This paper deals with the Keller–Segel–Navier–Stokes system nt + u ⋅∇n = Δn −∇⋅ (nS(x,n,c) ⋅∇c), ct + u ⋅∇c = Δc − c + n, ut + (u ⋅∇)u = Δu −∇P + n∇ϕ,∇⋅ u = 0, in a bounded domain Ω ⊂ ℝ3 with smooth boundary, where ϕ ∈ W2,∞(Ω) and S ∈ C2(Ω × [0,∞)2; ℝ3×3) are given functions. We shall develop a weak solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component n. Under the assumption that there exist S0 > 0 and α > 1 3 such that |S(x,n,c)|≤ S0 ⋅ (1 + n)−αfor all x ∈Ω, n ≥ 0 and c ≥ 0, it is finally shown that for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined weak solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on α here is optimal. This result extends previous studies on global solvability for this system in the two-dimensional domain and for the associated chemotaxis-Stokes system obtained on neglecting the nonlinea...

Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with subcritical sensitivity

In a bounded domain $$\Omega \subset \mathbb {R}^2$$
 , we consider the the chemotaxis-Stokes system $$\begin{aligned} \left\{ \begin{array}{ll} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot \Big (nS(x,n,c) \cdot \nabla c \Big ), \qquad &{} x\in \Omega , \ t>0,\\ c_t + u\cdot \nabla c = \Delta c - nf(c), \qquad &{} x\in \Omega , \ t>0,\\ u_t = \Delta u + \nabla P + n\nabla \phi , \qquad \nabla \cdot u=0, \qquad &{} x\in \Omega , \ t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
 which arises as a model for populations of aerobic bacteria swimming in a sessile water drop. In accordance with refined modeling approaches, we do not necessarily assume the chemotactic sensitivity S herein to be a scalar function, but rather allow S to attain values in $$\mathbb {R}^{2\times 2}$$
 . As compared to the well-studied case of scalar-valued sensitivities in which an analysis can be based on favorable energy-type inequalities, this modification brings about significant new challenges which require to adequately cope with only little a priori information on regularity of solutions of (
 $$\star $$
 ). The present work creates a functional setup which despite this allows for the construction of certain global mass-preserving generalized solutions to an associated initial-boundary value problem in planar convex domains with smooth boundary, provided that the initial data and the parameter functions S, f and $$\phi $$
 are sufficiently smooth, and that S is bounded and f is nonnegative with $$f(0)=0$$
 .

Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components

An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion

PDE systems describing chemotaxis, the directed motion of organisms in response to a chemical signal, contain a cross-diffusive term which in many cases causes the unavailability of strong regularity information. An important part of their mathematical analysis is thus concerned with their behavior in situations where solutions are known to blow-up or where singularities cannot be excluded a priori. In this note we review some results, as well as some underlying fundamental analytical ideas, from the context of rigorous blow-up detection, and discuss some approaches addressing the design of solution theories which are able to adequately cope with the possible destabilizing effects of chemotactic cross-diffusion.

Facing Low Regularity in Chemotaxis Systems

Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions

In this paper, we investigate the 3D Keller–Segel–Stokes (K–S–S) system with nonlinear diffusion term $$\Delta n^{m}$$
 (
 $$m>0$$
 ) and rotational flux posed in a bounded domain $$\Omega $$
 with smooth boundary Under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity S(x, n, c) satisfies $$|S(x,n,c)|\le C_{S}(1+n)^{-\alpha }$$
 , by seeking some new functionals and using the bootstrap arguments on the regularized system, we establish the existence and boundedness of global weak solutions to K–S–S system for arbitrarily large initial data under the assumption $$m+2\alpha >2$$
 and $$m>\frac{3}{4}$$
 , which includes both the degenerate $$(m>1)$$
 and the singular $$(m<1)$$
 case

Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux

In this paper, we investigate the global existence of solutions to a coupled chemotaxis-fluids system in a three-dimensional unbounded domain with boundary In the chemotaxis-Navier–Stokes case, we

Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary

In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid: \begin{equation*} \left\{ \begin{split} \partial_t n + \mathbf{u}\cdot\nabla n&= \Delta n - \nabla\cdot\left(\chi(c)n\nabla c\right) + \nabla\cdot(n\nabla\phi), \\ \partial_t c + \mathbf{u}\cdot\nabla c&= \Delta c - nf(c), \\ \partial_t\mathbf{u} + \kappa(\mathbf{u}\cdot\nabla)\mathbf{u} + \nabla P&= \Delta\mathbf{u} - n\nabla\phi + \chi(c)n\nabla c, \\ \nabla\cdot\mathbf{u}&= 0 \end{split} \right. \end{equation*} in $\mathbb{R}^d\times(0,T)\, (d=2,3)$. One of the novelties and difficulties here is that the coupling in this model is stronger and more nonlinear than the most-studied chemotaxis-fluid model. We will first establish several extensibility criteria of classical solutions, which ensure us to extend the local solutions to global ones in the three dimensional chemotaxis-Stokes case and in the two dimensional chemotaxis-Navier-Stokes version under suitable smallness assumption on $\|c_0\|_{L^{\infty}}$ with the help of a new entropy functional inequality. Some further decay estimates are also obtained under some suitable growth restriction on the potential $\nabla \phi$ at infinity. As a byproduct of the entropy functional inequality, we also establish the global-in-time existence of weak solutions to the three dimensional chemotaxis-Navier-Stokes system. To the best of our knowledge, this seems to be the first work addressing the global well-posedness and decay property of solutions to the Cauchy problem of self-consistent chemotaxis-fluid system.

Yingping Peng

Papers

Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux

Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions

Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary

Global existence and decay rates to self-consistent chemotaxis-fluid system