PROPAGATION OF SINGULARITIES, HAMILTON-JACOBI EQUATIONS AND NUMERICALAPPLICATIONS

We consider applications of Hamilton-Jacobi equations for which the in itial data is only assumed to be in L(infinity). Such problems arise f or example when one attempts to describe several characteristic singul arities of the compressible Euler equations such as contact and acoust ic surfaces, propagating from the same discontinuous initial front. Th ese surfaces represent the level sets of solutions to a Hamilton-Jacob i equation which belongs to a special class. For such Hamilton-Jacobi equations we prove the existence and regularity of solutions for any p ositive time and convergence to initial data along rays of geometrical optics at any point where the gradient of the initial data exists. Fi nally, we present numerical algorithms for efficiently capturing singu lar fronts with complicated topologies such as corners and cusps. The approach of using Hamilton-Jacobi equations for capturing fronts has b een used in [14] for fronts propagating with curvature-dependent speed .