Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary

• Published in: Communications in partial differential equations Vol. 46; no. 6; pp. 1058 - 1091
• Main Authors: Wang, Yulan, Winkler, Michael, Xiang, Zhaoyin
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.06.2021; Taylor & Francis Ltd
• Subjects: Boundary conditions; Boundary value problems; Chemotaxis-fluid; Computational fluid dynamics; Dirichlet problem; energy estimates; generalized solution; Primary: 92C17; Qualitative analysis; Secondary: 35Q30; Smooth boundaries
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2020.1870236

The chemotaxis-Stokes system is considered in a bounded domain with smooth boundary. The corresponding solution theory is quite well-developed in the case when ( ) is accompanied by homogeneous boundary conditions of no-flux type for n and c, and of Dirichlet type for u. In such situations, namely, a quasi-Lyapunov structure provides regularity features sufficient to facilitate not only a basic existence theory, but also a comprehensive qualitative analysis. However, if in line with what is suggested by the modeling literature the boundary condition for the signal is changed so as to become with some constant then such structures apparently cease to be present at spatially global levels. The present work reveals that such properties persist at least in a weakened form of suitably localized variants, and on the basis of accordingly obtained a priori estimates it is shown that for widely arbitrary initial data an associated initial-boundary value problem for ( ) admits a globally defined generalized solution.