Related papers: Vector Fields and Flows on Subcartesian Spaces
Given a continuous vector field $\lambda(t, \cdot)$ of Sobolev class $H^{\frac 32}$ on the unit circle $S^1$, the flow maps $\eta=g(t, \cdot)$ of the differential equation $$ \cases \frac{d\eta}{dt}=\lambda(t, \eta)\\ \eta(0,\zeta)=\zeta…
Let $M$ be a K\"ahler manifold with complex structure $J$ and K\"ahler metric $g$. A c-projective vector field is a vector field on $M$ whose flow sends $J$-planar curves to $J$-planar curves, where $J$-planar curves are analogs of what…
Given an initial $C^1$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean…
Rotation minimizing vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even the curvature vanishes. Nevertheless, many other geometric properties have…
We present a classification of hamiltonian vector fields on multisymplectic and polysymplectic fiber bundles closely analogous to the one known for the corresponding dual jet bundles that appear in the multisymplectic and polysymplectic…
We have found that the relation between the flow through campylotic (generically curved) media, consisting of randomly located curvature perturbations, and the average Ricci scalar of the system exhibits two distinct functional expressions…
New elementary, self-contained proofs are presented for the topological and the smooth classification theorems of linear flows on finite-dimensional normed spaces. The arguments, and the examples that accompany them, highlight the…
Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies…
In this paper we develop a formalism to describe a superfluid in a gravitational background. This formalism is based on a covariant generalization of the field description for a superconductor in terms of a U(1) spontaneous symmetry…
In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this…
Using Morita type stratifications, we establish a one-to-one correspondence between geometric vector fields on a separated differentiable stack and stratified vector fields on its orbit space. This correspondence enables us to derive a…
Let $g:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial of degree $p>1$, $G=(-g'_{y}, g'_{x})$ be its Hamiltonian vector field, and $G_t$ be the local flow generated by $G$. Denote by $E(G,O)$ the space of germs of $C^{\infty}$…
We consider a gradient flow associated to the mean field equation on $(M,g)$ a compact riemanniann surface without boundary. We prove that this flow exists for all time. Moreover, letting $G$ be a group of isometry acting on $(M,g)$, we…
We obtain a standard local presentation for a vector-valued multisymplectic form on a smooth manifold, generalizing the known proof for polysymplectic forms. We show that vector-valued multisymplectic forms on a finite-dimensional real…
The "Seifert Conjecture" asks, "Does every non-singular vector field on the 3-sphere ${\mathbb S}^3$ have a periodic orbit?" In a celebrated work, Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a plug, which is…
We study continuum-wise expansive flows with fixed points on metric spaces and low dimensional manifolds. We give sufficient conditions for a surface flow to be singular cw-expansive and examples showing that cw-expansivity does not imply…
Given a unit vector field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of topological invariants which combines the second…
We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal…
This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130,…
We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with…