scholarly journals On totally global solvability of controlled second kind operator equation

2020 ◽  
Vol 30 (1) ◽  
pp. 92-111
Author(s):  
A.V. Chernov
Keyword(s):  
Operator Equation ◽  
Stokes System ◽  
Sufficient Conditions ◽  
Global Solvability ◽  
Functional Integral ◽  
Navier Stokes ◽  
Volterra Type ◽  
Comparison System ◽  
Functional Integral Equations ◽  
Evolutionary Operator

We consider the nonlinear evolutionary operator equation of the second kind as follows $\varphi=\mathcal{F}\bigl[f[u]\varphi\bigr]$, $\varphi\in W[0;T]\subset L_q\bigl([0;T];X\bigr)$, with Volterra type operators $\mathcal{F}\colon L_p\bigl([0;\tau];Y\bigr)\to W[0;T]$, $f[u]$: $W[0;T]\to L_p\bigl([0;T];Y\bigr)$ of the general form, a control $u\in\mathcal{D}$ and arbitrary Banach spaces $X$, $Y$. For this equation we prove theorems on solution uniqueness and sufficient conditions for totally (with respect to set $\mathcal{D}$) global solvability. Under natural hypotheses associated with pointwise in $t\in[0;T]$ estimates the conclusion on univalent totally global solvability is made provided global solvability for a comparison system which is some system of functional integral equations (it could be replaced by a system of equations of analogous type, and in some cases, of ordinary differential equations) with respect to unknown functions $[0;T]\to\mathbb{R}$. As an example we establish sufficient conditions of univalent totally global solvability for a controlled nonlinear nonstationary Navier-Stokes system.

scholarly journals On Attractivity and Positivity of Solutions for Functional Integral Equations of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Xianyong Huang ◽  
Junfei Cao
Keyword(s):  
Banach Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Functional Integral ◽  
Measures Of Noncompactness ◽  
Positivity Of Solutions ◽  
Unbounded Intervals ◽  
Functional Integral Equations

We investigate a class of functional integral equations of fractional order given byx(t)=q(t)+f1(t,x(α1(t)),x(α2(t)))+(f2(t,x(β1(t)),x(β2(t)))/Γ(α))×∫0t(t−s)α−1f3(t,s,x(γ1(s)),x(γ2(s)))ds: sufficient conditions for the existence, global attractivity, and ultimate positivity of solutions of the equations are derived. The main tools include the techniques of measures of noncompactness and a recent measure theoretic fixed point theorem of Dhage. Our investigations are placed in the Banach space of continuous and bounded real-valued functions defined on unbounded intervals. Moreover, two examples are given to illustrate our results.

scholarly journals Global attractivity for Volterra type Hadamard fractional integral equations in Fréchet spaces

2018 ◽  
Vol 51 (1) ◽  
pp. 131-140
Author(s):  
Saïd Abbas ◽  
Ravi P. Agarwal ◽  
Mouffak Benchohra ◽  
Farida Berhoun
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Fractional Integral ◽  
Global Attractivity ◽  
Functional Integral ◽  
Fréchet Spaces ◽  
Volterra Type ◽  
Functional Integral Equations ◽  
Hadamard Fractional Integral

Abstract In this paper, we present some results concerning the existence and the attractivity of solutions for some functional integral equations of Hadamard fractional order. We use an extension of the Burton-Kirk fixed point theorem in Fréchet spaces.

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 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.

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 On the Existence of the Solutions for Some Nonlinear Volterra Integral Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
İsmet Özdemir ◽  
Ümit Çakan ◽  
Bekir İlhan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Measure Of Noncompactness ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Functional Integral ◽  
Space Of Continuous Functions ◽  
Nonlinear Functional ◽  
Functional Integral Equations

We present a theorem which gives sufficient conditions for existence of at least one solution for some nonlinear functional integral equations in the space of continuous functions on the interval[0,a]. To do this, we will use Darbo's fixed-point theorem associated with the measure of noncompactness. We give also an example satisfying the conditions of our main theorem but not satisfying the conditions described by Maleknejad et al. (2009).

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