Chemotaxis-fluid systems with logarithmic sensitivity and slow consumption global generalized solutions and eventual smoothness

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arXiv:2211.01019v1 [math.AP] 2 Nov 2022

Chemotaxis(-ﬂuid) systems with logarithmic sensitivity

and slow consumption: global generalized solutions and

eventual smoothness

Mario Fuest∗

Leibniz Universität Hannover,

Institut für Angewandte Mathematik,

Welfengarten 1, 30167 Hannover, Germany

Abstract

We consider the system







nt+u· ∇n= ∆n−χ∇ · (n

c∇c),

ct+u· ∇c= ∆c−nf(c),

ut+ (u· ∇)u= ∆u+∇P+n∇φ, ∇ · u= 0,

in smooth bounded domains Ω ⊂RN,N∈N, for given f≥0, φand complemented with initial and

homogeneous Neumann–Neumann–Dirichlet boundary conditions, which models aerobic bacteria in a ﬂuid

drop. We assume f(0) = 0 and f′(0) = 0, that is, that fdecays slower than linearly near 0, and construct

global generalized solutions provided that either N= 2 or N > 2 and no ﬂuid is present.

If additionally N= 2, we next prove that this solution eventually becomes smooth and stabilizes in the

large-time limit. We emphasize that these results require smallness neither of χnor of the initial data.

Key words: chemotaxis, ﬂuid, logarithmic sensitivity, generalized solutions, eventual smoothness

AMS Classiﬁcation (2020): 35K55 (primary); 35B40, 35B65, 35D99, 35Q35, 92C17 (secondary)

1 Introduction

This article is concerned with the chemotaxis–Navier–Stokes system with logarithmic sensitivity











nt+u· ∇n= ∆n−χ∇ · (n

c∇c) in Ω ×(0,∞),

ct+u· ∇c= ∆c−nf(c) in Ω ×(0,∞),

ut+ (u· ∇)u= ∆u+∇P+n∇φ, ∇ · u= 0 in Ω ×(0,∞),

∂νn=∂νc= 0, u = 0 on ∂Ω×(0,∞),

(n, c, u)(·,0) = (n0, c0, u0) in Ω,

(1.1)

∗e-mail: fuest@ifam.uni-hannover.de, corresponding author

variants of which have been proposed in [32] to model the behavior of aerobic bacteria interacting with a ﬂuid by

means of transportation and buoyancy. Apart from random motion (term ∆n) the bacteria with density nmay

also partially orient their movement towards higher concentrations of oxygen with density c(term −χ∇·(n

c∇c),

where χ > 0 is given), a mechanism called chemotaxis. The oxygen diﬀuses (term ∆c) and is consumed by

the bacteria (term −nf (c), where fis a given nonnegative function) and both the bacteria and the oxygen

are transported by a ﬂuid with velocity ﬁeld u(terms +u· ∇nand +u· ∇c), which in turn is given by the

Navier–Stokes equation with an inhomogeneity accounting for buoyancy eﬀects, i.e. the observation that water

with bacteria is heavier than without (term +n∇φ, where φis a given gravitational potential). The modelling

in [32] is motivated by experiments performed in [7]; for an alternative derivation of such systems based on a

multiscale approach see [1].

The mathematical analysis of such models has ﬁrst focussed on variants with linear taxis sensitivity, i.e. with

−χ∇·(n

c∇c) in the ﬁrst equation replaced by −χ∇·(n∇c), for which global classical [4] and weak solutions [8, 36]

have been constructed for small and large data, respectively. Beyond that, many more results on chemotaxis–

ﬂuid systems (for instance also covering situations where the signal is produced rather than consumed) are

available; we refer to [2, Section 4.6] for a recent overview.

The reason behind considering a logarithmic sensitivity in (1.1) is to account for the so-called Weber–Fechner

law of stimulus perception, an idea already present in earlier models by Keller and Segel [18] (see also [29] and

[15, Section 2.2]).

From a mathematical point of view, the logarithmic sensitivity poses two main challenges: First, it destroys

the (quasi-) energy structure crucially made use of for instance in [8] and [36] inter alia for linear sensitivity

functions as cf(c) cannot be concave near c= 0 for nonnegative smooth fwith f(0) = 0 (cf. the discussion in

[8, p. 1638]). Second, the consumption term in the second equation in (1.1) may force cto become very small,

which means that the factor 1

cin the taxis term may become very large. In fact, even local-in-time positive

lower bounds for cappear to be unavailable without relying on (generally not available) upper bounds for n—in

contrast to logarithmic chemotaxis systems with signal production where even time-independent lower bounds

have been derived, see for instance [10, Lemma 2.2].

