Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(n\nabla c)+\kappa n-\mu n^{2},\&x\in\Omega,&t>0,\\ c_{t}+u\cdot\!\nabla c&=\Delta c-nc,\&x\in\Omega,&t>0,\\ u_{t}&=\Delta u+\nabla P+n\nabla\phi,\&x\in\Omega,&t>0,\\ \nabla\cdot u&=0,\&x\in\Omega,&t>0,\\ \big(\nabla n-n\nabla c\big)\cdot\nu&=0,\quad c=c_{\star}(x),\quad u=0,&x\in\partial\Omega,&t>0, \end{array}\right. \end{align*} where $\kappa\geq0$, $\mu>0$ and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $\Omega\subset\mathbb{R}^N$ with $N\in\{2,3\}$, is a prescribed time-independent nonnegative function $c_{\star}\in C^{2}\!\big(\overline{\Omega}\big)$. Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.


Introduction
Chemotaxis, the oriented movement of bacteria and cells in response to a chemical substance in their surrounding environment, is an important motility scheme in nature. An interesting facet of colonies of such chemotactically active bacteria and cells consists of the possibility to spontaneously generate spatial patterns, as not only witnessed by the experimental findings on the aerobic Bacillus subtilis ([11, 24, 7]) but also in settings where the attracting signal is produced by the cells themselves ( [15,45]). This emergence of spatial structures, captivating biologists and mathematicians alike, has lead to an intensive study of chemotaxis systems in the past decades and is still garnering attention in the field of mathematical modeling and analysis. (See also the surveys [14,1,21].) In order to study the plume-like aggregation patterns observed to occur when a population of Bacillus subtilis is suspended in a drop of water, the authors of [33] proposed a model of the form x ∈ Ω, t > 0, c t + u ·∇c = ∆c − nc, x ∈ Ω, t > 0, u t + (u · ∇)u = ∆u + ∇P + n∇φ, x ∈ Ω, t > 0, ∇ · u = 0, x ∈ Ω, t > 0, (1.1) where n, c, u, P denote the density of the bacteria, the oxygen concentration, the velocity field of the incompressible fluid and the associated pressure, respectively, φ is a prescribed gravitational potential and Ω is a bounded domain in R N . While the authors of [33] suggest to augment the system with a non-zero Dirichlet boundary condition for the chemical at the stress-free fluid-air interface and a no-flux condition for the bacteria (in fact they even propose mixed boundary conditions distinguishing between the bottom layer of the drop and the fluid-air interface), a large part of the literature on chemotaxis-fluid systems only considers no-flux conditions for both n and c and a no-slip condition for u.
In this setting the global solvability of (1.1) is well studied and most of the remaining problems remain in the case of N = 3. Actually, for N = 2 global classical solutions and their stabilization properties have been established in [38] and [39], respectively. Whereas, in the higher dimensional setting it was shown in [42,43] that (1.1) possesses at least one global weak solution, which becomes smooth after some possibly large waiting time. A recent study by the same author also reveals that on small timescales (possible) singularities can only arise in a set of measure zero ( [44]). Similar results have also been established in models where the bacteria are assumed to obey a logistic population growth (i.e. including the term +κn − µn 2 on the right hand side of the first equation). In fact, existence of weak solutions was shown in [34] and [20] considers the eventual smoothness of weak solutions in 3D. Analytical results providing pattern formation as discovered in the experiments, however, are still missing, which raised the question whether the assumed boundary conditions should be adjusted for further advances. Under consideration of different boundary conditions, the knowledge of (1.1) is quite enigmatic, with most of the current results on existence theory only discussing the two-dimensional setting or relying on the inclusion of small changes to (1.1), like logistic growth terms, an enhanced diffusion rate for the bacteria