Local smoothing effect and existence for the one-phase Hele–Shaw problem with zero surface tension

Abstract

We study an initial value problem for the one-phase Hele–Shaw problem with zero surface tension. We establish local well-posedness for the initial value problem in Sobolev space. Furthermore, we obtain that, on average in time, the solution gains 1/2 derivative of smoothness in spatial variable compared to the initial data.

Keywords:

• unsteady Hele–Shaw problem
• Sobolev space, initial value problem
• well-posedness

AMS Subject Classifications::

• 35R35
• 76N05
• 80A22

Acknowledgement

The author thanks to Professor Robert Gilbert, for his encouragement and advice over the years.