Abstract
This paper deals with a chemotaxis-Navier–Stokes model with indirect signal production involving Dirichlet signal boundary condition in a bounded domain with smooth boundary. A recent literature has asserted that for all reasonably regular initial data, the associated no-flux/saturation/no-flux/no-slip problem possesses at least one globally defined weak solution in the logistic-type degradation here is weaker than quadratic case. But the knowledge on regularity properties of solution has not yet exceeded some information on fairly basic integrability features. The present study reveals that each of these weak solutions becomes eventually classical and bounded under some suitably strong sub-quadratic degradation assumption and an explicit smallness condition. Furthermore, in comparison with the related contributions in the case of the direct signal production, our findings inter alia rigorously reveal that the indirect signal production mechanism genuinely contributes to the global solvability and eventual smoothness of the chemotaxis-Navier–Stokes system.
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1 Introduction
Chemotaxis is the directed movement of cells or organisms in response to the gradients of concentration of the chemical stimuli, plays essential roles in various biological process such as aggregative patterns of bacteria, slime mold formation, angiogenesis in tumor progression and wound healing [9, 23, 34]. The classical Keller–Segel model proposed by Keller and Segel [26] to the following chemotaxis model
where \(\Omega \subset {\mathbb {R}}^N~(N\ge 1)\) is a bounded domain with smooth boundary, \(\chi >0\) is the chemotactic sensitivity and the unknown variables n(x, t) and v(x, t) represent the density of cell and the concentration of chemical signal, respectively. The system (1.1) have received considerable interest of mathematicians to develop a detailed qualitative analysis concentrating on the global existence, boundedness and asymptotic behavior of solutions for the corresponding homogeneous Neumann initial-boundary value problem (IBVP). To be specific, if \(N=1\), the corresponding solution is always globally bounded [33]; if \(N=2\), a critical mass blow-up phenomenon occurs in the radially symmetric setting, namely solution is globally bounded in the case \(\Vert n_{0}\Vert _{L^{1}(\Omega )}<\frac{8\pi }{\chi }\) [31], while the corresponding solution blows up in finite time for the case \(\Vert n_{0}\Vert _{L^{1}(\Omega )}>\frac{8\pi }{\chi }\) [22, 24]; if \(N\ge 3\), the finite-time blow-up of radial solutions may happen with arbitrarily small mass \(\Vert n_{0}\Vert _{L^{1}(\Omega )}\) [46], and the corresponding solution exists globally in time and converges to the constant steady state provided that \(\Vert n_{0}\Vert _{L^{\frac{N}{2}}(\Omega )}+\Vert v_{0}\Vert _{L^{N}(\Omega )}\) is sufficiently small [8, 40].
In the case that the time scale of chemotactic movement is rather long, the proliferation and death of cell should be taken into account. On the basis of this fact, a large body of work has been devoted to the following Keller–Segel-growth model
where \(r\in {\mathbb {R}}\), \(\mu >0\) and \(\alpha >1\). In contrast to the minimal Keller–Segel system (1.1) in which solutions of problem may blow up in the case \(N\ge 2\) [22, 24, 31, 40, 46], the associated IBVP of (1.2) with quadratic degradation (i.e., \(\alpha =2\)) possesses a global bounded classical solution for the case either \(N=2\) and arbitrary \(\mu >0\) [32], or \(N\ge 3\) and suitably large \(\mu >0\) [45], and this solution was further shown to approach the spatially homogeneous steady state [47]. Of note, only the global existence of weak solutions have been obtained for \(N\ge 3\) and any \(\mu >0\) [27], it is still unknown whether or not explosions may occur for small \(\mu >0\). After all, the above results indicate that the suitably strong logistic-type degradation exerts a somewhat stabilizing influence on the system in the sense of blow-up prevention. Some analytical findings have revealed that some weaker degradation may fail to prevent finite-time blow-up of solution on \(N\ge 2\), but exclude the possibility of collapse into persistent Dirac-type measures. Indeed, it was asserted in [51] that the no-flux IBVP of (1.2) admits at least one global very weak solution for any \(\alpha >1\) and \(\mu >0\), and that these solutions were shown to stabilize toward the nontrivial spatially homogeneous steady state in the large time limit for \(\alpha \ge 2-\frac{2}{N}\) and appropriately large \(\mu >0\) [43]. For more related results on (1.2) and its numerous variants, we refer to [28, 41] and two impressive surveys [1, 2].
In the Keller–Segel model (1.1) and Keller–Segel-growth model (1.2), the chemical signal is directly produced by cells themselves. Nevertheless, the signal may be produced indirectly in numerous biological circumstances such as the Mountain Pine Beetle spread and aggregation in a forest habitat [25], predator–prey interaction in marine ecosystem [36] and tumor invasion inside the body of cancer patient [15]. In recent years, much attention has been focused on the following Keller–Segel-growth system with indirect signal production
Note that the signal production mechanism in (1.3) is indirect, that is, the chemoattractant v is not produced by cells directly, but is governed by the quantity w arising from n. The prototypical model of (1.3) (i.e., \(r=\mu =0\) in (1.3)) was proposed and studied by Fujie and Senba [16]. Several recent studies indicate that the indirect signal production mechanism can give rise to different interactions of the cross-diffusion and the mass condition of initial data or the logistic source. When \(r=\mu =0\), in the radially symmetric setting, Fujie and Senba [16] proved that the corresponding homogeneous Neumann problem of (1.3) possesses a unique globally bounded classical solution if either \(N\le 3\) or \(N=4\) and \(\Vert n_{0}\Vert _{L^{1}(\Omega )}<\frac{(8\pi )^{2}}{\chi }\), and they claimed that the classical solution in will be blowing up in finite or infinite time if \(N=4\) and \(\Vert n_{0}\Vert _{L^{1}(\Omega )}\in \left( \frac{(8\pi )^{2}}{\chi },\infty \right) \left\{ j\cdot \frac{(8\pi )^{2}}{\chi }|j\in N\right\} \) [17]. These results revealed that the qualities \(\Vert n_{0}\Vert _{L^{1}(\Omega )}=\frac{(8\pi )^{2}}{\chi }\) and \(N=4\) are critical mass and dimension in distinguishing the global boundedness and blow-up of solutions, respectively.
Models (1.1)–(1.3) are supposed that there is no interplay between cells/chemicals and their ambient surroundings. However, some experimental observations indicate that the motion of cells also can be substantially influenced by the surrounding fluid [37]. For instance, populations of aerobic bacteria suspended in sessile drops of water may exhibit quite a complex but structured collective dynamics, inter alia involving the spontaneous formation of plume-like aggregates [10, 12]. Considering the interaction, some researchers lately have investigated the following chemotaxis model
where \(n=n(x,t)\) and \(v=v(x,t)\) are defined as above. Here, \(u=u(x,t)\), \(P=P(x,t)\), \(\phi =\phi (x)\) and \(\kappa \in {\mathbb {R}}\) represent the fluid velocity field, the associated pressure of the fluid, the potential of the gravitational field and the strength of the nonlinear fluid convection correspondingly. The fluid velocity evolution may be characterized by the Stokes equation (\(\kappa =0\)) instead of the full Navier–Stokes equation (\(\kappa =1\)) if the fluid flow is comparatively slow.
In the case of no-flux boundary condition for the signal: There have been quite a comprehensive analytical results on system (1.4) at the level of solvability and stabilization when it is posed on a bounded domain \(\Omega \). For instance, when the fluid flow remains small (i.e., \(\kappa =0\)), Lorz [30] firstly gave the analytical result involving the local existence of weak solution in two or three dimensional setting. However, when the fixed number \(\kappa \in {\mathbb {R}}/\{0\}\), the fluid motion is governed by Navier–Stokes equations with nonlinear convection, and Winkler [48] proved the system admits a unique global classical solution in the case of \(N= 2\). While in the case of \(N=3\), global weak solutions have been established in [49], and the large time asymptotic behavior was discovered in [50]. Based on the literature [48], Winkler [53] showed that in the two-dimensional case such a classical solution will stabilize to a spatially uniform equilibrium.