or the consideration of Stokes fluid (i.e. dropping (u · ∇)u in the third equation) and even then solutions can often only be obtained with quite mild regularity. In this regard, the work [2] contains the most intricate result in this direction, with the treatment of (1.1) with logistic growth terms under the Robin boundary condition ∂c ∂ν = 1 − c on ∂Ω. The author proves the existence of global classical solutions in 2D and global weak solutions in 3D. Additional results featuring a Robin boundary condition in fluid-free (i.e. u ≡ 0) variants of (1.1) have been investigated in [3] and [10]. The former considers a stationary (and hence doubly elliptic) system and establishes existence and uniqueness of a classical solution for any prescribed mass M := Ω n > 0. The latter studies a parabolic-elliptic variant and attains results on global and bounded classical solutions and their long-term behavior. The recent result in [46] provides the existence of global weak solutions to the two dimensional version of (1.1) with superlinear diffusion (i.e. replacing ∆n by ∆n m with m > 1 in the first equation) and Robin boundary condition for c. Concerning non-zero Dirichlet data for c we are only aware of two unpublished works. The first proves global existing generalized solutions in 3D for the Stokes variant of (1.1) ( [36]) and the second provides global generalized solutions for N ≥ 2 in a Stokes variant of (1.1) with nonlinear diffusion satisfying m ≥ 1 for N = 2 and m > 3N −2 ). Results on more regular solutions and included logistic population growth appear to be missing for the Dirichlet boundary data case. (See also [23] and [25,26] for first analytical results concerning well-posedness of systems closely related to (1.1) with mixed boundary conditions, [16] for a more general fluid-free one-dimensional system with non-zero Dirichlet or Neumann boundary data and [33,5,22] for numerical studies related to (1.1).) Main results. Motivated by the observations above, we are going to consider a chemotaxis-Stokes system with logistic population growth of the form in a smoothly bounded domain Ω ⊂ R N with N ∈ {2, 3} and ν denoting the outward normal vector field on ∂Ω. We prescribe κ ≥ 0, µ > 0, a time constant function c * satisfying and initial data (n 0 , c 0 , u 0 ) satisfying and Ω ⊂ R N be a bounded domain with smooth boundary. Suppose that κ ≥ 0 and µ > 0 and that the functions c * and φ satisfy (1.3) and (1.4), respectively. Then, for any n 0 , c 0 and u 0 complying with (1.5), the system (1.2) admits at least one global weak solution (n, c, u) in the sense of Definition 2.1. Outline. In Section 2 we will recall the definition of a global weak solution. Section 3 will be devoted to the introduction of families of appropriately regularized systems and their time-global classical solvability. On the path toward time-global classical solvability of the approximating system, we will also establish a first set of basic a priori estimates. The commonly employed testing procedures in chemotaxis systems, however, rely heavily on the Neumann boundary conditions and hence adjustments in the treatment of c are necessary here. The substantial regularity information on n, as entailed by the quadratic decay present in the first equation, will be the driving force for the distillation of bounds on the gradient of c (see Lemma 3.5), which are an important cornerstone of our further analysis. In Section 4 we will concern ourselves with improving the bounds on n, where, in particular, time-space information on ∇n is the main objective of the section. In Section 5 we prepare estimates on the time-derivatives, which upon combination with boundedness results of previous sections allows for the construction of a limit object by means of an Aubin-Lions type argument at the start of Section 6. Finally, in the second part of Section 6, we will verify that the limit solution indeed satisfies the properties required of a global weak solution.

