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@MikeShulman I see, thanks for your comment. Actually, this was in some sense the point of my question. I understand that CT cannot be just a wizard or some miracle cure, solving all of (our) problems. I am indeed seeking some new point of view, e.g. on 1st order linear/non linear evolutionary PDEs and I was curious to see if CT could provide one. Maybe some of the results I know about this PDE's area that I consider "hard" are just "trivial", seen from the right perspective. Thanks again for your comment.
Ah yes, of course! I know that paper (albeit I have never seriously studied it). I have misunderstood your previous comment and I apologize for that. Actually I thought you were referring to something analogous to this concept of solution. I realized only now you were referring to viscosity solutions in the sense of Bressan-Bianchini.
Nope. Do you think there is some link between them and CT? I have heard people talking about them in conferences a few times, but mostly in talks about other kind of equations - I never saw them related to Hyp. Cons. Laws (linear or nonlinear). I'll give a look, thanks.
Thanks for your (in some sense expected) answer. My feeling is that CT might be potentially useful to (re)formulate (P)DE's in some abstract and neat language, but - as far as I got - it is not suitable to fully understand and detect the structure of solutions. E.g. entropy solutions are particular distributional solutions to a hyp cons law: while distributional solutions are generally infinite, they happen to be unique and to have some fine structure - some kind of BV regularity: I do not see a proper way to cast this in the language of CT, nor I see some kind of usefulness. Thanks.
Wow, looks interesting, thanks! I need of course some time to digest it. Rather than (what I understand for) Geometric Measure Theory, the paper you mention might be relevant for analysis on metric spaces and e.g. non smooth differential geometry. Do you know if something of this kind has ever been done (or tried)? Is it worth? Thanks a lot for your interesting answer.
