Abstract
We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form
where \(\kappa \ge 0\), \(\mu >0\) 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 \(c_*\in C^{2}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) \). 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.
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1 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 [6, 10, 23] but also in settings where the attracting signal is produced by the cells themselves [14, 45]. This emergence of spatial structures, captivating biologists and mathematicians alike, has led 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 [1, 13, 20].)
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
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, \(\phi \) is a prescribed gravitational potential and \(\Omega \) is a bounded domain in \(\mathbb {R}^N\). While the authors of [33] suggest to augment the system with a nonzero 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 time-scales (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 \(+\kappa n-\mu n^2\) on the right-hand side of the first equation). In fact, existence of weak solutions was shown in [34] and [19] 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 \((u\cdot \nabla )u\) 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 \(\frac{\partial c}{\partial \nu }=1-c\) on \({{\,\mathrm{\partial \Omega }\,}}\). 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., \(u\equiv 0\)) variants of (1.1) have been investigated in [3] and [9]. The former considers a stationary (and hence doubly elliptic) system and establishes existence and uniqueness of a classical solution for any prescribed mass \(M:=\int \limits _{\Omega }\!n>0\). The latter studies a parabolic-elliptic variant and attains results on global and bounded classical solutions and their long-term behavior. Concerning nonzero Dirichlet data for c, we are only aware of two 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 \(N\ge 2\) in a Stokes variant of (1.1) with nonlinear diffusion satisfying \(m\ge 1\) for \(N=2\) and \(m>\frac{3N-2}{2N}\) if \(N\ge 3\) [35]. Results on more regular solutions and included logistic population growth appear to be missing for the Dirichlet boundary data case.
(See also [22] and [25, 26] for first analytical results concerning well-posedness of systems closely related to (1.1) with mixed boundary conditions, [15] for a more general fluid-free one-dimensional system with nonzero Dirichlet or Neumann boundary data and [5, 21, 33] 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 \(\Omega \subset \mathbb {R}^N\) with \(N\in \{2,3\}\) and \(\nu \) denoting the outward normal vector field on \({{\,\mathrm{\partial \Omega }\,}}\). We prescribe \(\kappa \ge 0\), \(\mu >0\), a time constant function \(c_*\) satisfying
a gravitational potential function \(\phi \) fulfilling
and initial data \((n_0,c_0,u_0)\) satisfying
with \(q>N\), \(\varrho \in (\frac{N}{4},1)\). Herein, \(A:=-\mathcal {P} \Delta \) denotes the Stokes operator with its domain \(D(A):=W^{2,2}\left( \Omega ; \mathbb {R}^{N}\right) \cap W_{0}^{1,2}\left( \Omega ; \mathbb {R}^{N}\right) \cap L_{\sigma }^{2}(\Omega )\) with \(L_{\sigma }^{2}(\Omega ):=\left\{ \varphi \in L^{2\;\!}\!\left( \Omega ;\mathbb {R}^N\right) \,\vert \,\nabla \cdot \varphi =0\right\} \) and \(\mathcal {P}\) stands for the Helmholtz projection of \(L^{2\;\!}\!\left( \Omega ;\mathbb {R}^{N}\right) \) onto \(L_{\sigma }^{2}(\Omega ).\)
Theorem 1.1
Let \(N\in \{2,3\}\) and \(\Omega \subset \mathbb {R}^N\) be a bounded domain with smooth boundary. Suppose that \(\kappa \ge 0\) and \(\mu >0\) and that the functions \(c_*\) and \(\phi \) satisfy (1.3) and (1.4), respectively. Then, for any \(n_0,c_0\) and \(u_0\) complying with (1.5), system (1.2) admits at least one global weak solution (n, c, u) in the sense of Definition 2.1.
Outline. In Sect. 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 Sect. 4, we will concern ourselves with improving the bounds on n, where, in particular, time-space information on \(\nabla n\) is the main objective of the section. In Sect. 5, we prepare estimates on the time derivatives, which upon combination with boundedness results of previous sections allow for the construction of a limit object by means of an Aubin–Lions-type argument at the start of Sect. 6. Finally, in the second part of Sect. 6, we will verify that the limit solution indeed satisfies the properties required of a global weak solution.
2 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 \(W_{0,\sigma }^{1,p}\big (\Omega ;\mathbb {R}^N\big ):=W_0^{1,p}\big (\Omega ;\mathbb {R}^N\big )\cap L_\sigma ^2(\Omega )\) for \(p\ge 1\).
