Reaction-Driven Relaxation in Three-Dimensional Keller–Segel–Navier–Stokes Interaction

The Keller–Segel–Navier–Stokes system nt+u·∇n=Δn-χ∇·(n∇c)+ρn-μn2,ct+u·∇c=Δc-c+n,ut+(u·∇)u=Δu+∇P+n∇ϕ+f(x,t),∇·u=0,(⋆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{rcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \chi \nabla \cdot (n\nabla c) + \rho n-\mu n^2,\\ c_t + u\cdot \nabla c &{}=&{} \Delta c-c+n, \\ u_t + (u\cdot \nabla )u &{}=&{} \Delta u + \nabla P + n \nabla \phi + f(x,t), \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$\end{document}is considered in a smoothly bounded convex domain Ω⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^3$$\end{document}, with ϕ∈W2,∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \in W^{2,\infty }(\Omega )$$\end{document} and f∈C1(Ω¯×[0,∞);R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^1(\bar{\Omega }\times [0,\infty );\mathbb {R}^3)$$\end{document}, and with χ>0,ρ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi >0, \rho \in \mathbb {R}$$\end{document} and μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >0$$\end{document}. As recent literature has shown, for all reasonably mild initial data a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem possesses a global generalized solution, but the knowledge on its regularity properties has not yet exceeded some information on fairly basic integrability features. The present study reveals that whenever ω>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega >0$$\end{document}, requiring that ρmin{μ,μ32+ω}0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =\eta (\omega )>0$$\end{document}, and that f satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical. Furthermore, under these hypotheses (⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\star $$\end{document}) is seen to admit an absorbing set with respect to the topology in L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega )$$\end{document}. By trivially applying to the case when μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >0$$\end{document} is arbitrary and ρ≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \le 0$$\end{document}, these results especially assert essentially unconditional statements on eventual regularity in taxis-reaction systems interacting with liquid environments, such as arising in contexts of models for broadcast spawning discussed in recent literature.


( )
is considered in a smoothly bounded convex domain ⊂ R 3 , with φ ∈ W 2,∞ ( ) and f ∈ C 1 (¯ × [0, ∞); R 3 ), and with χ > 0, ρ ∈ R and μ > 0. As recent literature has shown, for all reasonably mild initial data a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem possesses a global generalized solution, but the knowledge on its regularity properties has not yet exceeded some information on fairly basic integrability features. The present study reveals that whenever ω > 0, requiring that ρ min{μ, μ 3 2 +ω } < η with some η = η(ω) > 0, and that f satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical. Furthermore, under these hypotheses ( ) is seen to admit an absorbing set with respect to the topology in L ∞ ( ). By trivially applying to the case when μ > 0 is arbitrary and ρ ≤ 0, these results especially assert essentially unconditional statements on eventual regularity in taxis-reaction systems interacting with liquid environments, such as arising in contexts of models for broadcast spawning discussed in recent literature.

