Small-data solutions of chemotaxis-fluid system with indirect signal production

2021 ◽  
pp. 125908
Author(s):  
Xi Liu ◽  
Yichen Zhang ◽  
Yongjie Han
Keyword(s):  
Small Data ◽  
Fluid System ◽  
Signal Production ◽  
Indirect Signal Production

scholarly journals Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

2020 ◽  
pp. 1-34
Author(s):  
CHRISTINA SURULESCU ◽  
MICHAEL WINKLER
Keyword(s):  
Blow Up ◽  
Initial Boundary ◽  
Tumour Microenvironment ◽  
Cancer Invasion ◽  
Two Dimensional ◽  
Classical Solvability ◽  
Differential Equation Models ◽  
Signal Production ◽  
Initial Boundary Value ◽  
Indirect Signal Production

We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Global solvability and asymptotic stabilization in a three-dimensional Keller–Segel–Navier–Stokes system with indirect signal production

2021 ◽  
pp. 1-73
Author(s):  
Feng Dai ◽  
Bin Liu
Keyword(s):  
Stokes System ◽  
Three Dimensional ◽  
Global Solvability ◽  
Navier Stokes ◽  
Homogeneous State ◽  
Direct Signal ◽  
Signal Production ◽  
Indirect Signal Production ◽  
Quadratic Case ◽  
Uniformly Bounded

This paper deals with the Keller–Segel–Navier–Stokes model with indirect signal production in a three-dimensional (3D) bounded domain with smooth boundary. When the logistic-type degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the associated no-flux/no-flux/no-flux/Dirichlet problem possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in [Formula: see text] with any [Formula: see text]. Moreover, under an explicit condition on the chemotactic sensitivity, these solutions are shown to stabilize toward the corresponding spatially homogeneous state in the sense of some suitable norms. We underline that the same results were established for the corresponding system with direct signal production in a well-known result if the degradation is quadratic. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the 3D Keller–Segel–Navier–Stokes system.

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