In the case of realistic boundary condition for the signal: As pointed in Tuval et al. [37], the boundary conditions on n, v and u are central to the global flows and possible singularities. Some recent analytical results intended to introduce suitably nontrivial boundary conditions, in particular, for the signal concentration v. For instance, taking into account for the exchange of oxygen between the drop and its environment, the Robin boundary condition for v was proposed by Braukhoff [6] and then the corresponding solvability was further investigated by [7, 54]. The inhomogeneity of Robin boundary conditions restrained the use of the standard semigroup argument and the maximal Sobolev regularity. However, these obstacles have been overcome in these works by introducing a Lions–Magenes-type transformation, which in particular transformed the Robin signal boundary condition into the usual no-flux boundary value at the cost of some extra terms appeared in the equations. On the other hand, on the water-air surface, the oxygen concentration outside the drop can be assumed equal to its saturation value \(v_{*}\) inside the water to match the experiment in Tuval et al. [37]. This assumption motivated no-flux/saturation/no-slip boundary conditions
Under such boundary conditions, Wang et al. [38] developed a local energy method to show the existence of globally defined generalized solution to the three-dimensional system (1.4) with \(\kappa =0\). When the system (1.4) involves logistic source (i.e., \(f(n)=rn-\mu n^{2}\)), Black and Wu [5] showed the Stokes variant of the system (1.4) has at least one global weak solution for any suitably regular triplet of initial data. Very recently, they [4] extended the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting and investigated the eventual regularity properties in the slightly more restrictive setting of \(v_{*}\) being also constant in space. They showed that sufficiently strong logistic influence, in the sense that for \(\omega >0\) and \(\mu _{0}>0\), there is some \(\eta =\eta (\omega ,\mu _{0},v_{*})>0\) with the property that whenever
are satisfied the global weak solution eventually becomes a smooth and classical solution waiting time depending on \(\omega ,\mu _{0},\eta ,v_{*}\) and the initial data. In the nonlinear cell diffusion case (i.e., \(n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n\chi (v)\nabla v)\) is replaced by \(n_{t}+u\cdot \nabla n=\nabla \cdot (n^{m-1}\nabla n)-\nabla \cdot (n\chi (v)\nabla v)\) in (1.4)), the boundary conditions (1.5) becomes the following setting
In the above boundary conditions, Wang et al. [39] constructed the global mass-preserving solutions for system (1.4) with \(\kappa =0\) and \(m>\frac{3N-2}{2N}\). Wu and Xiang [55] established the existence of globally bounded weak solutions in the same setting. Recently, Black and Winkler [3] also investigated such boundary conditions on the full chemotaxis-Navier–Stokes system and obtained global weak solutions which either may be unbounded for \(m>\frac{7}{6}\) and \(N=3\). Very recently, the positive effect of the indirect signal production mechanism on the global solvability of the three-dimensional (3D) chemotaxis-Navier–Stokes system under Dirichlet signal boundary condition has been considered in our work [29]. Nevertheless, the knowledge on regularity properties of solution for the 3D chemotaxis-Navier–Stokes system with indirect signal production mechanism under Dirichlet signal boundary condition has not yet exceeded some information on fairly basic integrability features as derived in [29]. This inspires us to ask the following interesting and significant question: Will the indirect signal production mechanism genuinely contribute to the eventual smoothness and boundedness of solution to the 3D chemotaxis-Navier–Stokes system involving Dirichlet signal boundary condition?
Motivations and main results
Inspired by the outstanding works [3, 11, 42, 49, 52], in this paper, we investigate the following chemotaxis-Navier–Stokes with indirect signal production
in a bounded domain \(\Omega \subset {\mathbb {R}}^3\) with smooth boundary \(\partial \Omega \), where \(\nu \) denotes the outward normal vector on \(\partial \Omega \), \(\chi >0\), \(r\in {\mathbb {R}}\), \(\mu >0\) and \(\alpha \in (1,2)\). To prepare a precise presentation of our main results, throughout this work, we assume that the given gravitational potential function \(\phi \) fulfills
and the time constant function \(v_{*}\) satisfies
as well as that the initial data \(n_{0}\), \(v_{0}\), \(w_{0}\), \(u_{0}\) satisfy
where \(A=-P\Delta \) denotes the realization of the Stokes operator in \(L^{2}(\Omega ;{\mathbb {R}}^3)\) defined on its domain \(D(A):=W^{2,2}(\Omega ;{\mathbb {R}}^3)\cap W_{0}^{1,2}(\Omega ;{\mathbb {R}}^3)\cap L_{\sigma }^{2}(\Omega ;{\mathbb {R}}^3)\) with \(L_{\sigma }^{2}(\Omega ;{\mathbb {R}}^3):=\{\varphi \in L^{2}(\Omega ;{\mathbb {R}}^3)|\nabla \cdot \varphi =0\}\), and P represents the Helmholtz projection of \(L^{2}(\Omega ;{\mathbb {R}}^3)\) onto \(L_{\sigma }^{2}(\Omega ;{\mathbb {R}}^3)\). Since \(\Omega \) is bounded, we know that A is self-adjoint and sectorial in \(L_{\sigma }^{2}(\Omega ;{\mathbb {R}}^3)\), and possesses densely defined self-adjoint fractional powers \(A^{\delta }\) for arbitrary \(\delta \in {\mathbb {R}}\) [35].
Within this setting, the following result on global existence of weak solution for problem (1.8) has been established in our recent work [29].
Proposition 1.1
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^3\) with smooth boundary, and let \(\chi >0\), \(r\in {\mathbb {R}}\), \(\mu >0\), \(\alpha \in \left( \frac{4}{3},2\right) \) and \(\kappa \ne 0\). Then for given \(\phi \) fulfilling (1.9), \(v_{*}\) satisfies (1.10) and initial data \((n_0,v_0,w_0,u_0)\) complying with (1.11), the system (1.8) admits at least one global weak solution (n, v, w, u) in the sense of Definition 2.1 below.
Remark 1.1
In comparison with the existing result on the chemotaxis-Navier–Stokes system with direct signal production,Proposition 1.1 rigorously confirms that the indirect signal production mechanism genuinely facilitates the global solvability of the 3D chemotaxis-Navier–Stokes system. Indeed, the global existence of a weak solution to the 3D chemotaxis-Navier–Stokes system with quadratic degradation (i.e., \(\alpha =2\)) was obtained in [4, Theorem 1.1], while Proposition 1.1 in this work established the global solvability in the same sense as that of [4] to the 3D chemotaxis-Navier–Stokes system (1.8) with suitably weak logistic-type degradation (i.e., \(\alpha \in (\frac{4}{3},2)\)). In addition, Proposition 1.1 also holds for \(N=2\).
