Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations

Jesús Bonilla; Juan Vicente Gutiérrez-Santacreu

doi:10.1515/jnma-2023-0016

Published by De Gruyter October 16, 2023

Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations

Jesús Bonilla and Juan Vicente Gutiérrez-Santacreu

From the journal Journal of Numerical Mathematics

https://doi.org/10.1515/jnma-2023-0016

Showing a limited preview of this publication:

Abstract

The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality.

Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L¹(Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures.

Our findings show that the existence threshold value 2π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4π may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.

MSC 2010: 35Q35; 65N30; 92C17

Keywords: Keller–Segel equations; Navier–Stokes equations; stabilized finite-element approximation; shock detector; lower and a priori bounds; blowup

Acknowledgments

JVGS would like to thank Dr. Juan Carlos Dana and Dr. Víctor Álvarez from University of Seville for their valuable discussions on existence and uniqueness of approximate solutions.

References

[1] Adams, R. A.; Fournier, J. J. F. Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.Search in Google Scholar

[2] Badia, S; Bonilla, J. Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization. Comput. Methods Appl. Mech. Engrg. 313 (2017), 133–158.10.1016/j.cma.2016.09.035Search in Google Scholar

[3] Badia, S; Bonilla, J.; Gutiérrez-Santacreu, J, V. Bound-preserving finite element approximations of the Keller–Segel equations, Math. Models Methods Appl. Sci. 33 (2023), no. 3, 609–642.10.1142/S0218202523500148Search in Google Scholar

[4] Brenner, S. C.; Scott, L. R., The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008.10.1007/978-0-387-75934-0Search in Google Scholar

[5] Chertock, A.; Kurganov, A., A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111 (2008), no. 2, 169–20510.1007/s00211-008-0188-0Search in Google Scholar

[6] Chertock, A.; Epshteyn, Y.; Hu, H.; Kurganov, A., High-order positivity-preserving hybrid finitevolume-finite-difference methods for chemotaxis systems. Adv. Comput. Math. 44 (1) (2018) 327–350.10.1007/s10444-017-9545-9Search in Google Scholar

[7] Girault, V.; Lions, J.-L., Two-grid finite-element schemes for the transient Navier-Stokes problem. M2AN Math. Model. Numer. Anal. 35 (2001), no. 5, 945–980.10.1051/m2an:2001145Search in Google Scholar

[8] Guillén-González, F.; Gutiérrez-Santacreu, J. V., From a cell model with active motion to a Hele-Shaw-like system: a numerical approach Numer. Math. 143 (2019), no. 1, 107–137.10.1007/s00211-019-01053-7Search in Google Scholar

[9] Gutiérrez-Santacreu, J. V.; Rodríguez-Galván, J. R. Analysis of a fully discrete approximation for the classical Keller-Segel model: Lower and a priori bounds. Comput. Math. Appl. 85 (2021), 69–81.10.1016/j.camwa.2021.01.009Search in Google Scholar

[10] Horstmann, D.; Wang, G., Blow-up in a chemotaxis model without symmetry assumptions. European J. Appl. Math. 12 (2001), 159–177.10.1017/S0956792501004363Search in Google Scholar

[11] Keller, E. F.; Segel, L. A., Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.10.1016/0022-5193(70)90092-5Search in Google Scholar PubMed

[12] Keller; E. F.; Segel, L. A., Model for chemotaxis. J. Theor. Biol. 30 (1971), 225–234.10.1016/0022-5193(71)90050-6Search in Google Scholar PubMed

[13] Kiselev, A.; Ryzhik, L., Biomixing by chemotaxis and enhancement of biological reactions. Comm. Partial Differential Equations, 37 (2012) 298–318.10.1080/03605302.2011.589879Search in Google Scholar

[14] Kiselev, A.; Ryzhik, L., Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case. J. Math. Phys., 53 (2012), 115609.10.1063/1.4742858Search in Google Scholar

[15] Li, X. H.; Shu, C.-W.; Yang, Y., Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model., J. Sci. Comput. 73 (2017), no. 2-3, 943–967.10.1007/s10915-016-0354-ySearch in Google Scholar

