Is there any theory of Hamilton-Jacobi system? I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems unavailable. So is there any way to define a solution (a solution with singularity) properly?
(A local classical solution seems possible by using characteristic method. But I hope to obtain a kind of 'weak' solution which can co-exist with singularities. )
Any suggestions or comments are welcome. Thank you. 
 A: Unfortunately, there is (to the best of my knowledge) no general theory for systems of Hamilton-Jacobi equations. That is to say, it is easy to come up with various notions of viscosity solution for systems, but no general uniqueness results are known. The main obstacle is how to extend the maximum principle to systems. 
However, there are some special cases of weakly coupled systems for which there is some suitable theory. Generally, speaking, the coupling is required to be only in the zeroth order terms, and must satisfy a monotonicity condition so that the maximum principle holds. This is likely quite restrictive for applications. Below is a, certainly not exhaustive, list of papers with results along these lines. 
"Optimal Switching for Ordinary Differential Equations", Capuzzo-Dolcetta and Evans
http://epubs.siam.org/doi/abs/10.1137/0322011
"Viscosity Solutions for Weakly Coupled Systems of Hamilton-Jacobi Equations", Engler and Lenhart
http://plms.oxfordjournals.org/content/s3-63/1/212.short
"Perron's method for monotone systems of second-order elliptic partial differential equations", Ishii
http://projecteuclid.org/euclid.die/1371086978
"Viscosity solutions for monotone systems of second–order elliptic PDES", Ishii and Koike
http://www.tandfonline.com/doi/abs/10.1080/03605309108820791?journalCode=lpde20
A: If you are interested in the case of HJ systems for a single unknown function, it could be a good idea to look at the papers on multitime HJ systems, in particular Hopf Formula and Multitime Hamilton-Jacobi Equations and The taxation principle and multi-time Hamilton-Jacobi equations as well as later works that refer to them; in particular, for the nonsmooth case see e.g. Nonsmooth multi-time Hamilton-Jacobi systems. In this connection note that there is a somewhat nonstandard application for pairs of HJ equations for a single unknown function in the theory of integrable, in the sense of soliton theory, partial differential systems: such pairs are employed as nonlinear Lax pairs for many integrable dispersionless systems, see e.g. this article and references therein.