Even without ﬂuid interaction, global classical solutions of (1.1) have only been constructed under smallness

conditions [35] or for related, more regular systems (in [20] with nonlinear diﬀusion enhancement, in [23] with

saturated taxis sensitivity, and in [21] with weaker consumption terms, i.e. for −nf(c) replaced by −nβcfor

some β∈(0,1)).

Main result I: Global generalized solutions. The lack of unconditional global existence results for classical

solutions is not entirely surprising. Systems including cross-diﬀusive terms such as −χ∇·(n

c∇c) in (1.1) generally

have quite low regularity properties. Indeed, the probably most prominent representative of such models, the

classical Keller–Segel system

(nt= ∆n− ∇ · (n∇c),

ct= ∆c−c+n

admits classical solutions blowing up in ﬁnite time in both two- [14, 26] and higher-dimensional [37] settings—as

do many variants of this system, see for instance [28] for blow-up in a parabolic–elliptic system with logarithmic

sensitivity and the recent survey [22] for further examples. This makes it necessary to consider weaker solution

concepts when studying global solvability of cross-diﬀusive systems and while sometimes switching to usual

weak formulations suﬃces, one often needs to consider even more general concepts; see [9, Section 1.2] for a

more thorough discussion of various notions of solvability.

Such generalized solutions have also been obtained for (1.1) with f(c) = cboth in the two-dimensional setting

with ﬂuid ([24]; see also the precedents [33] for the chemotaxis–Stokes system, i.e. (1.1) without the nonlinear

convection term (u· ∇)uin the ﬂuid equation, and [40] for the ﬂuid-free case) and higher dimensional ﬂuid-free

radially symmetric settings ([43]).

Our ﬁrst main result expands on these ﬁndings and states that (1.1) possesses global generalized solutions if

either N= 2 or if N≥3 and there is no ﬂuid present, and if fdoes not only fulﬁll

f∈C1([0,∞)) with f(0) = 0 and f > 0 in (0,∞) (1.2)

but also

f′(0) = 0.(1.3)

By (1.2), the condition (1.3) is equivalent to limsց0f(s)

s= 0; that is, (1.3) requires superlinear decay of fnear

0, which reﬂects that oxygen is consumed more slowly if nearly none is left. We remark that (1.3) is not fulﬁlled

for the most typical choice f(c) = cbut, for instance, for f(c) = c2.

Theorem 1.1. Let Ω⊂RN,N≥2, be a smooth, bounded domain, χ > 0and φ∈W2,∞(Ω), and assume

(1.2),(1.3) and











n0∈L1(Ω) with n0>0a.e. and ln n0∈L1(Ω),

c0∈L∞(Ω) with c0≥δfor some δ > 0,

u0∈L2(Ω; RN)with ∇ · u0= 0 in D′(Ω).

(1.4)

N= 2 or (u0≡0and φ≡0),(1.5)

then there exists a global generalized solution (n, c, u)of (1.1) in the sense of Deﬁnition 3.1 below which has the

property that u≡0if u0≡0and φ≡0.

The main novelty of Theorem 1.1 is the ﬂuid-free higher dimensional case. Indeed, for χ= 1, more regular initial

data and f(c) = cthe two-dimensional setting with ﬂuid has already been treated in [24] and the challenges

caused by large χ, less regular initial data and more general fare relatively limited. (In fact, the main reason

to include the two-dimensional existence result here is so that we can refer to it in Theorem 1.2 below.)

On the other hand, new ideas are necessary for N≥3. While quasi-energy functionals quickly give a priori

bounds for (weighted) gradients of all components of solutions to approximative systems (see Lemma 3.5 and

Lemma 3.6), they are only strong enough to imply uniform integrability of nεf(cε) in the two-dimensional

setting (where such an estimate follows for instance from Heihoﬀ’s inequality [13, (1.2)]). However, such an

a priori estimate turns out to be crucial for undertaking the limit processes in the weak formulations for the

ﬁrst and second solution components; see the discussion at the beginning of Subsection 3.4 for a more thorough

discussion of this point. One way to obtain uniform integrability of nε(and hence also of nεf(cε)) is to modify

the system by adding a superlinear degradation term to the ﬁrst equation in (1.1). This observation stands as

the core of [9, Section 7], where recently global generalized solution for the corresponding problem have been

constructed (cf. also the precedent [19] for quadratic dampening terms).

