An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system

In a smoothly bounded two‐dimensional domain Ω$\Omega$ and for a given nondecreasing positive unbounded ℓ∈C0([0,∞))$\ell \in C^0([0,\infty))$ , for each K>0$K>0$ and η>0$\eta >0$ the inequality ∫Ωφlnφ⩽η∫Ω|∇φ|2φ2+C(K,η)$$\begin{eqnarray*} \hspace*{140pt} \int _\Omega \varphi \ln \varphi \leqslant \eta \int _\Omega \frac{|\nabla \varphi|^2}{\varphi ^2} + C(K,\eta) \end{eqnarray*}$$is shown to hold for any positive φ∈C1(Ω¯)$\varphi \in C^1(\overline{\Omega })$ fulfilling ∫Ωφℓ(φ)⩽K.$$\begin{eqnarray*} \hspace*{100pt}\int _\Omega \varphi \ell (\varphi) \leqslant K. \end{eqnarray*}$$This is thereafter applied to nonglobal solutions of the Keller–Segel system coupled to the incompressible Navier–Stokes equations through transport and buoyancy, and it is seen that in any such blow‐up event the corresponding population density cannot remain uniformly integrable over Ω$\Omega$ near its explosion time.


INTRODUCTION
The main purpose of this manuscript is to derive information on the qualitative behavior of nonglobal solutions to a Keller-Segel-Navier-Stokes system near their blow-up time.
As our approach toward this will rely on an apparently novel interpolation inequality to a major extent, however, a detailed discussion of the latter is provided as a preparation for this.
Estimating zero-order expressions by small multiples of weighted Dirichlet integrals.Interpolation inequalities involving Orlicz classes different from classical Lebesgue spaces are of significant importance in a number of application contexts in the analysis of nonlinear partial differential equations.One recurrent theme in this regard consists in the ambition to appropriately estimate functions, for instance in terms of suitable Dirichlet integrals, under the prerequisite that bounds for their size in certain Orlicz spaces are available; situations of this type can typically be found when Gibbs-type expressions involving logarithmic deviations from linearity arise as parts of global entropy-like structures, such as in large classes of reaction diffusion and especially Fokker-Planck equations [9,12,19,20], or also in cross-diffusion systems ( [2,5,28,34]; cf. also the discussion in [32]).
A classical result in this direction ( [2]) now concentrates on suitably regular bounded planar domains Ω and implies that given  > 0 and  > 0 one can find (, ) > 0 such that for all nonnegative  ∈  1 (Ω) fulfilling with () ∶= ln( + ),  ⩾ 0 (see [2, p.1199] for a derivation of (1.1), and to [33,Appendix] for some close relatives).Through the appearance of arbitrarily small numbers  > 0 here, as being of relevance when superlinear nonlinearities are to be controlled by diffusive action in parabolic problems (cf.[2,24,28,30,33] and [9] for some examples), this statement can be viewed as quantifying a certain advantage of the framework described by (1.2) over, e.g., the neighboring situation in which instead of (1.2) only bounds of the form are presupposed; in such cases, namely, a standard Gagliardo-Nirenberg interpolation guarantees (1.1)only with some not necessarily small  = () > 0, the minimal size of which is in an evident manner restricted by positivity of the best constant   associated with the Sobolev embedding inequality ‖‖  2 (Ω) ⩽   ‖‖  1,1 (Ω) in the two-dimensional domain Ω [29].An extension of (1.1)-(1.2) that does not only allow for more general choices of the integrability power on the right-hand side therein, but which moreover covers unbounded  of arbitrarily slow growth, can now be obtained as a fairly straightforward consequence of an interpolation result recorded in [34]: Proposition 1.1.Let Ω ⊂ ℝ 2 be a bounded domain with smooth boundary, let  ∈  0 ([0, ∞)) be nondecreasing and positive with () → +∞ as  → ∞, and let  ∈ (−∞, 2).Then for all  > 0 and any  > 0, one can find (, ) > 0 such that for each  ∈  1 (Ω) fulfilling  > 0 in Ω as well as (1.2), we have + (, ).(1.4)This statement, a brief proof of which is included in the Appendix A below for completeness, gives rise to the question whether also in the borderline case  = 2 some genuine advantage of comparable flavor can be expected to result from (1.2).Indeed, a hint in this respect can be obtained by recalling a by-product of the two-dimensional Moser-Trudinger inequality, according to which given a smoothly bounded domain Ω ⊂ ℝ 2 and a constant  > 0, one can fix () > 0 such that for each positive  ∈  1 (Ω) fulfilling (1.3) ( [28,40]; cf. also [3] for a discussion of related limit cases in interpolation inequalities of Gagliardo-Nirenberg type).
