Trajectory attractor and global attractor for Keller-Segel-Stokes model with arbitrary porous medium diffusion

Wenlong Sun, Yeping Li



We investigate long-time behavior of weak solutions for the Keller-Segel-Stokes model with arbitrary porous medium diffusion in 2D bounded domains. We first prove the existence of the trajectory attractor $\mathcal{A}^{\text{\rm tr}}$ for the translation semigroup in the trajectory space. Further, we construct the global attractor $\mathcal{A}$ in a generalized sense. The results are shown by the definition of trajectory attractor and global attractor, and energy estimates.


Keller-Segel-Stokes model; trajectory attractor; global attractor; translation semigroup

Full Text:



R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

J.M. Ball, Continuity properties of global attractors of generalized semiflows and the Navier–Stokes equations, J. Nonlinear Sci. 7 (1997), 475–502.

T. Caraballo, P.E. Kloeden and P.M. Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl. 288 (2003), 692–707.

T. Caraballo, J. Langa, V. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multiva ued dynamical systems, Set-Valued Anal. 11 (2003), 153–201.

T. Caraballo, J. Langa and J. Valero, Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise, J. Math. Anal. Appl. 260 (2001), 602–622.

V.V. Chepyzhov and M.I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl. 76 (1997), 913–664.

V.V. Chepyzhov and M.I. Vishik, Attractors for Equations of Mathematical Physics, Amercian Mathematical Society, Providence, 2002.

V.V. Chepyzhov, E.S. Titi and M.I. Vishik, On the convergence of solutions of the Leray-α model to the trajectory attractor of the 3D Navier–Stokes system, Discrete Contin. Dynam. Syst. A 17 (2007), 481–500.

M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dynam. Syst. A 28 (2010), 1437–1453.

C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Goldstein and J.O. Kessler, Selfconcentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett. 93 (2004), 098103-1-4.

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

A. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier–Stokes system, J. Differential Equations 240 (2007), 249–278.

J.L. Lion, Quelques Méthode de Résolution des Probèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann Inst. H. Poincarè Anal. Non Linèaire 28 (2011), 643–652.

A. Lorz, Coupled chemotaxis fluid model, Math. Mod. Meth. Appl. Sci. 20 (2012), 987–1004.

V. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal. 6 (1998), 83–111.

V. Melnik and J. Valero, On global attractors of multivalued semiprocess and nonautonomous evolution inclusions, Set-Valued Anal. 8 (2000), 375–403.

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm Sup. Pisa 20 (1966), 733–737.

H. Sohr, The Navier–Stokes Equations: An Elementary Functional Analytic Approach, Birkäuser, Basel, 2001.

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

I. Tuval, L. Cisnerous, C. Dombrowski, C. Wolgemuth, J.O. Kessler and R.E. Goldstein, Bacterial swimming and oxygen transport near contact limes, Proc. Natl. Acad. Sci. USA 102 (2005), 2277–2282.

M.I. Vishik and V.V. Chepyzhov, Trajectory and global attractors of 3D Navier–Stokes systems, Math. Notes 77 (2002), 177–193.

M.I. Vishik and V.V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys 66 (2011), 637–731.

M.I. Vishik, E.S. Titi and V.V. Chepyzhov, On the convergence of trajectory attractor of 3D Navier–Stokes-α model as α approaches 0, Sb. Math. 198 (2007), 1703–1736.

Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations 232 (2007), 573–622.

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.

C.-D. Zhao, L. Kong, G.-W. Liu and M. Zhao, The trajectory attractor and its limiting behavior for the convective Brinkman–Forchheimer equations, Topol. Methods Nonlinear Anal. 44 (2014), 413–433.

C.-D. Zhao, W.-L. Sun and C. Hsu, Pullback dynamical behaviors of the nonautonomous micropolar fluid flows, Dyn. Partial Difer. Equ. 12 (2015), 265–288.


  • There are currently no refbacks.

Partnerzy platformy czasopism