Global classical solutions in a chemotaxis(-Navier)-Stokes system with indirect signal production

2021 ◽  
pp. 125299
Author(s):  
Ying Dong ◽  
Yingping Peng
Keyword(s):  
Stokes System ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Global Classical Solutions ◽  
Signal Production ◽  
Indirect Signal Production

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.

Singular limit of a dispersive Navier–Stokes system with an entropy structure

2014 ◽  
Vol 13 (01) ◽  
pp. 77-99
Author(s):  
C. David Levermore ◽  
Weiran Sun ◽  
Konstantina Trivisa
Keyword(s):  
Stokes System ◽  
Singular Limit ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Third Order ◽  
Fluid System ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Whole Space ◽  
Entropy Structure

We prove a low Mach number limit for a dispersive fluid system [3] which contains third-order corrections to the compressible Navier–Stokes. We show that the classical solutions to this system in the whole space ℝn converge to classical solutions to ghost-effect systems [7]. Our analysis follows the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and to simplify the analysis in [4].

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