a nicer
mathematical structure. The resemblance between this model and the Euler
equations of inviscid incompressible fluids inspired us to adapt the techniques
and ideas introduced for the two-dimensional and the three-dimensional Euler
equations to prove the global existence and uniqueness of solutions for our
model. Moreover, we prove the continuous dependence on initial data of
solutions for the pseudo-Hasegawa-Mima model. These are the first results on
existence and uniqueness of solutions for a model that is related to the
three-dimensional inviscid Hasegawa-Mima equations
- …