Definition 2.1
A triple (n, c, u) of functions
with \(n\ge 0\) and \(c\ge 0\) in \({{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty )\) will be called a global weak solution of (1.2) if
if
holds for all \(\varphi \in C_0^\infty \big ({{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty )\big )\), if
is valid for all \(\hat{\varphi }\in C_0^\infty \big (\Omega \times [0,\infty )\big )\), and if
is fulfilled for all \(\psi \in C_{0}^\infty \big (\Omega \times [0,\infty );\mathbb {R}^N\big )\) with \(\nabla \cdot \psi \equiv 0\).
3 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 \((\rho _\varepsilon )_{\varepsilon \in (0,1)}\subset C_0^\infty (\Omega )\) of smooth cutoff functions satisfying
we introduce the corresponding family of approximating problems to (1.2) given by
where \(f_\varepsilon (s):=\frac{1}{(1+\varepsilon s)^3}\) and \(g_\varepsilon (s):=\frac{s}{1+\varepsilon s}\) for \(s\ge 0\) and \(\varepsilon \in (0,1)\).
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 substitute \(\hat{c}_{\varepsilon }:=c_*-c_\varepsilon \) 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 \(\varepsilon \in (0,1)\) 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.
Lemma 3.1
Let \(\Omega \subset \mathbb {R}^N\) be a bounded domain with smooth boundary, \(q>N\), \(\varrho \in (\frac{N}{4},1)\), \(\kappa \ge 0\), \(\mu >0\). Suppose that \(c_*\) and \(\phi \) satisfy (1.3) and (1.4), respectively, and that \(n_0\), \(c_0\) and \(u_0\) comply with (1.5). Then for any \(\varepsilon \in (0,1)\), there exist \(T_{max,\;\!\varepsilon }\in (0,\infty ]\) and a uniquely determined triple \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of functions
which, together with some \(P_\varepsilon \in C^{1,0}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\times (0,T_{max,\;\!\varepsilon })\right) \), solve (3.1) in the classical sense and satisfy \(n_\varepsilon \ge 0\) and \(c_\varepsilon \ge 0\) in \({{\,\mathrm{\overline{\Omega }}\,}}\times [0,T_{max,\;\!\varepsilon })\). Moreover, either \(T_{max,\;\!\varepsilon }=\infty \) or
Proof
Augmenting well-established fixed point arguments as, e.g., presented in [38, 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 \(c_\varepsilon =c_*-\hat{c}_{\varepsilon }\), 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 \(T\in (0,1]\), to be specified later, we define the Banach space \(X:=L^{\infty \;\!}\!\big ((0, T) ; C^{0}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) \times W_0^{1,q}(\Omega ) \times D\!\left( A^{\varrho }\right) \big )\) and its subset
Next, denoting by \(\big (e^{t\Delta }\big )_{t\ge 0}\), \(\big (e^{t\Delta '}\big )_{t\ge 0}\) and \(\big (e^{-tA}\big )_{t\ge 0}\) the Neumann heat semigroup, the Dirichlet heat semigroup and the Stokes semigroup with Dirichlet boundary data, respectively, we introduce the mapping \(\Phi :=(\Phi _1,\Phi _2,\Phi _3):X\rightarrow X\) given by
and
We will now show that \(\Phi \) acts as a contracting self-map on S, provided R and T are suitably fixed beforehand. Dropping the \(\varepsilon \)-subscript for readability, we pick \((n_1,\hat{c}_1,u_1)\), \((n_2,\hat{c}_2,u_2)\in S\) and observe that according to (3.4)
Hence, drawing on semigroup estimates as, e.g., provided by [37, Lemma 1.3], [4, Lemma 2.1] and [18, Lemma 3.1], we can find \(C_1=C_1(\Omega )>0\) such that
where we also used the facts that \(\rho _\varepsilon \le 1\) in \(\Omega \), \(f_\varepsilon \le 1\) in \([0,\infty )\). Moreover, we have \(|f_\varepsilon (a)-f_\varepsilon (b)|\le |a-b||a^2+b^2+ab+3a+3b+3|\) for \(a,b\in [0,\infty )\) and all \(\varepsilon \in (0,1)\) and \(q>N\) as well as \(D(A^\varrho )\hookrightarrow C^{\theta }\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) \) for any \(\theta \in (0,2\varrho -\tfrac{N}{2})\) (e.g., [30, Lemma III.2.4.3] and [8, Thm. 5.6.5]) so that we can find \(C_2=C_2(c_*,\kappa ,\mu ,\varrho ,N,q,R,\Omega )\) such that
