Global Solvability and Eventual Smoothness in a Two-Species Chemotaxis–Navier–Stokes System with Lotka–Volterra Type Competitive Kinetics

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].

In this work, we study global existence, eventual smoothness and large time behavior of positive solutions for the following two-species chemotaxis consumption model:$ \left\{ \begin{array}{lll} &u_t = \Delta u-\chi_1\nabla \cdot ( u\nabla w), &\quad x\in \Omega, t>0, \\[0.2cm] & v_t = \Delta v-\chi_2\nabla \cdot (v\nabla w), &\quad x\in \Omega, t>0, \\[0.2cm] & w_t = \Delta w -(\alpha u+\beta v)w, &\quad x\in \Omega, t>0, \end{array}\right. $in a bounded and smooth domain $ \Omega\subset \mathbb{R}^n (n = 2,3,4,5) $ with nonnegative initial data $ u_0, v_0, w_0 $ and homogeneous Neumann boundary data. Here, the parameters $ \chi_1,\chi_2 $ are positive and $ \alpha,\beta $ are nonnegative.In such setup, for all reasonably regular initial data and for all parameters, we show global existence and uniform-in-time boundedness of classical solutions in 2D, global existence of weak solutions in $ n $D $ (n = 3,4,5) $, and, finally, we show eventual smoothness and uniform convergence of global weak solutions in $ 3 $D convex domains. Our 2D boundedness removes a smallness condition required in [50] and other findings improve and extend the existing knowledge about one-species chemotaxis-consumption models in the literature.

Eventual smoothness and stabilization in a three-dimensional Keller–Segel–Navier–Stokes system with rotational flux

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.

We consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form for parameter choices kappa ge 0 and mu >0. Herein, we moreover impose a nonnegative and time-constant prescribed concentration c_star in C^2({overline{Omega }}) for the signal chemical on the boundary of the domain Omega subset {mathbb {R}}^{mathcal {N}} with {mathcal {N}}in {2,3}. After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of c_star being also constant in space. We show that sufficiently strong logistic influence, in the sense that for omega >0 and mu _0>0 there is some eta =eta (omega ,mu _0,c_star )>0 with the property that whenever μ0≤μandκmin{μ,μN+66+ω}<η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mu _0\\le \\mu \\quad \ ext {and}\\quad \\frac{\\kappa }{\\min \\{\\mu ,\\mu ^{\\frac{{\\mathcal {N}}+6}{6}+\\omega }\\}}<\\eta \\end{aligned}$$\\end{document}are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on omega ,mu _0,eta ,c_star and the initial data.

In this paper, we consider a Keller-Segel-Navier–Stokes system involving subquadratic logistic degradation: nt+u·∇n=Δn-∇·(n∇c)+ρn-μnα,ct+u·∇c=Δc-c+n,ut+(u·∇)u=Δu+∇P+n∇ϕ,∇·u=0in a three-dimensional smoothly bounded domain along with reasonably mild initial conditions and no-flux/no-flux/Dirichlet boundary conditions, where ρ∈R and μ>00$$\\end{document}]]>. The purpose of the present work is to firstly establish the generalized solvability for the model under the subquadratic exponent restriction α≥43, which indicates that persistent Dirac-type singularities can be ruled out, and to secondly exhibit the eventual smoothness of these solutions under the stronger restriction α>53 \\frac{5}{3}$$\\end{document}]]> whenever ρ is not too large in the sense of (ρ++1)α-1ρ+≤δ0μα,(ρ++1)min{1,α-1}ρ+max{1,3-α}≤δ0μ2,ρ+≤δ0μfor some δ0=δ0(α)>00$$\\end{document}]]>. These results especially extend the precedent works due to Winkler (J Functional Anal 276: 1339-1401, 2019; Comm Math Phys 367: 439–489, 2022), where, among other things, the corresponding studies focus on the case α=2 of quadratic degradation.