Definition of global weak solutions
Before we start with our analysis let us briefly recount the necessary properties for a global weak solution in the following definition, where here and below we set W 1,1 0,σ Ω; with n ≥ 0 and c ≥ 0 in Ω ×[0, ∞), will be called a global weak solution of (1.2) if

Global existence of approximate solutions and essential regularity estimates
The global weak solution asserted by Theorem 1.1 will be obtained as a limit object of solutions to certain regularized problems. To this end, for a fixed family (ρ ε ) ε∈(0,1) ⊂ C ∞ 0 (Ω) of smooth cut-off functions satisfying 0 ≤ ρ ε (x) ≤ 1 for all x ∈ Ω such that ρ ε 1 as ε 0, we introduce the corresponding family of approximating problems to (1.2) given by where f ε (s) := 1 (1+εs) 3 and g ε (s) := s 1+εs for s ≥ 0 and ε ∈ (0, 1). Due to the non-homogeneous boundary condition, this form of the second equation of (3.1), however, is not easily accessible for Dirichlet heat semigroup estimates we will draw on in our following analysis and hence, we substituteĉ ε := c * − c ε and accordingly rewrite the system into the equivalent formulation where, in light of the assumed regularity of c * , all important properties can be easily transferred back to (3.1). The transformed system will only play a role in the proof of time local existence of solutions (Lemma 3.1) and in the proof that the maximal existence time for fixed ε ∈ (0, 1) is actually infinite (Lemma 3.7), as otherwise our analysis in the latter will not necessarily require semigroup arguments for the second component of the systems. Now, let us begin by establishing time-local existence of solutions to (3.2) (and in turn (3.1)) by means of well-established fixed point arguments.
Suppose that c * and φ satisfy (1.3) and (1.4), respectively, and that n 0 , c 0 and u 0 comply with (1.5). Then for any ε ∈ (0, 1), there exist T max, ε ∈ (0, ∞] and a uniquely determined triple (n ε , c ε , u ε ) of functions Proof: Augmenting well-established fixed point arguments as e.g. presented in [40, Lemma 2.1] and [1, Lemma 3.1] we will first establish time-local existence for the transformed system (3.2), which afterwards, in view of the substitution c ε = c * −ĉ ε , can be easily transferred back to the corresponding statement for (3.1). For the sake of completeness let us specify the main steps involved: First, for some large R > 0 and T ∈ (0, 1], to be specified later, we define the Banach space X : Next, denoting by e t∆ t≥0 , e t∆ t≥0 and e −tA t≥0 the Neumann heat semigroup, the Dirichlet heat semigroup and the Stokes semigroup with Dirichlet boundary data, respectively, we utilize introduce the mapping Φ : and We will now show that Φ acts as a contracting self map on S, provided R and T are suitably fixed beforehand. Dropping the ε-subscript for readability, we pick (n 1 ,ĉ 1 , u 1 ), (n 2 ,ĉ 2 , u 2 ) ∈ S and observe that according to (3.4) Hence, drawing on semigroup estimates as e.g. provided by [ where we also used the facts that and [9, Thm. 5.6.5]) so that we can find Similarly, noting that |g ε (a) − g ε (b)| ≤ |a − b| for a, b ∈ [0, ∞) we can draw on semigroup theory for the Dirichlet heat semigroup (see [28,Proposition 48.4] and [13]) and (3.5) to conclude the existence of so that collecting (3.7), (3.8) and (3.9) yields we find that for some Hence, by first taking R > 3 max (n 0 , c * − c 0 , u 0 ) X , 2C 7 and then T ∈ (0, 1] sufficiently small such , we see from (3.11) and (3.10) that indeed Φ is a contraction map on S and aided by Banach's fixed point theorem we obtain a unique (n ε ,ĉ ε , u ε ) ∈ S with Φ(n ε ,ĉ ε , u ε ) = (n ε ,ĉ ε , u ε ). In light of standard bootstrapping procedures drawing on regularity theories for parabolic equations and the Stokes semigroup [27,31,17] one can verify that (n ε ,ĉ ε , u ε ) actually satisfies the claimed regularity properties, which then entails the existence of a corresponding P ε such that (n ε ,ĉ ε , u ε , P ε ) solves (3.2) classically in Ω × (0, T ). Uniqueness of (n ε ,ĉ ε , u ε ) can be verified by standard L 2 testing procedures for the differences of two assumed solutions. Noticing that the choice of T only depends on fixed system parameters and the initial data, we may iterate the arguments (with different initial data and possibly larger R) to extend the solution on a maximal time interval (0, T max, ε ) such that either T max, ε = ∞ or Clearly, by substituting c ε = c * −ĉ ε (and recalling (1.3)), we immediately obtain the desired results for (3.1), where, finally, the nonnegativity of n ε and c ε is entailed by two applications of the maximum principle to the first and second equation of (3.1).
For the remainder of the work we will now assume that 3) and (1.4), respectively, and initial data n 0 , c 0 , u 0 obeying (1.5) are fixed and, accordingly, for ε ∈ (0, 1) denote by (n ε , c ε , u ε ) the triple of functions provided by Lemma 3.1 and by T max, ε the corresponding maximal existence time. Time-local existence at hand, we can now proceed with a first set of a priori properties obtained by straightforward integration and an application of the maximum principle.