[16] Moser, J. A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077–1092.10.1512/iumj.1971.20.20101Search in Google Scholar

[17] Nagai, T.; Senba T.; K. Yoshida Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac., 40 (1997), pp. 411–433.Search in Google Scholar

[18] Nochetto, R. H. Finite element methods for parabolic free boundary problems. Advances in numerical analysis, Vol. I (Lancaster, 1990), 34–95, Oxford Sci. Publ., Oxford Univ. Press, New York, 199110.1093/oso/9780198534389.003.0002Search in Google Scholar

[19] Rivière, B; Wheeler, M. F; Girault, V., A priori error estimates for finite element methods based on discontinuous approximations spaces for elliptic problems. SIAM J. Numer. Anal. 39 (2001), no. 3, 902–93110.1137/S003614290037174XSearch in Google Scholar

[20] Saito, N., Error analysis of a conservative finite-element approximation for the Keller–Segel system of chemotaxis, Commun. Pure Appl. Anal. 11 (2012), no. 1, 339–364.10.3934/cpaa.2012.11.339Search in Google Scholar

[21] Strehl, R., Sokolov A., Kuzmin, D., Turek, S., A flux-corrected finite element method for chemotaxis problems, Comput. Methods Appl. Math. 10 (2010), no. 2, 219–232.10.2478/cmam-2010-0013Search in Google Scholar

[22] Strehl R., Sokolov, A., Kuzmin D., Horstmann D., Turek, A positivity-preserving finite element method for chemotaxis problems in 3D, J. Comp. Appl. Math. 239 (2013), no. 1, 290–303.10.1016/j.cam.2012.09.041Search in Google Scholar

[23] Sulman M., Nguyen T., A positivity preserving moving mesh finite element method for the Keller–Segel chemotaxis model, J. Sci. Comput. 80 (2019), no. 1, 649–666.10.1007/s10915-019-00951-0Search in Google Scholar

[24] Scott, L.R.; Zhang, S. Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493.10.1090/S0025-5718-1990-1011446-7Search in Google Scholar

[25] Senba, T; Suzuki, T. Parabolic system of chemotaxis: blowup in a finite and the infinite time. Methods Appl. Anal. 8 (2001) 349–367.10.4310/MAA.2001.v8.n2.a9Search in Google Scholar

[26] Temam, T. Navier-Stokes equations. Theory and numerical analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001.10.1090/chel/343Search in Google Scholar

[27] 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. USA, 102 (2005) 2277–2282.10.1073/pnas.0406724102Search in Google Scholar PubMed PubMed Central

[28] Trudinger, N. S. On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473–483.10.1512/iumj.1968.17.17028Search in Google Scholar

[29] Winkler, M., Small-mass solutions in the two-dimensional Keller-Segel system coupled to the Navier-Stokes equations. SIAM J. Math. Anal. 52 (2020), no. 2, 2041–2080.10.1137/19M1264199Search in Google Scholar

Appendix A. Existence and uniqueness of numerical solutions

For the sake of completeness we sketch the proof of the existence and uniqueness of numerical solutions defined by the nonlinear system (11)–(14). In order to simplify the argument it is assumed that uhm is given, since including the Navier–Stokes subsystem does not introduce any additional difficulty. Additionally, it is considered that (44), (45) and (61) hold.

From now on we will make use of the notation introduced throughout this paper and the inverse inequalities (7) without being previously mentioned.

Appendix A.1 Existence

Let us consider the continuous mapping P: N_h × C_h → N_h × C_h defined as follows. Given (nhm,chm,uhm)∈Nh×Ch×Uh, we want to find P(n_h, c_h) ∈ N_h × C_h such that

(P(nh,ch),(n¯h,c¯h))=(P1(nh,ch),n¯h)+((P2(nh,ch),c¯h)),

where

(P1(nh,ch),n¯h)=1k(nh−nhm,n¯h)h−(nhuhm,∇n¯h)∗+(∇nh,∇n¯h)−(nh∇ch,∇n¯h)∗+(Bn(nh,uhm)nh,n¯h),

and

(P2(nh,ch),c¯h)=1k(ch−chm,c¯h)+(uhm⋅∇ch,c¯h)+12(∇⋅uhmch,c¯h)+(∇ch,∇c¯h)+(ch,c¯h)+(Bc(ch,uhm)ch,c¯h)−(nh,c¯h)h.

