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Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

  • * Corresponding author: Hai-Yang Jin

    * Corresponding author: Hai-Yang Jin
The research of H.Y. Jin was supported by the NSF of China No. 11501218, and the Fundamental Research Funds for the Central Universities (No. 2017MS107).
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  • This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

    $\begin{cases}\tag{*}n_t+u·\nabla n = \nabla ·(d(c)\nabla n)-\nabla ·(χ (c) n\nabla c)+a n-bn^2, &x∈ Ω, ~~t>0, \\ c_t+u·\nabla c = Δ c+ n-c,&x∈ Ω, ~~t>0, \\ u_t+ u·\nabla u = Δ u-\nabla P+n\nabla φ,&x∈ Ω, ~~t>0, \\\nabla · u = 0& x∈ Ω, \ t>0, \end{cases}$

    in a bounded smooth domain $Ω\subset \mathbb{R}^2$ with homogeneous Neumann boundary conditions, where $a≥0$ and $b>0$ are constants, and the functions $d(c)$ and $χ(c)$ satisfy the following assumptions:

    ● $(d(c), χ (c))∈ [C^2([0, ∞))]^2$ with $d(c), χ(c)>0$ for all $c≥0$, $d'(c)<0$ and $\lim\limits_{c\to∞}d(c) = 0$.

    ● $\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$ and $\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$ exist.

    The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition $\lim\limits_{c\to∞}d(c) = 0$. In this paper, we will use function $d(c)$ as weight function and employ the method of energy estimate to establish the global existence of classical solutions of (*) with uniform-in-time bound. Furthermore, by constructing a Lyapunov functional, we show that the global classical solution $(n, c, u)$ will converge to the constant state $(\frac{a}{b}, \frac{a}{b}, 0)$ if $b>\frac{K_0}{16}$ with $K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$.

    Mathematics Subject Classification: Primary: 35A01, 35B40, 35K55, 35Q92, 92C17.


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  • [1] N. BellomoA. BellouquidY. S. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.
    [2] J. P. Bourguignon and H. Brezis, Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.
    [3] M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.
    [4] M. ChaeK. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.
    [5] A. ChertockK. FellnerA. KurganovA. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190.  doi: 10.1017/jfm.2011.534.
    [6] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.
    [7] T. Ciéslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.
    [8] T. Ciéslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.
    [9] M. DiFrancescoA. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.  doi: 10.3934/dcds.2010.28.1437.
    [10] R. J. DuanA. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.
    [11] R. J. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 7 (2014), 1833-1852.  doi: 10.1093/imrn/rns270.
    [12] M. Eisenbach, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Chemotaxis, Imperial College Press, London, 2004.
    [13] E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001.
    [14] X. FuH. TangC. LiuJ. D. HuangT. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.  doi: 10.1103/PhysRevLett.108.198102.
    [15] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solution of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212.  doi: 10.1016/0022-0396(86)90096-3.
    [16] Y. Giga and H. Sohr, Abstract Lp estimate for the Cauchy problem with application to the Navier-Stokes system in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.
    [17] M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. 
    [18] T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.
    [19] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569.
    [20] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165. 
    [21] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien., 106 (2004), 51-69. 
    [22] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.
    [23] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.
    [24] H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, to appear.
    [25] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. 
    [26] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.
    [27] O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968.
    [28] J. G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.  doi: 10.1016/j.anihpc.2011.04.005.
    [29] A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.
    [30] A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.  doi: 10.4310/CMS.2012.v10.n2.a7.
    [31] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 
    [32] T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 
    [33] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 
    [34] K. Painter and J. A. Sherratt, Modeling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.  doi: 10.1016/S0022-5193(03)00258-3.
    [35] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.
    [36] T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.
    [37] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.
    [38] Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.
    [39] Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.
    [40] Y. S. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.
    [41] Y. S. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2016), 2555-2573.  doi: 10.1007/s00033-015-0541-y.
    [42] Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.  doi: 10.1142/S0218202517500282.
    [43] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci., USA, 102 (2005), 2277–2282 doi: 10.1073/pnas.0406724102.
    [44] M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.
    [45] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.
    [46] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.
    [47] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.
    [48] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.
    [49] M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.
    [50] M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b.
    [51] C. Yoon and Y. J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.  doi: 10.1007/s10440-016-0089-7.
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