Vol. 9, No. 1, 2016

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Global regularity of chemotaxis equations with advection

Saad Khan, Jay Johnson, Elliot Cartee and Yao Yao

Vol. 9 (2016), No. 1, 119–131

We study the Patlak–Keller–Segel (PKS) equations in 2D that describe chemotaxis with an additional advection term. We show that solutions are globally regular for smooth initial data with subcritical mass as long as the flow has nonpositive divergence. For initial data with supercritical mass, numerical simulations suggest that blow-up might be prevented by imposing some strong incompressible advection term.

chemotaxis, partial differential equations, PDE, symmetric decreasing rearrangement, advection, Keller–Segel
Mathematical Subject Classification 2010
Primary: 35A01
Received: 25 August 2014
Revised: 8 January 2015
Accepted: 9 January 2015
Published: 17 December 2015

Communicated by Behrouz Emamizadeh
Saad Khan
Department of Mathematics
New York University
New York, NY 10012-1185
United States
Jay Johnson
Department of Mechanical Engineering
University of Texas at Austin
Austin, TX 78712
United States
Elliot Cartee
Department of Mathematics
Cornell University
Ithaca, NY 14850
United States
Yao Yao
Department of Mathematics
University of Wisconsin-Madison
Madison, WI 53703
United States