Introduction
Regularity issues form a central aspect in the literature concerned with the analysis of evolution systems accounting for taxis mechanisms. In fact, well-known findings on the occurrence of spontaneous singularity formation already in simple Keller-Segel type systems form quite unambiguous caveats which indicate that including chemotactic cross-diffusion as a model element may go along with substantial limitations of solution regularity ( [22,39,48,54]; cf. also the survey [34]). Corresponding mathematical questions naturally become yet more sophisticated when taxis processes are embedded into more intricate models, and understanding the singularity-supporting potential of chemotactic cross-diffusion in complex frameworks has accordingly attracted considerable attention in the past years ( [1,42]). The present work addresses this problem area in the context of a model for the interaction of a chemotactically active population with a liquid environment, as found to be of relevance not only in experimental setups involving populations of swimming bacteria ( [8,37,52]), but moreover also in descriptions of spatio-temporal evolution in processes of broadcast spawning during coral fertilization ( [7,25,26,38]). Specifically, we shall be concerned with the Keller-Segel-Navier-Stokes system ⎧ ⎨ ⎩ n t + u · ∇n = n − χ ∇ · (n∇c) + ρn − μn 2 , x ∈ , t > 0, c t + u · ∇c = c − c + n, x ∈ , t > 0, u t + (u · ∇)u = u + ∇ P + n∇φ + f (x, t), ∇ · u = 0, x ∈ , t > 0, (1.1) for an unknown population density n in an N -dimensional domain , and for a signal concentration c and an incompressible fluid represented through its velocity field u and an associated pressure P. By requiring χ to be positive, (1.1) models attractive tactic motion of individuals toward increasing signal concentrations, additionally affected by transport of both these quantities through the surrounding fluid which, in turn, is influenced not only by an external force f but also by cells through buoyancy. Under the assumptions μ > 0 and ρ ∈ R considered here, (1.1) moreover accounts for quadratic degradation in the population density, and hence both addresses chemotaxis-growth processes in which populations undergo natural logistic-type proliferation and death ( [18,42]), and also covers situations in which quadratic absorption, then mainly accompanied by the choice ρ = 0 or even ρ < 0, is due to the inclusion of reaction mechanisms ( [26]). With regard to solution regularity, the interplay of chemotactic cross-diffusion with such zero-order dissipation seems quite delicate, though yet far from completely understood, already in contexts of corresponding fluid-free Keller-Segel systems. Indeed, in the resulting version of (1.1) with u ≡ 0 any choice of μ > 0 is sufficient to suppress any blow-up phenomenon in two-dimensional initial value problems in the sense that for widely arbitrary initial data, global bounded solutions always exist ( [41,45]); in associated three-and higher-dimensional counterparts, however, similar findings on exclusion of explosions to date seem to rely on the stronger hypothesis that μ > μ 0 with some μ 0 = μ 0 ( ) > 0 ( [53,61]), while for small values of μ > 0 only some weak solutions are known to exist globally ( [32]). Although some studies concerned with simplified model variants have revealed some considerably strong singularity-counteracting effects of logistic damping in the sense of immediate regularization of strongly singular distributions ( [33,60]), not only results on possibly transient emergence of high population densities ( [24,56]), but moreover especially some detections of genuine blow-up both in high-dimensional systems with quadratic zero-order dissipation ( [12]), and in three-dimensional models involving some subquadratic but yet superlinear absorption ( [12]), indicate some persistence of taxis-driven destabilization also in the presence of such degradation mechanisms.
In light of these prerequisites, it seems far from surprising that the knowledge on corresponding issues in coupled chemotaxis-fluid systems of the form (1.1) is yet quite thin. After all, results on smooth global solvability could be established for the twodimensional version of (1.1) whenever μ is positive ( [10,50]), while a similar statement could be derived when N = 3 and μ ≥ 23 at least for a Stokes simplification of (1.1) in which the nonlinear convective term (u · ∇)u is neglected ( [49]). For the fully coupled three-dimensional Keller-Segel-Navier-Stokes system (1.1) with arbitrary μ > 0, however, merely a statement on global existence of certain generalized solutions seems available, asserting quite poor regularity properties only (see Proposition 1.1 and Definition 9.2 below). In this sense, (1.1) seems much less understood than its wellstudied relative ⎧ ⎨ ⎩ n t + u · ∇n = n − χ ∇ · (n∇c), x ∈ , t > 0, c t + u · ∇c = c − nc, x ∈ , t > 0, u t + (u · ∇)u = u + ∇ P + n∇φ, ∇ · u = 0, x ∈ , t > 0, (1.2) and logistic variants thereof, in which the decisive difference in signal evolution, here accounting for signal consumption through individuals rather than production as in (1.1), has facilitated energy-based analytical approaches to establish results not only on global existence of solutions in fairly natural frameworks of weak solvability ( [9,31,36,57]), but also on qualitative aspects such as eventual regularization and large-time stabilization toward homogeneous states ( [6,31,55,58]; see also [28] and [4] for an analysis of smalldata solutions).
