Vortex filament dynamics for Gross-Pitaevsky type equations

Robert L. Jerrard

Displaying similar documents to “Vortex filament dynamics for Gross-Pitaevsky type equations”

On the motion of a curve by its binormal curvature

Jerrard, Robert L., Didier Smets (2015)

Journal of the European Mathematical Society

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We propose a weak formulation for the binormal curvature flow of curves in $ℝ^{3}$ . This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.

A diffused interface whose chemical potential lies in a Sobolev space

Yoshihiro Tonegawa (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms $W^{1, p}$ of the associated chemical potential fields are bounded uniformly, where $p > \frac{n}{2}$ and $n$ is the dimension of the domain. We show that the limit interface as $ε$ tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.

A new proof of the rectifiable slices theorem

Robert L. Jerrard (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper gives a new proof of the fact that a $k$ -dimensional normal current $T$ in $ℝ^{m}$ is integer multiplicity rectifiable if and only if for every projection $P$ onto a $k$ -dimensional subspace, almost every slice of $T$ by $P$ is $0$ -dimensional integer multiplicity rectifiable, in other words, a sum of Dirac masses with integer weights. This is a special case of the Rectifiable Slices Theorem, which was first proved a few years ago by B. White.

Tangency properties of sets with finite geometric curvature energies

Sebastian Scholtes (2012)

Fundamenta Mathematicae

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We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature $ℳ_{p}^{α} (X) : = \int_{X} \int_{X} \int_{X} κ^{p} (x, y, z) d_{X}^{α} (x) d_{X}^{α} (y) d_{X}^{α} (z)$ , where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that $ℳ_{p}^{α} (X) < \infty$ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant...

Curvature measures and fractals

Steffen Winter

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Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ’classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets $F \subseteq ℝ^{d}$ (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic...

A measure theoretic approach to higher codimension mean curvature flows

Luigi Ambrosio, Halil Mete Soner (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Convexity estimates for flows by powers of the mean curvature

Felix Schulze (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study the evolution of a closed, convex hypersurface in $ℝ^{n + 1}$ in direction of its normal vector, where the speed equals a power $k \geq 1$ of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvatures at every point is close enough to $1$ , depending only on $k$ and $n$ , then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere. ...

The inverse mean curvature flow and $p$ -harmonic functions

Roger Moser (2007)

Journal of the European Mathematical Society

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We consider the level set formulation of the inverse mean curvature flow. We establish a connection to the problem of $p$ -harmonic functions and give a new proof for the existence of weak solutions.