arXiv subjects

Ilya A. Bogaevsky

Publications and source records attributed to Ilya A. Bogaevsky.

3 recordsLinked to original sources

arXiv2004

Matter evolution in Burgulence

In inviscid solutions of the forced Burgers equation the matter accumulates in the shock discontinuities. We describe the limit motion of particles everywhere including the shocks as the trajectories of a discontinuous velocity field being a generalization of the gradient of the limit potential. The latter is not differentiable but satisfies some convexity properties which guarantee the existence of the gradient. It turns out that for such discontinuous gradient ordinary differential equations there are natural existence, uniqueness, and continuity theorems. The general results are applied for investigation of formation and motion in plane of massive points which are interpreted as various clusters in the adhesion model of the Universe.

math-ph↗

arXiv2003

Singularities of convex hulls of smooth hypersurfaces

We describe singularities of the convex hull of a generic compact smooth hypersurface in four-dimensional affine space up to diffeomorphisms. It turns out there are only two new singularities (in comparison with the previous dimension case) which appear at separate points of the boundary of the convex hull and are not removed by a small perturbation of the original hypersurface. The first singularity does not contain functional, but has at least nine continuous number invariants. A normal form which does not contain invariants at all is found for the second singularity.

math.MG↗

arXiv2002

Perestroikas of Shocks and Singularities of Minimum Functions

The shock discontinuities, generically present in inviscid solutions of the forced Burgers equation, and their bifurcations happening in the course of time (perestroikas) are classified in two and three dimensions -- the one-dimensional case is well known. This classification is a result of selecting among all the perestroikas occurring for minimum functions depending generically on time, the ones permitted by the convexity of the Hamiltonian of the Burgers dynamics. Topological restrictions on the admissible perestroikas of shocks are obtained. The resulting classification can be extended to the so-called viscosity solutions of a Hamilton--Jacobi equation, provided the Hamiltonian is convex.

math.AP↗