The first intention of this study now consists in making sure that also in this more subtle relative of (1.4), arbitrarily small multiples in the corresponding first-order expression can be observed if instead of (1.3) a slightly stronger assumption of the form (1.2) is imposed on the function to be estimated.In fact, in Section 2, a combination of the Moser-Trudinger inequality with a suitably arranged zero-order interpolation will lead to the following replacement of (1.4) in the limit case  = 2. Proposition 1.2.Let Ω ⊂ ℝ 2 be a bounded domain with smooth boundary, and suppose that  ∈  0 ([0, ∞)) is a nondecreasing positive function fulfilling Then for all  > 0 and any  > 0, there exists (, ) > 0 such that whenever  ∈  1 (Ω) is positive in Ω with we have An essentially explicit construction of counterexamples indicates that the option of choosing  > 0 arbitrary here indeed is inherently linked to the unboundedness of : Proposition 1.3.Let Ω ⊂ ℝ 2 be a bounded domain with smooth boundary, and let  > 0. Then there exists  0 =  0 () > 0 with the property that for any  > 0, one can find   ∈  ∞ (Ω) such that   > 0 in Ω and but that But also with regard to the growth of the expression on its right-hand side with respect to the function , the estimate (1.8) seems the best possible that can be expected to result from (1.7) when assumed to hold for arbitrary  fulfilling (1.6).The following statement in this direction will result from a second construction of explicit examples.Proposition 1.4.Let Ω ⊂ ℝ 2 be a bounded domain with smooth boundary, let  0 > 0, and let ℎ ∈  0 ([0, ∞)) be a positive and nondecreasing function with the property that given any strictly increasing positive  ∈  0 ([0, ∞)) fulfilling () → +∞ as  → ∞, for each  > 0 one can find (, ) > 0 such that for all  ∈  1 (Ω) fulfilling (1.7).Then sup >0 ℎ() < ∞. (1.12) Characterizing blow-up behavior in a chemotaxis-Navier-Stokes system.Next returning to our original objective, in the second part of this manuscript we will consider the model for chemotactic bacterial movement interacting with liquid environments through buoyant forces.
Here,  and  represent the respective population density and attractant concentration, where  and  denote the fluid velocity and the associated pressure, and the gravitational potential Ξ is a given suitably regular function.
Since the experimental and modeling-oriented works by Goldstein et al. [8,35], problems of this and some closely related forms have received considerable interest in the mathematical literature [4,10,11,14,21,25,39,42].Especially in settings involving chemoattractant production through cells, as underlying (1.13), a predominant focus has been on the question how far fluid interaction can influence the explosion-supporting properties known as a core characteristic of classical two-component Keller-Segel systems ( [18,38]; see [23] for a sufficient criterion for the occurrence of blow-up in (1.13) under additional assumptions on fluid regularity).In fact, both simulationbased evidence and also some rigorous analytical findings indeed indicate a substantial influence of fluid couplings on the mere possibility of blow-up phenomena and the time of their occurrence [15,16,22,26,41]; understanding qualitative effects of fluid interaction on solution behavior near the onset of singularities seems yet at a rather early stage, however; indeed, precedent results in this regard seem to reduce in essence to information on loss of certain regularity properties undergone by the whole triple (, , ) near collapse ( [6,7]; cf. also [1] and [37]).In particular, the blow-up behavior of the biologically relevant quantity  yet seems to lack characterizations as detailed as that available for the two-dimensional Keller-Segel system obtained on letting  = ∇Ξ = 0 in (1.13), for which it is known, namely, that classical solutions (, ) in Ω × (0,  max ) that cease to exist beyond  max ∈ (0, ∞) must satisfy arguments leading to such far-reaching descriptions up to now crucially rely on the fact that this fluid-free variant of (1.13) possesses quite a fragile energy structure ( [31]) that seems no longer present in the chemotaxis-fluid system (1.13).