In comparison to (1.2), the system (1.1) apparently lacks any similarly meaningful energy-like global dissipative features; in fact, the well-known gradient structure of the classical proliferation-free Keller-Segel system ( [40]) seems to disappear already when only one of the extra model elements in (1.1) is added to the latter, that is when either logistic contributions are included without any fluid interplay, or when, alternatively, a coupling to the (Navier-)Stokes equations is considered in the absence of such zeroorder terms. This can be viewed as indicating the possibility of dynamics considerably far from spatial homogeneity, and hence remarkably different from that in (1.2): Indeed, the combined action of self-enhancing chemoattraction with logistic proliferation is not only known to generate spatially structured equilibria ( [29]), but may beyond this bring about some quite colorful dynamical facets, as detected partially in the course of numerical simulations ( [19]), and partially even by means of rigorous analysis ( [24,56]). Apart from this, several findings in the recent analytical literature have revealed some nontrivial qualitative and even quantitative effects that fluid interaction may have on the solution behavior in various types of chemotaxis systems ( [20,[25][26][27]).
Main results. The purpose of the present study now consists in developing an approach that, despite the challenges accordingly resulting from a lack of favorable structural properties, is capable of identifying situations in which solutions to the fully coupled system (1.1) exhibit regular behavior at least eventually. To formulate this more precisely, let us assume henceforth that ⊂ R 3 be a bounded convex domain with smooth boundary, that χ > 0, ρ ∈ R and μ > 0, and that We shall then consider (1.1) along with the initial conditions (1.4) and under the boundary conditions ∂n ∂ν = ∂c ∂ν = 0 and u = 0 on∂ , (1.5) where our standing assumptions are that In order to appropriately recall from [59] a basic result on existence and approximation, for ε ∈ (0, 1) we furthermore introduce the regularized variant of (1.1), (1.4), (1.5) given by with the family (Y ε ) ε∈(0,1) of Yosida approximations determined by where here and below, A represents the Stokes operator under homogeneous Dirichlet boundary conditions in , with its respective realization in L p ( ; and with the corresponding fractional powers thereof denoted by A β = A β p , β ∈ R, in the sequel. Within this setting, the following result on global existence and approximation has been obtained in [59].
In line with the above discussion, in general it seems unclear how far regularity properties beyond those documented in (1.9) can be expected, especially in view of the fact that the considered coupling to the full three-dimensional Navier-Stokes system apparently limits possible extensions of knowledge on boundedness features in the fluidrelated part. The main results of the present manuscript now make sure that at least when the reproduction parameter ρ lies below a certain positive number, increasing with the degradation coefficient μ, all the solutions obtained in Proposition 1.1 become smooth and bounded after some individual relaxation time, and that in this case even a bounded absorbing set within a convenient topology can be identified: Theorem 1.2. Let ⊂ R 3 be a bounded convex domain with smooth boundary, and let χ > 0 and φ ∈ W 2,∞ ( ). Then for all ω > 0 there exist η = η(ω) > 0 and κ = κ(ω) > 0 with the following property: Suppose that ρ ∈ R, μ > 0 and f ∈ C 1 (¯ × [0, ∞); R 3 ) are such that ρ < η · min μ , μ as well as hold with some p > 3 2 and q > 2p 2p−3 , that n 0 , c 0 and u 0 satisfy (1. 6), and that (n, c, u) denotes the corresponding global generalized solution of (1.1), (1.4), (1.5) from Proposition 1.1. Then one can find t 0 = t 0 (ω, η, f, n 0 , c 0 , u 0 ) > 0 such that (1.13) and such that with some P ∈ C 1,0 (¯ ×[t 0 , ∞)), the quadruple (n, c, u, P) is a classical solution of (1.1), (1.5) in¯ × [t 0 , ∞). Moreover, under these assumptions on f the problem (1.1) possesses a bounded absorbing set in (L ∞ ( )) 5 in the sense that there exists C = C(ω, f ) > 0 such that any such solution satisfies (1.14) Remark. i) We emphasize that since (1.10) is trivially satisfied whenever ρ ≤ 0, the conclusion of Theorem 1.2 fully covers situations in which the considered population does not spontaeously proliferate, as naturally present in contexts merely involving reaction-like quadratic degradation in the zero-order part, such as in the modeling framework addressed in [25] and [26] to describe broadcast spawning. ii) Apart from those specified above, further possible dependencies on , and χ influence the choice of the quantities η, κ, t 0 and C in Theorem 1.2, as would, in a natural manner, the additional inclusion of non-normalized further system parameters such as diffusivities or rates of signal production and decay. In order to avoid further expansion of the already considerable technicalities in the arguments to be developed in the sequel, here and below we refrain from precisely tracking these explicitly.
