389 research outputs found
Solvability via viscosity solutions for a model of phase transitions driven by configurational forces
In the present article, we are interested in an initial boundary value
problem for a coupled system of partial differential equations arising in
martensitic phase transition theory of elastically deformable solid materials,
e.g., steel. This model was proposed and investigated in previous work by Alber
and Zhu in which the weak solutions are defined in a standard way, however the
key technique is not applicable to multi-dimensional problem. Intending to
solve this multi-dimensional problem and to investigate the sharp interface
limits of our models, we thus define weak solutions in a different way by using
the notion of viscosity solution, then prove the existence of weak solutions to
this problem in one space dimension, yet the multi-dimensional problem is still
open.Comment: 21 page
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A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calderón
We prove a Plancherel theorem for a nonlinear Fourier transform in two
dimensions arising in the Inverse Scattering method for the defocusing
Davey-Stewartson II equation. We then use it to prove global well-posedness and
scattering in for defocusing DSII. This Plancherel theorem also implies
global uniqueness in the inverse boundary value problem of Calder\'on in
dimension , for conductivities \sigma>0 with .
The proof of the nonlinear Plancherel theorem includes new estimates on
classical fractional integrals, as well as a new result on -boundedness of
pseudo-differential operators with non-smooth symbols, valid in all dimensions
Finite element methods for surface PDEs
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples
Flow-plate interactions: Well-posedness and long-time behavior
We consider flow-structure interactions modeled by a modified wave equation
coupled at an interface with equations of nonlinear elasticity. Both subsonic
and supersonic flow velocities are treated with Neumann type flow conditions,
and a novel treatment of the so called Kutta-Joukowsky flow conditions are
given in the subsonic case. The goal of the paper is threefold: (i) to provide
an accurate review of recent results on existence, uniqueness, and stability of
weak solutions, (ii) to present a construction of finite dimensional,
attracting sets corresponding to the structural dynamics and discuss
convergence of trajectories, and (iii) to state several open questions
associated with the topic. This second task is based on a decoupling technique
which reduces the analysis of the full flow-structure system to a PDE system
with delay.Comment: 1 figure. arXiv admin note: text overlap with arXiv:1208.5245,
arXiv:1311.124
Periodic Homogenization for Inertial Particles
We study the problem of homogenization for inertial particles moving in a
periodic velocity field, and subject to molecular diffusion. We show that,
under appropriate assumptions on the velocity field, the large scale, long time
behavior of the inertial particles is governed by an effective diffusion
equation for the position variable alone. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit . This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos
corrected. One reference adde
On the splash singularity for the free-surface of a Navier-Stokes fluid
In fluid dynamics, an interface splash singularity occurs when a locally
smooth interface self-intersects in finite time. We prove that for
-dimensional flows, or , the free-surface of a viscous water wave,
modeled by the incompressible Navier-Stokes equations with moving
free-boundary, has a finite-time splash singularity. In particular, we prove
that given a sufficiently smooth initial boundary and divergence-free velocity
field, the interface will self-intersect in finite time.Comment: 21 pages, 5 figure
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