3,291 research outputs found
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher
This review is dedicated to recent results on the 2d parabolic-elliptic
Patlak-Keller-Segel model, and on its variant in higher dimensions where the
diffusion is of critical porous medium type. Both of these models have a
critical mass such that the solutions exist globally in time if the mass
is less than and above which there are solutions which blowup in finite
time. The main tools, in particular the free energy, and the idea of the
methods are set out. A number of open questions are also stated.Comment: 26 page
Optimal Control for a Steady State Dead Oil Isotherm Problem
We study the optimal control of a steady-state dead oil isotherm problem. The
problem is described by a system of nonlinear partial differential equations
resulting from the traditional modelling of oil engineering within the
framework of mechanics of a continuous medium. Existence and regularity results
of the optimal control are proved, as well as necessary optimality conditions.Comment: This is a preprint of a paper whose final and definitive form will
appear in Control and Cybernetics. Paper submitted 24-Sept-2012; revised
21-March-2013; accepted for publication 17-April-2013. arXiv admin note: text
overlap with arXiv:math/061237
Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
We construct finite-dimensional approximations of solution spaces of
divergence form operators with -coefficients. Our method does not
rely on concepts of ergodicity or scale-separation, but on the property that
the solution space of these operators is compactly embedded in if source
terms are in the unit ball of instead of the unit ball of .
Approximation spaces are generated by solving elliptic PDEs on localized
sub-domains with source terms corresponding to approximation bases for .
The -error estimates show that -dimensional spaces
with basis elements localized to sub-domains of diameter (with ) result in an
accuracy for elliptic, parabolic and hyperbolic
problems. For high-contrast media, the accuracy of the method is preserved
provided that localized sub-domains contain buffer zones of width
where the contrast of the medium
remains bounded. The proposed method can naturally be generalized to vectorial
equations (such as elasto-dynamics).Comment: Accepted for publication in SIAM MM
The Theory of Quasiconformal Mappings in Higher Dimensions, I
We present a survey of the many and various elements of the modern
higher-dimensional theory of quasiconformal mappings and their wide and varied
application. It is unified (and limited) by the theme of the author's
interests. Thus we will discuss the basic theory as it developed in the 1960s
in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore
the connections with geometric function theory, nonlinear partial differential
equations, differential and geometric topology and dynamics as they ensued over
the following decades. We give few proofs as we try to outline the major
results of the area and current research themes. We do not strive to present
these results in maximal generality, as to achieve this considerable technical
knowledge would be necessary of the reader. We have tried to give a feel of
where the area is, what are the central ideas and problems and where are the
major current interactions with researchers in other areas. We have also added
a bit of history here and there. We have not been able to cover the many recent
advances generalising the theory to mappings of finite distortion and to
degenerate elliptic Beltrami systems which connects the theory closely with the
calculus of variations and nonlinear elasticity, nonlinear Hodge theory and
related areas, although the reader may see shadows of this aspect in parts
Optimal Control of the Thermistor Problem in Three Spatial Dimensions
This paper is concerned with the state-constrained optimal control of the
three-dimensional thermistor problem, a fully quasilinear coupled system of a
parabolic and elliptic PDE with mixed boundary conditions. This system models
the heating of a conducting material by means of direct current. Local
existence, uniqueness and continuity for the state system are derived by
employing maximal parabolic regularity in the fundamental theorem of Pr\"uss.
Global solutions are addressed, which includes analysis of the linearized state
system via maximal parabolic regularity, and existence of optimal controls is
shown if the temperature gradient is under control. The adjoint system
involving measures is investigated using a duality argument. These results
allow to derive first-order necessary conditions for the optimal control
problem in form of a qualified optimality system. The theoretical findings are
illustrated by numerical results
Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems
We analyze the inverse problem of identifying the diffusivity coefficient of
a scalar elliptic equation as a function of the resolvent operator. We prove
that, within the class of measurable coefficients, bounded above and below by
positive constants, the resolvent determines the diffusivity in an unique
manner. Furthermore we prove that the inverse mapping from resolvent to the
coefficient is Lipschitz in suitable topologies. This result plays a key role
when applying greedy algorithms to the approximation of parameter-dependent
elliptic problems in an uniform and robust manner, independent of the given
source terms. In one space dimension the results can be improved using the
explicit expression of solutions, which allows to link distances between one
resolvent and a linear combination of finitely many others and the
corresponding distances on coefficients. These results are also extended to
multi-dimensional elliptic equations with variable density coefficients. We
also point out towards some possible extensions and open problems
- …