iii) Already in the fluid-free case u ≡ 0 trivially included, Theorem 1.2 provides some progress in comparison with the existing literature: In contrast to the precedent finding on eventual smoothness in the corresponding taxis-only version of (1.1) stated in [32, Theorem 1.1], namely, the above result reveals ultimate regularity under an assumption which, besides being independent of the initial data, relates the required smallness condition on ρ to μ through (1.10) in an essentially explicit manner. iv) A natural problem arising in the interpretation of Theorem 1.2 appears to consist in providing information about the large time behavior of solutions which is more detailed than that contained in (1.14). In fact, a preliminary result in this direction asserts that under the explicit assumption that μ > ds → 0 as t → ∞, then also ess lim t→∞ u(·, t) L 2 ( ) = 0 ( [59]). Upon a straightforward interpolation, it evidently follows from Theorem 1.2 that if these requirements are fulfilled beyond those in (1.10)-(1.12), then actually for all p ∈ [1, ∞) we have (1. 15) apart from that, if even ρ = 0 then the results from [23] and [5] become directly applicable so as to assert the quantitative estimate for all t > t 0 (1. 16) with some appropriately large C > 0, and with t 0 > 0 as in Theorem 1.2 (cf. also [23] for a bound on sup t>t 0 (t + 1) ∇c(·, t) L p ( ) for arbitrary p ≥ 1). In more general parameter frameworks, and especially for positive ρ and small values of μ, however, in view of known analytical results on multiplicity in corresponding steady state problems ( [29]) and of simulation-based indications for the possibility of quite chaotic solution behavior ( [19]) we do not expect simple asymptotics as in (1.15) to prevail the dynamics in (1.1). v) Our approach will make essential use of the boundedness assumption on the physical domain ; in fact, inter alia due to a lack of comparison principles for (1.1) this requirement appears to be crucial already in the derivation of very basic boundedness features especially -but not exclusively -when ρ > 0 (cf. Sect. 2). We therefore have to leave open here the interesting question how far conclusions similar to those from Theorem 1.2 can be drawn in cases of unbounded domains, and particularly when = R 3 .

Strategy.
Our analysis can be viewed as being composed of a first part in which the action of chemotaxis is yet essentially faded out, and a second level in which full tribute is paid to the whole complexity of (1.1). In particular, basic eventual smallness properties of n ε , as obtained for suitably small ρ from mere integration and subsequent interpolation, will be summarized in Sect. 2 and thereafter used in Sect. 3 to derive corresponding smallness features of A β u ε with respect to the norm in L 2 ( ) for some β in the range ( 1 4 , ∞) throughout which D(A β 2 ) continuously embeds into L 3 ( ; R 3 ). By means of a standard zero-order testing procedure and an application of maximal Sobolev regularity theory to the second Eq. in (1.7), in Sect. 4 this will in provide some basic knowledge on eventual smallness of ∇c ε in an appropriately integrated sense.
The second stage of our argument will then be entered in Sect. 5, where by inter alia explicitly using the first Eq. in (1.7) for a second time, a coupled quantity of the form ψ(n ε − ρ + μ ) + |∇c ε | 2 p will be seen to enjoy some quasi-Lyapunov property for some p > 3 2 and a suitably designed function ψ on R satisfying s − p ψ(s) → 1 as s → +∞; we remark already here that only from this point on, dependencies of the obtained estimates on χ appear. In Sects. 6 and 7, the improved eventual smallness properties of ∇c ε thereby implied will be developed into L ∞ bounds for n ε and a Hölder estimate for u ε , whereupon standard regularity theories for parabolic and Stokes evolution problems become applicable so as to confirm the statement from Theorem 1.2 in Sect. 8.