However, as we aim to prove a global existence result for (1.1) without any dampening terms and also for

N≥3, these ideas alone are evidently insuﬃcient for our purpose. Fortunately, we can adapt an idea recently

introduced in [16]: The crucial a priori estimate, uniform boundedness of RT

0RΩnεf(cε)|ln(nεf(cε))|, follows

eventually from considering the time evolution of the functional RΩ(ln nε)cε, see Lemma 3.9 and the discussion

directly preceding that lemma.

Additionally relying on (1.3), we are then also able to favourably bound ∇ln cεon sets where cεis small (cf.

Lemma 3.11), which allows us to obtain strong convergence of ∇ln cεin Lemma 3.12. Combined, these bounds

and convergence properties then make it possible to pass to the limit in each term of the weak formulations for

the approximate problems, allowing us to prove Theorem 1.1 in Subsection 3.5.

Main result II: Eventual smoothness and stabilization in the two-dimensional setting. A natural next step

is to analyze the behavior of the solution given by Theorem 1.1 for large times with respect to both eventual

regularization and convergence in the large-time limit. These two points are actually related: Both bounds

in comparatively strong topologies and smallness of certain quantities may be key in favourably estimating

worrisome terms in energy functionals, which, depending on the functional, in turn may imply stronger a priori

estimates or convergence (see also the discussion after Theorem 1.2 below).

Indeed, for certain relatives of (1.1) with scalar nonsingular taxis sensitivity, both convergence to homoge-

neous steady states (see for instance [17, 38] for the two-dimensional as well as [4, 5] for the small-data three-

dimensional setting and [41] for results regarding weak solutions) and eventual smoothness properties (see e.g.

[30] for the ﬂuid-free three-dimensional case and [34] for a three-dimensional chemotaxis–Navier–Stokes system

with superlinear degradation) have been shown. Moreover, for tensor-valued taxis sensitivities, global general-

ized solutions have been constructed in planar domains ([12]; see also [39] for a precedent dealing with a Stokes

ﬂuid), whose large-time and eventual regularity properties have been analysed in [13].

Regarding chemotaxis-ﬂuid systems with logarithmic sensitivity, the results appear to be limited to two-

dimensional settings: For Stokes ﬂuids, [33] shows convergence towards homogeneous steady states while [3]

asserts eventual smoothness provided RΩn0is suﬃciently small. Moreover, under a similar smallness condition,

both convergence and eventual smoothness are obtained in [24] for the full Navier–Stokes equation.

Our second main result is then able to give an aﬃrmative answer to the question whether similar relaxation

properties are also exhibited by the global generalized solution (potentially with large mass) constructed in

Theorem 1.1.

Theorem 1.2. Suppose in addition to the assumption of Theorem 1.1 that N= 2. Then the solution given by

Theorem 1.1 eventually becomes smooth in the sense that there are t⋆>0and P∈C1,0(Ω×[t⋆,∞)) such that

n, c ∈C2,1(Ω×[t⋆,∞)), u ∈C2,1(Ω ×[t⋆,∞); R2)

and that (n, c, u, P )is a classical solution of (1.1) in Ω×[t⋆,∞); that is, that the ﬁrst four equations therein

are fulﬁlled pointwise in Ω×[t⋆,∞).

Moreover, this solution stabilises in the large time limit. More precisely,

lim

t→∞ kn(·, t)−n0kC2(Ω) +kc(·, t)kC2(Ω) +ku(·, t)kC2(Ω)= 0,(1.6)

where n0=1

|Ω|RΩn0.

This theorem relates to [24, Theorem 1.2] which proves eventual smoothness by requiring smallness of RΩn0

instead of (1.3). A key ingredient to both results is the energy functional (4.1), which is conditional in the sense

that it only decreases throughout evolution if certain conditions are met, namely that the functional is already

suﬃciently small at some time t0≥0. That this functional indeed dissipates under certain conditions is veriﬁed

in Lemma 4.1, where we already rely on (1.3).

