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. 



Is there any theory of Hamilton-Jacobi system?
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>