Without explicit further mentioning, throughout the sequel we shall suppose that φ ∈ W 2,∞ ( ) is fixed, and given χ > 0, ρ ∈ R, μ > 0, f ∈ C 1 ( × [0, ∞); R 3 ) and initial data fulfilling (1.6), for ε ∈ (0, 1) we let (n ε , c ε , u ε , P ε ) denote the corresponding solution of (1.7) obtained in Proposition 1.1. Apart from that, let us announce already here that within each of our proofs, constants will be labeled as C 1 , C 2 , ..., and that in order to avoid an abundant globally consecutive numbering involving high indices, constants such as C 1 may attain different values in different proofs.

Basic Eventual Bounds for n ε
A starting point for our asymptotic analysis is formed by the following basic information on global and eventual bounds for the first solution component, as resulting from a simple integration of the first Eq. in (1.7) in a standard manner.
Through appropriate interpolation, the latter shows how the particular form of the assumption in (1.10) enters a further eventual boundedness property of n ε which involves topological information somewhat weaker than that in (2.4), but which on the other hand apparently yields genuine ultimate smallness, and which does so also in some cases when in Lemma 2.1 the expression ρ 0 μ 2 is large.

Smallness of u ε in D(
The purpose of this key section is to make sure that when applied to suitably small δ > 0, the bounds obtained in Lemma 2.2 provide sufficient eventual smallness properties of the coupling-induced contribution n ε ∇φ to the forcing term in the Navier-Stokes subsystem of (1.7), so as to warrant ultimate estimates for u ε with respect to the norm in D(A β 2 ) for some β exceeding the number 1 4 quite commonly encountered in regularity analysis of three-dimensional Navier-Stokes problems ( [46]).
As a means to appropriately derive upper estimates for functions satisfying linearly damped ODIs involving sources for which certain averaged bounds are known, from [59, Lemma 3.4] let us recall the following observation that will be referred to not only in this section, but also in Lemma 4.1 below. Then

3)
and in particular Through an analysis of the energy inequality associated with the approximate Navier-Stokes Eq. in (1.7), a first conclusion of Lemma 3.1 now asserts eventual smallness of a temporally averaged Dirichlet integral associated with the fluid velocity field, provided that the external force f satisfies a smallness assumption slightly weaker than that in (1.11).
Proceeding similarly and then using the Cauchy-Schwarz inequality, (3.25) and (3.26), we see that the convective term can be controlled according to for all t > 0. Since A β 0 and Y ε commute on e.g. D(A 2 ), and since it can easily be checked that Y ε ϕ L 2 ( ) ≤ ϕ L 2 ( ) for all ϕ ∈ L 2 σ ( ), by definition of T we herein have so that recalling the definition of C 9 , from (3.52) we obtain In conjunction with (3.49), (3.50) and (3.51), this shows that which in light of (3.30) implies that (3.44) and (3.45) combined with (3.42), (3.43) and the fact that κ ≤ κ 1 guarantee that Lemma 3.1 says that (3.53) implies the inequality which according to (3.48) and (3.40) in particular warrants that As therefore y(t) ≤ 1 8C 9 for all t ∈ (t ε , T ) due to (3.38), by continuity of y this firstly shows that indeed T cannot be finite, and thereupon we secondly conclude from (3.54) that by (3.32) and (3.39), and that by (3.29) and (3.37), By means of the Hölder inequality, for arbitrary r ∈ [3, 3 + θ 3 ] we hence obtain from the definition (3.36) of C 11 that because 0 ≤ 3+θ 3 −r (3+θ 3 )r ≤ θ 3 9 for any such r . Since t ε < t 3 + 1 = t 0 , this completes the proof.

