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Yes, in pure/applied math, we know a lot about various cases of approximation. But in practice there are more cases of approximation, and, right, the Euler equations are another such case. Or, to be a little flippant, generally in applications to real problems, we look at a lot of the features and throw out some, modify some, and actually honor some!! So, a question is, can we improve our ability to make such approximations and know something about the accuracy of the solutions we will get? E.g., for the Euler equations, will that approximation of an "incompressible" fluid ever work in practice and, if so, when and, there, how accurate can/will it be? Or, what about, hmm, just to be picky and pick something, friction on the side of the tube? What if the tube is not a perfect tube? A few grains of dirt: What if the liquid is water but, like most real water, has some solids floating around in it? Right, we can say, so there are a few grains of dirt floating around in the water, and they won't matter -- to be picky, that's an approximation, and we are likely correct, but where is an actual math theorem that says we are correct or how correct, i.e., accurate, are we? Right, a few grains of dirt -- we don't much care. But that's practical judgment and not really theorem/proof math. And similarly for other approximations we get as we throw out, modify, or honor real features? So, as stated, this is too difficult as a pure/applied math research direction. Okay, ..., then, is there anything at all in that direction that might be not absurdly difficult as a research direction? Or, to be simplistic, we work hard and get a numerical solution to a boundary value problem. Now someone tweaks the boundary. Can we say that our numerical solution is only tweaked? Or, when can we say that small changes in the problem statement will result in only small changes in the solution? Right, we are into some topology and looking for a case of continuity .... Hmm .... If we had some linearity ...!!! Right, the two pillars of analysis are continuity and linearity ...! But here with Euler we were considering nonlinear partial differential equations! Again I ask, is there any hope we can do anything for some corresponding math?? |