Editing Stefan problem (section)

== Premises to the mathematical description ==

From a mathematical point of view, the phases are merely regions in which the solutions of the underlying PDE are continuous and differentiable up to the order of the PDE. In physical problems such solutions represent properties of the medium for each phase. The moving boundaries (or [[Interface (chemistry)|interface]]s) are infinitesimally thin [[surface (mathematics)|surface]]s that separate adjacent phases; therefore, the solutions of the underlying PDE and its derivatives may suffer discontinuities across interfaces.

The underlying PDEs are not valid at the phase change interfaces; therefore, an additional condition—the '''Stefan condition'''—is needed to obtain [[Well-posed problem|closure]]. The Stefan condition expresses the local [[velocity]] of a moving boundary, as a function of quantities evaluated at either side of the phase boundary, and is usually derived from a physical constraint. In problems of [[heat transfer]] with phase change, for instance, [[conservation of energy]] dictates that the discontinuity of [[heat flux]] at the boundary must be accounted for by the rate of [[latent heat]] release (which is proportional to the local velocity of the interface).

The regularity of the equation has been studied mainly by [[Luis Caffarelli]]<ref>{{Cite journal|last=Caffarelli|first=Luis A.|date=1977|title=The regularity of free boundaries in higher dimensions|journal=Acta Mathematica|volume=139|issue=none|pages=155–184|doi=10.1007/BF02392236|s2cid=123660704|issn=0001-5962|doi-access=free}}</ref><ref>{{Cite journal|last=CAFFARELLI|first=LUIS A.|date=1978|title=Some Aspects of the One-Phase Stefan Problem|journal=Indiana University Mathematics Journal|volume=27|issue=1|pages=73–77|doi=10.1512/iumj.1978.27.27006|jstor=24891579|issn=0022-2518|doi-access=free}}</ref> and further refined by work of [[Alessio Figalli]], [[Xavier Ros-Oton]] and Joaquim Serra<ref>{{cite journal
 | last1 = Figalli | first1 = Alessio | author1-link = Alessio Figalli
 | last2 = Ros-Oton | first2 = Xavier | author2-link = Xavier Ros-Oton
 | last3 = Serra | first3 = Joaquim
 | arxiv = 2103.13379
 | doi = 10.1090/jams/1026
 | issue = 2
 | journal = [[Journal of the American Mathematical Society]]
 | mr = 4695505
 | pages = 305–389
 | title = The singular set in the Stefan problem
 | volume = 37
 | year = 2024}}</ref><ref>{{Cite web|last=Rorvig|first=Mordechai|date=2021-10-06|title=Mathematicians Prove Melting Ice Stays Smooth|url=https://www.quantamagazine.org/mathematicians-prove-melting-ice-stays-smooth-20211006/|access-date=2021-10-08|website=Quanta Magazine|language=en}}</ref>