Lemma 3.2.
There is C > 0 such that for any ε ∈ (0, 1) the solution (n ε , c ε , u ε ) of (3.1) satisfies Proof: Making use of the fact that u ε is divergence free, by integrating the first equation of (3.1) over Ω and utilizing integration by parts as well as Young's inequality we deduce that for all ε ∈ (0, 1) Employing Young's inequality once more to estimate the quadratic term on the left from below we obtain which, when combined with the nonnegativity of n ε and an ODE comparison argument, implies Ω n ε (·, t) ≤ C 1 := max Ω n 0 , κ 2 2µ 2 + 1 |Ω| for all t ∈ (0, T max, ε ) and all ε ∈ (0, 1).
In order to distill further uniform bounds from the somewhat sparse (yet sufficiently powerful) spacetime information on n 2 ε provided by Lemma 3.2, we state the following comparison result for ordinary differential equations. This lemma is copied from [18,Lemma 3.4], whereto we refer the reader for details of the proof. Lemma 3.3.
for all t ∈ (0, T ). Then y ≤ y(0) + C 1−e −a throughout (0, T ). With the comparison lemma above, we can make now turn to obtain some uniform bounds for the third solution component.
Proof: First, we test the third equation in (3.1) against u ε , integrate by parts over Ω, and employ the Young and Poincaré inequalities as well as (1.4) to conclude the existence of C 1 > 0 such that for all ε ∈ (0, 1) is valid on (0, T max, ε ). Then, again denoting by P the Helmholtz projection and by A the Stokes operator, we multiply the projected third equation by Au ε to obtain C 2 > 0 such that for all ε ∈ (0, 1) the inequality holds on (0, T max, ε ), where we once more made use of the boundeness of ∇φ and Young's inequality. In light of the Poincaré inequality, a combination of (3.13) and (3.14) entails the existence of C 3 , C 4 > 0 such that d dt on (0, T max, ε ) for all ε ∈ (0, 1). Drawing on Lemmas 3.2 and 3.3, we conclude that there is C 5 > 0 satisfying Ω |∇u ε (·, t)| 2 ≤ C 5 for all t ∈ (0, T max, ε ) and ε ∈ (0, 1).
The uniform bounds on u ε in L ∞ (0, T max, ε ); L 6 (Ω) and n ε in L 2 (Ω × (0, T max, ε )) will now be the key ingredient in obtaining information on ∇c ε . We start by exploiting the fact that c * is constant in time to establish an ordinary differential inequality for Ω |∇c ε (·, t)| 2 on (0, T max, ε ).

Lemma 3.5.
There exists C > 0 such that for all ε ∈ (0, 1) the solution (n ε , c ε , u ε ) of (3.1) satisfies Proof: Since the boundary conditions in (3.1) imply that ∂ ∂t c ε ∂Ω = 0 on (0, T max, ε ), we can multiply the second equation of (3.1) by −∆c ε and integrate by parts to find that for all ε ∈ (0, 1) . Employing Young's inequality to the last two terms on the right and making use of the fact that |g ε (s)| ≤ s for all s ≥ 0 we obtain that for all ε ∈ (0, 1) Furthermore, relying on the Hölder inequality and Lemma 3.4, we find C 2 > 0 such that for all ε ∈ (0, 1) we have The Gagliardo-Nirenberg inequality and Lemma 3.2, moreover, imply the existence of C 3 , C 4 > 0 satisfying on (0, T max, ε ) for all ε ∈ (0, 1), so that an application of Young's inequality entails the existence of C 5 > 0 such that for all ε ∈ (0, 1) we have Ω |u ε · ∇c ε | 2 ≤ 1 4 Ω |∆c ε | 2 + C 5 on (0, T max, ε ). (3.17) A combination of (3.15)-(3.17) finally shows that with C := max{C 1 , C 5 } we obtain Next, we combine the recently established differential inequality with the Gagliardo-Nirenberg inequality, the comparison Lemma 3.3 and the space-time bound for n ε from Lemma 3.2 to obtain the following. Lemma 3.6.