In order to actually make use of this conditional energy structure we then need to show that various quantities

become small at some point in time. The starting point is the quasi-energy inequality (3.15) holding for

approximate solutions (nε, cε, uε), ε∈(0,1). Its right-hand side χ2RΩnεc−1

εf(cε) becomes small if f(cε) = cε

and RΩnε=RΩn0is small; this is a core idea of [24, Lemma 5.2]. Not wanting to impose a smallness condition

of the initial data, we are forced to argue diﬀerently. A key observation is that nεc−1

εf(cε) can be controlled

favourably for cεbounded away from 0 while for small cεthe key condition (1.3) shows that the respective term

becomes small as well. Thus, splitting the right-hand side of (3.15) in integrals over suitable subdomains allows

us to obtain smallness of the ﬁrst part of the conditional energy functional, see Lemma 4.2 and Lemma 4.3.

For the remaining term, we crucially rely on the inequalities (2.3) and (2.4) recently derived by Heihoﬀ in [13].

They improve on related inequalities by getting rid of an additional additive L1term on the right-hand side,

which make them favourably applicable also for large RΩn0, see Lemma 4.4.

Finally, the bounds implied by the conditional energy functional serve as a starting point for a bootstrap

procedure sketched in Lemma 4.6, which provides further estimates suﬃciently strong to arrive at Theorem 1.2

in Subsection 4.4.

Main result III: Global existence of classical solutions under smallness conditions. As a byproduct of the

arguments developed for proving Theorem 1.2, we immediately obtain global classical solutions emanating from

initial data already satisfying the conditions for the conditional energy functional.

Theorem 1.3. Let Ω⊂R2be a smooth, bounded domain, χ > 0and φ∈W2,∞(Ω), suppose that fcomplies

with (1.2) and (1.3) and let m0>0. There exists η > 0such that whenever

(n0, c0, u0)∈C0(Ω) ×W1,q(Ω) × D(Aβ)for some q > 2and β∈(1

2,1),

where Adenotes the Stokes operator, are such that n0>0and c0>0in Ω,

ZΩ

n0=m0kc0kL∞(Ω) ≤m0

and

ZΩ

n0ln n0

+ZΩ

0+ZΩ

|∇c0|2

+ZΩ

|u0|2≤η,

then there exists a global classical solution (n, c, u, P )of (1.1), which moreover satisﬁes (1.6).

Plan of the paper. We ﬁrst collect some useful general inequalities in Section 2 before proving Theorem 1.1

in Section 3 and Theorem 1.2 as well as Theorem 1.3 in Section 4; we refer to the beginning of the later two

sections for a discussion of the ﬁner structure. Let us here just point directly to the most crucial new steps:

Lemma 3.9 asserts uniform integrability of nεf(cε) and key ideas relying on (1.3) are employed in Lemma 3.11,

Lemma 4.1 and Lemma 4.2.

Notation. Let Ω ⊂RN,N∈N, be a smooth, bounded domain. Throughout the article, we abbreviate

ϕ:=1

|Ω|RΩϕfor ϕ∈L1(Ω), set L2

σ(Ω; RN):={ϕ∈L2(Ω) : ∇ · ϕ= 0 in D′(Ω) }, where D′(Ω) is the

space of distributions on Ω, as well as W1,2

0,σ (Ω; RN):=W1,2

0(Ω; RN)∩L2

σ(Ω; RN). We further denote by

Pthe Helmholtz projection on L2(Ω; RN) and by Athe Stokes operator −P∆ on L2(Ω; RN) with domain

W2,2(Ω; RN)∩W1,2

0,σ (Ω; RN), and often write L2(Ω) instead of L2(Ω; RN) etc. when the codomain can be

inferred from the context. Moreover, uppercase constants Ciare unique throughout the article while lowercase

constants ciare “local” to each proof.

2 Preliminaries: The Csiszár–Kullback and Heihoﬀ inequalities

This preliminary section is concerned with multiple estimates regarding RΩϕln ϕ

ϕand RΩ

|∇ϕ|2

ϕ2for positive

functions ϕ. First, we recall the classical Csiszár–Kullback inequality which gives a nontrivial lower bound for

the former term.

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摘要：

arXiv:2211.01019v1[math.AP]2Nov2022Chemotaxis(-ﬂuid)systemswithlogarithmicsensitivityandslowconsumption:globalgeneralizedsolutionsandeventualsmoothnessMarioFuest∗LeibnizUniversitätHannover,InstitutfürAngewandteMathematik,Welfengarten1,30167Hannover,GermanyAbstractWeconsiderthesystemnt+u·∇n=∆n−χ∇·...

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Chemotaxis-fluid systems with logarithmic sensitivity and slow consumption global generalized solutions and eventual smoothness

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