### Existence, uniqueness and decay estimates on mild solution to a fractional chemotaxis-fluid system

DOI: http://dx.doi.org/10.12775/TMNA.2020.029

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Y. Ascasibar, R. Granero-Belinchón and J.M. Moreno, An approximate treatment of gravitational collapse, Phys. D 262 (2013), 71–82.

F. Bartumeus, F. Peters, S. Pueyo, C. Marraśe and J. Catalan, Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton, Proc. Nat. Acad. Sci. 100 (2003), 12771–12775.

P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup in twodimensional chemotaxis models, Discrete Contin. Dynam. Syst. Ser. A 37 (2017), 1841–1856.

P. Biler, T. Funaki and W.A. Woyczyński, Interacting particle approximation for nonlocal quadratic evolution problems, Probab. Math. Statist. 19 (1999), 267–286.

P. Biler and G. Karch, Blowup of solutions to generalized Keller–Segel model, J. Evol. Equations 10 (2009), 247–262.

P. Biler and W.A. Woyczyński, Nonlocal quadratic evolution problems, Banach Center Publ. 52 (2000), 11–24.

P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Method Appl. Sci. 32 (2009), 112–126.

T. Black, Sublinear signal production in a two-dimensional Keller–Segel–Stokes system, Nonlinear Anal. Real World Appl. 31 (2016), 593–609.

J. Burczak and R. Granero-Belinchón, Global solutions for a supercritical driftdiffusion equation, Adv. Math. 295 (2016), 334–367.

J. Burczak and R. Granero–Belinchón, On a generalized doubly parabolic Keller–Segel system in one dimension, Math. Models Methods Appl. Sci. 26 (2016), 111–160.

J. Burczak and R. Granero-Belinchón, Critical Keller–Segel meets Burgers on S1 : large-time smooth solutions, Nonlinearity 29 (2016), 3810–3836.

J. Burczak and R. Granero-Belinchón, Suppression of blow up by a local source in 2D Keller–Segel system with fractional dissipation, J. Differential Equations 263 (2017), 6115–6142.

J. Burczak and R. Granero-Belinchón, Boundedness and homogeneous asymptotics for a fractional logistic Keller–Segel equations, Discrete Contin. Dynam. Syst. Ser. S 13 (2020), 139–164.

M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dynam. Syst. Ser. A 33 (2013), 2272–2297.

M. Chae, K. 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.

M. Chae, K. Kang and J. Lee, Asymptotic behaviors of solutions for an aerotaxis model coupled to fluid equations, J. Korean Math. Soc. 53 (2016), 127–146.

M. Chae, K. Kang, J. Lee and K. Lee, A regularity condition and temporal asymptotics for chemotaxis-fluid equations, Nonlinearity 31 (2018), 351–387.

L. Caffarelli and Y. Sire, On Some Pointwise Inequalities Involving Nonlocal Operators, Appl. Numer. Harmon. Anal., Birkhäuser, Cham, 2017.

X. Cao, Global classical solutions in chemotaxis-Navier–Stokes system with rotational flux term, J. Differential Equations 261 (2016), 6883–6914.

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis-Navier–Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations 55 (2016), Art. 107, 39 pp.

X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two species chemotaxis-Stokes system with competitive kinetics, Math. Methods Appl. Sci. 41 (2018), 3138–3154.

R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations 35 (2010), 1635–1673.

C. Escudero, Chemotactic collapse and mesenchymal morphogenesis, Phys. Rev. E 72 (2005), 022903.

C. Escudero, The fractional Keller–Segel model, Nonlinearity 19 (2006), 2909–2918.

A. Garfinkel, Y. Tintut, D. Petrasek, K. Boström and L.L. Demer, Pattern formation by vascular mesenchymal cells, Proc. Natl. Acad. Sci. USA 101 (2004), 9247–9250.

R. Granero-Belinchón, Global solutions for a hyperbolic-parabolic system of chemotaxis, J. Math. Anal. Appl. 449 (2017), 872-883.

