Elsevier

Journal of Differential Equations

Volume 272, 25 January 2021, Pages 164-202
Journal of Differential Equations

A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

https://doi.org/10.1016/j.jde.2020.09.029Get rights and content

Abstract

This paper deals with the following quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization(*){nt+un=Δn(nS(x,n,c)c)nm,xΩ,t>0,ct+uc=Δcc+m,xΩ,t>0,mt+um=Δmnm,xΩ,t>0,ut+κ(u)u+P=Δu+(n+m)ϕ,xΩ,t>0,u=0,xΩ,t>0 under no-flux boundary conditions in a bounded domain ΩR3 with smooth boundary, where ϕW2,(Ω). Here S(x,n,c) denotes the rotational effect which satisfies SC2(Ω¯×[0,)2;R3×3) and |S(x,n,c)|S0(c)(1+n)α with α0 and some nonnegative nondecreasing function S0. Based on a new weighted estimate and some careful analysis, if α>0, then for any κR, system () possesses a global weak solution. Furthermore, for any p>1, this solution is uniformly bounded with respect to the norm in Lp(Ω)×L(Ω)×L(Ω)×L2(Ω;R3). Moreover, if κ=0, then system () admits a classical solution which is global in time and bounded.

Introduction

This work is concerned with the following chemotaxis-fluid system modelling coral fertilization:{nt+un=Δn(nS(x,n,c)c)nm,xΩ,t>0,ct+uc=Δcc+m,xΩ,t>0,mt+um=Δmnm,xΩ,t>0,ut+κ(u)u+P=Δu+(n+m)ϕ,xΩ,t>0,u=0,xΩ,t>0(nnS(x,n,c))ν=cν=mν=0,u=0,xΩ,t>0,n(x,0)=n0(x),c(x,0)=c0(x),m(x,0)=m0(x),u(x,0)=u0(x),xΩ, where ΩR3 is a bounded domain with smooth boundary ∂Ω and the matrix-valued function S(x,n,c) indicates the rotational effect which satisfiesSC2(Ω¯×[0,)2;R3×3) as well as|S(x,n,c)|(1+n)αS(c)for all(x,n,c)Ω×[0,)2withS(c)nondecreasing on[0,) and α0. As described in [17], [16], [8], [9], problems of this type arise in the modeling of the phenomenon of coral broadcast spawning, where the sperm n chemotactically moves toward the higher concentration of the chemical c released by the egg m, while the egg m is merely affected by random diffusion, fluid transport and degradation upon contact with the sperm (see also [20]). Here κ,u,P and ϕ denote, respectively, the strength of nonlinear fluid convection, the velocity field, the associated pressure of the fluid and the potential of the gravitational field. We further note that the sensitivity tensor S(x,n,c) may take values that are matrices possibly containing nontrivial off-diagonal entries, which reflects that the chemotactic migration may not necessarily be oriented along the gradient of the chemical signal, but may rather involve rotational flux components (see [54], [53] for the detailed model derivation).

Chemotaxis is the directed movement of the cells as a response to gradients of the concentration of the chemical signal substance in their environment, where the chemical signal substance may be produced or consumed by cells themselves (see e.g. Hillen and Painter [10] and [1]). The classical chemotaxis system was introduced in 1970 by Keller and Segel ([15]), which is called Keller-Segel system. Since then, the Keller-Segel model has attracted more and more attention, and also has been constantly modified by various authors to characterize more biological phenomena (see Cieślak and Stinner [4], Cieślak and Winkler [6], Ishida et al. [13], Painter and Hillen [25], Hillen and Painter [10], Wang et al. [33], [32], Winkler et al. [6], [11], [52], [41], [43], [42], [45], Zheng [56] and references therein for detailed results). For related works in this direction, we mention that a corresponding quasilinear version (see e.g. [29], [52], [58], [56], [59]), the logistic damping or the signal consumed by the cells, has been deeply investigated by Cieślak and Stinner [4], [5], Tao and Winkler [29], [40], [52], and Zheng et al. [56], [65], [58], [66].

