Uniform L p estimates for solutions to the inhomogeneous 2D Navier–Stokes equations and application to a chemotaxis–ﬂuid system with local sensing

The chemotaxis-Navier–Stokes system


Introduction
The two-dimensional Navier-Stokes equations with a force term in L ∞ -L 1 .In the first part of the present paper, we are concerned with the regularity of solutions (v, Q) of the two-dimensional incompressible Navier-Stokes equations x ∈ ∂Ω, t ∈ (0, T ), where Ω ⊂ R 2 is a smoothly bounded domain and f is a given external force.In contrast to the threedimensional setting where strong a priori estimates remain elusive even if f ≡ 0 (cf.[21], [30]), the 2D case is quite well understood even for forces f with rather weak integrability and regularity properties.
As observed in [44,Theorem 1.2], this problem can be overcome by additionally requiring a bound for T 0 Ω |f | ln(|f | + 1) as well.Although this additional assumption is rather mild, it turns out to be too strong for the system considered in the second part of the present paper (see below), the main problem being that the bounds derived in [44,Theorem 1.2] could depend in an unfavorable way on the space-time L log L norm of f .In contrast to this, our first main result establishes uniform L p bounds which may also depend on the space-time L 2 norm of f , but only raised to an arbitrary small power.That is, while we require a bound for f in a stronger topology compared to [44,Theorem 1.2], we are able to control the influence of this bound in a convenient quantitative manner.To be precise, in Section 2 we shall see the following.Theorem 1.1 Let Ω ⊂ R 2 be a bounded domain with smooth boundary, and let 3) holds and that (1.1) is solved in the classical sense.Then for any p ∈ [1, ∞) and each θ > 0 there exists Similarly as in [44], our proof starts by splitting v into the solution v 1 of the inhomogeneous Stokes equations and a solution v 2 of the Navier-Stokes equations containing certain error terms but not explicitly depending on f .Testing the latter with A 2β 2 v 2 for β ∈ (0, 1 2 ) close to 1 2 (see Lemma 2.4) and making use of the energy identity (cf.Lemma 2.2) yield L p estimates for v essentially depending on a quantity exponential in T t 0 Ω |f | ln(|f | + 1).For t 0 sufficiently close to T , we can then conclude (1.4) for t ∈ (t 0 , T ), while the regularity of v entails (1.4) for t ∈ [0, t 0 ], see Lemma 2.5.Of course, this approach necessitates that the constant K in (1.4) may depend on v and not just on the parameters and some norm of the data.
Application: A chemotaxis-Navier-Stokes system with local sensing.Next, we consider in a smoothly bounded domain Ω ⊂ R 2 , where α > 0 is arbitrary and Φ ∈ W 2,∞ (Ω) is a given function.The system (1.5) models the behavior of bacteria (with density n) which may partially orient their movement towards oxygen (with density c) and who interact with a fluid (with velocity u) through buoyancy and transportation.The right-hand side of the first equation in (1.5) expands to ∇ • (D(c)∇n − nχ(c)∇c) with D(c) = c −α and χ(c) = αc −α−1 , c > 0, and for general choices of D and χ such a system (without fluid) has already been proposed by Keller and Segel in the 1970s [18], while the coupling to a Navier-Stokes system goes back to [35].
Such systems have received considerable attention in the last decade; especially for results regarding the fluid-free setting we refer to the recent survey [19].Here, we mainly confine ourselves with twodimensional chemotaxis-fluid systems.While in the arguably simplest case, namely when both D and χ are positive constants, global classical solutions always exist ( [15]), the situation becomes much more delicate when χ is singular at zero, as the second solution component may become small and hence strengthen the potentially destabilizing chemotaxis term over time; in fact, to underline this we recall that already in the classical Keller-Segel-production system the size of χ (relative to Ω n 0 ) determines if all solutions remain bounded or if singularities can form for some initial data (see, e.g., [14], [28], [13]).
This challenge is reflected by the fact that unconditional global existence results of classical solutions (or, alternatively, proofs of finite-time blow-up) of chemotaxis-consumption models with fluid and logarithmic sensitivity (i.e., D ≡ 1 and χ(c) = 1 c for c > 0), hence accounting for stimulus perception in accordance with the Weber-Fechner law ( [29]), are apparently yet unavailable.Indeed, up to now only some global generalized solutions have been constructed ( [37], [27]) which become smooth after some finite time if the mass of the bacteria is sufficiently small ([1], [27]) or if the chemical is consumed sufficiently slowly ( [7]).As by-products, these latter works also obtain global classical solvability under certain smallness conditions.
