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Local existence in free interface problems with underlying second-order Stefan condition. (English) Zbl 1424.35363
This survey paper discusses the local in time existence and uniqueness of a solution for some free interface problems in which there is no specific condition on the velocity of the free interface, at least near some equilibrium. The authors revisit two models from combustion theory, namely a simple one-dimensional one-phase problem, and the near-equidiffusional flames system in the whole space. Then they investigate a general overdetermined problem in a domain \(\Omega_t\) of \(\mathbb{R}^N\) (\(N\geq 1\)) with its boundary \(\partial \Omega_t\) the free interface. The interface’s velocity is related to a combination of spatial derivatives up to the second-order, called second-order Stefan condition, and the problem is reformulated as a fully nonlinear problem.

35R35 Free boundary problems for PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35R37 Moving boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
80A25 Combustion