RESEARCH IN PAIRS
KellerSegel Fluid Systems on NonSmooth Domains (2578)
Systèmes de fluides de KellerSegel sur les domaines non lisses
Dates: 2125 February 2022
Place: CIRM (Marseille Luminy, France)
KellerSegel Fluid Systems on NonSmooth Domains (2578)
Systèmes de fluides de KellerSegel sur les domaines non lisses
Dates: 2125 February 2022
Place: CIRM (Marseille Luminy, France)
DESCRIPTION
State of Research
The KellerSegel 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 blowup 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 chemotaxisfluid models, see e.g. [4] for NavierStokes type models. It is also very interesting to consider quasilinear Kellersegel 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 KellerSegel 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 NavierStokes type problems on Lipschitz domains important steps towards the understanding of KellerSegelfluid models in convex or Lipschitz domains is expected. In particular, we think here of • an extension of the iteration scheme for NavierStokes equations on Lipschitz domains given in [8] to the coupled KellerSegelFluid model within the L3setting. • results on periodic solutions extending recent approaches in [2] and [1] to KellerSegelFluid models. References
[1] M. Hieber, N. Kashiwara, K. Kress, P. Tolksdorf, The periodic version of the Da PratoGrisvard theorem and applications to the bidomain equations with FitzHughNagumo transort. Ann. Math. Pura Appl., to appear, DOI 10.1007/s10231020009756. [2] M. Hieber, Ch. Stinner, Strong time periodic solutions to KellerSegel systems: an approach by the quasilinear ArendtBu 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), 183217. [4] H. Kozono, M. Miura, Y. Sugiyama, Time global existence and finite blowup criterion for solutions to the KellerSegel system coupled with the NavierStokes fluid. J. Diff. Equ., 267 (2019), 54105492. [5] M. Mitrea, S. Monniaux, M. Wright, The Stokes operator with Neumann booundary conditions on Lipschitz domains J. Math. Sci. 176 (2011) 409417. [6] S. Monniaux, Z. Shen, Stokes problems in irregular domains with various boundary conditions. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 207240, Springer, 2018. [7] B. Perthame, Parabolic Equations in Biology. Springer LN Math.Model. in Life Sciences, Springer, 2015. [8] P. Tolksdorf. On the Lptheory of the NavierStokes equations on three dimensional Lipschitz domains. Math. Ann., 371:445460, 2018. 
PARTICIPANTS

SPONSORS OF THIS SEMESTER (to be completed)