Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in higher dimensions

Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in one space variable

Global Propagation of Singularities for Time Dependent Hamilton-Jacobi Equations

We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations of the form ut(t, x) +H(∇u(t, x)) = 0, a.e. (t, x) ∈ (0,+∞)×Ω ⊂ R n+1 . (1) It is well known that the singularities of such solutions propagate locally along generalized characteristics. Special generalized characteristics, satisfying an energy condition, can be constructed, under som...

A Level Set Method for the Computation of Multivalued Solutions to Quasi-linear Hyperbolic Pdes and Hamilton-jacobi Equations

We develop a level set method for the computation of multivalued solutions to quasi-linear hyperbolic partial differential equations and Hamilton-Jacobi equations in any number of space dimensions. We use the classic idea of Courant and Hilbert to define the solution of the quasi-linear hyperbolic PDEs or the gradient of the solution to the Hamilton-Jacobi equations as zero level sets of level ...

Simple Singularities for Max-concave Hamilton-jacobi Equations and Generalized Characteristics

We consider piecewise smooth viscosity solutions to a Hamilton-Jacobi equation, for which the Hamiltonian is the maximum of a finite number of smooth concave functions. We describe the possible types of “basic” singularities, namely jump discontinuities in the derivative of either the solution or the Hamiltonian which occur across a smooth hypersurface. Each such type of singularity is describe...

Convex ENO Schemes for Hamilton-Jacobi Equations

In one dimension, viscosity solutions of Hamilton-Jacobi (HJ) equations can be thought as primitives of entropy solutions for conservation laws. Based on this idea, both theoretical and numerical concepts used for conservation laws can be passed to HJ equations even in multi dimensions. In this paper, we construct convex ENO (CENO) schemes for HJ equations. This construction is a generalization...