On the global existence of weak solutions for the Cucker-Smale-Navier-Stokes system with shear thickening
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We study the large-time dynamics of Cucker-Smale (C-S) flocking particles interacting with non-Newtonian incompressible fluids. Dynamics of particles and fluids were modeled using the kinetic Cucker-Smale equation for particles and non-Newtonian Navier-Stokes system for fluids, respectively and these two systems are coupled via the drag force, which is the main flocking (alignment) mechanism between particles and fluids. We present a global existence theory for weak solutions to the coupled Cucker-Smale-Navier-Stokes system with shear thickening. We also use a Lyapunov functional approach to show that sufficiently regular solutions approach flocking states exponentially fast in time.
Keywordsemergence flocking Cucker-Smale model Navier-Stokes equation non-Newtonian fluid
MSC(2010)70F45 76A05 76D05
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The work of Seung-Yeal Ha was supported by the Samsung Science and Technology Foundation (Grant No. SSTF-BA1401-03). The work of Hwa Kil Kim was supported by the National Research Foundation of Korea (Grant No. NRF2015R1D1A1A01056696). The work of Jae-Myoung Kim was supported by BK21 PLUS SNU Mathematical Sciences Division and the National Research Foundation of Korea (Grant No. NRF-2016R1D1A1B03930422).
- 24.Neunzert H. An Introduction to the Nonlinear Boltzmann-Vlasov Equation. Kinetic Theories and the Boltzmann Equation. Lecture Notes in Mathematics, vol. 1048, Berlin: Springer, 1984Google Scholar
- 28.Whitaker S. Introduction to Fluid Mechanics. Upper Saddle River: Prentice-Hall, 1986Google Scholar