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

2017 ◽  
Vol 27 (14) ◽  
pp. 2745-2780 ◽  
Author(s):  
Yulan Wang
Keyword(s):  
Weak Solution ◽  
Stokes System ◽  
Three Dimensional ◽  
Solution Concept ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Solvability ◽  
Navier Stokes ◽  
Fluid Equation ◽  
Free System

This paper deals with the Keller–Segel–Navier–Stokes system [Formula: see text] in a bounded domain [Formula: see text] with smooth boundary, where [Formula: see text] and [Formula: see text] are given functions. We shall develop a weak solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component [Formula: see text]. Under the assumption that there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] 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 [Formula: see text] 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 nonlinear convective term in the fluid equation.

Pull-back attractors for three-dimensional Navier—Stokes—Voigt equations in some unbounded domains

2013 ◽  
Vol 143 (2) ◽  
pp. 223-251 ◽  
Author(s):  
Cung The Anh ◽  
Pham Thi Trang
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Three Dimensional ◽  
Unbounded Domains ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Uniform Attractor ◽  
Finite Dimensional ◽  
Initial Boundary Value

We study the first initial–boundary-value problem for the three-dimensional non-autonomous Navier–Stokes–Voigt equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Faedo–Galerkin method. We then show the existence of a unique minimal finite-dimensional pull-back $\smash{\mathcal D_\sigma}$-attractor for the process associated with the problem, with respect to a large class of non-autonomous forcing terms. We also discuss relationships between the pull-back attractor, the uniform attractor and the global attractor.

scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

Attractors for the modifications of the three-dimensional Navier-Stokes equations

1994 ◽  
Vol 346 (1679) ◽  
pp. 173-190 ◽  
Keyword(s):  
Viscous Fluid ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Initial Boundary Value ◽  
Dimensional Domain

The modifications of the three-dimensional Navier-Stokes equations, which I suggested earlier for the description of viscous fluid flows with large gradients of velocities, are considered. It is proved that the first initial-boundary value problem for these equations in any bounded three-dimensional domain has a compact minimal global B-attractor. Some properties of the attractor are established.

scholarly journals Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties

2020 ◽  
Vol 30 (06) ◽  
pp. 1075-1117 ◽  
Author(s):  
Tobias Black
Keyword(s):  
Solution Concept ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Global Solvability ◽  
Generalized Solutions ◽  
Food Density ◽  
Nutrient Density ◽  
Regularity Properties ◽  
Eventual Regularity

In this paper, we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the forager group. Specifically, we will investigate an initial boundary-value problem for a prototypical forager–exploiter model of the form [Formula: see text] in a smoothly bounded domain [Formula: see text], where [Formula: see text], [Formula: see text] is nonnegative and the functions [Formula: see text] are assumed to satisfy [Formula: see text], [Formula: see text] as well as [Formula: see text] respectively, with constants [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text]. Assuming that [Formula: see text], [Formula: see text] and that [Formula: see text] satisfies certain structural conditions, we establish the global solvability of this system with respect to a suitable generalized solution concept and then, for the more restrictive case of [Formula: see text] and [Formula: see text], investigate an eventual regularity effect driven by the decay of the nutrient density [Formula: see text].

scholarly journals Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jiayi Han ◽  
Changchun Liu
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Stokes System ◽  
Three Dimensional ◽  
Global Weak Solution ◽  
Navier Stokes ◽  
Bounded Domains

<p style='text-indent:20px;'>We consider a two-species chemotaxis-Navier-Stokes system with <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian in three-dimensional smooth bounded domains. It is proved that for any <inline-formula><tex-math id="M3">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>, the problem admits a global weak solution.</p>

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.

Global solvability and eventual smoothness in a chemotaxis-fluid system with weak logistic-type degradation

2020 ◽  
Vol 30 (06) ◽  
pp. 1217-1252 ◽  
Author(s):  
Yulan Wang
Keyword(s):  
Weak Solution ◽  
Stokes System ◽  
Source Term ◽  
Dirichlet Boundary ◽  
Global Solvability ◽  
Navier Stokes ◽  
Initial Value ◽  
Flux Boundary Conditions ◽  
Bounded Smooth ◽  
Quadratic Case

We consider the coupled chemotaxis–Navier–Stokes system with logistic source term [Formula: see text] in a bounded, smooth domain [Formula: see text], where [Formula: see text] and where [Formula: see text], [Formula: see text] and [Formula: see text] are given parameters. Although the degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the initial-value problem for this system under no-flux boundary conditions for [Formula: see text] and [Formula: see text] and homogeneous Dirichlet boundary condition for [Formula: see text] possesses at least one globally defined weak solution. And this weak solution becomes smooth after some waiting time provided [Formula: see text].

scholarly journals Global Existence of the Cylindrically Symmetric Strong Solution to Compressible Navier-Stokes Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jian Liu ◽  
Ruxu Lian
Keyword(s):  
Strong Solution ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cylindrically Symmetric ◽  
Initial Boundary Value ◽  
Dependent Viscosity

This paper is concerned with the initial boundary value problem for the three-dimensional Navier-Stokes equations with density-dependent viscosity. The cylindrically symmetric strong solution is shown to exist globally in time and tend to the equilibrium state exponentially as time grows up.

The initial value problem for the non-homogeneous Navier—Stokes equations with general slip boundary condition

2000 ◽  
Vol 130 (4) ◽  
pp. 827-835 ◽  
Author(s):  
S. Itoh ◽  
A. Tani
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Solid Boundary ◽  
Dimensional Problem ◽  
Navier Stokes Equations ◽  
Slip Boundary

The initial-boundary value problem for the non-homogeneous Navier-Stokes equations including the slipping on the solid boundary is considered. The unique solvability is established locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.

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