Before stating our main result, let us introduce some positive constants which will usually appear in the statement of our main result and its proof. First of all, since \(\alpha \in (\frac{5}{3},2)\), we have \(\frac{3\alpha -4}{4}\in (\frac{1}{4},\frac{1}{2})\) and \(\frac{3\alpha -3}{8}\in (\frac{1}{4},\frac{3}{8})\) as well as \(\frac{3\alpha -4}{4}-\frac{3\alpha -3}{8}=\frac{3\alpha -5}{8}>0\). This enables us to conclude that for some \(\beta _0\in (\frac{3\alpha -3}{8},\frac{3\alpha -4}{4})\subset (\frac{1}{4},\frac{1}{2})\),
and
as well as
The inequality (1.12) allows us to use [44, Lemma 3.2] to find \(M_1=M_1(\beta _0,\alpha ,\Omega )>0\) such that
Conjunction with the inequalities (1.13) and (1.14) as well as some embedding properties of the Stokes operator and its fractional powers (see [19, 20]) warrants that \(D(A^{\beta _0}_2)\hookrightarrow L^\frac{4}{3-\alpha }(\Omega )\) and \(D\left( A^{\frac{1+2\beta _0}{2}}_2\right) \hookrightarrow W^{1,\frac{4}{3-\alpha }}(\Omega )\). This amounts to say, we have
and
with \(M_2=M_2(\beta _0,\alpha ,\Omega )>0\) and \(M_3=M_3(\beta _0,\alpha ,\Omega )>0\). In addition, for the above \(\alpha \in (\frac{5}{3},2)\) and \(\beta _0\in (\frac{3\alpha -3}{8},\frac{3\alpha -4}{4})\), it is easy to see that \(\frac{1}{4}<\beta _0<\frac{1+2\beta _0}{2}<1\), which enables us to apply the fact that \(D(A^{\alpha _1}_2)\hookrightarrow D(A^{\alpha _2}_2)\) for \(0<\alpha _2\le \alpha _1<1\) (see [21, p.25]) to conclude that \(D\left( A^{\frac{1+2\beta _0}{2}}_2\right) \hookrightarrow D(A^{\beta _0}_2)\). That is to say, there exists \(M_4=M_4(\beta _0,\Omega )>0\) such that
Apart from that, the fact that the Helmholtz projection \({\mathcal {P}}\) is bounded on \(L^p(\Omega ;{\mathbb {R}}^3)\) for \(p>1\) (cf. [18, Theorems 1 and 2]) provides \(M_5=M_5(\alpha ,\Omega )>0\) such that
Similarly to (1.18), recalling \(\Vert A^\frac{1}{2}\varphi \Vert _{L^2(\Omega )}=\Vert \nabla \varphi \Vert _{L^2(\Omega )}\), we once more infer from \(\beta _0\in (\frac{3\alpha -3}{8},\frac{3\alpha -4}{4})\subset (\frac{1}{4},\frac{1}{2})\) that there exists \(M_6=M_6(\beta _0,\Omega )>0\) such that
Next, relying on (1.9), we let
Finally, an application of the Poincaré inequality and the imbedding \(W_0^{1,2}(\Omega )\hookrightarrow L^6(\Omega )\) warrants the existence of \(M_P=M_P(\Omega )>0\) and \(M_S=M_S(\Omega )>0\) such that
and
In this context, our main result reads as follows.
Theorem 1.1
Let the conditions of Proposition 1.1 hold. Furthermore, if \(v_{*}\ge 0\) is constant, \(\alpha \) and r satisfy
with
and
where constants \(M_i\;(i=1,2,...,6),M_\phi ,M_P,M_S>0\) are specified in (1.15)–(1.23), then one can find \(t_0=t_0(r,\mu ,\alpha ,n_0,v_0,w_0, u_0,\Omega )>0\) such that the global weak solution (n, v, w, u) of the problem (1.8) provided by Proposition 1.1 enjoys the following properties satisfy
and such that with some \(P\in C^{1,0}({\bar{\Omega }}\times [t_0,\infty ))\), the quintuple (n, v, w, u, P) forms a classical solution of (1.8) in \({\bar{\Omega }}\times [t_0,\infty )\). In addition, the problem (1.8) possesses a bounded absorbing set in \(L^\infty (\Omega )\times W^{1,\infty }(\Omega )\times W^{1,\infty }(\Omega )\times L^\infty (\Omega ;{\mathbb {R}}^3)\) in the sense that there exists \(C=C(\chi ,r,\mu ,\alpha ,\kappa ,\Omega )>0\) such that any such solution fulfills
Remark 1.2
-
(i)
Apart from the positive effect of indirect signal production mechanism on global solvability, Theorem 1.1 also shows that the indirect signal production mechanism genuinely facilitates the global eventual smoothness and bounedness of solution to the 3D chemotaxis-Navier–Stokes system. To be specific, it was asserted by a recent analytic progress (see [4, Theorem 1.2) that the global weak solution of the 3D Navier–Stokes system (1.4) with quadratic degradation (i.e., \(\alpha =2\)) involving the boundary condition (1.5) constructed in [4, Theorem 1.2] will eventually become smooth if \(\mu \ge \mu _{0}\) and \(r<\eta \cdot \min \{\mu ,\mu ^{\frac{3}{2}+\omega }\}\) for all \(\omega >0\) with suitably small \(\eta =\eta (\omega ,\Omega )>0\) (i.e., (1.6) holds), whereas in Theorem 1.1 of this work we have shown the eventual smoothness and boundedness of the weak solution gained in Theorem 1.1 for the 3D chemotaxis-Navier–Stokes problem (1.8) with sub-quadratic degradation (i.e., \(\alpha \in (\frac{5}{3},2)\)) if \(r\le \min \{{\tilde{\eta }}^{*},{\tilde{\eta }}^{**}\}\cdot \min \{\mu ,\mu ^\frac{2}{3-\alpha }\}\) for some \({\tilde{\eta }}^{*}={\tilde{\eta }}^{*}(\alpha ,\Omega )>0\) and \({\tilde{\eta }}^{**}={\tilde{\eta }}^{**}(\alpha ,\Omega )>0\). Of note, unlike the appropriately small constant \(\eta >0\) and \(\mu \ge \mu _{0}\) required in [4, Theorem 1.2], our result shows that the positive constant \(\min \{{\tilde{\eta }}^{*},{\tilde{\eta }}^{**}\}\) is not necessary to be properly small and \(\mu \ge \mu _{0}\) is not necessary, and that we can determine the explicit expression of \({\tilde{\eta }}^{*}\) and \({\tilde{\eta }}^{**}\).
-
(ii)
) To the best of our knowledge, the global solvability of the 3D chemotaxis-Navier–Stokes system with indirect signal production involving Dirichlet signal boundary condition has never been touched before, thus our work fills this gap to some extent. However, there is a gap between \(\alpha \in (\frac{4}{3},2)\) and \(\alpha \in (1,2)\). Accordingly, we have to leave an open question here whether or not the range of can be relaxed to \(\alpha \in (1,2)\), and we expect that we are able to solve it in future work.
Plan of this paper
In Sect. 2, we introduce a family of regularized problems (2.21) of system (1.8) and state the local-in-time solvability in the framework of classical solution (Lemma 2.1). In addition, some useful lemmas are summarized (Lemmas 2.2–2.5). Section 3 is dedicated to deriving the eventual smoothness and boundedness of weak solution to the 3D chemotaxis-Navier–Stokes system (1.8).
Notations. Throughout this paper, we abbreviate the integrals \(\int \limits _{\Omega }f(x)dx\) and \(\int \limits _{0}^{t}\int \limits _{\Omega }f(x,s)dxds\) as \(\int \limits _{\Omega }f(x)\) and \(\int \limits _{0}^{t}\int \limits _{\Omega }f(x,s)\) for simplicity, and adapt standard notation by abbreviating \({\bar{\rho }}:=\frac{1}{|\Omega |}\int \limits _{\Omega }\rho \) for all \(\rho \in L^{1}(\Omega )\). Moreover, for the convenience of notation, we use symbols \(M_{i},C,C_{i}\) \((i=1,2,3...)\) to denote some universal positive constants which may vary in context, and denote them by \(M_{i}(a,b...),C(a,b...),C_{i}(a,b...)\) when we need emphasize their dependence on parameters a, b...