In settings where the motion of the organism is starvation-driven, also D(c) may drastically increase as c approaches zero, see [6], [26], [4] for recent modelling considerations.Mathematically, one might hope that when also the regularizing effect of the diffusion term is enhanced for small c, the (provable) regularity of solutions improves compared to settings where only χ is singular at zero.On a technical level, this is in particular the case when D ′ = −χ, i.e., when ∇ • (D(c)∇n − nχ(c)∇c) = ∆(nD(c)), as then techniques based on duality arguments become available.This is the setting we consider here and which has also been proposed in the modelling works referenced above.This hope is indeed justified: Global classical solutions for the fluid-free counterpart of (1.5) have very recently been constructed in [43] for all α > 0, while global weak solutions are known to exist also in the higher-dimensional setting ( [32]).For precedents with non-singular motility, where c −α in the first equation in (1.5) is replaced by (c + ε) −α for positive ε, we refer to, e.g., [25] and [44].Let us also briefly mention that chemotaxis systems with local sensing have also been studied in various related situations, see for instance [22], [20], [36], [23], [41], [42] for the non-singular but potentially degenerate setting and [9], [8], [2], [17], [24], [5] among others for the case of signal production.
In contrast to [43], however, the coupling to the Navier-Stokes equations in (1.5) requires to control the terms stemming from the transportation term u • ∇n in order to gain any useful information from duality arguments.This eventually comes down to estimating u in some L p norm, but as apart from mass conservation no a priori information for the first solution component appears to be available at this point -not even a space-time L log L bound -the L p estimates derived in [44] are not applicable.Hence, we make use of Theorem 1.1 instead.If we choose θ > 0 therein sufficiently small, the righthand side in (1.4) results in only a small power of T 0 Ω n 2 which can be absorbed into a dissipative term (cf.Lemma 3.2).
After this crucial first step, the additional presence of a fluid only requires minor modifications of the ideas from [43], so that we are able to bootstrap the bounds obtained in Lemma 3.2 to estimates so strong that they exclude the possibility of finite-time blow-up.
Our main results concerning (1.5) then reads as follows.
Theorem 1.2 Let Ω ⊂ R 2 be a bounded domain with smooth boundary, let α > 0 and Φ ∈ W 2,∞ (Ω), and suppose that and that Then there exist functions such that n ≥ 0 and c > 0 in Ω × [0, ∞), and that (1.5) is solved in the classical sense.
2 Estimates for solutions to (1.1).Proof of Theorem 1.1 Throughout this section, for p ∈ (1, ∞) we denote the Stokes operator on and the Helmholtz projection from L p (Ω; R 2 ) to L p σ (Ω) by P p for p ∈ (1, ∞).Since these operators coincide on C ∞ 0 (Ω; R 2 ) and thus on the intersection of their domains, we may often write A and P instead of A p and P p , respectively, without specifying p.Moreover, we henceforth suppose that the hypotheses of Theorem 1.1 hold, and set v 0 := v(•, 0).
We first note that letting by relying on (1.3) in quite a straightforward manner we can obtain bounds for v 1 , the solution of a Stokes system with force term f .Lemma 2.1 Under the assumptions of Theorem 1.1, we have Proof.This follows by applying standard smoothing estimates for the Stokes semigroup ([12, p. 201]) to (2.1) in the considered two-dimensional framework, see [38,Lemma 2.5] for details.
Next, we record an estimate resulting from a well-known combination of the Navier-Stokes energy identity combined with the Trudinger-Moser inequality, which is implicitly contained in [40,Lemma 2.7] or [44, Lemma 2.5], for instance.Unlike in these works, however, we do not have a space-time L log L bound for f at our disposal, and hence the bounds in the following lemma do not yet provide any unconditional estimates but need to depend on f .(That is, the right-hand side in (2.2) below may be unbounded for t ր T .) for all t ∈ (t 0 , T ). (2.2) and according to a Poincaré inequality we can find C 2 > 0 fulfilling We next rely on a consequence of the Moser-Trudinger inequality in the planar domain Ω (see [40, Lemma 2.2]) to pick C 3 > 0 in such a way that so that with C 4 := C 3 C 1 + |Ω| e , thanks to the Cauchy-Schwarz inequality we obtain that Therefore, and hence for all t 0 ∈ [0, T ) and t ∈ (t 0 , T ) and thus (2.2) upon evident choices of K 1 and (k 1 (t 0 )) t 0 ∈(0,T ) .