Take n¯_h = n_h and c¯_h = c_h to get

(P1(nh,ch),nh)=1k∥nh∥h2−1k(nhm,nh)h−(nhuhm,∇nh)∗+∥∇nh∥L2(Ω)2−(nh∇ch,∇nh)∗+(Bn(nh,uhm)nh,nh)

and

(P2(nh,ch),ch)=1k∥ch∥L2(Ω)2−1k(chm,ch)+∥∇ch∥L2(Ω)2+∥ch∥L2(Ω)2+(Bc(ch,uhm)ch,ch)−(nh,ch)h.

We want to prove that there exists L > 0 such that (P(n_h, c_h), (n_h, c_h)) > 0 holds for all (n_h, c_h) ∈ N_h × C_h satisfying ∥(nh,ch)∥2:=∥nh∥L2(Ω)2+∥ch∥L2(Ω)2=L2. To bound P₁, we write

(nhuhm,∇nh)∗=∑i∈I∑j∈I∗(Ωai)−(nj+1)(ni+1)log⁡(nj+1)−log⁡(ni+1)nj−ni+nj+1)(ni−nj)(φajuhm,∇φai)−∑i∈I∑j∈I∗(Ωai)(nj+1)(ni−nj)(φajuhm,∇φai)=∑i∈I∑j∈I∗(Ωai)−(nj+1)(ni+1)log⁡(nj+1)−log⁡(ni+1)nj−ni+nj+1)(ni−nj)(φajuhm,∇φai)−((nh+1)uhm,∇nh).

The mean value theorem implies that

−(nj+1)(ni+1)log⁡(nj+1)−log⁡(ni+1)nj−ni+nj+1≤|nj−ni|

and hence

(nhuhm,∇nh)∗≤∑i∈I∑j∈I∗(Ωai)(ni−nj)2|(φajuhm,∇φai)|−(nhuhm,∇nh):=S1+S2.

Now let T_ij ∈ 𝓣_h be such that a_i, a_j ∈ T_ij and denote h_ij = |a_i ™ a_j|. Then

nj−ni=hij∇nh|Tij⋅rij=∫Eij∇nh|Tij⋅rijdσ,

where E_ij ∈ 𝓔_h such that a_i, a_j ∈ E_ij. Cauchy–Schwarz' inequality and an inverse inequality [19, lm. 2.1] give

|nj−ni|≤C∥∇nh∥L2(Tij).

Thus

S1≤∑i∈I∑j∈I∗(Ωai)(ni−nj)2|(φajuhm,∇φai)|≤Ch∑i∈I∑j∈I∗(Ωai)∥∇nh∥L2(Tij)2∥φaj∥L∞(Ω)∥uhm∥L∞(Ω)∥φai∥L∞(Ω)≤Ch∥uhm∥L∞(Ω)∥∇nh∥L2(Ω)2≤Ch2∥uhm∥L2(Ω)∥nh∥L2(Ω)2.

Furthermore

S2=(nhuhm,∇nh)≤∥uhm∥L∞(Ω)∥nh∥L2(Ω)∥∇nh∥L2(Ω)≤Ch2∥uhm∥L2(Ω)∥nh∥L2(Ω)2.

Therefore

(nhuhm,∇nh)∗≤Ch2∥uhm∥L2(Ω)∥nh∥L2(Ω)2.

Additionally, on noting that rε(s)≤2s for all s > 0, it follows that

(nh∇ch,∇n¯h)∗=∑i<j∈Irε(ni)rε(nj)(cj−ci)(n¯i−n¯j)(∇φaj,∇φai)≤∥nh∥L∞(Ω)∥∇nh∥L2(Ω)∥∇ch∥L2(Ω)≤Ch4∥nh∥L1(Ω)∥nh∥L2(Ω)∥ch∥L2(Ω).

