554 research outputs found
Laplacian Growth and Whitham Equations of Soliton Theory
The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in
the case of zero surface tension is proven to be equivalent to an integrable
systems of Whitham equations known in soliton theory. The Whitham equations
describe slowly modulated periodic solutions of integrable hierarchies of
nonlinear differential equations. Through this connection the Laplacian growth
is understood as a flow in the moduli space of Riemann surfaces.Comment: 33 pages, 7 figures, typos corrected, new references adde
Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential
The Schwarz function has played an elegant role in understanding and in
generating new examples of exact solutions to the Laplacian growth (or "Hele-
Shaw") problem in the plane. The guiding principle in this connection is the
fact that "non-physical" singularities in the "oil domain" of the Schwarz
function are stationary, and the "physical" singularities obey simple dynamics.
We give an elementary proof that the same holds in any number of dimensions for
the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17]
(1989). A generalization is also given for the so-called "elliptic growth"
problem by defining a generalized Schwarz potential. New exact solutions are
constructed, and we solve inverse problems of describing the driving
singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n -
techniques can be used to locate the singularity set of the Schwarz potential.
One of our methods is to prolong available local extension theorems by
constructing "globalizing families". We make three conjectures in potential
theory relating to our investigation
Bubble break-off in Hele-Shaw flows : Singularities and integrable structures
Bubbles of inviscid fluid surrounded by a viscous fluid in a Hele-Shaw cell
can merge and break-off. During the process of break-off, a thinning neck
pinches off to a universal self-similar singularity. We describe this process
and reveal its integrable structure: it is a solution of the dispersionless
limit of the AKNS hierarchy. The singular break-off patterns are universal, not
sensitive to details of the process and can be seen experimentally. We briefly
discuss the dispersive regularization of the Hele-Shaw problem and the
emergence of the Painlev\'e II equation at the break-off.Comment: 27 pages, 9 figures; typo correcte
A Duality Exact Sequence for Legendrian Contact Homology
We establish a long exact sequence for Legendrian submanifolds L in P x R,
where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that
displaces the projection of L off of itself. In this sequence, the singular
homology H_* maps to linearized contact cohomology CH^* which maps to
linearized contact homology CH_* which maps to singular homology. In
particular, the sequence implies a duality between the kernel of the map
(CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this
duality is compatible with Poincare duality in L in the following sense: the
Poincare dual of a singular class which is the image of a in CH_* maps to a
class \alpha in CH^* such that \alpha(a)=1.
The exact sequence generalizes the duality for Legendrian knots in Euclidean
3-space [24] and leads to a refinement of the Arnold Conjecture for double
points of an exact Lagrangian admitting a Legendrian lift with linearizable
contact homology, first proved in [6].Comment: 57 pages, 10 figures. Improved exposition and expanded analytic
detai
Weak vorticity formulation for the incompressible Euler equations in domains with boundary
In this article we examine the interaction of incompressible 2D flows with
compact material boundaries. Our focus is the dynamic behavior of the
circulation of velocity around boundary components and the possible exchange
between flow vorticity and boundary circulation in flows with vortex sheet
initial data We begin by showing that the velocity can be uniquely
reconstructed from the vorticity and boundary component circulations, which
allows to recast 2D Euler evolution using vorticity and the circulations as
dynamic variables. The weak form of this vortex dynamics formulation of the
equations is called the weak vorticity formulation. The main result in this
article is the equivalence between the weak velocity and weak vorticity
formulations, without sign assumptions. Next, we focus on weak solutions
obtained by mollifying initial data and passing to the limit, with the portion
of vorticity singular with respect to the Lebesgue measure assumed to be
nonnegative. For these solutions we prove that the circulations around each
boundary component cannot be smaller than the initial data circulation, so that
nonnegative vorticity may be absorbed by the boundary, but not produced by the
boundary. In addition, we prove that if the weak solution conserves circulation
at the boundary components it is a boundary coupled weak solution, a stronger
version of the weak vorticity formulation. We prove existence of a weak
solution which conserves circulation at the boundary components if the initial
vorticity is integrable. In addition, we discuss the definition of the
mechanical force which the flow exerts on material boundary components and its
relation with conservation of circulation. Finally, we describe the
corresponding results for a bounded domain with holes, and the adaptations
required in the proofs.Comment: 37 page
Weak and strong fillability of higher dimensional contact manifolds
For contact manifolds in dimension three, the notions of weak and strong
symplectic fillability and tightness are all known to be inequivalent. We
extend these facts to higher dimensions: in particular, we define a natural
generalization of weak fillings and prove that it is indeed weaker (at least in
dimension five),while also being obstructed by all known manifestations of
"overtwistedness". We also find the first examples of contact manifolds in all
dimensions that are not symplectically fillable but also cannot be called
overtwisted in any reasonable sense. These depend on a higher-dimensional
analogue of Giroux torsion, which we define via the existence in all dimensions
of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor
edits. v3: exposition improved using referee's comments. Published by Invent.
Mat
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