In various situations, however, the interaction of chemotactic movement of the gametes and the surrounding fluid is not negligible (see Tuval et al. [31]). In 2005, Tuval et al. ([31]) proposed the following prototypical signal consuming model (with tensor-valued sensitivity):{nt+un=Δn(nS(x,n,c)c),xΩ,t>0,ct+uc=Δcnf(c),xΩ,t>0,ut+κ(u)u+P=Δu+nϕ,xΩ,t>0,u=0,xΩ,t>0, where f(c) denotes the consumption rate of the oxygen by the cells. Here S is a tensor-valued function or a scalar function which is the same as (2.1). The model (1.4) describes the interaction of oxygen-taxis bacteria with a surrounding incompressible viscous fluid in which the oxygen is dissolved. After this, assume that the chemotactic sensitivity S(x,n,c):=S(c) is a scalar function. This kind of models have been studied by many researchers by making use of energy-type functionals (see e.g. Chae et al. [3], Duan et al. [7], Liu and Lorz [23], [24], Tao and Winkler [30], [44], [46], [48], Zhang and Zheng [55] and references therein). In fact, if S(x,n,c):=S(c), Winkler ([44] and [46]) proved that in two-dimensional domains (1.4) admits a unique global classical solution which stabilizes to the spatially homogeneous equilibrium (n¯0,0,0) in the large time limit, where n¯0:=1|Ω|Ωn0>0 (see also [48] for a partial extension to the three-dimensional analogue). While in three-dimensional setting, he (see [48]) also showed that there exists a globally defined weak solution to (1.4).

Experiment [54] show that the chemotactic movement could be not directly along the signal gradient, but with a rotation, so that, the corresponding chemotaxis-fluid system with tensor-valued sensitivity loses entropy-like functional structure, which gives rise to considerable mathematical difficulties during the process of analysis. The global solvability of corresponding initial value problem for chemotaxis-fluid system with tensor-valued sensitivity have been deeply investigated by Cao, Lankeit [2], Ishida [12], Wang et al. [34], [37] and Winkler [47].

If nf(c) in the c-equation is replaced by c+n, and the u-equation is a (Navier-)Stokes equation, then (1.4) becomes the following chemotaxis-(Navier-)Stokes system in the context of signal produced other than consumed by cells (see [50], [36], [37], [38], [62], [14]){nt+un=Δn(nS(x,n,c)c),xΩ,t>0,ct+uc=Δcc+n,xΩ,t>0,ut+κ(u)u+P=Δu+nϕ,xΩ,t>0,u=0,xΩ,t>0. Due to the presence of the tensor-valued sensitivity S(x,n,c) as well as the strongly nonlinear term (u)u and lower regularity for n, the analysis of (1.5) with tensor-valued sensitivity began to flourish (see [50], [36], [37], [38], [62], [14]). In fact, the global boundedness of classical solutions to the Stokes-version (κ=0 in the third equation of system (1.5)) of system (1.5) with the tensor-valued S satisfying |S(x,n,c)|CS(1+n)α with some CS>0 and α>0 which implies that the effect of chemotaxis is weakened when the cell density increases has been proved for any α>0 in two dimensions (see Wang and Xiang [37]) and for α>12 in three dimensions (see Wang and Xiang [38]). Then Wang-Winkler-Xiang ([36]) further shows that when α>0 and ΩR2 is a bounded convex domain with smooth boundary, system (1.5) possesses a global-in-time classical and bounded solution. Recently, Zheng ([60]) extends the results of [36] to the general bounded domain by some new entropy-energy estimates. More recently, by using new entropy-energy estimates, Zheng and Ke ([14]) presented the existence of global and weak solutions for the system (1.5) under the assumption that S satisfies (1.2) and|S(x,n,c)|(1+n)αfor all(x,n,c)Ω×[0,)2 with α>13, which, in light of the known results for the fluid-free system (see Horstmann and Winkler [11] and Bellomo et al. [1]), is an optimal restriction on α. For more works about the chemotaxis-(Navier-)Stokes models (1.5), we mention that a corresponding quasilinear version or the logistic damping has been deeply investigated by Zheng [63], [57], Wang and Liu [22], Tao and Winkler [30], Wang et al. [37], [38].

