Propagation of singularities, Hamilton-Jacobi equations and numerical applications

Abstract:

We consider applications of Hamilton-Jacobi equations for which the initial data is only assumed to be in ${L^\infty }$. Such problems arise for example when one attempts to describe several characteristic singularities of the compressible Euler equations such as contact and acoustic surfaces, propagating from the same discontinuous initial front. These surfaces represent the level sets of solutions to a Hamilton-Jacobi equation which belongs to a special class. For such Hamilton-Jacobi equations we prove the existence and regularity of solutions for any positive time and convergence to initial data along rays of geometrical optics at any point where the gradient of the initial data exists. Finally, we present numerical algorithms for efficiently capturing singular fronts with complicated topologies such as corners and cusps. The approach of using Hamilton-Jacobi equations for capturing fronts has been used in [14] for fronts propagating with curvature-dependent speed.