2 Preliminaries
Firstly, Let us specify the notion of weak solution to which we will refer in the sequel. Throughout the sequel, for vectors \(v\in {\mathbb {R}}^3\) and \(w\in {\mathbb {R}}^3\), we let v \(\otimes \) w denote the matrix \((a_{ij})_{i,j\in \{1,2,3\}}\in {\mathbb {R}}^{3\times 3}\) defined on setting \(a_{ij}:=v_{i}w_{j}\) for \(i,j\in \{1,2,3\}\). In addition, we introduce the divergence-free spaces \(W_{0,\sigma }^{1,1}(\Omega ;{\mathbb {R}}^N):=W_{0}^{1,1}(\Omega ;{\mathbb {R}}^N)\cap L_{\sigma }^{2}(\Omega )\), which appears a few times during our investigations.
Definition 2.1
Let \(\alpha \in (\frac{4}{3},2)\). By a global weak solution of system (1.8) we mean a quadruple (n, v, w, u) of functions
such that \(n\ge 0\), \(v\ge 0\) and \(w\ge 0\) a.e. in \(\Omega \times (0,\infty )\),
that \(\nabla \cdot u=0\) a.e. in \(\Omega \times (0,\infty )\), and that
for all \(\varphi \in C_{0}^{\infty }({\bar{\Omega }}\times [0,\infty ))\),
for all \(\varphi \in C_{0}^{\infty }({\bar{\Omega }}\times [0,\infty ))\), and
for all \(\varphi \in C_{0}^{\infty }({\bar{\Omega }}\times [0,\infty ))\), as well as
for all \(\varphi \in C_{0}^{\infty }(\Omega \times [0,\infty );{\mathbb {R}}^3)\) satisfying \(\nabla \cdot \varphi \equiv 0\).
In order to gain a global weak solutions of (1.8) through a suitable approximation procedure, we employ the approaches used in [38, 44] to regularize the system (1.8). Namely, we fix a family \((\rho _{\varepsilon })_{\varepsilon \in (0,1)}\subset C_{0}^{\infty }(\Omega )\) of smooth cut-off functions satisfying
and introduce the corresponding family of approximating problems to (1.8) given by
with \(f_{\varepsilon }(s):=\frac{1}{(1+\varepsilon s)^{3}}\) and \(g_{\varepsilon }(s):=\frac{s}{1+\varepsilon s}\) for \(s\ge 0\) and \(\varepsilon \in (0,1)\), where \(Y_{\varepsilon }\) denotes the standard Yosida approximation [35] defined by
Applying the well-established arguments involving the fixed point theorem, the standard regularity theory of parabolic and Stokes equation and the maximum principle, we are able to prove that the regularized system (2.9) is locally solvable in a classical sense for each \(\varepsilon \in (0,1)\). We omit the proof for simplicity and refer the reader to [48, Lemma 2.1] and [5, Lemma 3.1] for its more details.
Lemma 2.1
Let \(\Omega \subset {\mathbb {R}}^3\) be a bounded domain with smooth boundary, \(r\in {\mathbb {R}}\), \(\mu >0\), \(\chi >0\) and \(\kappa \ne 0\). Assume that \(v_{*}\) and \(\phi \) satisfy (1.10) and (1.9), respectively. Then for each \(\varepsilon \in (0,1)\) and \((n_{0},v_{0},w_{0},u_{0})\) fulfilling (1.11), there exists a maximal existence time \(T_{max,\varepsilon }\in (0,\infty ]\) and a uniquely determined quintuple \((n_{\varepsilon },v_{\varepsilon },w_{\varepsilon },u_{\varepsilon },P_{\varepsilon })\) of functions such that
which classically solves (2.9) and fulfills \(n_{\varepsilon }\ge 0\), \(v_{\varepsilon }\ge 0\) and \(w_{\varepsilon }\ge 0\) in \({\bar{\Omega }}\times (0,T_{max,\varepsilon })\). Moreover, if \(T_{max,\varepsilon }<\infty \), then for all \(\beta \in (\frac{3}{4},1)\), we have
Then, we recall the following well-known Gagliardo–Nirenberg interpolation inequality which will be frequently used in the forthcoming proofs (see [14]).
Lemma 2.2
Let \(\Omega \subset {\mathbb {R}}^N~(N\ge 1)\) be a bounded domain with smooth boundary. Suppose that \(1\le p,q\le \infty \) satisfying \(p(N-q)<Nq\) and \(r\in (0,p)\). Then, there exists \(C_{GN}=C_{GN}(p,q,r,N,\Omega )>0\) such that
where \(\theta =\frac{\frac{N}{r}-\frac{N}{p}}{1+\frac{N}{r}-\frac{N}{q}}\in (0,1)\).
In addition, we state the following auxiliary lemma on boundedness in a linear differential inequality which will be utilized in several places below. For its proof, we refer to [42, Lemma 3.4].
Lemma 2.3
Let \(t_{0}\in {\mathbb {R}}\), \(T\in (t_{0},\infty ]\) and assume that \(y\in C^{0}([t_{0},T))\cap C^{1}((t_{0},T))\) fulfills
with any \(a>0\) and the nonnegative function \(h\in L_{loc}^{1}({\mathbb {R}})\) for which there exist \(\tau \) and \(b>0\) such that
then
Moreover, as a straightforward consequence of the Hölder inequality, the following \(L^{p}((0,T);L^{q}(\Omega ))\) interpolation result can be deduced (see [13]).
Lemma 2.4
Let \(T>0\), and let \(p_{1},p_{2},q_{1},q_{2},p,q\ge 1\) be such that
with \(\theta \in [0,1]\). Then,
holds for all \(z\in L^{p_{1}}((0,T);L^{q_{1}}(\Omega )\cap L^{p_{2}}((0,T);L^{q_{2}}(\Omega )\).
Finally, we give the following lemma which plays an important role in later proof. For its detailed proofs, we refer to [44, Lemma 3.8].
Lemma 2.5
Let \(\Omega \subset {\mathbb {R}}^3\) be a bounded domain with smooth boundary. Assume that \(q\ge 1\) and
Then, there exists \(C=C(\lambda ,q,\Omega )>0\) such that for all \(\varphi \in C^2({\bar{\Omega }})\) fulfilling \(\varphi \cdot \frac{\partial \varphi }{\partial \nu }=0\) on \(\partial \Omega \), we have
3 Eventual smoothness
In this section, we devote to improving the regularity properties of the weak solutions after a large waiting time under an explicit smallness condition, in which we assume \(v_{*}\ge 0\) to not only be constant in time, but also constant in space.
First of all, we give the following estimate, which is the cornerstone of the later proof.
Lemma 3.1
Let \(r\in {\mathbb {R}}\), \(\mu >0\), \(\chi >0\), \(\kappa \ne 0\) and \(\alpha \in (\frac{5}{3},2)\). If \({\tilde{r}}>0\) is such that \({\tilde{r}}\ge r\), then we have
and
Proof
Since the proof is similar to [4, Lemma 5.1] and [52, Lemma 2.1], we omit its details to avoid repetition. \(\square \)
With the help of Lemma 3.1, the following eventual smallness of \(n_{\varepsilon }\) can be obtained by suitable use of Hölder inequality.
Lemma 3.2
Let \(r\in {\mathbb {R}}\), \(\mu >0\), \(\chi >0\), \(\kappa \ne 0\) and \(\alpha \in (\frac{5}{3},2)\). Assume that
and
Then, there exists \(t_{0}=t_{0}({\tilde{\eta }}_{1},\delta ,\mu ,\alpha )>0\) such that for all \(\delta >0\) we have
Proof
For fixed \(\delta >0\), we let
so that we have
In addition, we assume that
From Lemma 3.1, we know that
and
with \(t_{0}=t_{0}({\tilde{\eta }}_{1},\delta ,\mu ,\alpha )=\frac{\ln 2}{{\tilde{r}}(\alpha -1)}\). By means of Lemma 2.4, (3.9) and (3.10), we have
\(\square \)
Assume that \(\mu \in (0,1]\). From (3.8) and \(0<{\tilde{\eta }}_{1}\le 1\) we know that \({\tilde{r}}={\tilde{\eta }}_{1}\mu ^{\frac{2}{3-\alpha }}\) and \({\tilde{r}}+1={\tilde{\eta }}_{1}\mu ^{\frac{2}{3-\alpha }}+1\le {\tilde{\eta }}_{1}+1\le 2\) and
As for the case \(\mu >1\), proceeding in a same as proving Lemma 2.2 in [14], we can achieve it. Here, we omit its details to avoid repetition.