Other variants of the model (1.5) have been used in the mathematical study of coral broad-cast spawning. In fact, Kiselev and Ryzhik ([17] and [16]) introduced the following Keller-Segel type system to model coral fertilization:{ρt+uρ=Δρχ(ρc)ερq,0=Δc+ρ, where ρ,u,χ and ερq, respectively, denote the density of egg (sperm) gametes, the smooth divergence free sea fluid velocity as well as the positive chemotactic sensitivity constant and the reaction (fertilization) phenomenon. In fact, under suitable conditions, the global-in-time existence of the solution to (1.6) is presented by Kiselev and Ryzhik in [17]. Moreover, they proved that the total mass m0(t)=R2ρ(x,t)dx approaches a positive constant whose lower bound is C(χ,ρ0,u) as t when q>2. In the critical case of N=q=2, a corresponding weaker but yet relevant effect within finite time intervals is detected (see [16]).

In order to analyze a further refinement of the model (1.6) which explicitly distinguishes between sperms and eggs, Espejo and Winkler ([9]) have recently considered the Navier-Stokes version of (2.1):{nt+un=Δn(nc)nm,xΩ,t>0,ct+uc=Δcc+m,xΩ,t>0,mt+um=Δmnm,xΩ,t>0,ut+κ(u)u+P=Δu+(n+m)ϕ,xΩ,t>0,u=0,xΩ,t>0 in a bounded domain ΩR2. If N=2, Espejo and Winkler ([9]) established the global existence of classical solutions to the associated initial-boundary value problem (1.7), which tend towards a spatially homogeneous equilibrium in the large time limit. Furthermore, if S(x,n,c) satisfying (1.2) and (1.3) with α13 or α0 and the initial data satisfy a certain smallness condition, Li-Pang-Wang ([20]) proved the same result for the three-dimensional Stokes (κ=0 in the fourth equation of (1.1)) version of system (1.1). From [20], we know that α13 is enough to warrant the boundedness of solutions to system (2.1) for any large data (see Li-Pang-Wang [20]). We should point that the core step of [20] is to establish the estimates of the functionaln(,t)L2(Ω)2+c(,t)L2(Ω)2+u(,t)W1,2(Ω)2, which strongly relies on α13 and κ=0 (see the proof of Lemma 3.1 of [20]). To the best of our knowledge, it is yet unclear whether for α<13 or κ0, the solutions of (2.1) exist (or even bounded) or not. Recently, relying on the functionaly(t):={Ωnε4α+23(,t)+a1Ω|cε(,t)|2+b1Ω|uε(,t)|2ifα112,Ωnε(,t)lnnε(,t)+a2Ω|cε(,t)|2+b2Ω|uε(,t)|2ifα=112, we ([61]) presented the existence of global weak solutions for the system (1.1) under the assumption that S satisfies (1.2) and (1.3) with α>0. However, the existence of global (stronger than the result of [61]) weak solutions is still open. In this paper, by using a new weighted estimate (see Lemma 3.2), we try to obtain enough regularity and compactness properties (see Lemma 3.2, Lemma 3.3, Lemma 3.5), then show that system (1.1) possesses a globally defined weak solution, which improves the result of [61]. Therefore, collecting the above results, it is meaningful to analyze the following question:

Whether or not the assumption of α is optimal? Can we further relax the restriction on α, say, to α>0? Moreover, can we consider the regularity of global solution for system (1.1)?

