Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?

In a bounded planar domain Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varOmega $\end{document} with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system {nt+∇⋅(nu)=Δn−∇⋅(n∇c),x∈Ω,t>0,0=Δc−c+n,x∈Ω,t>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} n_{t} + \nabla \cdot (nu) = \Delta n - \nabla \cdot (n\nabla c), & x\in \varOmega , \ t>0, \\ 0 = \Delta c -c+n, & x\in \varOmega , \ t>0, \end{array}\displaystyle \right . \end{aligned}$$ \end{document} is considered, where u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u$\end{document} is a given sufficiently smooth velocity field on Ω‾×[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {\varOmega }\times [0,\infty )$\end{document} that is tangential on ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\partial \varOmega $\end{document} but not necessarily solenoidal. It is firstly shown that for any choice of n0∈C0(Ω‾)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n_{0}\in C^{0}(\overline {\varOmega })$\end{document} with ∫Ωn0<4π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int _{\varOmega}n_{0}<4\pi $\end{document}, this problem admits a global classical solution with n(⋅,0)=n0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n(\cdot ,0)=n_{0}$\end{document}, and that this solution is even bounded whenever u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u$\end{document} is bounded and ∫Ωn0<2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int _{\varOmega}n_{0}<2\pi $\end{document}. Secondly, it is seen that for each m>4π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m>4\pi $\end{document} one can find a classical solution with ∫Ωn(⋅,0)=m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int _{\varOmega}n(\cdot ,0)=m$\end{document} which blows up in finite time, provided that Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varOmega $\end{document} satisfies a technical assumption requiring ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\partial \varOmega $\end{document} to contain a line segment. In particular, this indicates that the value 4π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$4\pi $\end{document} of the critical mass for the corresponding fluid-free Keller-Segel system is left unchanged by any fluid interaction of the considered type, thus marking a considerable contrast to a recent result revealing some fluid-induced increase of critical blow-up masses in a related Cauchy problem in the entire plane.