Based on the eventual smallness of \(n_{\varepsilon }\) in Lemma 3.2 and the standard Navier–Stokes energy inequality, we can prove the following inequality.
Lemma 3.3
Let \(r\in {\mathbb {R}}\), \(\mu >0\), \(\chi >0\), \(\kappa \ne 0\) and \(\alpha \in (\frac{5}{3},2)\). Assume that \(M_{\phi },M_{P}\), and \(M_{S}\) satisfy (1.21)–(1.23). For all \(\delta >0\) we let
and
Thus, we have
with \(t_0=t_0({\tilde{\eta }}_{2},\delta ,r,\mu ,\alpha ,n_0,u_0,\Omega )>0\).
Proof
We test the fourth equation of (2.9) by \(2u_\varepsilon \), and use the Hölder inequality, Young’s inequality, (1.21) and (1.23), we have
which in conjunction with (1.22) directly implies
By means of Lemmas 3.1 and 2.4, we obtain
Moreover, since \(\alpha \in (\frac{5}{3},2)\), we have \(\frac{2}{3-\alpha }>\frac{3}{2}>\frac{6}{5}\). Furthermore, we apply the Hölder inequality and (3.18) to find that for all \(t>0\),
Thus, applying Lemma 2.3 in (3.17) yields
For any fixed \(\delta >0\) and \(\alpha \in (\frac{5}{3},2)\) as well as \(C_P,C_S>0\) specified in (1.22) and (1.23), we let
and yields we obtain directly
Moreover, we set
and
and assume that
Noticing that \(\eta _2:=\eta _2(\delta ,\alpha ,\Omega ):=\overline{\eta _1}(\delta _1,\alpha ,\Omega )\), from Lemma 3.2, we can know that there exists \(t_1=t_1(\eta _2,\delta ,\mu ,\alpha ,\Omega )>0\) such that
which implies
by using the Hölder inequality and \(\alpha \in (\frac{5}{3},2)\). In addition, we can choose \(t_0=t_0(\eta _2,\delta ,r,\mu ,\alpha ,n_0,u_0,\Omega )>t_1\) large enough satisfying
Based on (3.24) and (3.25), employing Lemma 2.3 again in (3.17) entails
Consequently, we integrate (3.17) in time and use (3.26), (3.24) and (3.22), we can conclude that
This readily accomplishes the proof of (3.15). \(\square \)
Indeed, the nonlinear convection term involving in the fluid equation is highly intractable due to the typical difficulty in studying 3D Navier–Stokes equation. For the sake of overcoming this obstacle, inspired largely by the innovative and technical approach used in the proof of Lemma 3.3 in [52], we can establish the eventual smallness bound for \(u_{\varepsilon }\) in \(L^{p}(\Omega )\).
Lemma 3.4
Let \(r\in {\mathbb {R}}\), \(\mu >0\), \(\chi >0\), \(\kappa \ne 0\), \(\alpha \in (\frac{5}{3},2)\) and \({\tilde{\beta }}_{0}\in \left( \frac{3\alpha -3}{8},\frac{3\alpha -4}{4}\right) \) and \(\kappa \ne 0\). Assume that
with
and
where positive constants \(M_i\;(i=1,2,...,6)\) and \(M_\phi ,M_p,M_S\) are specified in (1.15)–(1.23). Then, there exists \(t_0=t_0(r,\mu ,\alpha ,{\tilde{\beta }}_0,n_0,u_0,\Omega )>0\) such that
and
with \(C=C({\tilde{\beta }}_0,\alpha ,\kappa ,\Omega )>0\). In particular, for the above \(t_0\) we have
and
with \({\widetilde{C}}={\widetilde{C}}(\alpha ,\kappa ,\Omega )>0\).
Proof
Using the Helmholtz projector \({\mathcal {P}}\) to the fourth equation of (2.9), testing the resulting equation \(u_{\varepsilon t}+ Au_\varepsilon ={\mathcal {P}}[-\kappa (Y_\varepsilon u_\varepsilon \cdot \nabla )u_\varepsilon +n_\varepsilon \nabla \phi ]\) against \(A^{2\tilde{\beta _0}}u_\varepsilon \), we have
Employing the self-adjointness of fractional powers of A, Young’s inequality, (1.15), (1.19) and (1.21) to obtain
where \(M_1, M_5, M_\phi >0\) are determined in (1.15), (1.19) and (1.21), respectively. Utilizing the Cauchy–Schwarz inequality, (1.15), (1.16), (1.17) and (1.19), we obtain that
where \(M_2, M_3>0\) are specified in (1.16) and (1.17), respectively. Substituting (3.33) and (3.34) into (3.32) directly shows that for each \(\varepsilon \in (0,1)\) and all \(t>0\),
\(\square \)
For \(\kappa \ne 0\) and constants \(M_1,M_2,M_3,M_4,M_5,M_6,M_\phi >0\) determined in (1.15)–(1.21), we define
and
Furthermore, we choose
and
For the above \({\tilde{\eta }}_{1},{\tilde{\eta }}_{2}>0\), we suppose that
so that we derive from Lemma 3.2 that there exists \(t_1=t_1({\tilde{\eta }}_{1},\mu ,\alpha ,\Omega )>0\) satisfying
and find from Lemma 3.3 that there exists \(t_2=t_2({\tilde{\eta }}_{1},{\tilde{\eta }}_{2},r,\mu ,\alpha ,\tilde{\beta _{0}},n_0,u_0,\Omega )>t_1\) satisfying
Now, we shall claim that (3.28)–(3.31) for \(t_0=t_0({\tilde{\eta }}_{1},{\tilde{\eta }}_{2},r,\mu ,\alpha ,\tilde{\beta _{0}},n_0,u_0,\Omega ):=t_2+1\). We first deduce from (3.40) that for each \(\varepsilon \in (0,1)\) there exists \(t_\varepsilon =t_\varepsilon ({\tilde{\eta }}_{1},{\tilde{\eta }}_{2},r,\mu ,\alpha ,\tilde{\beta _{0}},n_0,u_0,\Omega )\in (t_2,t_2+1)\) such that
which combined with (1.23) and (3.37) shows
Let
Here we point out that (3.42) and the continuity of \(\Vert A^{{\tilde{\beta }}_0}u_\varepsilon (\cdot ,t)\Vert _{L^2(\Omega )}\) with respect to t asserted by Lemma 2.1 and Proposition 1.1 guarantee that S is indeed not empty and thus \(T_\varepsilon :=\sup S\in (0,\infty ]\) is well-defined. Clearly, it remains to make sure that \(T_\varepsilon =\infty \). To achieve it, using the facts that \(A^{{\tilde{\beta }}_0}\) commutes with \(Y_\varepsilon \) on \(D(A^{{\tilde{\beta }}_0}_2)\), and that \(\Vert Y_\varepsilon \varphi \Vert _{L^2(\Omega )}\le \Vert \varphi \Vert _{L^2(\Omega )}\) for all \(\varphi \in L^2_\sigma (\Omega )\), we infer from the definition of \(T_\varepsilon \) that for all \(t\in (t_\varepsilon ,T_\varepsilon )\),
and hence by (3.35), we have
which together with (1.18) gives
Thus, applying Lemma 2.3 along with (3.39), (3.36) and (3.42) immediately reveals
Suppose that \(T_\varepsilon <\infty \), once more making use of the continuity of \(\int \limits _\Omega |A^{{\tilde{\beta }}_0}u_\varepsilon (\cdot ,t)|^2\) with respect to t ensured by Lemma 2.1, we infer from (3.47) that we can find appropriately small \(\varrho =\varrho (\kappa ,\alpha ,\Omega )>0\) such that \(\int \limits _\Omega |A^{{\tilde{\beta }}_0}u_\varepsilon (\cdot ,t)|^2\le \frac{1}{4\kappa ^2M_1^2M_2^2M_3^2M_5^2}\) for all \(t\in [t_\varepsilon ,T_\varepsilon +\delta )\), which contradicts our definition of \(T_\varepsilon \) evidently. Therefore, we conclude that we must have \(T_\varepsilon =\infty \), and that (3.28) is valid upon evident choice of \(t_0:=t_2+1>t_\varepsilon \). Moreover, on the basis of (3.28) and (3.39), the estimate (3.29) results from a straightforward time integration of (3.45). Having (3.28) and (3.29) at hand, we easily derive (3.30) and (3.31) through the use of (1.16) and (1.17). Consequently, the proof is completed.
