Geometrical Solutions of Hamilton-jacobi Equations

The concept of the geometrical solution of Hamilton-Jacobi equations in arbitrary space dimension is introduced. The characterization of such solution is based on the intersection of several invariant hyper-surfaces in the space of 1-jets. This solution notion allows not only for the smooth evolution beyond the usual singularity formation but also for superposition of underlying geometrical solutions. The need to consider such solutions occurs, for example, when the underlying Hamilton-Jacobi equation arises in the high frequency wave approximation problems. Classical characteristic analysis is limited to the local considerations owing to the crossing of characteristics. The notion of the classical viscosity solution leads to the global well-posedness, but excludes possible multi-valued solutions. The notion of the geometrical solution advocated here serves as an intrinsic identification for possible multivalued solutions. The geometrical solution is first described for general first-order PDEs followed by illustrations on Hamilton-Jacobi equations and quasi-linear hyperbolic equations. The consistency, existence and uniqueness results are all established. Furthermore, we clarify the relations of the geometrical solution to the entropy shock solution of scalar conservation laws as well as the viscosity solution of Hamilton-Jacobi equations. It is shown that the entropy shock in one dimensional space can be realized by compressing geometrical solution branches following an equal area principle; and the viscosity solution determined by the Lax-Hopf formula is just the smallest of branches of the geometrical solution.