In Section 3, it will turn out that nevertheless a comparably far-reaching description of cell aggregation can be gained also in the fully coupled system (1.13), even to an extent yet somewhat more thorough than in (1.14).Indeed, in the first and most pivotal step of our analysis in this regard a suitably designed exploitation of the interpolation result from Proposition 1.2 will enable us to detect a second quasi-energy property that operates at levels of regularity mild enough to cope with widely arbitrary smoothness features of the unknown fluid field.In Lemma 3.3, namely, we shall thereby establish an inequality of the form under the mere assumption that (, , , ) blowups up at  max ∈ (0, ∞) but is such that ((⋅, )) ∈(0, max ) is uniformly integrable over Ω.A subsequent bootstrap-type procedure will turn this primal information into knowledge on higher regularity features, and in the course of a contradiction-based argument we will finally conclude that no such equi-integrability feature can be observed for any nonglobal solution of (1.13).In formulating this and throughout the sequel, we let  = −Δ denote the realization of the Stokes operator in  2 (Ω; ℝ 2 ), with its domain given by () =  2,2 (Ω; ℝ 2 ) ∩  1,2 0, (Ω), and with  representing the Helmholtz projection on  2 (Ω; ℝ 2 ), and for  > 0 we let   denote the corresponding sectorial fractional powers.Theorem 1.5.Let Ω ⊂ ℝ 2 be a bounded domain with smooth boundary, let Ξ ∈  2,∞ (Ω), and suppose that ) and such that the corresponding solution (, , , ) of (1.13) blows up in finite time in the sense that there exist  max ∈ (0, ∞) and uniquely determined functions

A logarithmic end-point extension of (1.4): Proof of Proposition 1.2
An elementary but important preparation for our argument in Proposition 1.2 makes sure that for an arbitrary given nondecreasing positive unbounded  ∈  0 ([0, ∞)), one can find an unbounded minorant, the growth of which can be controlled, in essence, by that of certain logarithmic expressions.
In particular, so that the intended inequality results from the fact that due to the upper bound for  in (2.16) we have  ⩽  2 − 0 and therefore ln  ⩽ 2 − ln  0 for any such .□ Remark.As kindly pointed out to us by one of the reviewers, as a nearby attempt to design an unbounded increasing  ∈  0 ([0, ∞)) fulfilling (2.1) and an inequality of the form in (2.15) one might think of circumventing the above discrete construction by alternatively choosing  as a suitable extension to [0, ∞) of the solution to with appropriate  > 1 and   > 0, and with l as defined by (2.3).In fact, it can readily be seen that adequately adjusting parameters here will indeed yield a strictly increasing  ∈  1 ([0, ∞)) that satisfies (2.1), and for which exists  > 0 such that ln  () ⩽  ⋅ ln  0 ( 0 ) for all  ∈ (0,  0 ) and each  0 > 1.That this simple approach in general fails to assert unboundedness of  as a property essential for our subsequent reasoning (cf.(2.24)), however, will be indicated by a counterexample detailed in the Appendix B below.
In drawing a second conclusion from (2.2), we shall make use of the elementary observation recorded in the following.
In fact, using this we can infer from Lemma 2.2, yet at an elementary level of real numbers, an interpolation inequality that will play a key role in our verification of Proposition 1.2.
Lemma 2.4.Let  ∈  0 ([0, ∞)) be positive and nondecreasing and such that (2.2) holds.Then for each  ∈ (0, 1  16 ),  ln  ⩽ due to the fact that the validity of ln  ⩽  for all  > 0 implies that ln ) according to the monotonicity of .□ We are now prepared for the derivation of our main result on interpolation on the basis of the latter lemma and the Moser-Trudinger inequality in the two-dimensional domain Ω: Proof of Proposition 1.2.We first employ the Moser-Trudinger inequality to fix  1 > 0 and  2 > 0 such that ) noting that the latter is possible thanks to the unboundedness of .We thereupon fix an arbitrary positive  ∈  1 (Ω) fulfilling (1.7), and observe that if incidentally ∫ Ω  2 ⩽ , then using that ln  ⩽  for all  > 0, we trivially estimate If, conversely, ∫ Ω  2 > , then letting defines a positive number that has the property that with  as in Lemma 2.3, we have where in order to control the first summand on the right we note that an application of (2.21) to  ∶= 2 ln( + 1) reveals that Since ln( + 1) ⩽  for all  > 0 and thus due to the monotonicity of  and again (1.7), twice estimating  2 ⩽ ( + 1) 2 we obtain that Our two results concerned with optimality of Proposition 1.2 will both utilize the radially symmetric functions defined and characterized as follows.
and that thus also (1.10) holds.□ Apart from that, a slightly more subtle use of the functions from Lemma 2.5 reveals that also on the right-hand side of (1.8), no substantial improvement can be achieved.

ANALYSIS OF (1.13): PROOF OF THEOREM 1.5
The starting point of our analysis concerning (1.13) is formed by the following essentially wellknown basic statement on local solvability and extensibility.