Introduction
Understanding possible effects of fluid interaction on chemotaxis systems has been the objective of considerable efforts in the mathematical literature during the past decade. Motivated by experimentally obtained results reporting significant influences of corresponding B M. Winkler michael.winkler@math.uni-paderborn.de 1 Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany transport mechanisms on the structure-enhancing potential of aggregation due to attractive cross-diffusion ( [23]), a noticeable literature has been concerned with various types of associated chemotaxis-fluid systems. Even in the most complex case in which, according to the modeling approach presented in [23], the fluid velocity itself is an unknown system variable according to buoyancy-induced feedback effects of cells on the fluid flow, beyond establishing basic solution theories ( [3,4,7,8,12,21,24,25,27,29,31]) it has been possible to address aspects related to qualitative solution behavior in some situations ( [4,5,14,28,30,32,33,35]).
While the latter class of findings seems yet limited to results identifying conditions under which spatial homogeneity ultimately prevails due to dominance of various dissipative mechanisms, somewhat deeper insight with regard to genuine structure formation could be gained upon renouncing any feedback of the considered population on the fluid evolution, thus considering the fluid flow as an externally given system ingredient. Examples in this direction address corresponding variants of classical Keller-Segel systems which in their fluid-free two-and higher-dimensional versions are known to exhibit blow-up phenomena due to the aggregation-enhancing interplay of chemotactic attraction to a signal produced by the cells themselves ( [1,10,17,19]). The results reported in [11], for instance, indicate that in two-and three-dimensional cases, any such explosion can be suppressed by an appropriately chosen, and hence data-dependent, incompressible fluid velocity field.
Even a class of very simple and explicit fluid fields has recently been found to substantially influence the critical mass phenomenon known as the probably most striking characteristic feature of the Cauchy problem for the unperturbed parabolic-elliptic Keller-Segel system in the whole plane: Namely, it has been shown in [9] that for each m ∈ (0, 16π) there exist A > 0 and some initial data n 0 fulfilling possesses a globally defined smooth solution. As the corresponding fluid-free analogue with u ≡ 0 is well-known to allow for global solutions only when R 2 n 0 ≤ 8π , whereas any choice of reasonably regular n 0 with R 2 n 0 > 8π enforces finite-time blow-up of the associated solution ( [19,20]), this demonstrates that fluid transport in fact can increase the value of the critical mass in this Cauchy problem, which appears to be in quite good accordance with the predictions formulated as conclusions from the numerical simulations in [16]. The intention of the present work is to show that this considerably changes when the considered physical region is a bounded domain Ω ⊂ R 2 with smooth boundary, rather than the entire plane. For this purpose, let us recall that in the case u ≡ 0, the Neumann initialboundary value problem still exhibits a critical mass phenomenon with respect to finite-time blow-up, but that this slightly differs from the above in its nature: It is well-known, namely, that whenever n 0 is sufficiently regular with Ω n 0 < m c := 4π , then (1.2) admits a global classical solution, while for any m > 4π it is possible to find at least some smooth n 0 such that Ω n 0 = m, but that (1.2) possesses a solution blowing up in finite time with respect to the spatial L ∞ norm of the component n ( [1,17,20]). Indeed, a role of m c equally strict to that of the number 8π in the context of (1.1) cannot be expected in (1.2) due to the presence of the constant steady states (n, c) ≡ ( m |Ω| , m |Ω| ) at arbitrary mass levels m > 0. As we shall see below, in this slightly modified form the criticality of m c = 4π remains untouched when allowing for widely arbitrary u in (1.2), not even requiring solenoidality. To substantiate this, throughout the sequel we shall assume that and that Then the first of our main results shows that any such fluid interaction cannot decrease the critical mass in the sense described above: and that n > 0 and c > 0 in Ω × (0, ∞).
All these solutions are even bounded whenever u is bounded and Ω n 0 < 2π : for some r > 0, and that u satisfies (1.3). Then for all m > 4π one can find μ(m) > 0 with the property that whenever n 0 complies with (1.4) and is such that Ω n 0 = m and and for which we have We remark that at the cost of additional technical efforts based on the refined analysis in [20,Chap. 5] it is possible to remove the restriction (1.7) on ∂Ω; in order to keep the presentation conveniently simple, however, we refrain from detailing this here.

Local Existence and Upper Bounds for the Increase of Energy
To begin with, let us state a basic result on local existence and extensibility that can be obtained by adapting standard arguments from the existence theories of parabolic-elliptic Keller-Segel type systems to the present situation; we may therefore refrain from giving details here, and rather refer to the literature (see [13,15] or [6], for instance).

Moreover, this solution has the property that
3) The following observation provides some information on how the evolution of the natural Lyapunov functional associated with the unperturbed Keller-Segel system is influenced by the presence of a fluid flow. Here a control of the corresponding additional contribution will be achieved by making appropriate use of the dissipation rate functional Ω | ∇n √ n − √ n∇c| 2 which, due to its complicated coupling of both solution components, is only trivially estimated in most places in the literature.

Deriving L ∞ Estimates from Bounds in L log L
In order to substantiate our goals in connection with Theorem 1.1 and Theorem 1.2, let us next perform a variant of a standard bootstrap procedure to make sure that similar to the situation in the original parabolic-elliptic Keller-Segel system, also in the presence of a suitably regular fluid flow a supposedly available bound for Ω n ln n already implies a corresponding L ∞ estimate.
Proof Proceeding in a standard manner, we first use n 3 as a test function in the first equation from (1.2) to see that since ∂n we may next invoke well-known smoothing estimates for the Neumann heat semigroup (e t ) t≥0 on Ω ( [26]) to see that with some C 5 > 0, once more due to (2.6) we have

Two Functional Inequalities Resulting from the Moser-Trudinger Inequality
Now in subsequently deriving estimates for Ω n ln n on the basis of Lemma 2.2, we shall rely on the following consequence of the Moser-Trudinger inequality observed in [ where ϕ := 1 |Ω| Ω ϕ.
As a first and quite well-known consequence thereof, as usual we can make sure that the crucial quantity Ω n ln n is essentially dominated by the energy functional from (2.4) whenever Ω n 0 < 4π (cf. also [18,Lemma 3.4]). (1.3) and (1.4) with Ω n 0 < 4π . Then there exists C > 0 such that
As documented in [34, Lemma 2.3], Lemma 2.4 furthermore entails the following functional inequality, to be used in Lemma 4.1, which solely involves a single function.

