Timeline for reference request : "Solutions in the large for nonlinear hyperbolic systems of equations"
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 12, 2016 at 10:03 | comment | added | Rajesh D | But the hopes are still there for the scalar case $k=1$ which keeps me interested in BV estimates. Thanks for the discussion and also your paper throwing light on the subject in a crisp and short way. | |
| Dec 12, 2016 at 8:56 | comment | added | Rajesh D | I see Rauch paper dashes hopes for all the inviscid equations of compressible fluid dynamics. Wonder what the case for incompressible inviscid equations. | |
| Dec 12, 2016 at 3:51 | comment | added | Rajesh D | By starting point, this seems to be a base paper for going to higher dimensions. It is demonstrated that if initial data is small and with jump discontinuties, then shocks are formed in the solutions. But when we try to go to 2-dimensions, to begin with, how do you define a jump discontinuity in 2-dimensional initial data? request your comment on this attempt : fourierkingdom.wordpress.com/2016/03/06/… | |
| Dec 12, 2016 at 3:40 | comment | added | Rajesh D | ^ This seems to be a counter example for the converse. | |
| Dec 12, 2016 at 3:39 | comment | added | Rajesh D | 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]." | |
| Dec 12, 2016 at 3:38 | comment | added | Rajesh D | 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. | |
| Dec 11, 2016 at 21:58 | comment | added | Willie Wong | 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. | |
| Dec 11, 2016 at 12:59 | comment | added | Rajesh D | Are there any attempts to prove the converse of Glimm existence theorem? | |
| Dec 11, 2016 at 7:39 | vote | accept | Rajesh D | ||
| Dec 11, 2016 at 7:24 | comment | added | Rajesh D | 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 | |
| Dec 10, 2016 at 21:31 | history | answered | Willie Wong | CC BY-SA 3.0 |