Inspired by the above works, the first result of paper is to prove the existence of global (and bounded) solution for any α>0.

Throughout this paper, we assume thatϕW2,(Ω) and the initial data (n0,c0,m0,u0) fulfills{n0C(Ω¯)withn00andn00,c0W1,(Ω)withc00inΩ¯,m0C(Ω¯)withm00andm00,u0D(Arγ)for someγ(34,1)and anyr(1,), where Ar denotes the Stokes operator with domain D(Ar):=W2,r(Ω)W01,r(Ω)Lσr(Ω), and Lσr(Ω):={φLr(Ω)|φ=0} for r(1,) ([28]).

In the context of these assumptions, our main result can be read as follows.

Theorem 1.1

Let ΩR3 be a bounded domain with smooth boundary. Suppose that the assumptions (1.2)(1.3) and (1.8)(1.9) hold. Ifα>0, then for any κR, there exist{nLloc1(Ω¯×[0,)),cL(Ω×(0,))Lloc1([0,);W1,1(Ω)),mL(Ω×(0,))Lloc1([0,);W1,1(Ω)),uLloc1([0,);W1,1(Ω)), such that (n,c,m,u) is a global weak solution of the problem (2.1) in the natural sense as specified in [61] (cf. Appendix A). This solution can be obtained as the pointwise limit a.e. in Ω×(0,) of a suitable sequence of classical solutions to the regularized problems (2.1) below. Moreover, (n,c,m,u) has the additional properties that{nL([0,);Lp(Ω))Lloc2([0,),W1,2(Ω))for anyp1,cL(Ω×(0,))Lloc2([0,),W2,2(Ω))Lloc4([0,),W1,4(Ω)),mL(Ω×(0,))Lloc2([0,),W2,2(Ω))Lloc4([0,),W1,4(Ω)),uLloc2([0,),W0,σ1,2(Ω))Lloc([0,),L2(Ω)). Furthermore, if κ=0, the problem (2.1) possesses at least one global classical solution (n,c,m,u,P). In addition, this solution is bounded in Ω×(0,) in the sense thatn(,t)L(Ω)+c(,t)W1,(Ω)+m(,t)W1,(Ω)+u(,t)L(Ω)Cfor allt>0.

Remark 1.1

(i) Theorem 1.1 indicates that α>0 and κ=0 is enough to ensure the global existence and uniform boundedness of solution of the three-dimensional Keller-Segel-Stokes system (1.1), which improves the result obtained in [20], therein α13 is required.

(ii) This result also improves the result of our recent paper ([61]), where the more weak solution than our result was obtained (by using a different method). In fact, here, the higher regularity for n is found, that is, for any t>0, there exists K>0 independent of t such thatΩnp(x,t)Kfor allt>0andp>1.

Mathematical challenges for the regularity and stabilization of the solution for system (1.1). System (1.1) incorporates fluid and rotational flux, which involves more complex cross-diffusion mechanisms and brings about many considerable mathematical difficulties. Firstly, even when posed without any external influence, that is, n=c=m0, the corresponding Navier-Stokes system (1.1) does not admit a satisfactory solution theory up to now (see Leray [19] and Sohr [28], Wiegner [39]). As far as we know that the question of global solvability in classes of suitably regular functions yet remains open except in cases when the initial data are appropriately small (see e.g. Wiegner [39]). Moreover, the tensor-valued sensitivity functions result in new mathematical difficulties, mainly linked to the fact that a chemotaxis system with such rotational fluxes thereby loses an energy-like structure (see e.g. [54]). In [20] and [9], relying on globally bounded for the solution, Espejo-Winkler ([9]) Li-Pang-Wang ([20]) proved that all these solutions of problem (1.1) are shown to approach a spatially homogeneous equilibrium in the large time limit when N=2 or N=3 and κ=0, respectively. As already mentioned in the above, in the case N=3, it is not only unknown whether the incompressible Navier-Stokes equations possess global smooth solutions for arbitrarily large smooth initial data (see e.g. Wiegner [39] and Sohr [28]). Therefore, when κ0 and N=3, we can not use the idea of [20] and [9] to discuss the large time behavior to problem (1.1), since, the globally bounded for the solutions are needed in [20] and [9].

