How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?

By Michael Winkler. 2015

290 citations recorded by the source corpus.

The chemotaxis-Navier-Stokes system    nt + u · ∇n = ∆n−∇ · (nχ(c)∇c), ct + u · ∇c = ∆c− nf(c), ut + (u · ∇)u = ∆u+∇P + n∇Φ, ∇ · u = 0, (⋆) (0.1) is considered under boundary conditions of homogeneous Neumann type for n and c, and Dirichlet type for u, in a bounded convex domain Ω ⊂ R with smooth boundary, where Φ ∈ W (Ω) and χ and f are sufficiently smooth given functions generalizing the prototypes χ ≡ const. and f(s) = s for s ≥ 0. It is known that for all suitably regular initial data n0, c0 and u0 satisfying 0 6≡ n0 ≥ 0, c0 ≥ 0 and ∇ · u0 = 0, a corresponding initial-boundary value problem admits at least one global weak solution which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. The present paper shows that after some relaxation time, this solution enjoys further regularity properties and thereby complies with the concept of eventual energy solutions which is newly introduced here, and which inter alia requires that two quasi-dissipative inequalities are ultimately satisfied. Moreover, it is shown that actually for any such eventual energy solution (n, c, u) there exists a waiting time T0 ∈ (0,∞) with the property that (n, c, u) is smooth in Ω× [T0,∞), and that n(x, t) → n0, c(x, t) → 0 and u(x, t) → 0 hold as t → ∞, uniformly with respect to x ∈ Ω. This resembles a classical result on the three-dimensional Navier-Stokes system, asserting eventual smoothness of arbitrary weak solutions thereof which additionally fulfill the associated natural energy inequality. In consequence, our results inter alia indicate that under the considered boundary conditions, the possibly destabilizing action of chemotactic cross-diffusion in (⋆) does not substantially affect the regularity properties of the fluid flow at least on large time scales.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/dcb6d5855a59585695892b44db84423e2bc5e18a

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Citations in the corpus

Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary (2021, background): “1) it is at least possible to construct global weak solutions [11] which eventually become smooth and classical [12].”
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, background): “2) has been identified to be sufficient for solutions to become eventually smooth and to approach spatially homogeneous equilibria in several cases ([60], [32]).”
Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D (2017, background): “In particular, for the fluid free case eventual smoothness of solutions was shown in [27] for N = 3 and a result including fluid is contained in [40], where certain weak eventual energy solutions are considered.”
Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components (2018, methodology): “Combined with appropriate regularization properties of the fluid flow, these may be used to derive a priori estimates which may form the starting point for global existence theories (see [10], [49], [39]) and also [29], for instance) as well as for the description of the large time asymptotics of so”