Similarly, noting that \(|g_\varepsilon (a)-g_\varepsilon (b)|\le |a-b|\) for \(a,b\in [0,\infty )\) we can draw on semigroup theory for the Dirichlet heat semigroup (see [28, Proposition 48.4] and [12]) and (3.5) to conclude the existence of \(C_3=C_3(c_*,\varrho ,N,q,R,\Omega )>0\) satisfying
For (3.6), we rely on semigroup estimates for the Stokes equation (cf. [4, Lemma 2.3] and [40, Lemma 3.1]) to obtain \(C_4=C_4(\phi ,\varrho ,N,R,\Omega )>0\) such that
so that collecting (3.7), (3.8) and (3.9) yields
with some \(C_5=C_5(c_*,\kappa ,\mu ,\phi ,\varrho ,N,q,R,\Omega )>0\). Moreover, since the Dirichlet heat-semigroup estimates provide \(C_6=C_6(\Omega )>0\) such that
we find that for some \(C_7=C_7(c_*,\Omega )>0\)
Hence, by first taking \(R>3\max \big \{\big \Vert (n_0,c_*-c_0,u_0)\big \Vert _{X},2C_7\big \}\) and then \(T\in (0,1]\) sufficiently small such that \(C_5\big (T+T^\frac{1}{2}+T^{\frac{1}{2}-\frac{N}{2q}}+T^{1-\varrho }\big )<\frac{1}{3}\), we firstly see from (3.11) that \(\Phi \) maps S onto itself and secondly from (3.10) that indeed \(\Phi \) also acts as contraction on S. Aided by Banach’s fixed point theorem, we therefore obtain a unique \((n_\varepsilon ,\hat{c}_{\varepsilon },u_\varepsilon )\in S\) with \(\Phi (n_\varepsilon ,\hat{c}_{\varepsilon },u_\varepsilon )=(n_\varepsilon ,\hat{c}_{\varepsilon },u_\varepsilon )\). In light of standard bootstrapping procedures drawing on regularity theories for parabolic equations and the Stokes semigroup [16, 27, 31], one can verify that \((n_\varepsilon ,\hat{c}_{\varepsilon },u_\varepsilon )\) actually satisfies the claimed regularity properties, which then entails the existence of a corresponding \(P_{\varepsilon }\) such that \((n_\varepsilon ,\hat{c}_{\varepsilon },u_\varepsilon ,P_{\varepsilon })\) solves (3.2) classically in \(\Omega \times (0,T)\). Uniqueness of \((n_\varepsilon ,\hat{c}_{\varepsilon },u_\varepsilon )\) 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 \((0,T_{max,\;\!\varepsilon })\) such that either \(T_{max,\;\!\varepsilon }=\infty \) or
Clearly, by substituting \(c_\varepsilon =c_*-\hat{c}_{\varepsilon }\) (and recalling (1.3)), we immediately obtain the desired results for (3.1), where, finally, the nonnegativity of \(n_\varepsilon \) and \(c_\varepsilon \) is entailed by two applications of the maximum principle to the first and second equation of (3.1). \(\square \)
For the remainder of the work, we will now assume that \(N\in \{2,3\}\), \(\Omega \subset \mathbb {R}^N\), \(\kappa \ge 0\), \(\mu >0\), \(q>N\), \(\varrho \in (\frac{N}{4},1)\), \(c_*,\phi \) satisfying (1.3) and (1.4), respectively, and initial data \(n_0,c_0,u_0\) obeying (1.5) are fixed and, accordingly, for \(\varepsilon \in (0,1)\) denote by \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) the triple of functions provided by Lemma 3.1 and by \(T_{max,\;\!\varepsilon }\) 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 \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) satisfies
Proof
Making use of the fact that \(u_\varepsilon \) is divergence free, by integrating the first equation of (3.1) over \(\Omega \) and utilizing integration by parts as well as Young’s inequality, we deduce that for all \(\varepsilon \in (0,1)\)
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 \(n_\varepsilon \) and an ODE comparison argument, implies
Furthermore, integration of (3.12) over \(\big ((t-1)_+,t\big )\) now provides,
for all \(t\in (0,T_{max,\;\!\varepsilon })\) and all \(\varepsilon \in (0,1)\). Finally, by the maximum principle, we instantly obtain that
which completes the proof. \(\square \)
In order to distill further uniform bounds from the somewhat sparse (yet sufficiently powerful) space–time information on \(n_\varepsilon ^2\) provided by Lemma 3.2, we state the following comparison result for ordinary differential equations. This lemma is copied from [17, Lemma 3.4], whereto we refer the reader for details of the proof.