Proof:
According to the Gagliardo-Nirenberg inequality and Lemma 3.2, there are C 1 , C 2 > 0 such that for all ε ∈ (0, 1) on (0, T max, ε ), which upon combination with the differential inequality for c ε in Lemma 3.5, the bounds obtained in Lemma 3.2 and the ODE-comparison of Lemma 3.3 entails the existence of C 3 > 0 such that for all ε ∈ (0, 1) we have Ω |∇c ε | 2 ≤ C 3 on (0, T max, ε ). (3.18) Then, returning to the differential inequality for c ε from Lemma 3.5, we obtain C 4 > 0 such that for all ε ∈ (0, 1) we have t (t−1)+ Ω |∆c ε | 2 ≤ C 4 on (0, T max, ε − τ ) from straightforward integration of said inequality in light of (3.18) and Lemma 3.2. Finally, once again in view of the Gagliardo-Nirenberg inequality, we we find C 5 > 0 such that for all ε ∈ (0, 1) completing the proof by drawing on the previous parts of this lemma and Lemma 3.2.
The boundedness property of ∇c ε in L ∞ (0, T max, ε ); L 2 (Ω) was the last missing piece of information necessary for proving time-global existence of solution to (3.1). Augmenting the bounds we established in this Section with additional ε-dependent bounds in the proof below, we will be able to draw on a Moser-Alikakos-type iteration procedure (see [32,Lemma A.1]) to finally conclude that for fixed ε the maximal existence time T max, ε provided by Lemma 3.1 is indeed not finite.

Refined a priori information on n ε
While the uniform bounds for c ε and u ε provided by Section 3 would already be strong enough for our limit procedure, we still lack sufficiently good uniform bounds for n ε . As it turns out, the spacetime bound for ∇c ε of Lemma 3.6, however, can be exploited when considering the functional y ε (t) := Ω n ε ln n ε (·, t), which has often been a good resource for information in chemotaxis settings ([8, 20, 38, 42]). We start by formulating a corresponding functional inequality. Lemma 4.1. There exists C > 0 such that for all ε ∈ (0, 1) the solution (n ε , c ε , u ε ) of (3.1) satisfies Proof: In light of the first equation of (3.1), the fact that u ε is divergence free and two integrations by parts we see that on (0, ∞) for all ε ∈ (0, 1). Observing that again both boundary integrals disappear due to the prescribed boundary conditions and the fact that ρ ε (x) = 0 for x ∈ ∂Ω and noting that there is some C 1 > 0 satisfying κs − µs 2 ≤ C 1 for all s ≥ 0 and such that (κs − µ 2 s 2 ) ln(s) ≤ C 1 for all s > 0, we may hence on (0, ∞) for all ε ∈ (0, 1). To further estimate the integral on the right, we make use of the fact that |ρ ε (x)f ε (s)| = ρε(x) (1+εs) 3 ≤ 1 for all s ≥ 0, x ∈ Ω and ε ∈ (0, 1) and two applications of Young's inequality to obtain d dt Ω n ε ln n ε + on (0, ∞) for all ε ∈ (0, 1), which concludes the proof upon the choice of C := max{ 1 4 , 2C 1 }. Clearly, we can draw on previously established space-time bounds to extract additional space-time information on ∇ √ n ε from the previous Lemma, which in a second interpolation step can also be refined to a bound on ∇n ε in L 4 3 (Ω × (0, ∞)).

Lemma 4.2. For any
Proof: Integration of the differential inequality featured in Lemma 4.1 over (0, T ) provides C 1 > 0 such that for all ε ∈ (0, 1) Recalling the bounds provided by Lemma 3.2, Lemma 3.6 and (1.5) as well as the obvious estimate − 1 e ≤ s ln s for all s ≥ 0, the conclusion is immediate.

Lemma 4.3.
For all T > 0 there exists C(T ) > 0 such that for all ε ∈ (0, 1) the solution (n ε , c ε , u ε ) of (3.1) fulfills Proof: Rewriting the integral under consideration and employing Young's inequality twice, we find that for all ε ∈ (0, 1). Reordering and making use of the bounds provided by Lemmas 3.2 and 4.2 we obtain the asserted bound.

Regularity estimates for the time derivatives
As final element for an Aubin-Lions type argument we are going to undertake in Section 6, we now prepare uniform bounds for the time derivatives in suitable spaces. Lemma 5.1.

Existence of a limit solution. The proof of Theorem 1.1
Collecting the uniform bounds presented in Sections 2-5, we can now construct a limit object which satisfies all the regularity requirements present in Definition 2.1. Proposition 6.1.
Finally, it remains to be checked that the limit objected provided by Proposition 6.1 indeed satisfies the integral identities (2.1), (2.2) and (2.3) of Definition 2.1. This, however, is a straightforward procedure, as the convergence properties of Proposition 6.1 already cover everything we need to pass to the limit in the corresponding equations of (3.1).