Timeline for Open problems in PDEs, dynamical systems, mathematical physics
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2023 at 22:48 | comment | added | Sidharth Ghoshal | the link seems broken, heres a wayback machine link from 2022: web.archive.org/web/20220708215457/https://… | |
| Jun 8, 2015 at 0:40 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Mar 20, 2013 at 16:33 | comment | added | Terry Tao | Another major research program in this subject (which, for many equations, is actually more or less complete) is the task of establishing wellposedness for subcritical or critical equations (with the latter requiring either a defocusing equation or for the data to be smaller than the ground state in some sense, the latter assertion being known nowadays as the "threshold conjecture"). See e.g. web.math.princeton.edu/~seri/homepage/papers/telaviv.pdf | |
| Mar 20, 2013 at 15:48 | comment | added | Willie Wong | In the context of, say, nonlinear waves with power nonlinearities, (a) is the characterisation of ground states (b) is stability of ground states (see e.g. Nakanishi and Schlag's monograph Invariant Manifolds and ...; Walter Strauss's review of which is great for your purpose ams.org/journals/bull/2013-50-02/S0273-0979-2012-01389-7/… ) and (c) is the "soliton resolution conjecture". (Incidentally, for equations which do not admit bound states/stationary solutions, the analogue of "soliton resolution conjecture" is just the "global wellposedness/scattering".) | |
| Mar 20, 2013 at 15:41 | comment | added | Willie Wong | What Igor describes is one facet of a general theme in evolutionary PDEs nowadays. A lot of energy is now focussed on PDEs modelling attractive self-interaction (such as gravity). For these "focussing" type equations, it makes sense to ask (a) do there exist stationary solutions, and if so, are they unique? (b) are these stationary e solutions stable? (c) do these stationary solutions describe asymptotic behaviour? In the context of relativity, (a) is the "black hole uniqueness theorems", (b) is "stability of Kerr", and (c) is "final state conjecture". | |
| Mar 20, 2013 at 12:49 | history | edited | Igor Khavkine | CC BY-SA 3.0 |
added 1 characters in body
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| Mar 20, 2013 at 12:42 | history | answered | Igor Khavkine | CC BY-SA 3.0 |