Lemma 3.3
For some \(T\in (0,\infty ]\), let \(y\in C^{1}\!\left( (0,T)\right) \cap C^{0}\!\left( [0,T)\right) \), \(h\in C^{0}\!\left( [0,T)\right) \), \(h\ge 0\), \(C>0\), \(a>0\) satisfy
for all \(t\in (0,T)\). Then \(y\le y(0)+\frac{C}{1-e^{-a}}\) throughout (0, T).
With the comparison lemma above, we can make now turn to obtain some uniform bounds for the third solution component.
Lemma 3.4
There is \(C>0\) such that for any \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) satisfies
Proof
First, we test the third equation in (3.1) against \(u_\varepsilon \), integrate by parts over \(\Omega \), and employ the Young and Poincaré inequalities as well as (1.4) to conclude the existence of \(C_1>0\) such that for all \(\varepsilon \in (0,1)\)
is valid on \((0,T_{max,\;\!\varepsilon })\). Then, again denoting by \(\mathcal {P}\) the Helmholtz projection and by A the Stokes operator, we multiply the projected third equation by \(Au_\varepsilon \) to obtain \(C_2>0\) such that for all \(\varepsilon \in (0,1)\) the inequality
holds on \((0,T_{max,\;\!\varepsilon })\), where we once more made use of the boundedness of \(\nabla \phi \) and Young’s inequality. Since \(A=-\mathcal {P}\Delta \), we have \(\Vert A^\frac{1}{2}u_\varepsilon \Vert _{L^{2}(\Omega )}=\Vert \nabla u_\varepsilon \Vert _{L^{2}(\Omega )}\), so that 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
on \((0,T_{max,\;\!\varepsilon })\) for all \(\varepsilon \in (0,1)\). Drawing on Lemmas 3.2 and 3.3, we conclude that there is \(C_5>0\) satisfying
Finally, relying on the Sobolev embedding theorem, we find \(C_6>0\) such that
The uniform bounds on \(u_\varepsilon \) in \(L^{\infty \;\!}\!\left( (0,T_{max,\;\!\varepsilon });L^{6}(\Omega )\right) \) and \(n_\varepsilon \) in \(L^{2\;\!}\!\left( \Omega \times (0,T_{max,\;\!\varepsilon })\right) \) will now be the key ingredient in obtaining information on \(\nabla c_\varepsilon \). We start by exploiting the fact that \(c_*\) is constant in time to establish an ordinary differential inequality for \(\int \limits _{\Omega }\!|\nabla c_\varepsilon (\cdot ,t)|^2\) on \((0,T_{max,\;\!\varepsilon })\).
Lemma 3.5
There exists \(C>0\) such that for all \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) satisfies
on \((0,T_{max,\;\!\varepsilon })\).
Proof
Since the boundary conditions in (3.1) imply that \(\frac{\partial }{\partial t}c_{\varepsilon }\big \vert _{{{\,\mathrm{\partial \Omega }\,}}}=0\) on \((0,T_{max,\;\!\varepsilon })\), we can multiply the second equation of (3.1) by \(-\Delta c_\varepsilon \) and integrate by parts to find that for all \(\varepsilon \in (0,1)\)
on \((0,T_{max,\;\!\varepsilon })\). Employing Young’s inequality to the last two terms on the right and making use of the fact that \(|g_\varepsilon (s)|\le s\) for all \(s\ge 0\) we obtain that for all \(\varepsilon \in (0,1)\)
In view of Lemma 3.2, there is \(C_1>0\) satisfying
Furthermore, relying on the Hölder inequality and Lemma 3.4, we find \(C_2>0\) such that for all \(\varepsilon \in (0,1)\) we have
The Gagliardo–Nirenberg inequality and Lemma 3.2, moreover, imply the existence of \(C_3>0\) and \(C_4>0\) satisfying
on \((0,T_{max,\;\!\varepsilon })\) for all \(\varepsilon \in (0,1)\), so that an application of Young’s inequality entails the existence of \(C_5>0\) such that for all \(\varepsilon \in (0,1)\) we have
A combination of (3.15)–(3.17) finally shows that with \(C:=\max \{C_1,C_5\}\) we obtain
on \((0,T_{max,\;\!\varepsilon })\) for all \(\varepsilon \in (0,1)\). \(\square \)
Next, we combine the recently established differential inequality with the Gagliardo–Nirenberg inequality, the comparison Lemma 3.3 and the space–time bound for \(n_\varepsilon \) from Lemma 3.2 to obtain the following.
Lemma 3.6
There is \(C>0\) such that for any \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) fulfills
for all \(t\in (0,T_{max,\;\!\varepsilon })\).
