References for well-posedness of weak solutions to Stefan problem Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)?
I wish to avoid classical theory because I prefer Sobolev spaces and the like.
Thank you
 A: There is a huge literature in free boundaries. Here I collected some papers (in no particular order) addressing different physical systems (coming mainly from fluid dynamics) and questions (so, not only well-posedness). Hopefully you find these papers interesting.
First of all a very nice, in my opinion, review: http://arxiv.org/pdf/1005.5329
1) Stefan
http://arxiv.org/pdf/1212.1422
http://arxiv.org/pdf/1112.5817
2) Water waves
http://arxiv.org/pdf/0910.2473
http://arxiv.org/pdf/1212.0632
http://arxiv.org/pdf/1305.4090
http://arxiv.org/pdf/0810.5340
http://arxiv.org/pdf/1106.2120
http://arxiv.org/pdf/1208.2726
http://arxiv.org/pdf/1201.4919
http://arxiv.org/pdf/1312.2917
http://arxiv.org/pdf/1401.1252
http://arxiv.org/pdf/1110.5155
http://arxiv.org/pdf/1005.4565
http://arxiv.org/pdf/math/0702015
3) Hele-Shaw & Muskat
http://arxiv.org/pdf/1102.1902
http://arxiv.org/pdf/0806.2258
http://arxiv.org/pdf/1311.7653
http://arxiv.org/pdf/1208.6213
http://arxiv.org/pdf/1303.1769
http://arxiv.org/pdf/1311.0430
4) Vortex sheet
http://arxiv.org/pdf/math/0502215
5) SQG
http://arxiv.org/pdf/math/0701447
A: A brief introduction can be found in M. Taylor's PDEs, vol. 3., Chapter 15.6