Eventual smoothness in a chemotaxis-Navier–Stokes system with indirect signal production involving Dirichlet signal boundary condition

Eventual smoothness and asymptotic stabilization in a two-dimensional logarithmic chemotaxis-Navier–Stokes system with nutrient-supported proliferation and signal consumption

The following chemotaxis-fluid system $ \begin{equation*} \left\{ \begin{split} &amp;n_t = \Delta n-\chi\nabla\cdot\left(n\nabla{c}\right)+nc, &amp;\;\;x\in \Omega, t&gt;0, \\ &amp; c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &amp;\;\;x\in \Omega, t&gt;0, \\ &amp;{\bf u}_t+\kappa({\bf u}\cdot\nabla){\bf u}+\nabla{P} = \Delta{ {\bf u}}+n\nabla\phi , &amp; \;\;x\in \Omega, \, t&gt;0, \\ &amp;\nabla\cdot {\bf u} = 0, &amp;\;\; x\in \Omega, \, t&gt;0\ \end{split} \right. \end{equation*} $ is considered under homogeneous Neumann boundary conditions in a bounded convex domain $ \Omega\subset \mathbb{R}^d (d\in \{2, 3\}) $ with smooth boundary. For $ d = 2 $, it is shown that the corresponding chemotaxis-Navier-Stokes system possesses a globally bounded classical solution which stabilizes toward a spatially homogeneous equilibrium in the sense that$ \begin{equation*} \begin{split} n(\cdot, t)\rightarrow n_\infty, \quad c(\cdot, t)\rightarrow 0 \text{ and } {\bf u}(\cdot, t)\rightarrow {\bf 0} \text{ in } L^\infty(\Omega) \end{split} \end{equation*} $as $ t\rightarrow \infty $, where $ n_\infty $ is a constant satisfying $ n_\infty\geq\frac{1}{|\Omega|}\int_{\Omega}n_0 $. For $ d = 3 $, it is seen that the corresponding chemotaxis-Stokes system possesses a globally defined weak solution.

We consider the global classical solvability of a mixed initial-boundary value problem for semilinear hyperbolic systems with nonlinear reaction of Lotka-Volterra type. The reaction nonlinearity is not globally Lipschitz in L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , but has Lipschitz properties depending on an L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sup> -norm bound. We reformulate the problem in an abstract setting as a modified Cauchy problem with homogeneous boundary conditions and solve it based on Banach contraction mapping theorem. Based on Sobolev and Moser-type inequalities we prove regularity of the local solutions in Sobolev spaces. We show that global existence of classical solutions holds if a uniform a-priori bound on the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sup> -norm of the solution and boundary term exists.

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.

The time-dependent Navier–Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a unique solution with prescribed fluxes over cross-sections of outlets to infinity which tends in each outlet to the corresponding time-dependent Poiseuille flow. The obtained results are proved for arbitrary large norms of the data (in particular, for arbitrary fluxes) and globally in time.

In this work, we consider the two-species chemotaxis system with Lotka–Volterra competitive kinetics in a bounded domain with smooth boundary. We construct weak solutions and prove that they become smooth after some waiting time. In addition, the asymptotic behavior of the solutions is studied. Our results generalize some well-known results in the literature.

Global existence and stabilization in a two-dimensional chemotaxis-Navier-Stokes system with consumption and production of chemosignals

We consider the chemotaxis-Navier–Stokes system with gradient-dependent flux limitation and nonlinear production: [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in a bounded domain [Formula: see text], where the flux limitation function [Formula: see text] and the signal production function [Formula: see text] generalize the prototypes [Formula: see text] and [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. For the linear production case of [Formula: see text], the global boundedness of solutions has been verified in the related literature for [Formula: see text]. In this paper, we expand to prove that the corresponding initial-boundary value problem possesses a unique globally bounded solution if [Formula: see text] for [Formula: see text], or if [Formula: see text] for [Formula: see text], which shows that when [Formula: see text], that is, the self-enhancement ability of chemoattractant is weak, the solutions still remain globally bounded even though the flux limitation is relaxed to permit proper [Formula: see text]; however, if [Formula: see text], it is necessary to impose the stronger flux limitation than that in the case [Formula: see text] to inhibit the possible finite-time blow-up. This seems to be the first result on the global solvability in the chemotaxis-Navier–Stokes model with nonlinear production.