Proof
According to the Gagliardo–Nirenberg inequality, Hölder’s inequality and Lemma 3.2, there are \(C_1,C_2>0\) such that for all \(\varepsilon \in (0,1)\)
on \((0,T_{max,\;\!\varepsilon })\), which upon combination with the differential inequality for \(c_\varepsilon \) 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 \(\varepsilon \in (0,1)\) we have
Then, returning to the differential inequality for \(c_\varepsilon \) from Lemma 3.5, we obtain \(C_4>0\) such that for all \(\varepsilon \in (0,1)\) we have \(\int \limits _{(t-1)_+}^t\!\int \limits _{\Omega }\!|\Delta c_\varepsilon |^2 \le C_4\) on \((0,T_{max,\;\!\varepsilon })\) 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 find \(C_5>0\) such that for all \(\varepsilon \in (0,1)\)
completing the proof by drawing on the previous parts of this lemma and Lemma 3.2. \(\square \)
The boundedness property of \(\nabla c_\varepsilon \) in \(L^{\infty \;\!}\!\left( (0,T_{max,\;\!\varepsilon });L^{2}(\Omega )\right) \) 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 \(\varepsilon \)-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 \(\varepsilon \) the maximal existence time \(T_{max,\;\!\varepsilon }\) provided by Lemma 3.1 is indeed not finite.
Lemma 3.7
For all \(\varepsilon \in (0,1)\), the solution of (3.1) is global in time, i.e., \(T_{max,\;\!\varepsilon }=\infty \).
Proof
We fix \(\varepsilon \in (0,1)\) and assume for contradiction that \(T:=T_{max,\;\!\varepsilon }<\infty \). Subsequently, we will consider estimates for the quantities appearing in the extensibility criterion (3.3), and, as our estimation process relies on employing semigroup arguments to the second component, we will once more return to working in the transformed system (3.2). Since an immediate estimation of \(n_\varepsilon \) in \(L^{\infty \;\!}\!\left( \Omega \times (0,T)\right) \) is out of our reach, we will first establish the boundedness of \(n_\varepsilon \) in \(L^{\infty \;\!}\!\left( (0,T);L^{6}(\Omega )\right) \). To this regard, we multiply the first equation of (3.2) by \(n_\varepsilon ^5\) and integrate by parts to find that due to \(u_\varepsilon \) being divergence-free
on (0, T). Here, the last two integrals disappear because of the boundary conditions and the fact that \(\rho _\varepsilon =0\) on \({{\,\mathrm{\partial \Omega }\,}}\). Moreover, noticing that \(|f_\varepsilon (s)s^3|\le \frac{1}{\varepsilon ^3}\) for all \(s\ge 0\) and that \(|\rho _\varepsilon |\le 1\) on \(\Omega \), we make use of two applications of Young’s inequality to find \(C_1:=C_1(\kappa ,\mu )>0\) such that
Since \(|\nabla \hat{c}_{\varepsilon }-\nabla c_*|^2=|\nabla c_\varepsilon |^2\), we conclude from Lemma 3.6 and a straightforward comparison argument that there is \(C_2=C_2(\varepsilon )>0\) satisfying
This bound at hand, we pick \(\varrho \in (\frac{N}{4},1)\) as in (1.5) and then rely on smoothing properties of the Stokes semigroup (e.g., [11, p.201] and [40, Lemma 3.1]), (1.5), (1.4) and (3.19) to obtain \(C_3=C_3(\varepsilon )>0\) satisfying
for all \(t\in (0,T)\), which, due to the embedding \(D(A^\varrho )\hookrightarrow C^{\theta }\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) \) for any \(\theta \in (0,2\varrho -\tfrac{N}{2})\) (cf. [30, Lemma III.2.4.3] and [8, Thm. 5.6.5]), also entails that there is some \(C_4=C_4(T,\varepsilon )>0\) such that
Next, drawing on the Dirichlet heat-semigroup representation of \(\hat{c}_{\varepsilon }\) we find that
for all \(t\in (0,T)\). Picking \(q\in (N+2,6)\) and letting \(t_0:=\min \{1,\frac{T}{2}\}\), in light of well-known semigroup estimates ([12]), we can hence obtain \(C_5>0\) satisfying
where we also relied on the estimate \(g_\varepsilon (s)\le s\) for \(s\ge 0\). Here, due to \(c_*-\hat{c}_{\varepsilon }=c_\varepsilon \), we can make use of (3.20) and Lemma 3.6 for the first integral and (3.19) combined with \(q<6\), Lemma 3.2 and (1.3) for the second integral, to find \(C_6=C_6(T,\varepsilon )>0\) and, since \(\frac{2N}{N-2}\ge 6>q\) entails \(-\frac{1}{2}-\frac{N}{2}(\frac{1}{2}-\frac{1}{q})>-1\), also \(C_7=C_7(T,\varepsilon )\) satisfying
