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Consider the metric space BV(0,1) with the following metric $$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing that is stopping us from using this space for DE's and PDE problems!

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    $\begingroup$ So the "obvious thing" would be that the solution schemes used by people studying DE and PDE require certain properties that is not satisfied by the space listed. That said, bounded variation function spaces (in general) are used to study 1+1 dimensional conservation laws. A large part of the theory of shocks that originated with Lax and Glimm is built upon BV spaces. So I don't really get your question. $\endgroup$
    – Willie Wong
    Commented Dec 1, 2015 at 14:23
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    $\begingroup$ books.google.com/books/about/… onlinelibrary.wiley.com/doi/10.1002/cpa.3160180408/abstract for your last comment. To your original question: what makes you think that your metric space is good/not good for PDEs? $\endgroup$
    – Willie Wong
    Commented Dec 1, 2015 at 14:54
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    $\begingroup$ To be precise: 1) This distance comes from a Banach space norm. 2) It Is NOT compact as claimed, and not even locally compact (for TVS, LC=finite dimension. 3) It is a dual space, so that its closed unit ball is compact in the w* topology by Banach-Alaoglu, and it is also metrizable due to separability (the distance being strictly weaker than the one you wrote). 4) However on the whole space, the w* topology is not metrizable. $\endgroup$
    – Pietro Majer
    Commented Dec 1, 2015 at 16:17
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    $\begingroup$ @RajeshDachiraju This metric actually seems weird to me: it is not equivalent to the usual metric $d(u,v)=TV(u-v)$ on BV spaces. To wit, let $u$ and $v$ be bounded by $\epsilon$ and have the same total variation, but $u$ oscillates wildly on $[0,1/3]$ and $v$ oscillates like crazy on $[2/3,1]$. Then $u$ and $v$ will be $2\epsilon$ close in your metric, but very far away in the standard BV metric. $\endgroup$
    – Fan Zheng
    Commented Dec 1, 2015 at 18:51
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    $\begingroup$ @RajeshDachiraju If that is your question, then I think you should read the first comment of Willie Wong, in particular this line: "in solving PDEs, one does not choose a function space first. Instead, one chooses a method and finds a function space in which the method can be applied." If your metric doesn't show up much in the literature, that is probably just a sign that it isn't a good one for expressing "closeness" in the BV sense. $\endgroup$
    – Fan Zheng
    Commented Dec 1, 2015 at 18:54

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You should look at Ambrosio's paper "Transport equation and Cauchy problem for BV vector fields", Invent. Math. 158 (2004), no. 2, 227–260 and at the references therein. The author is proving that bounded $BV$ vector fields with bounded divergence do have a flow, in an Eulerian sense, that is the IVP defined by the vector field has unique weak solutions (in $L^\infty$).

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