Proof.This is a direct consequence of the de La Vallée-Poussin theorem.□ Indeed, having this function  at hand we can draw on the interpolation result from Proposition 1.2 to establish the integrability property announced in (1.15) on the basis of a variational approach that does not rely on fluid regularity in any quantitative sense.Lemma 3.3.Suppose that (1.16) holds, that  max < ∞, and that ((⋅, )) ∈(0, max ) is uniformly integrable over Ω.Then Proof.According to (1.13) and Young's inequality, and Here we recall a functional inequality stated in [40,Lemma 2.2] to fix  1 > 0 such that for each  > 0, all nonnegative  ∈  0 (Ω) and any  ∈  1 (Ω), writing  ∶= ∫ Ω  we have Since ∫ Ω  = ∫ Ω  0 > 0 for all  ∈ (0,  max ) and ∫ Ω  ⩽ max } for all  ∈ (0,  max ) by simple integration in (1.13), an application of this to  ∶= for all  ∈ (0,  max ), so that from (3.5) we infer that Now Proposition 1.2 enters so as to show that thanks to our hypothesis and Lemma 3.2, we can find  3 > 0 fulfilling and that thus a combination of (3.4) with (3.7) leads to the inequality for all  ∈ (0,  max ).
As 0 ⩽ ∫ Ω ln( + 1) ⩽ ∫ Ω  = ∫ Ω  0 for all  ∈ [0,  max ), this shows that from which (3.3) follows upon another application of (3.8).□ In turning this into two basic integrability features of the fluid field, let us recall some general results on how regularity of a given external force  influences regularity of solutions  in the Navier-Stokes system for  > 0.
(i) This can be obtained by straightforward adaptation of an argument based on the standard Navier-Stokes energy inequality in contexts of  log  forces, as performed in [40, Sect.2.3], for instance.(ii) The inequality claimed here can be obtained by first establishing time-independent bounds for ∫ Ω |∇| 2 on the basis of a variational methods tracing the evolution of this functional, and by thereupon, second, using this to control the nonlinear convective term in (3.9) when estimating    through an application of smoothing features of the Stokes semigroup to a Duhamel representation associated with (3.9) (cf.[36, Lemma 3.3 and Lemma 3.4] for details).□ Now Lemma 3.3 facilitates a first application of this, and hence a derivation of the following as a simple by-product.Lemma 3.5.Assume (1.16) and suppose that  max < ∞ and that ((⋅, )) ∈(0, max ) is uniformly integrable over Ω.Then there exists  > 0 such that ) and moreover we have Proof.This directly follows from an application of Lemma 3.4 (i) on the basis of Lemma 3.3.□ We next utilize this to obtain an integrability property of the taxis gradient by means of a second, and now more standard, set of testing procedures.In accomplishing this, we once again explicitly rely on (3.2) in a second interpolation argument that, in contrast to that from Lemma 3.3, operates within the framework of Proposition 1.1 far from the borderline situation addressed by Proposition 1.2.Lemma 3.6.Suppose that (1.16) holds, that  max < ∞, and that ((⋅, )) ∈(0, max ) is uniformly integrable over Ω.Then there exists  > 0 such that Proof.From the first two equations in (1.13) we obtain using integration by parts that and for all  ∈ (0,  max ), so that due to Young's inequality and the Cauchy-Schwarz inequality, for all  ∈ (0,  max ).(3.16)Here in view of (3.2) we may employ Proposition 1.1 to find  1 > 0 such that Proof.We integrate by parts in the first two equations from (1.13) and use the identity ∇ ⋅  = 0 to see that for all  ∈ (0,  max ) so that by Young's inequality, . Here according to the known fact that with some  2 > 0 we have ,2 (Ω) we can find  3 > 0 such that due to Lemma 3.6 we have whereas the Gagliardo-Nirenberg inequality in conjunction with Lemma 3.6, Young's inequality, Proposition 1.1 and (3.2) ensures the existence of positive constants  4 ,  5 ,  6 , and  7 fulfilling As another application of the Gagliardo-Nirenberg inequality together with Young's inequality shows that with some  8 > 0 and  9 > 0 we have for all  ∈ (0,  max ), for all  ∈ (0,  max ).□ In view of Lemma 3.1, our standing assumption on uniform integrability of ((⋅, )) ∈(0, max ) must thus actually have been false.for all  ∈  1,2 (Ω).
Using suitable power-type transformations of the functions to be estimated, from this we readily obtain our counterpart of Proposition 1.2 in the off-borderline situation under consideration.Then assuming that  > 1 and   > 0, and that  solves (B.2), we take   > 1 large enough such that   >  for all  ⩾   , and observe that (B.4) ensures that for all  > 1 we have  ⋆  >   and  +1 −  ⋆  =   2 +2+1 −