Compiling the above bounds, we find

(P1(nh,ch),nh)≥1k(1−C[∥nh∥L1(Ω)h4+∥uhm∥L2(Ω)h2])∥nh∥L2(Ω)2−1k(nhm,nh)h+Ch4∥nh∥L1(Ω)∥ch∥L2(Ω)2,

where we used the bounds ∥n_h∥_L²(Ω) ≤ ∥n_h∥_h ≤ C ∥n_h∥_L²(Ω) from [8, Prop. 2.3].

For P₂, we have:

(P2(nh,ch),ch)≥1k(1−kC2)∥ch∥L2(Ω)2−1k(chm,ch)−C2∥nh∥L2(Ω)2.

Finally,

(P(nh,ch),(n¯h,c¯h))≥1k(1−kC[1+∥nh∥L1(Ω)h4+∥uhm∥L2(Ω)h2])∥nh∥L2(Ω)2+1k(1−Ck[1+∥nh∥L1(Ω)h4])∥ch∥L2(Ω)2−1k(nhm,nh)h−1k(chm,ch).

Letting k be small enough such that

1−kC[1+∥nh∥L1(Ω)h4+∥uhm∥L2(Ω)h2]≤12,

we find

(P(nh,ch),(n¯h,c¯h))≥12k∥(nh,ch)∥2−1kC∥(nhm,chm)∥∥(nh,ch)∥=12k∥(nh,ch)∥(∥(nh,ch)∥−2C∥(nhm,chm)∥).

Thus, selecting L=3C∥(nhm,chm)∥ and applying Brouwer’s fixed point theorem [26, Lm 4.1] is enough to insure existence.

Appendix A.2 Uniqueness

It is assumed that there exist two different pair solutions (n_h, c_h) and (ñ_h, ñ_h). For concreteness we will only focus on the two more troublesome terms : (nhuhm,∇n¯h)∗ and (n_h ∇ c_h, ∇ n¯_h)_*, and (Bn(nh,uhm)nh,n¯h).

We first compare the convective terms:

(nhuhm,∇n¯h)∗−(n~huhm,∇n¯h)∗=∑i,j∈I(γijc(nh)−γjic(n~h))(n¯i−n¯j)(φajuhm,∇φai).

From the definition of γijc in (23), we have four possible combinations. The case when n_i ≠ n_j and ñ_i ≠ ñ_j is only treated because the others have quite a similarity. By the mean value theorem, we write

γijc(nh)−γjic(n~h)=(n~j+1)(n~i+1)θ~(n~i+1)+(1−θ~)(n~j+1)−(nj+1)(ni+1)θ(ni+1)+(1−θ)(nj+1)[5pt]≤(n~j+1)(n~i+1)θ~(n~i+1)+(1−θ~)(n~j+1)−(ni+1)+|ni−n~i|+(n~i+1)−(nj+1)(ni+1)θ(ni+1)+(1−θ)(nj+1):=T1+T2+T3,

for θ, θ̃ ∈ (0, 1). The term T₁ is bounded as

T1=(n~j+1)(n~i+1)θ~(n~i+1)+(1−θ~)(n~j+1)−(ni+1)(nj+1)θ~(nj+1)+(1−θ~)(nj+1)≤1θ~(n~i+1)+(1−θ~)(n~j+1)((n~j+1)|n~i−ni|+(ni+1)|n~j−nj|)+(nj+1)(ni+1)1θ~(n~i+1)+(1−θ~)(n~j+1)−1θ~(nj+1)+(1−θ~)(nj+1)≤max{∥nh+1∥L∞(Ω),∥n~h+1∥L∞(Ω)}(|n~i−ni|+|n~j−nj|)+∥nh+1∥L∞(Ω)(|n~i−ni|+|n~j−nj|)≤Ch2∥nh0+1∥L1(Ω)(|n~i−ni|+|n~j−nj|),

where in the second inequality we used Young’s inequality a^θ̃ b^1™θ̃ ≤ θ̃ a + (1 ™ θ̃) b. Thus

and hence

(nhuhm,∇n¯h)∗−(n~huhm,∇n¯h)∗≤Ch3∥uhm∥L∞(Ω)∥nh0+1∥L1(Ω)∑i,j∈I∥∇n¯h∥L2(Tij)∥nh−n~h∥L2(Tij).=Ch3∥uhm∥L∞(Ω)∥nh0+1∥L1(Ω)∥nh−n~h∥L2(Ω)∥∇n¯h∥L2(Ω)≤Ch5∥uhm∥L2(Ω)∥nh0+1∥L1(Ω)∥nh−n~h∥L2(Ω)∥n¯h∥L2(Ω).

