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Uniqueness and existence results on the Hele-Shaw and the Stefan problems. (English) Zbl 1044.76019
Summary: We introduce a notion of viscosity solutions for the one-phase Hele-Shaw and Stefan problems when there is no surface tension. We prove the uniqueness and existence of the solutions for both problems, and establish the uniform convergence of solutions of porous medium equation related the to Hele-Shaw equation.

MSC:
76D27 Other free boundary flows; Hele-Shaw flows
76S05 Flows in porous media; filtration; seepage
80A22 Stefan problems, phase changes, etc.
35Q35 PDEs in connection with fluid mechanics
35Q72 Other PDE from mechanics (MSC2000)
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