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

By Yulan Wang, Michael Winkler, Zhaoyin Xiang. Communications in Partial Differential Equations, 2021

64 citations recorded by the source corpus.

Abstract 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.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/7e83ef49cd59365f5f07d9055d68e44f23db6008

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Citations in the corpus

Depleting the signal: Analysis of chemotaxis‐consumption models—A survey (2023, methodology): “In [134], a localized energy functional (cutting off the functions near the boundary) served as key ingredient in the construction of global generalized solutions to the Dirichlet variant of the system, whereas [66] relied on the de facto one-dimensionality of said problem in a radially symmetric se”