Lemma 2.6
Let 0 ≡ ϕ ∈ C 0 (Ω) be nonnegative. Then for any choice of ε > 0, so that In light of Corollary 2.5 and Lemma 2.1, this would imply that with some C 2 > 0, Ω n ln n ≤ C 2 for all t ∈ (0, T max ), so that Lemma 2.3 would apply so as to provide C 3 > 0 fulfilling which however contradicts (2.2).
We have thereby already established the first of our main results: In order to next address the boundedness property claimed in Theorem 1.2, let us now make use of the functional inequality from Lemma 2.6 in discovering a second quasi-dissipative structure under the more restrictive assumption that Ω n 0 < 2π . Here unlike our analysis of (2.5), our subsequent exploitation of (4.1) will exclusively refer to the dissipation rate Ω n ln n appearing therein.
Proof We use (1.2) and integrate by parts to see that for all t > 0, where another integration by parts shows that Ω n (n + 1) 2 ∇n · ∇c = Ω ∇ ln(n + 1) because − c ≤ n by (1.2). Now since our hypotheses warrant that m := Ω n 0 satisfies m < 2π , we can pick ε ∈ (0, 1) suitably small such that C 1 := (1−ε)·2π m − 1 is positive, and apply Lemma 2.6 along with (2.3) to obtain C 2 > 0 such that so that from (4.3) and (4.2) we infer that for all t > 0. As, by Young's inequality, for all t > 0, in view of the assumed boundedness property of u this establishes (4.1).
Indeed, this implies a spatio-temporal L log L estimate for n: Proof This directly follows on integrating (4.1) in time and using that 0 ≤ ln(n + 1) ≤ Ω n = Ω n 0 for all t > 0 by (2.3).
As a consequence, under such smallness conditions the energy from (2.4) is bounded in its temporal average in the following sense.
Proof Relying on (1.5), from Lemma 2.2 we infer the existence of C 1 > 0 such that whereas Lemma 4.3 asserts that t+1 t F (s)ds ≤ C 2 for all t > 0 with some C 2 > 0. Therefore, given any t > 1 we can find t (t) ∈ (t − 1, t) such that F (t (t)) ≤ C 2 , and that thus, by (4.7), indeed Again thanks to Corollary 2.5 and Lemma 2.3, this implies our main result on boundedness under the assumption that Ω n 0 < 2π : Finally concerned with the blow-up result announced in Theorem 1.3, we shall see that the present framework involving a given fluid flow in fact allows for an appropriate adaptation of the classical argument from [17] based on the analysis of functionals that can be viewed as certain localized variants of second moments. In order to keep our presentation compact here, let us import from [17] two preparatory observations, the first of which summarizes some elementary features of said localization procedure. (5.1) Proof Noting that ϕ(r) := − r 1 , is nonincreasing with ϕ(r 1 ) = 1 and ϕ(r 2 ) = r 1 r 2 ∈ (0, 1), we readily verify the inequalities Φ(x) ≤ |x| 2 and Φ(x) ≥ 0 for all x ∈ R 2 . The observations concerning regularity and the estimate |∇Φ| ≤ 2 √ Φ have been documented in [17, p. 41] already.
We next recall some information on the behavior of the diffusive and cross-diffusive contributions to the first equation in (1.2) when tested against one particular among these functions, with the latter being chosen in such a way that, inter alia, its center of symmetry is located at the line segment of the boundary of Ω addressed in the hypothesis (1.7) from Theorem 1.3. As a consequence of this and an adequate estimation of the respective contribution due to the fluid field, the evolution of the corresponding moment-type functional can be described as follows. (1.4) hold, the function y ∈ C 0 ([0, T max )) ∩ C 1 ((0, T max )) defined by letting has the property that for any choice of ε > 0,