On the well-posedness of incompressible flow in porous media with supercritical diffusion

Abstract

In this article we study the heat transfer equation with a supercritical diffusion term of an incompressible fluid in porous media governed by Darcy's law. We obtain the global well-posedness for small initial data belonging to critical Besov spaces and the local well-posedness for arbitrary initial data. We further show the pointwise blowup criterion.

Keywords:

• porous media
• well-posedness
• Besov space
• blowup criterion

AMS Subject Classifications::

• 76S05
• 76D03

Acknowledgements

The author would like to thank Prof. Changxing Miao for helpful comments and suggestions. The author is also grateful to the referees for valuable advice.