In fact, the Dirichlet signal boundary condition for \(v_{\varepsilon }\) is highly intractable. In order to solve this obstacle, motivated by the innovative and technical approach used in the proof of Lemma 3.4 in [3], we can deal with the normal derivative \(\frac{\partial |\nabla v_{\varepsilon }|^{2}}{\partial \nu }\).
Lemma 3.5
Let \(N\ge 2\) and \(\Omega \subset {\mathbb {R}}^{N}\) be a bounded domain with \(C^{2}\)-boundary and let \(L\in {\mathbb {R}}\) denote the maximum of the curvatures on \(\partial \Omega \). Then, whenever \(\varphi \in C^{2}({\bar{\Omega }})\) and \(\varphi _{*}\in R\) are such that \(\varphi =\varphi _{*}\) on \(\partial \Omega \),
An application of Lemma 3.5, Hölder inequality and Young’s inequality directly, we can obtain the following inequality, which is taken from [4, Lemma 5.9].
Lemma 3.6
Let \(p\in (1,2)\). There is \(C=C(p)>0\) such that for all \(\gamma >0\), \(\vartheta >0\) and each \(\varphi \in C^{2}({\bar{\Omega }})\) with \(\varphi =0\) on \(\partial \Omega \) the inequality
holds.
Using Young’s inequality, Hölder inequality and Gagliardo–Nirenberg inequality, we can improve the regularity of \(w_{\varepsilon }\).
Lemma 3.7
Let \(r\in {\mathbb {R}}\), \(\mu >0\), \(\chi >0\), \(\kappa \ne 0\) and \(\alpha \in (\frac{5}{3},2)\). Then for all initial data fulfilling (1.11), we have
with \(C=C(r,\mu ,\alpha ,n_{0},w_{0},\Omega )>0\).
Proof
Testing the third equation of (2.9) against \(w_\varepsilon ^\frac{5(\alpha -1)}{5-2\alpha }\) with \(\alpha \in (\frac{4}{3},2)\) and use the Hölder inequality to prove that
for all \(t\in (0,T_{max})\). By means of (4.1), we can know that \(\Vert w_{\varepsilon }\Vert _{L^{1}(\Omega )}\le {\tilde{C}}\) with some constants \({\tilde{C}}>0\). Furthermore, applying the Gagliardo–Nirenberg inequality and Young’s inequality to find that
with \(C_1=C_1(\Omega )>0\), \(C_2=C_2(\alpha ,\Omega )>0\) and \(C_3=C_3(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\). By means of Young’s inequality implies
with \(C_4=C_4(\alpha ,\Omega )>0\). Substituting (3.52) into (3.51) and rearranging the resulting inequality provide \(C_5=C_5(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\) such that for all \(t>0\),
\(\square \)
Employing the Gagliardo–Nirenberg inequality and Young’s inequality again to show that there exist \(C_6=C_6(\alpha ,\Omega )>0\) and \(C_7=C_7(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\) fulfilling
which, by means of an elementary inequality \((a-b)_+^\xi \ge 2^{1-\xi }a^\xi -b^\xi \) with \(a,b\ge 0\) and \(\alpha \ge 1\), enables us to take \(a:=\frac{1}{C_7}\left( \int \limits _\Omega w_\varepsilon ^\frac{3\alpha }{5-2\alpha }+1\right) \), \(b:=1\) and \(\xi :=\frac{11\alpha -5}{15\alpha -15}>\frac{17}{15}>1\) because of \(\alpha \in (\frac{4}{3},2)\) and to arrive at
This in conjunction with the basic inequality \(y^p\ge ep\ln y\) for all \(y>0\) and \(p>0\) further entails
with \(C_{8}:=\frac{\alpha e(10-4\alpha )(2C_9)^\frac{5-11\alpha }{15\alpha -15}}{(5-2\alpha )^2}>0\). Dividing the both sides of (4.53) by \(\int \limits _\Omega w_\varepsilon ^\frac{3\alpha }{5-2\alpha }+1\) and recalling (3.54) we have
In view of (4.2), we get
with \(C_{9}=C_{9}(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\). Applying Lemma 2.3 to (3.55) and recalling (3.56), we conclude that
Applying the exponential function to the above inequality immediately gives rise to
with \(C_{10}=C_{10}(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\). Integrating (3.53) in time and recurring to (3.57), we can find \(C_{11}=C_{11}(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\) fulfilling
and further deduce from the Gagliardo–Nirenberg inequality that for some \(C_{12}=C_{12}(\Omega )>0\) and \(C_{13}=C_{13}(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\),
In order to derive the time-ultimate \(L^{\infty }\) bounds for \(n_{\varepsilon }\), we need to establish suitable regularity properties of \(\nabla v_{\varepsilon }\).
Lemma 3.8
Let \(r\in {\mathbb {R}}\), \(\mu >0\), \(\chi >0\), \(\kappa \ne 0\) and \(\alpha \in (\frac{5}{3},2)\). Suppose that the assumption on r prescribed in Lemma 3.4 remains valid. Then, for all initial data fulfilling (1.11), one can find \(t_0=t_0(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega )>0\) such that
with \(C=C(r,\mu ,\alpha ,\kappa ,\Omega )>0\).
