Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion

By Qingshan Zhang, Yuxiang Li. 2015

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We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model $ \quad n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\chi(c)\nabla c), $ $ \quad c_t+u\cdot\nabla c=\Delta c-nf(c), $ $ \quad u_t+\kappa(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\Phi, $ $ \quad \nabla\cdot u=0, $ in a bounded convex domain $\Omega\subset\mathbb{R}^3$. It is proved that if $m\geq\frac{2}{3}$, $\kappa\in\mathbb{R}$, $0<\chi\in C^2([0,\infty))$, $0\leq f\in C^1([0,\infty))$ with $f(0)=0$ and $\Phi\in W^{1,\infty}(\Omega)$, then for sufficiently smooth initial data $(n_0, c_0, u_0)$ the model possesses at least one global weak solution.

Subfields: Cell Biology, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/88e3c36c4a23a163c76190c3cee6961de12f59c1

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Citations in the corpus

A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization (2019, background): “For such models, numerous results have been achieved in the past few years, addressing topics ranging from the mere existence of global solutions ([13], [73], [3], [6]), also in the context of nonlinear cell diffusion ([14], [55], [60], [9], [77]), to global boundedness ([66], [5], [24], [4]) and ev”
Long-term behaviour in a chemotaxis-fluid system with logistic source (2016, background): “Other variants of the model that are commonly treated include nonlinear (porous medium type) diffusion of bacteria, where ∆n is replaced by ∆n for some m > 1 (see [34, 35, 10, 7, 49]), thereby improving chances for finding bounded solutions, or, exchanging χ∇ · (n∇c) for ∇ · (nS(n, c, x)∇c), more co”
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”
The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system (2017, unknown): “2) in the linear case D ≡ 1 [40], but also in presence of nonlinear diffusion of the form D(n) = nm−1, n > 0, form ≥ (2)3 [44]; in the case D ≡ 1 these solutions become smooth and classical after a certain time T > 0 [41], but possibly irregular behavior prior to this relaxation time can not yet be ”
Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities (2016, background): “1) with large initial data has been demonstrated for bounded three-dimensional domains in [54], see also [67] for even milder diffusion effects, followed by studies of the long-term behaviour of any such “eventual energy solution” [55], which, namely, become smooth on some interval [T,∞) and uniform”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “5) with κ = 1 for any m > 1 (see also [7,9,24,28,29,31,45] and references therein).”