Smallness of t+1
t ∇c ε (·, s) 2 L q ( ) ds for Some q > 3 We next address the taxis gradient as the quantity of apparently most crucial influence on regularity in the cross-diffusion interplay in (1.7), in this section aiming at the derivation of a spatio-temporal boundedness feature thereof. In a preliminary step toward this, we pursue a standard L 2 testing strategy for the equation determining c ε , hence obtaining some basic result on ultimate smallness which, apart from mere solenoidality, does not rely on any quantitative information about fluid regularity: , ρ ∈ R and μ > 0 are such that (1.10) holds with some η < η 4 , then one can pick t 0 = t 0 (δ, η, n 0 , c 0 ) > 0 such that and any ε ∈ (0, 1).
With this information at hand, we can appropriately control the lower-order contributions to the linear inhomogeneous heat equation, as satisfied by c ε , in the course of an estimation procedure based on maximal Sobolev regularity theory for the latter, thereby obtaining higher order and especially gradient estimates.

Smallness of ∇c ε in L 2 p ( ) for Some p > 3 2
In this section of key importance we shall next aim at deriving appropriate temporally uniform eventual smallness properties of n ε , and especially of ∇c ε , with respect to norms which can be viewed supercritical in the sense that their control will quite directly imply L ∞ bounds for n ε . Indeed, as seen in Lemma 6.1 below, the space L 3 ( ) will retain some threshold character with regard to taxis gradient regularity, quite elaborately analyzed in contexts of fluid-free Keller-Segel systems ( [3]), at least to a certain extent also in the present setting, and accordingly the main objective of this section will be to ultimately bound ∇c ε with respect to the norm in L 2 p ( ) for some p > 3 2 . This will be accomplished on the basis of the observation that if, in dependence of the parameter ω in (1.10) and (1.11) the number p > 3 2 is chosen suitably close to 3 2 , then for some appropriately constructed function ψ = ψ(s) on R vanishing at s = 0 and essentially growing like s p as s → +∞, the quantity plays the role of a quasi-entropy functional in the sense of satisfying a superlinearly forced ODI with an eventually small source (cf. (5.52)), and hence remaining conveniently controllable beyond times at which this functional is small. This conclusion, to be drawn in Lemma 5.4, will be prepared by Lemma 5.1 and Lemma 5.3 which separately describe the time evolution of the summands appearing in (5.1).
In order to prepare an appropriate control of the first summand on the right-hand side of (5.7) in the course of a testing procedure associated with the identity (5.5), let us note the following immediate consequence of the definitions in (5.3) and (5.4). ∈ (1, 2), γ ≥ 0 and ψ = ψ p,γ be defined by (5.3)
On splitting the latter integral and recalling the definition of ψ, we see that in the case γ > 0, for all t > 0 and ε ∈ (0, 1), which shows that regardless of the sign of ρ we have for all t > 0 and ε ∈ (0, 1), (5.23) where by Young's inequality we obtain C 1 = C 1 (δ, p) > 0 such that for all t > 0 and ε ∈ (0, 1).

Boundedness of n ε
Now thanks to the fact that in (5.30) we have 2p > 3, we may rely on known smoothing properties of the heat semigroup to assert eventual L ∞ bounds for n ε in the following sense.
Proof. All statements directly result from Lemma 8.1 upon an application of the Arzelà-Ascoli theorem and thereafter taking ε = ε j k → 0 in each of the expressions in the PDE system in (1.7) separately, finally constructing the associated pressure P by means of a standard procedure ( [46,51]).
The derivation of our main results hence reduces to suitably collecting the essence of the above: .
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Appendix: The Underlying Solution Concept
For completeness, we finally import from [59] the framework of generalized solvability that underlies the existence statement from Proposition 1.1. Here with regard to the requirements imposed on candidates for solutions, the main relaxation consists in considering the first sub-problem in (1.  and that ∇ · u ≡ 0 in D ( × (0, ∞)). Then we say that (n, c, u) is a global weaksubsolution (resp., a global weak -supersolution) of the first Eqs. in (1. holds for all nonnegative ϕ ∈ C ∞ 0 (¯ × [0, ∞)).