reference request : "Solutions in the large for nonlinear hyperbolic systems of equations"

Question

I am studying the paper, "Solutions in the large for nonlinear hyperbolic systems of equations", and I'd like to know a few references for following questions :

This seems to be an important theorem, does this theorem he proved has got any famous name?
Is there a simplified version or more refined proof for this theorem? As it is in the paper it is very hard to follow.
Has anyone created a high dimensional version, with utmost importance to preserve the essence in the 1-d version?

Willie Wong · Accepted Answer · 2016-12-10 21:31:58Z

3

It is sometimes referred to as "Glimm's existence theorem". Though in some ways the proof is "more famous" than the theorem, and is frequently referred to as "Glimm's difference scheme".
Not that I know of. Glimm proves the existence of a specific class of weak solutions; the class is pretty tied to the scheme he used to its construction, and it is hard to imagine a significantly different way of obtaining the same result. If you are just looking for a different presentation: there's a shorter version of the proof in Hormander's Lectures on Nonlinear Hyperbolic Equations; there's also a longer version with lots of pictures in Constantine Dafermos' Hyperbolic conservation laws in continuum physics (basically the whole of Chapter 13 is devoted to this and extensions). There are also lots of other ones in various textbooks.
No. The method of Glimm relies on $L^1$ and $BV$ estimates; it is a result of Rauch (Comm. Math. Phys., 1986, 106, 481-484) that BV estimates usually fails for higher dimensional hyperbolic systems. In addition to this purely technical difficulty, there's also additional conceptual difficulties arising from the presence of dispersion in higher dimensional hyperbolic systems (one dimensional hyperbolic systems are not dispersive) and the possibilities of the presence of non-shock-type singularities. Some of these issues are discussed in the introduction section of this paper of my collaborators and me.

answered Dec 10, 2016 at 21:31

Willie Wong

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$\begingroup$ What I have been looking for! Is this a good starting point? (still in 1-D, but relativistic Euler equations!) math.ucdavis.edu/~temple/!!!PubsForWeb/cv33.pdf $\endgroup$
– Rajesh D
Commented Dec 11, 2016 at 7:24
$\begingroup$ Are there any attempts to prove the converse of Glimm existence theorem? $\endgroup$
– Rajesh D
Commented Dec 11, 2016 at 12:59
$\begingroup$ Re comment 1: Good starting point for what? Re commend 3: What do you mean by a "converse of Glimm existence theorem"? Theorems in analysis of PDEs generally have so many (implicit) assumptions in the hypotheses that the literal converse doesn't make much sense. $\endgroup$
– Willie Wong
Commented Dec 11, 2016 at 21:58
$\begingroup$ converse is a bit tricky. : I mean for a given pde, there is bound such that when the L1+BV of initial data is larger than that bound, then there is finite time blow up. $\endgroup$
– Rajesh D
Commented Dec 12, 2016 at 3:38
$\begingroup$ From internet search I found : "For genuinely nonlinear 2×2 systems, Glimm and Lax [38] proved that, if the initial data has small L∞ norm (but possibly large total variation), then cancellation effects dominate. Hence the Cauchy problem admits a weak solution with bounded variation for all times t > 0. An extension of these ideas to n × n systems can be found in [29]." $\endgroup$
– Rajesh D
Commented Dec 12, 2016 at 3:39

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