In order to derive these theorems, in Section 2, we introduce the regularized system of (1.1), establish some basic estimates of the solutions and recall a local existence result. In Section 3, a key step of the proof of our main results is to establish a bound for nε(,t) in Lp(Ω) for any p>1 and t>0, which will play a significant role in achieving the eventual smoothness and convergence of the solutions. The approach is based on the weighted estimate of Ωnεpg(cε) with some weight function g(cε) which is uniformly bounded both from above and below by positive constants. In fact, using some careful analysis, one could derive the following inequality1pddtΩnεpg(cε)+p12Ωnεp2g(cε)|nε|2γ0Ω|cε|2for allt>0 with some large γ0. Then applying the testing procedure, we can get the bounedneess of 0+Ω|cε|2 (see Lemma 2.4), so that, combined with the above inequality, one can finally gain the uniform boundedness (respect to time) for nε(,t) in Lp(Ω) for any p>1 and t>0, which is a new estimate for this system. Here nε and cε are components of the solutions to (2.1) below. On the basis of the previously established estimates and the compactness properties thereby implied, we shall pass to the limit along an adequate sequence of numbers ε=εj0 and thereby verify Theorem 1.1.

Section snippets

Preliminaries

As mentioned in the introduction, the chemotactic sensitivity S in the first equation in (1.1) and the nonlinear convective term κ(u)u in the Navier-Stokes subsystem of (1.1) bring about a great challenge to the study of system (1.1). To deal with these difficulties, according to the ideas in [48] (see also [51], [14], [49]), we first consider the approximate problems given by{nεt+uεnε=Δnε(nεSε(x,nε,cε)cε)nεmε,xΩ,t>0,cεt+uεcε=Δcεcε+mε,xΩ,t>0,mεt+uεmε=Δmεnεmε,xΩ,t>0,uεt+Pε=Δuε

A-priori estimates

In this section we want to ensure that the time-local solutions obtained in Lemma 2.1 are in fact global solutions. To this end, for any p>1, under the assumption that α>0, we firstly obtain the uniformly boundedness of nε in Lp(Ω) respect to time, which plays an important role in showing the eventual smoothness and convergence of the solutions. Inspired by the weighted estimate argument developed in [42] (see also [46], [49]), we shall invoke a weight function g(cε) which is uniformly bounded

Passing to the limit: the proof of Theorem 1.1

With the help of a priori estimates, in this subsection, by means of a standard extraction procedure we can now derive the following lemma which actually contains our main existence result (Theorem 1.1) already.

Lemma 4.1

Assume that α>0. Then for any κR, there exists a sequence (εj)jN(0,1) such that εj0 as j and thatnεna.e.inΩ×(0,)and inLloc2(Ω¯×[0,))nεnweak star inL([0,);Lp(Ω))for anyp>1,nεninLloc2(Ω¯×[0,)),cεcinLloc2(Ω¯×[0,))anda.e.inΩ×(0,),mεminLloc2(Ω¯×[0,))anda.e.inΩ×(0,),cεc

Acknowledgements

This work is partially supported by Shandong Provincial Science Foundation for Outstanding Youth (No. ZR2018JL005), the National Natural Science Foundation of China (No. 11601215) and Project funded by China Postdoctoral Science Foundation (No. 2019M650927, 2019T120168).

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      By means of the above a priori estimates, we now show that (1.1) indeed possesses a globally bounded weak solution as asserted in Theorem 1.1. Recalling Lemma 2.3, Lemma 3.1 and Lemma 3.2, Lemma 3.3 can be proved by following the proof of Lemma 4.1 in [60]. We omit the details here.

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