Next we handle the chemotaxis terms. In doing so, we use the property rε′(s)≤32ε for all s > 0 to the coefficients γjich:

Then

(nh∇ch,∇n¯h)∗−(n~h∇c~h,∇n¯h)∗=∑i<j∈Iγjich(nh)(cj−ci)(n¯i−n¯j)(∇φaj,∇φai)−∑i<j∈Iγjich(n~h)(c~j−c~i)(n¯i−n¯j)(∇φaj,∇φai)=∑i<j∈Iγjich(n~h)(cj−c~j+ci−c~i)(n¯i−n¯j)(∇φaj,∇φai)+∑i<j∈I(γjich(nh)−γjich(n~h))(cj−ci)(n¯i−n¯j)(∇φaj,∇φai)≤Cεh4∥nh∥L1(Ω)∥ch−c~h∥L2(Ω)∥n¯h∥L2(Ω)+Cεh5∥nh∥L1(Ω)∥ch∥L1(Ω)∥nh−n~h∥L2(Ω)∥n¯h∥L2(Ω).

For the stabilising terms we need to face the difference of the coefficients νijn. Indeed,

νijn(nh)−νijn(n~h)=max{αai(nh)fijn(nh),αaj(nh)fjin(nh),0}−max{αai(n~h)fijn(n~h),αaj(n~h)fjin(n~h),0}≤max{αai(nh)fijn(nh)−αai(n~h)fijn(n~h),αaj(nh)fjin(nh)−αaj(n~h)fjin(n~h),0}.

We now proceed in this way. Define fijn=f¯ijn/(nj−ni) and write

αai(nh)fijn(nh)−αai(n~h)fijn(n~h)=fij(nh)(αai(nh)−αai(n~h))+αai(n~h)(fijn(nh)−fijn(n~h))=fijn(nh)(αai(nh)−αai(n~h))+αai(n~h)nj−ni(f¯ijn(nh)−f¯ijn(n~h)).+αai(n~h)f¯ijn(n~h)(1nj−ni−1n~i−n~j).

Following the proof of [2, Th. 6.1] and noting that |fijn|≤1, by the mean value theorem, leads to

(Bn(nh,uhm)nh,n¯h)−(Bn(n~h,uhm)n~h,n¯h)≤Cq1h2(1+h∥uhm∥L∞(Ω))∥nh−n~h∥L2(Ω)∥n¯h∥L2(Ω)≤Cq1h2(1+∥uhm∥L2(Ω))∥nh−n~h∥L2(Ω)∥n¯h∥L2(Ω).

Then we have:

∥nh−n~h∥L2(Ω)≤kCh5∥uhm∥L2(Ω)∥nh+1∥L1(Ω)∥nh−n~h∥L2(Ω)+kCεh4∥nh∥L1(Ω)∥ch−c~h∥L2(Ω)+kCεh5∥ch∥L1(Ω)∥nh−n~h∥L2(Ω)+kCq1h2(1+∥uhm∥L2(Ω))∥nh−n~h∥L2(Ω).

Furthermore,

∥ch−c~h∥L2(Ω)≤Cqk(1+Ch2∥uhm∥L2(Ω)+Ch2)∥ch−c~h∥L2(Ω).

Summing the above two inequalities and choosing k to be small enough yields uniqueness.

Published Online: 2023-10-16

Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations

Abstract

Acknowledgments

References

Appendix A. Existence and uniqueness of numerical solutions

Appendix A.1 Existence

Appendix A.2 Uniqueness

Journal and Issue

Articles in the same Issue