R. Granero-Belinchón, On a drift-diffusion system for semiconductor devices, Ann. Henri Poincaré 17 (2016), 3473–3498.

B.L. Guo, X.K. Pu and F.H. Huang, Fractional Partial Differential Equations and their Numerical Solutions, Science Press, 2011.

H. He and Q. Zhang, Global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations, Nonlinear Anal. Real World Appl. 35 (2017), 336-349.

M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier–Stokes system with competitive kinetics, J. Differential Equations 263 (2017), 470–490.

H. Huang and J.G. Liu, Well-posedness for Keller–Segel equation with fractional Laplacian and the theory of propagation of chaos, Kinet. Relat. Models 9 (2016), 715-748.

S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete Contin. Dynam. Syst. Ser. A 35 (2015), 3463–3482.

J. Klafter, B. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walk, Biological Motion 89 (1990), 281–296.

H. Kozono, M. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid, J. Funct. Anal. 270 (2016), 1663–1683.

H. Kozono, M. Miura and Y. Sugiyama, Time global existence and finite time blow-up criterion for solutions to the Keller–Segel system coupled with the Navier–Stokes fluid, J. Differential Equations 267 (2019), 5410–5492.

J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci. 26 (2016), 2071–2109.

J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a highdimensional chemotaxis system with consumption, Discrete Contin. Dynam. Syst. Ser. A 37 (2017), 6099–6121.

M. Levandowsky, B. White and F. Schuster, Random movements of soil amebas, Acta Protozoologica 36 (1997), 237–248.

Y. Li and Y. Li, Global boundedness of solutions for the chemotaxis-Navier–Stokes system in R2 , J. Differential Equations 261 (2016), 6570–6631.

D. Li, J. L. Rodrigo, and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoamericana 26 (2010), 295–332.

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci. 20 (2010), 987–1004.

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.

Z. Tan and X. Zhang, Decay estimates of the coupled chemotaxis-fluid equations in R3 , J. Math. Anal. Appl. 410 (2014), 27–38.

Z. Tan and J. Zhou, Decay estimates of solutions to the coupled chemotaxis-fluid equations in R3 , Nonlinear Anal. Real World Appl. 43 (2018), 323–347.

Y. Tao and M. Winkler, Global existence and boundedness in a Keller–Segel–Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dynam. Syst. A 32 (2012), 1901–1914.

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys. 66 (2015), 2555–2573.

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a twodimensional Keller–Segel–Navier–Stokes system, Z. Angew. Math. Phys. 67 (2016), Art. 138, 23 pp.

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemottractant, J. Differential Equations 252 (2012), 2520–2543.

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. Natl. Acad. Sci. 102 (2005), 2277–2282.

Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discrete Contin. Dynam. Syst. Ser. B 20 (2015), 3235–3254.

X. Wang, Z. Liu and L. Zhou, Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions, Discrete Contin. Dynam. Syst. Ser. B 23 (2018), 4003–4020.

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.

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier–Stokes system, Arch. Rational Mech. Anal. 211 (2014), 455–487.

M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Trans. Amer. Math. Soc. 369 (2017), 3067–3125.

G. Wu and X. Zheng, On the well-posedness for Keller–Segel system with fractional diffusion, Math. Method Appl. Sci. 34 (2011), 1739–1750.

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier–Stokes system with nonlinear diffusion, J. Differential Equations 259 (2015), 3730–3754.

Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional chemotaxisNavier–Stokes system, Discrete Contin. Dynam. Syst. Ser. B 20 (2015), 2751–2759.

Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier–Stokes equations, SIAM J. Math. Anal. 46 (2014), 3078–3105.

S. Zhu, Z. Liu and L. Zhou, Decay estimates for the classical solution of Keller–Segel system with fractional Laplacian in higher dimensions, Appl. Anal. 99 (2020), 447–461.

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