RESEARCH IN PAIRS
Keller-Segel Fluid Systems on Non-Smooth Domains (2578)
Systèmes de fluides de Keller-Segel sur les domaines non lisses
Dates: 21-25 February 2022
Place: CIRM (Marseille Luminy, France)
DESCRIPTION
State of Research
​The Keller-Segel model  for  chemotaxis  was  first  introduced  by Keller  and  Segel  in  1970.  It  describes the movement of cells in response to chemical gradients. There are many results concerning local, global existence as well as blow-up of solutions; we refer here only to the survey articles [3] and the book [7] by
B.  Perthame.
It is very interesting to study such chemotaxis models not only by diffusion mechanism but also to include transport phenomena by a fluid, in which the cells are immersed. This leads to coupled chemotaxis-fluid models, see e.g. [4] for Navier-Stokes type models. It is also very interesting to consider quasilinear Keller-segel models with degenerate diffusion.
The existing analysis of models of the above type is mostly restricted to the case of domains with smooth boundaries. The described diffusion processes in fluids take often place, however, in domains with edges and corners, or more generally in convex or Lipschitz domains.
This is where the Research in Pairs stays comes into its own: combining the expertise of H. Kozono (Tokyo) working on Keller-Segel models and fluid dynamics [4] with the ones of S. Monniaux (Marseille) [6] [5] and P. Tolksdorf (Mainz) [8] on the analysis of elliptic, parabolic and Navier-Stokes type problems on Lipschitz domains important steps towards the understanding of Keller-Segel-fluid models in convex or Lipschitz domains is expected. In particular, we think here of
•  an extension of the iteration scheme for Navier-Stokes equations on Lipschitz domains given in [8] to the coupled Keller-Segel-Fluid model within the L3-setting.
•  results on periodic solutions extending recent approaches in [2] and [1] to Keller-Segel-Fluid models.
References
[1] M. Hieber, N. Kashiwara, K. Kress, P. Tolksdorf, The periodic version of the Da Prato-Grisvard theorem and applications to the bidomain equations with FitzHugh-Nagumo transort. Ann. Math. Pura Appl., to appear, DOI 10.1007/s10231-020-00975-6.
[2] M. Hieber, Ch. Stinner, Strong time periodic solutions to Keller-Segel systems: an approach by the quasilinear Arendt-Bu theorem. J. Diff. Equ., to appear, DOI 10.106/j.jde.2020.01.020.
[3] T. Hillen, K. Painter, A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009), 183-217.
[4] H. Kozono, M. Miura, Y. Sugiyama, Time global existence and finite blow-up criterion for solutions to the Keller-Segel system coupled with the Navier-Stokes fluid. J. Diff. Equ., 267 (2019), 5410-5492.
[5] M. Mitrea, S. Monniaux, M. Wright, The Stokes operator with Neumann booundary conditions on Lipschitz domains J. Math. Sci. 176 (2011) 409-417.
[6] S. Monniaux, Z. Shen, Stokes problems in irregular domains with various boundary conditions. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 207-240, Springer, 2018.
[7] B. Perthame, Parabolic Equations in Biology. Springer LN Math.Model. in Life Sciences, Springer, 2015.   [8] P. Tolksdorf. On the Lp-theory of the Navier-Stokes equations on three dimensional Lipschitz domains. Math.
Ann., 371:445-460, 2018.
PARTICIPANTS
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SPONSORS OF THIS SEMESTER (to be completed)