for all \(t\in (t_0,T)\). With these bounds, the facts \(|f_\varepsilon (s)s|\le \frac{1}{3\varepsilon }\) for all \(s\ge 0\) and that \(|\rho _\varepsilon |\le 1\) on \(\Omega \), we can easily check that a Moser–Alikakos-type iteration procedure (e.g., [32, Lemma A.1] employed with \((p_0,q_1,q_2)=(6,q,3)\)) becomes applicable to (3.2) and that hence there is \(C_8=C_8(T,\varepsilon )>0\) such that \(\Vert n_\varepsilon \Vert _{L^{\infty }(\Omega )}\le C_8\) on (0, T). Hence, we find
contradicting (3.3) and therefore proving \(T_{max,\;\!\varepsilon }=\infty \). \(\square \)
4 Refined a priori information on \(n_\varepsilon \)
While the uniform bounds for \(c_\varepsilon \) and \(u_\varepsilon \) provided by Sect. 3 would already be strong enough for our limit procedure, we still lack sufficiently good uniform bounds for \(n_\varepsilon \). As it turns out, the space–time bound for \(\nabla c_\varepsilon \) of Lemma 3.6, however, can be exploited when considering the functional \(y_\varepsilon (t):=\int \limits _{\Omega }\!\big (n_\varepsilon \ln n_\varepsilon \big )(\cdot ,t)\), which has often been a good resource for information in chemotaxis settings ( [7, 19, 38, 42]). We start by formulating a corresponding functional inequality.
Lemma 4.1
There exists \(C>0\) such that for all \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) satisfies
on \((0,\infty )\).
Proof
In light of the first equation of (3.1), the fact that \(u_\varepsilon \) is divergence-free and two integrations by parts we see that
on \((0,\infty )\) for all \(\varepsilon \in (0,1)\). Observing that again both boundary integrals disappear due to the prescribed boundary conditions and the fact that \(\rho _\varepsilon (x)=0\) for \(x\in {{\,\mathrm{\partial \Omega }\,}}\) and noting that there is some \(C_1>0\) satisfying \(\kappa s-\mu s^2\le C_1\) for all \(s\ge 0\) and such that \((\kappa s-\frac{\mu }{2}s^2)\ln (s)\le C_1\) for all \(s>0\), we may hence estimate
on \((0,\infty )\) for all \(\varepsilon \in (0,1)\). To further estimate the integral on the right, we make use of the fact that \(|\rho _\varepsilon (x)f_\varepsilon (s)|=\frac{\rho _\varepsilon (x)}{(1+\varepsilon s)^3}\le 1\) for all \(s\ge 0\), \(x\in \Omega \) and \(\varepsilon \in (0,1)\) and two applications of Young’s inequality to obtain
on \((0,\infty )\) for all \(\varepsilon \in (0,1)\), which concludes the proof upon the choice of \(C:=\max \{\frac{1}{4},2C_1\}\). \(\square \)
Clearly, we can draw on previously established space–time bounds to extract additional space–time information on \(\nabla \sqrt{n_\varepsilon }\) from the previous Lemma, which in a second interpolation step can also be refined to a bound on \(\nabla n_\varepsilon \) in \(L^{\frac{4}{3}\;\!}\!\left( \Omega \times (0,\infty )\right) \).
Lemma 4.2
For any \(T>0\) there is \(C(T)>0\) such that for all \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) satisfies
Proof
Integration of the differential inequality featured in Lemma 4.1 over (0, T) provides \(C_1>0\) such that for all \(\varepsilon \in (0,1)\)
Recalling the bounds provided by Lemma 3.2, Lemma 3.6 and (1.5) as well as the obvious estimate \(-\frac{1}{e}\le s\ln s\) for all \(s\ge 0\), the conclusion is immediate. \(\square \)
Lemma 4.3
For all \(T>0\), there exists \(C(T)>0\) such that for all \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) fulfills
Proof
Rewriting the integral under consideration and employing Young’s inequality twice, we find that
for all \(\varepsilon \in (0,1)\). Reordering and making use of the bounds provided by Lemmas 3.2 and 4.2, we obtain the asserted bound.
5 Regularity estimates for the time derivatives
As final element for an Aubin–Lions-type argument we are going to undertake in Sect. 6, we now prepare uniform bounds for the time derivatives in suitable spaces.