Proof
From (3.30) and (3.31) we know that there exist \(C_1=C_1(\kappa ,\alpha ,\Omega )>0\) and \(t_1=t_1(r,\mu ,\alpha ,n_0,u_0,\Omega )>0\) satisfying
Moreover, by the similar way in [11, Proposition 3.1] and (3.50), we can find \(C_2=C_2(r,\mu ,\alpha ,\Omega )>0\) and \(t_2=t_2(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega )>t_1\) such that
which enables us to find \(t_3=t_3(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega )\in (t_2,t_2+1)\) such that
\(\square \)
In addition, applying \(\nabla \) to the second equation in (2.9), multiplying the resulting identity by \(|\nabla v_\varepsilon |^{2p-2}\nabla v_\varepsilon \) for all \(p\in (\frac{3}{2},2)\) and integrating over \(\Omega \) by parts, we obtain that
By means of Lemma 3.6, Lemma 3.7 and the similar way in [3, Lemma 3.5], we have
with \(C_{3}=C_{3}(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\). Then, by Young’s inequality, Hölder inequality and the fact that \(|\Delta v_{\varepsilon }|\le \sqrt{3}|D^{2}v_{\varepsilon }|\), we obtain
and
as well as
with \(C_{4}=C_{4}(p)>0\) and \(C_{5}=C_{5}(p,v_{0},v_{*})>0\). Substituting (3.64)–(3.67) into (3.63) implies
with \(C_{6}=C_{6}(p)>0\), \(C_{7}=C_{7}(p,v_{0},v_{*})>0\) and \(C_{8}=C_{8}(r,\mu ,\alpha ,n_0,w_0,\Omega )>0\). In view of \(\alpha \in (\frac{5}{3},2)\), we have \(\frac{3\alpha }{5-2\alpha }>3\). We can apply (3.61), Hölder inequality and Gagliardo–Nirenberg inequality to obtain that there exist \(C_9=C_9(p,r,\mu ,\alpha ,\Omega )>0\) and \(C_{10}=C_{10}(p,r,\mu ,\alpha ,\Omega )>0\) satisfying
where \(\theta _2:=\frac{3p^2-4p}{3p^2-4p+1}\in (0,1)\) thanks to \(p>\frac{3}{2}>\frac{4}{3}\). This means \(\frac{2(p-1)}{p}\theta _2=\frac{6p-8}{3p-1}\in (0,2)\), which allows us to invoke Young’s inequality in (3.69) to find \(C_{11}=C_{11}(p,r,\mu ,\alpha ,\Omega )>0\) such that
Moreover, by the Hölder inequality, we see that
where an invocation of the Gagliardo–Nirenberg inequality warrants the existence of \(C_{12}=C_{12}(p,\Omega )>0\) satisfying
and an application of Lemma 2.5 and (3.61) provides \(C_{13}=C_{13}(p,\Omega )>0\) and \(C_{14}=C_{14}(p,r,\mu ,\alpha ,\Omega )>0\) such that
Moreover, we have
Inserting (3.72) and (3.73) into (3.71), and use Young’s inequality to show that
Substituting (3.69), (3.74) and (3.75) into (3.68), we deduce that for all \(t\ge t_3\),
Noticing that \(\frac{2}{3-\alpha }\in (\frac{3}{2},2)\) due to \(\alpha \in (\frac{5}{3},2)\), and that (3.75) holds for any \(p>\frac{3}{2}\), we can particularly take \(p=\frac{2}{3-\alpha }\) in (3.75) and apply the first assertion of (3.60) to find \(C_{15}=C_{15}(r,\mu ,\alpha ,\kappa ,\Omega )>0\) such that
Thanks to \(\alpha \in \left( \frac{5}{3},2\right) \), we easily see that
and thus
This enables us to make use of Young’s inequality in (3.77) so as to find \(C_{16}=C_{16}(r,\mu ,\alpha ,\kappa ,\Omega )>0\) with the property that
and thereby employ Lemma 2.3, the second assertion of (3.60), (3.62), the fact that \(\frac{4}{3-\alpha }\in (3.4)\) due to \(\alpha \in (\frac{5}{3},2)\) and the Hölder inequality to achieve the desired result.
Based on above lemmas, we can obtain eventual \(L^{\infty }\) bound for \(n_{\varepsilon }\).
Lemma 3.9
Under the assumption of Lemma 3.4. Then, for all initial data complying with (1.8), there exists \(t_0=t_0(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega )>0\) such that
with \(C=C(\chi ,r,\mu ,\alpha ,\kappa ,\Omega )>0\).
Proof
From Lemma 3.1, we know
Moreover, based on Lemmata 3.4 and 3.8, we can find \(C_2=C_2(r,\mu ,\alpha ,\kappa ,\Omega )>0\) and \(t_1=t_1(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega )\ge \frac{\ln 2}{(r_++1)(\alpha -1)}\) such that
Now, we shall prove that (3.79) is true for \(t_0=t_0(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega ):=t_1+1\). For any fixed \(T\in (t_0,\infty )\), we define
Of note, \(M_\varepsilon (T)\) is finite due to Lemma 2.1. By Duhamel’s principle, we express the solution component \(n_\varepsilon \) of (2.9) according to
where \(h_\varepsilon (x,t):=\chi \rho _{\varepsilon }f_{\varepsilon }(n_{\varepsilon })\nabla v_\varepsilon (x,t)+u_\varepsilon (x,t)\). \(\square \)
Firstly, by means of the well-known smoothing estimate for \(\{e^{t\Delta }\}_{0\le t<2}\) (see [40]) and (3.81), we can find \(C_3=C_3(\Omega )>0\) such that
Secondly, thanks to Young’s inequality, we have \(rn_\varepsilon -\mu n_\varepsilon ^\alpha \le \frac{(\alpha -1)(r_+)^\frac{\alpha }{\alpha -1}}{\mu ^\frac{1}{\alpha -1}\alpha ^\frac{\alpha }{\alpha -1}}\). This allows us to utilize the maximum principle to arrive at
Finally, a combination of (3.81) and the fact that \(|sf_{\varepsilon }(s)|\le 1\) for \(x\in \Omega \) and \(s\ge 0\) entails
with \(C_4=C_4(\chi ,r,\mu ,\alpha ,\kappa ,\Omega )>0\). Since \(\alpha \in (\frac{5}{3},2)\), we see that \(\frac{4}{3-\alpha }\in (3,4)\). This guarantees us to take \(q>3\) such that \(q\in (3,\frac{4}{3-\alpha })\), and take advantage of the smoothing \(L^p\)-\(L^q\) estimates of \(\{e^{t\Delta }\}_{t\ge 0}\) (see [40, Lemma 1.3) and the Hölder inequality as well as the interpolation inequality for \(L^p\)-norms to deduce that
where \(C_5=C_5(q,\Omega )>0\), and \(C_6:=C_1^\frac{4-(3-\alpha )q}{4q}C_4C_5\int \limits _{t-1}^t(t-s)^{-\frac{1}{2}-\frac{3}{2q}}ds =\frac{2qC_1^\frac{4-(3-\alpha )q}{4q}C_4C_5}{q-3}\) due to \(q>3\). Inserting (3.84), (3.85) and (3.87) into (3.83) and recalling (3.82) immediately yield
with \(C_7:=C_1C_3+ \frac{(\alpha -1)(r_+)^\frac{\alpha }{\alpha -1}}{\mu ^\frac{1}{\alpha -1}\alpha ^\frac{\alpha }{\alpha -1}}\). Therefore, by direct calculation, we have
which together with (3.82) readily establishes (3.79) by taking \(T\rightarrow \infty \).
With the help of Lemma 3.9, the eventual Hölder regularity of fluid velocity field, the eventual \(W^{1,\infty }(\Omega )\) boundedness of \(v_{\varepsilon }\) and \(w_{\varepsilon }\) and the higher order time-ultimate estimates of solutions can be obtain. The proof is similar to [4, Lemma 6.1, 6.2 and Proposition 6.3], we omit its details to avoid repetition.
Lemma 3.10
Let the assumptions of Lemma 3.4 be true. Then, for all initial data satisfying (1.11), one can find \(t_0=t_0(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega )>0\) such that
Furthermore, we have
and
with some \(\gamma \in (0,1)\) and \(C=C(\chi ,r,\mu ,\alpha ,\kappa ,\Omega )>0\).
Lemma 3.11
Under the hypotheses of Lemma 3.4. Then there exist \(\gamma \in (0,1)\), \(t_0=t_0(r,\mu ,\alpha ,n_0, v_0,w_0,u_0,\Omega )>0\) and \(C=C(T,\chi ,r,\mu ,\alpha ,\kappa ,\Omega )>0\) such that
for each \(T>t_0\) and \(\varepsilon \in (0,1)\).
Now, we ready to prove Theorem 1.1.
