Abstract
We consider the chemotaxis-fluid system
in a bounded convex domain \(\Omega \subset {\mathbb {R}}^3\) with smooth boundary, where \(\phi \in W^{1,\infty }(\Omega )\) and D, f and S are given functions with values in \([0,\infty ), [0,\infty )\) and \({\mathbb {R}}^{3\times 3}\), respectively. In the existing literature, the derivation of results on global existence and qualitative behavior essentially relies on the use of energy-type functionals which seem to be available only in special situations, necessarily requiring the matrix-valued S to actually reduce to a scalar function of c which, along with f, in addition should satisfy certain quite restrictive structural conditions. The present work presents a novel a priori estimation method which allows for removing any such additional hypothesis: besides appropriate smoothness assumptions, in this paper it is only required that f is locally bounded in \([0,\infty )\), that S is bounded in \(\Omega \times [0,\infty )^2\), and that \(D(n)\ge k_{D}n^{m-1}\) for all \(n\ge 0\) with some \(k_{D}>0\) and some
It is shown that then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a globally defined weak solution. The method introduced here is efficient enough to moreover provide global boundedness of all solutions thereby obtained in that, inter alia, \(n\in L^\infty (\Omega \times (0,\infty ))\). Building on this boundedness property, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium \((\overline{n_0},0,0)\) in an appropriate sense, where \(\overline{n_0}:=\frac{1}{|\Omega |} \int _{\Omega }n_0\), provided that merely \(n_0\not \equiv 0\) and \(f>0\) on \((0,\infty )\). To the best of our knowledge, these are the first results on boundedness and asymptotics of large-data solutions in a three-dimensional chemotaxis-fluid system of type (0.1).
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The author would like to thank the anonymous reviewer for numerous helpful comments and suggestions.
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Communicated by Y. Giga.
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Winkler, M. Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. Calc. Var. 54, 3789–3828 (2015). https://doi.org/10.1007/s00526-015-0922-2
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DOI: https://doi.org/10.1007/s00526-015-0922-2