Lemma 5.1
For every \(T>0\) there exists \(C(T)>0\) such that for any \(\varepsilon \in (0,1)\) the solution \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) fulfills
and
Proof
Given \(T>0\), we fix \(\varphi \in L^{\infty \;\!}\!\big ((0,T);W_0^{1,4}(\Omega )\big )\) with \(\Vert \varphi \Vert _{L^{\infty \;\!}\!((0,T);W_0^{1,4}(\Omega ))}\le 1\) and test the first equation of (3.1) against \(\varphi \) to obtain
for all \(t>0\) and \(\varepsilon \in (0,1)\), where the boundary integrals again disappear due to the boundary conditions for \(n_\varepsilon \) and \(u_\varepsilon \) and the fact that \(\rho _\varepsilon =0\) on \({{\,\mathrm{\partial \Omega }\,}}\). Here, we deduce from multiple applications of Hölder’s inequality and the fact that for all \(\varepsilon \in (0,1)\) we have \(\rho _\varepsilon f(n_\varepsilon )\le 1\) on \(\Omega \times (0,\infty )\) that
Here, we make use of Lemma 3.4 and multiple uses of Young’s inequality to conclude that there is \(C_1>0\) such that for all \(\varepsilon \in (0,1)\)
holds for all \(t>0.\) Then, since \(\Vert \varphi \Vert _{L^{\infty \;\!}\!((0,T);W_0^{1,4}(\Omega ))}\le 1\) and \(W_0^{1,4}(\Omega )\hookrightarrow L^{\infty }(\Omega )\), an integration over (0, T) immediately entails (5.1) thanks to Lemmas 3.2, 3.6 and 4.3.
Similarly, fixing \(\hat{\varphi }\in L^{\infty \;\!}\!\big ((0,T);W_0^{1,4}(\Omega )\big )\) with \(\Vert \hat{\varphi }\Vert _{L^{\infty \;\!}\!((0,T);W_0^{1,4}(\Omega ))}\le 1\) and testing the second equation of (3.1) against \(\hat{\varphi }\), we find \(C_2>0\) satisfying
for all \(\varepsilon \in (0,1)\), where we again made use of Lemmas 3.4 and 3.2, Young’s inequality and the fact that \(g_\varepsilon (s)\le s\) for all \(s\ge 0\), so that (5.2) is an evident consequence of the spatiotemporal bounds provided by Lemmas 3.6 and 3.2.
Finally, for any fixed \(\psi \in L^{\infty \;\!}\!\big ((0,T);W_{0,\sigma }^{1,2}(\Omega )\big )\) with \(\Vert \psi \Vert _{L^{\infty \;\!}\!((0,T);W_{0,\sigma }^{1,2}(\Omega ))}\le 1\), we multiply the third equation in 3.1 by \(\psi \) and integrate the resulting equation to derive that
for all \(\varepsilon \in (0,1)\), in light of Lemma 3.4, (1.4) and Lemma 3.2, and from which we conclude (5.3). \(\square \)
6 Existence of a limit solution. The proof of Theorem 1.1
Collecting the uniform bounds presented in Sects. 2–5, we can now construct a limit object, which satisfies all the regularity requirements presented in Definition 2.1.
Proposition 6.1
There exists a sequence \((\varepsilon _j)_{j\in \mathbb {N}}\subset (0,1)\) with \(\varepsilon _j\searrow 0\) as \(j\rightarrow \infty \) and functions
and such that the solutions \((n_\varepsilon ,c_\varepsilon ,u_\varepsilon )\) of (3.1) fulfill
as \(\varepsilon =\varepsilon _j\searrow 0\).