The proof of Theorem 1.1 Firstly, we have known that the global existence of weak solution in the sense of Definition 2.1 for the problem (1.8). From Lemma 3.11, we can make use of the \(Arzel\grave{a}\)-Ascoli theorem to find \(t_0=t_0(r,\mu ,\alpha ,n_0,v_0,w_0,u_0,\Omega )>0\) with the property that for any given \(\{\varepsilon _j\}_{j\in {\mathbb {N}}}\subset (0,1)\) fulfilling \(\varepsilon _j\searrow 0\) as \(j\rightarrow \infty \) one can find \((n,v,w,u,P)\in \left[ C^{2,1}({\bar{\Omega }}\times [t_0,\infty ))\right] ^3\times C^{2,1}({\bar{\Omega }}\times [t_0,\infty );{\mathbb {R}}^3)\times C^{1,0}({\bar{\Omega }}\times [t_0,\infty ))\) and a subsequence \(\{\varepsilon _{j_k}\}_{k\in {\mathbb {N}}}\subset (0,1)\) such that
as \(\varepsilon _{j_k}\searrow 0\). Taking \(\varepsilon =\varepsilon _{j_k}\searrow 0\) in (2.9) and constructing the corresponding pressure P as performing in [4], we can conclude that the limit objects (n, v, w, u, P) indeed forms the classical solution of (1.8). This together with Proposition 1.1 means that the weak solution (n, v, w, u) actually becomes eventually smooth in the sense that (1.24). Obviously, the boundedness property shown in (1.25) is as a straightforward consequence of Lemmas 3.9–3.10 and the approximation procedure.
References
Arumugam, G., Tyagi, J.: Keller–Segel chemotaxis models: a review. Acta Appl. Math. 171, 6 (2021)
Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25, 1663–1763 (2015)
Black, T., Winkler, M.: Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary. Math. Models Methods Appl. Sci. 32, 137–173 (2022)
Black, T., Wu, C.: Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation. Calc. Var. Partial. Differ. Equ. 61, 96 (2022)
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)
Braukhoff, M.: Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 1013–1039 (2017)
Braukhoff, M., Tang, B.: Global solutions for chemotaxis-Navier–Stokes system with Robin boundary conditions. J. Differ. Equ. 269, 10630–10669 (2020)
Cao, X.: Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces. Discrete Contin. Dyn. Syst 35, 1891–1904 (2015)
Chaplain, M., Stuart, A.: A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149–168 (1993)
Chertock, A., Fellner, K., Kurganov, A., Lorz, A., Markowich, P.A.: Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach. J. Fluid Mech. 694, 155–190 (2012)
Dai, F., Liu, B.: How far do indirect signal production mechanisms influence regularity in the three-dimensional Keller–Segel–Navier–Stokes system. Math. Models Methods Appl. Sci. 33, 2823–2877 (2023)
Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R.E., Kessler, J.O.: Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103 (2004)
Evans, L.C.: Partial Differential Equations, vol. 19. American Mathematical Society, Providence (2010)
Friedman, A.: Partial Differential Equations. Holt. Rinehart & Winston, New York (1969)
Fujie, K., Ito, A., Winkler, M., Yokota, T.: Stabilization in a chemotaxis model for tumor invasion. Discrete Contin. Dyn. Syst, Ser. A 36, 151–169 (2016)
Fujie, K., Senba, T.: Application of an Adams type inequality to a two-chemical substances chemotaxis system. J. Differ. Equ. 263, 88–148 (2017)
Fujie, K., Senba, T.: Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension. J. Differ. Equ. 266, 942–976 (2019)
Fujiwara, D., Morimoto, H.: An \(L^{r}\)-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo 24, 685–700 (1977)
Giga, Y.: Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 61, 186–212 (1986)
Giga, Y.: The Stokes operator in \(L^r\) spaces. Proc. Jpn. Acad. S 2, 85–89 (1981)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin/Heidelberg (1981)
Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. Pisa Cl. Sci. 24, 633–683 (1997)
Höfer, H., Sherratt, J., Maini, P.: Cellular pattern formation during Dictyostelium aggregation. Phys. D 85, 425–444 (1995)
Horstmann, D.: On the existence of radially symmetric blow-up solutions for the Keller–Segel model. J. Math. Biol. 44, 463–478 (2002)
Hu, B., Tao, Y.: To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production. Math. Models Methods Appl. Sci. 26, 2111–2128 (2016)
Keller, E., Segel, L.: Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26, 399–415 (1970)
Lankeit, J.: Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. J. Differ. Equ. 258, 1158–1191 (2015)
Liu, C., Liu, B.: Boundedness in a quasilinear two-species chemotaxis system with nonlinear sensitivity and nonlinear signal secretion. J. Differ. Equ. 320, 206–246 (2022)
Liu, C., Liu, B.: Global solvability in a three-dimensional chemotaxis-Navier–Stokes system with indirect signal production involving Dirichlet signal boundary condition, submitted
Lorz, A.: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20, 987–1004 (2010)
Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj 40, 411–433 (1997)
Osaki, K., Tsujikawa, T., Yagi, A., Mimura, M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. 51, 119–144 (2002)
Osaki, K., Yagi, A.: Finite dimensional attractor for one-dimensional Keller–Segel equations. Funkc. Ekvacioj 44, 441–469 (2001)
Petter, G., Byrne, H., Mcelwain, D., Norbury, J.: A model of wound healing and angiogenesis in soft tissue. Math. Biosci. 136, 35–63 (2003)
Sohr, H.: The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001)
Tello, J.I., Wrzosek, D.: Predator-prey model with diffusion and indirect prey-taxis. Math. Models Methods Appl. Sci. 26, 2129–2162 (2016)
Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C.W., Kessler, J.O., Goldstein, R.E.: Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. U.S.A. 102, 2277–2282 (2005)
Wang, Y., Winkler, M., Xiang, Z.: Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary. Commun. Partial Differ. Equ. 46, 1058–1091 (2021)
Wang, Y., Winkler, M., Xiang, Z.: Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal. Anal. Appl. 20, 141–170 (2022)
Winkler, M.: Aggregation versus global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)
Winkler, M.: A result on parabolic gradient regularity in Orlicz spaces and application to absorption-induced blow-up prevention in a Keller–Segel-type cross-diffusion system. Int. Math. Res. Not. 19, 16336–16393 (2023)
Winkler, M.: A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: global weak solutions and asymptotic stabilization. J. Funct. Anal. 276, 1339–1401 (2019)
Winkler, M.: Attractiveness of constant states in logistic-type Keller–Segel systems involving subquadratic growth restrictions. Adv. Nonlinear Stud. 20, 795–817 (2020)
Winkler, M.: Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. Calc. Var. Partial. Differ. Equ. 54, 3789–3828 (2015)
Winkler, M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equ. 35, 1516–1537 (2010)
Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013)
Winkler, M.: Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with logistic dampening. J. Differ. Equ. 257, 1056–1077 (2014)
Winkler, M.: Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling cellular swimming in fluid drops. Commun. Partial Differ. Equ. 37, 319–352 (2012)
Winkler, M.: Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1329–1352 (2016)
Winkler, M.: How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Trans. Am. Math. Soc. 369, 3067–3125 (2017)
Winkler, M.: \(L^{1}\) solutions to parabolic Keller–Segel systems involving arbitrary superlinear degradation. Ann. Sc. Norm. Super. Pisa, Cl. Sci 24, 141–172 (2023)
Winkler, M.: Reaction-driven relaxation in three-dimensional Keller–Segel–Navier–Stokes interaction. Comm. Math. Phys. 389, 439–489 (2022)
Winkler, M.: Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch. Ration. Mech. Anal. 211(2), 455–487 (2014)
Wu, C., Xiang, Z.: Asymptotic dynamics on a chemotaxis-Navier–Stokes system with nonlinear diffusion and inhomogeneous boundary conditions. Math. Models Methods Appl. Sci. 30, 1325–1374 (2020)
Wu, C., Xiang, Z.: Saturation of the signal on the boundary: global weak solvability in a chemotaxis-Stokes system with porous-media type cell diffusion. J. Differ. Equ. 315, 122–158 (2022)
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Liu, C., Liu, B. Eventual smoothness in a chemotaxis-Navier–Stokes system with indirect signal production involving Dirichlet signal boundary condition. Z. Angew. Math. Phys. 75, 180 (2024). https://doi.org/10.1007/s00033-024-02324-6
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DOI: https://doi.org/10.1007/s00033-024-02324-6