As used multiple times in the sequel, we briefly state embedding results regarding the domains of fractional powers of the Stokes operator, including the case of zeroth power, that is, embeddings into p 1 ) ֒→ D(Λ β 2 p 2 ), upon which the statement results by [11,Theorem 3].With Lemma 2.3 at hand, we may now test the equation solved by v 2 -which, importantly, does not explicitly depend on the force f -with A 2β v 2 to obtain uniform-in-time L p estimates also for v 2 .However, they yet depend on the quantity in the left-hand side of (2.2) and will then be combined with Lemma 2.2 in Lemma 2.5 below.As we shall see in the proof of the latter, it turns out to be crucial that the constant K 2 appearing in (2.5) below does not depend on t 0 .Lemma 2.4 For each p ≥ 1 there exists K 2 (p) > 0 with the property that whenever t 0 ∈ [0, T ), one can fix k 2 (p, t 0 ) > 0 in such a way that Proof.Lemma 2.3 allows us to choose β = β(p) ∈ (0, 1 2 ) suitably close to 1 2 such that with some and then obtain on testing the identity (Ω × (0, T )) according to (1.1) and (2.1), by Here, since D(A ) by Lemma 2.3 and as P is continuous on L for all t ∈ (0, T ), where again by the Hölder inequality, for i ∈ {1, 2} and all t ∈ (0, T ).
Estimates of the form v(•, t) L p (Ω) ≤ k(t 0 )e ) are direct consequences of Lemma 2.2 and Lemma 2.4.At least on small time scales, i.e., if t 0 is sufficiently close to T , we can estimate the right-hand side therein against a small power of the space-time L 2 norm of f .Since K does not depend on t 0 , we can conclude Lemma 2.5 For any p ≥ 1 and each θ > 0 there exists K > 0 such that Proof.According to [39, Lemma 3.6], we have so that by Young's inequality, where C 2 := 2|Ω| + e, and where C 1 := sup t∈(0,T ) Ω |f (•, t)| + 1 + 1 is finite thanks to (1.3).We now fix t 0 ∈ [0, T ) sufficiently close to T such that with K 1 and K 2 taken from Lemma 2.2 and Lemma 2.4 we have and combine the latter two lemmata with (2.8) to see that with k 1 = k 1 (t 0 ) and k 2 = k 2 (p, t 0 ) introduced there we have for all t ∈ (t 0 , T ).
As the Jensen inequality together with the fact that ξ ln 1 ξ ≤ 1 e for all ξ > 0 implies that here for all t ∈ (t 0 , T ) due to (2.9), we therefore obtain that Since v is continuous and hence bounded in Ω × [0, t 0 ], this yields (1.4) with some suitably large Proof of Theorem 1.1.The desired estimate has been proven in Lemma 2.5.
3 Analysis of (1.5).Proof of Theorem 1.2 We begin our study of (1.5) by stating a local existence result as well as the basic bounds for n and c in L ∞ -L 1 and L ∞ -L ∞ , respectively.
Lemma 3.1 Let Ω ⊂ R 2 be a bounded domain with smooth boundary, let α > 0 and Φ ∈ W 2,∞ (Ω), and assume (1.6).Then there exist T max ∈ (0, ∞] as well as functions ) and as well as Proof.This can be seen by adapting a standard contraction mapping argument ([16, Lemma 2.1]) to the present situation.(We note that as long n is bounded, c is at least locally in time bounded from below by a positive constant, so that we do not need to include a term such as 1 c(t) L ∞ (Ω) in (3.1).)Without commenting on this explicitly any further, we shall below assume that the smoothly bounded domain Ω ⊂ R 2 , the number α > 0 and the function Φ ∈ W 2,∞ (Ω) have been fixed, and let (n, c, u, P ) and T max be as thereupon obtained in Lemma 3.1.
The most crucial part of our analysis is performed in the following lemma.Similarly as for instance in [32], [33] or [43], the basic idea is to make use of the structure of the first equation in (1.5) which allows for reasonings based on duality arguments.The key additional challenge lies in the fact that no helpful a prioi estimates for the fluid equation appear to be available only based on (3.2). (This stays in contrast to situations where at least space-time L log L bounds for n can be obtained in a rather straightforward manner, such as in [44,Lemma 3.2].)Instead, we rely on Theorem 1.1 and estimate terms stemming from the fluid transportation term u • ∇n essentially against the space-time L 2 norm of n, whose boundedness is not established prior to this lemma but which appears as a dissipative term of the functional considered.
Therefore, using the Hölder inequality, Theorem 1.1 and (3.2) we obtain C 4 > 0 and C 5 > 0 such that due to (3.3), by Young's inequality it follows that with some C 6 > 0 and C 7 > 0, for all t ∈ (0, T max ).Again making use of Young's inequality, we further obtain for all t ∈ (0, T max ) and some C 8 > 0, which is finite due to our assumption that T max is finite.Combined with (3.9), we conclude so that by the definition of y this establishes both (3.4) and (3.5) with some C > 0.
With (3.4) and (3.5) at hand, the global existence proof proceeds quite analogously to the fluid-free setting considered in [43,Lemma 3.3].However, for the sake of completeness and as some modifications are necessary, we choose to at least sketch the proofs for the bootstrap procedure.
Proof of Theorem 1.2.It is sufficient to combine Lemma 3.8 with Lemma 3.1.