Proof
A combination of an Aubin–Lions-type lemma ( [29, Sec. 8, Corollary 4]) with the bounds presented in Lemmas 3.2, 4.3 and 5.1 ensures that
and that hence we can find a subsequence \((\varepsilon _j)_{j\in \mathbb {N}}\) with \(\varepsilon _j\searrow 0\) as \(j\rightarrow \infty \) and \(n\in L_{loc}^{\frac{4}{3}}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty )\right) \) such that \(n_\varepsilon \rightarrow n\) in \(L_{loc}^{\frac{4}{3}}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty )\right) \) and a.e. in \(\Omega \times (0,\infty )\). Additionally, the spatiotemporal bound of Lemma 4.3 also allows us to conclude that for some \(z\in L_{loc}^{\frac{4}{3}}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty );\mathbb {R}^N\right) \) we have \(\nabla n_\varepsilon {{\,\mathrm{\rightharpoonup }\,}}z\in L_{loc}^{\frac{4}{3}}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty );\mathbb {R}^N\right) \) along another subsequence (still denoted by \((\varepsilon _j)_{j\in \mathbb {N}}\)). Then, the established convergence of \(n_\varepsilon \rightarrow n\) in \(L_{loc}^{\frac{4}{3}}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty )\right) \), however, necessarily entails that z coincides with \(\nabla n\) due to the uniqueness of the weak limit and thereby verifies (6.2). Moreover, noting that \(\Theta :[0,\infty )\rightarrow [0,\infty ]\), \(\Theta (s):=s\ln (s+1)\) is an increasing and convex function with
we observe that according to Lemma 4.2 for any \(T>0\) there is \(C>0\) satisfying
which, by a result of de la Vallée–Poussin (e.g., [24, II.T22]), entails that \(\left\{ n_\varepsilon ^2\right\} _{\varepsilon \in (0,1)}\) is equi-integrable. Thus, a combination of the equi-integrability with the a.e. convergence of \(n_\varepsilon \) and Vitali’s theorem yields (6.1) along a further subsequence. Then, since \(|\rho _\varepsilon f_\varepsilon (n_\varepsilon )|\le 1\) in \(\Omega \times (0,\infty )\) for all \(\varepsilon \in (0,1)\) and \(\rho _\varepsilon f_\varepsilon (n_\varepsilon )\rightarrow 1\) a.e. in \(\Omega \times (0,\infty )\) as \(\varepsilon =\varepsilon _j\searrow 0\), we find from (6.1) and arguments akin to, e.g., [41, Lemma A.4], that (6.3) holds as well. Likewise, we may also conclude (6.4) from (6.1). Working along similar lines for the second and third components, we can draw on the bounds of Lemmas 3.2, 3.6 and 5.1 to obtain an additional subsequence along which (6.5), (6.6) and (6.7) hold and iterating the arguments once more with the bounds of Lemmas 3.4 and 5.1 concerning \(u_\varepsilon \), finally, also (6.8), (6.9) and (6.10). The claimed regularity properties of (n, c, u) and \(c-c_*\) are clearly a direct consequence of (6.1), (6.2), (6.6), (6.7), (6.8), (6.10) and (1.3) and the fact that \(c_\varepsilon -c_*=0\) on \({{\,\mathrm{\partial \Omega }\,}}\times [0,\infty )\). \(\square \)
Finally, it remains to be checked that the limit objects provided by Proposition 6.1 indeed satisfy 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).
Proof of Theorem 1.1
Since the regularity requirements imposed on a weak solution by Definition 2.1 are already covered by the properties obtained in Lemma 6.1, we only have to verify that the limit objects obtained in said lemma also satisfy the integral identities (2.1)–(2.3). We pick \(\varphi \in C_0^\infty \big ({{\,\mathrm{\overline{\Omega }}\,}}\times [0,\infty )\big )\), \(\hat{\varphi }\in C_0^\infty \big (\Omega \times [0,\infty )\big )\) and \(\psi \in C_{0}^\infty \big (\Omega \times [0,\infty );\mathbb {R}^N\big )\) with \(\nabla \cdot \psi \equiv 0\) and then fix \(T>0\) such that \(\varphi ,\hat{\varphi },\psi \equiv 0\) in \(\Omega \times [T,\infty )\). Now, we test the first equation of (3.1) against \(\varphi \) and integrate by parts to obtain
for all \(\varepsilon \in (0,1)\), where we made use of the prescribed boundary conditions and the fact that \(u_\varepsilon \) is solenoidal. According to (6.1) and (6.2), we immediately find that
and
Drawing on (6.1), (6.3) combined with (6.7) and (6.1) together with (6.8), we also conclude
and
so that passing to the limit in (6.11) immediately entails (2.1) since \(T>0\) was chosen such that \(\varphi \equiv 0\) in \(\Omega \times [T,\infty )\).
Similarly, multiplying the second equation of (3.1) by \(\hat{\varphi }\) and integrating, we have
for all \(\varepsilon \in (0,1)\). Therefore, utilizing (6.5), (6.7), (6.4) and (6.8), we may also pass to the limit in this equation and obtain (2.2). Finally, testing the third equation in (3.1) against \(\psi \) and integrating by parts yields
where (6.8), (6.10) and (6.1) imply that we may, once more, pass to the limit and obtain (2.3), concluding the proof. \(\square \)
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Acknowledgements
Tobias Black acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (project no. 411007140). Chunyan Wu was supported by the China Scholarship Council (CSC NO. 201906070104) and the Applied Fundamental Research Program of Sichuan Province (no. 2020YJ0264) during her stay at the University of Paderborn.
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Black, T., Wu, C. Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation. Z. Angew. Math. Phys. 72, 135 (2021). https://doi.org/10.1007/s00033-021-01565-z
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DOI: https://doi.org/10.1007/s00033-021-01565-z