July 24, 2002

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The European Singularities Network.

Singularity Theory @ Isaac Newton Institute for Mathematical Sciences, July to December 2000 - Preprints.

Publications by I. Bogaevsky on Singularity Theory:

• Perestroikas of Shocks and Singularities of Minimum Functions, 2002.
• Singularities of Linear Waves in Plane and Space, 2000.
• Singularities of Viscosity Solutions of Hamilton-Jacobi Equation, 1999+.
• Metamorphoses of Singularities of Minimum Functions and Bifurcations of Shock Waves of the Burgers Equation with Vanishing Viscosity, 1990.

BibTeX references.

Generic transitions of shocks are topologically classified in plane and space. Shocks are discontinuities of limit potential solutions of Burgers equation with vanishing viscosity and external potential force. More general, shocks are singularities of viscosity solutions of Hamilton--Jacobi equation. The classification is applied to them if the Hamiltonian is convex. Besides, it describes transitions of singularities of generic solutions of the classical variational problem with a convex Lagrangian and free left end.

Download from here: http://arXiv.org/abs/math.AP/0204237

A viscosity solution of a Hamilton-Jacobi equation is the asymptotics of the solution with the same initial condition of the original Hamilton--Jacobi equation regularized by vanishing viscosity. Even if the initial condition is smooth, the viscosity solution can have singularities. If the dimension of the space does not exceed 3, there is a classification of these singularities and their perestroikas (=bifurcations, metamorphoses) in the case of a smooth convex Hamiltonian and a generic initial condition. This classification will be discussed in detail.

The situation is quite different if the Hamiltonian is not convex. In this case there exist new singularities which have been classified only in one-dimensional case. The point is that in the convex case a viscosity solution is a continuous branch of the many-valued solution of the Hamilton-Jacobi equation with the same initial condition. In the non-convex case this fact is not true. Nevertheless it is possible to choose a continuous branch and get the so called weak solution of the Hamilton-Jacobi equation.

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