Well-posedness for chemotaxis–fluid models in arbitrary dimensions*The author acknowledges the support of the DAAD through the program 'Graduate School Scholarship Programme, 2018' (Number 57395813) and the Hausdorff Centre for Mathematics at Bonn. (1st December 2022)

• Record Type: Journal Article
• Authors: Yomgne Diebou, Gael
• Is Part Of: Nonlinearity. Volume 35:Number 12(2022)
• Page Start: 6241
• Page End: 6283
• Publication Date: 2022-12-01
• Subjects: chemotaxis–Navier–Stokes equations -- Campanato space -- Keller–Segel model -- Carleson measures -- double chemotaxis system -- mass conservation -- nonnegativity preservation; 92C17 -- 35Q35 -- 35K55 -- 35A02 -- 42B35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac98ec
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We study the Cauchy problem for the chemotaxis Navier–Stokes equations and the Keller–Segel–Navier–Stokes system. Local-in-time and global-in-time solutions satisfying fundamental properties such as mass conservation and nonnegativity preservation are constructed for low regularity data in 2 and higher dimensions under suitable conditions. Our initial data classes involve a new scale of function space, that is L 2, N − 2 − 1 ( R N ) which collects divergence of vector-fields with components in the square Campanato space L 2, N − 2 ( R N ), N > 2 (and can be identified with the homogeneous Besov space B ˙ 22 − 1 ( R N ) when N = 2) and are shown to be optimal in a certain sense. Moreover, uniqueness criterion for global solutions is obtained under certain limiting conditions.