592 research outputs found
Nonlinear Hodge maps
We consider maps between Riemannian manifolds in which the map is a
stationary point of the nonlinear Hodge energy. The variational equations of
this functional form a quasilinear, nondiagonal, nonuniformly elliptic system
which models certain kinds of compressible flow. Conditions are found under
which singular sets of prescribed dimension cannot occur. Various degrees of
smoothness are proven for the sonic limit, high-dimensional flow, and flow
having nonzero vorticity. The gradient flow of solutions is estimated.
Implications for other quasilinear field theories are suggested.Comment: Slightly modified and updated version; tcilatex, 32 page
A Fresh Look at Entropy and the Second Law of Thermodynamics
This paper is a non-technical, informal presentation of our theory of the
second law of thermodynamics as a law that is independent of statistical
mechanics and that is derivable solely from certain simple assumptions about
adiabatic processes for macroscopic systems. It is not necessary to assume
a-priori concepts such as "heat", "hot and cold", "temperature". These are
derivable from entropy, whose existence we derive from the basic assumptions.
See cond-mat/9708200 and math-ph/9805005.Comment: LaTex file. To appear in the April 2000 issue of PHYSICS TODA
Existence, Regularity, and Properties of Generalized Apparent Horizons
We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the
existence of the outermost generalized apparent horizon in an initial data set
and that it is outer area minimizing.Comment: 16 pages, thoroughly revised, no major changes, to appear in Comm.
Math. Phy
Local behavior of p-harmonic Green's functions in metric spaces
We describe the behavior of p-harmonic Green's functions near a singularity
in metric measure spaces equipped with a doubling measure and supporting a
Poincar\'e inequality
Supercritical biharmonic equations with power-type nonlinearity
The biharmonic supercritical equation , where and
, is studied in the whole space as well as in a
modified form with as right-hand-side with an additional
eigenvalue parameter in the unit ball, in the latter case together
with Dirichlet boundary conditions. As for entire regular radial solutions we
prove oscillatory behaviour around the explicitly known radial {\it singular}
solution, provided , where
is a further critical exponent, which was introduced in a recent work by
Gazzola and the second author. The third author proved already that these
oscillations do not occur in the complementing case, where .
Concerning the Dirichlet problem we prove existence of at least one singular
solution with corresponding eigenvalue parameter. Moreover, for the extremal
solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue
problem, we prove smoothness as long as
Continuous and discrete Clebsch variational principles
The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics
Pontryagin´s principle for state-constrained boundary control problems of semilinear parabolic equations
This paper deals with state-constrained optimal control problems governed by semilinear parabolic equations. We establish a minimum principle of Pontryagin's type. To deal with the state constraints, we introduce a penalty problem by using Ekeland's principle. The key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are de ned. Conditions for normality of optimality conditions are given
A remark on an overdetermined problem in Riemannian Geometry
Let be a Riemannian manifold with a distinguished point and
assume that the geodesic distance from is an isoparametric function.
Let be a bounded domain, with , and consider
the problem in with on ,
where is the -Laplacian of . We prove that if the normal
derivative of along the boundary of is a
function of satisfying suitable conditions, then must be a
geodesic ball. In particular, our result applies to open balls of
equipped with a rotationally symmetric metric of the form
, where is the standard metric of the sphere.Comment: 8 pages. This paper has been written for possible publication in a
special volume dedicated to the conference "Geometric Properties for
Parabolic and Elliptic PDE's. 4th Italian-Japanese Workshop", organized in
Palinuro in May 201
Variational formulation of ideal fluid flows according to gauge principle
On the basis of the gauge principle of field theory, a new variational
formulation is presented for flows of an ideal fluid. The fluid is defined
thermodynamically by mass density and entropy density, and its flow fields are
characterized by symmetries of translation and rotation. The rotational
transformations are regarded as gauge transformations as well as the
translational ones. In addition to the Lagrangians representing the translation
symmetry, a structure of rotation symmetry is equipped with a Lagrangian
including the vorticity and a vector potential bilinearly. Euler's
equation of motion is derived from variations according to the action
principle. In addition, the equations of continuity and entropy are derived
from the variations. Equations of conserved currents are deduced as the Noether
theorem in the space of Lagrangian coordinate \ba. Without , the
action principle results in the Clebsch solution with vanishing helicity. The
Lagrangian yields non-vanishing vorticity and provides a source
term of non-vanishing helicity. The vorticity equation is derived as an
equation of the gauge field, and the characterizes topology of the
field. The present formulation is comprehensive and provides a consistent basis
for a unique transformation between the Lagrangian \ba space and the Eulerian
\bx space. In contrast, with translation symmetry alone, there is an
arbitrariness in the ransformation between these spaces.Comment: 34 pages, Fluid Dynamics Research (2008), accepted on 1st Dec. 200
Regularity and singularity in solutions of the three-dimensional Navier-Stokes equations
Higher moments of the vorticity field in the form of
-norms () are used to explore the regularity problem
for solutions of the three-dimensional incompressible Navier-Stokes equations
on the domain . It is found that the set of quantities provide a natural scaling in the problem resulting in a bounded set of time
averages on a finite interval of time . The behaviour of
is studied on what are called `good' and `bad' intervals of
which are interspersed with junction points (neutral) . For
large but finite values of with large initial data \big(\Omega_{m}(0) \leq
\varpi_{0}O(\Gr^{4})\big), it is found that there is an upper bound
\Omega_{m} \leq c_{av}^{2}\varpi_{0}\Gr^{4} ,\qquad\varpi_{0} = \nu L^{-2},
which is punctured by infinitesimal gaps or windows in the vertical walls
between the good/bad intervals through which solutions may escape. While this
result is consistent with that of Leray \cite{Leray} and Scheffer
\cite{Scheff76}, this estimate for corresponds to a length scale
well below the validity of the Navier-Stokes equations.Comment: 3 figures and 1 tabl
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