# American Institute of Mathematical Sciences

October  2021, 20(10): 3583-3604. doi: 10.3934/cpaa.2021122

## Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain

 Department of Mathematics, Nanjing University, Nanjing, 210093, China

* Corresponding author

Received  April 2021 Revised  June 2021 Published  October 2021 Early access  July 2021

Fund Project: This work is supported by NSFC (Grant Nos. 12071212, 11971234) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions

We study a kinetic-fluid model in a $3D$ bounded domain. More precisely, this model is a coupling of the Vlasov-Fokker-Planck equation with the local alignment force and the compressible Navier-Stokes equations with nonhomogeneous Dirichlet boundary condition. We prove the global existence of weak solutions to it for the isentropic fluid (adiabatic coefficient $\gamma> 3/2$) and hence extend the existence result of Choi and Jung [Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain, arXiv: 1912.13134v2], where the velocity of the fluid is supplemented with homogeneous Dirichlet boundary condition.

Citation: Fucai Li, Yue Li. Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3583-3604. doi: 10.3934/cpaa.2021122
##### References:
 [1] S. Berres, R. Bürger, K. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80. [2] D. Bresch and P. E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. Math., 188 (2018), 577-684. [3] R. Bürger, W. L. Wendland and F. Concha, Model equations for gravitational sedimentation-consolidation processes, Z. Angew. Math. Mech., 80 (2000), 79-92. [4] J. A. Carrillo, Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system, Math. Methods Appl. Sci., 21 (1998), 907-938. [5] J. A. Carrillo, Y. P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, Ann. Inst. H. Poincaré Anal., 33 (2016), 273-307. [6] T. Chang, B. J. Jin and A. Novotný, Compressible Navier-Stokes system with general inflow-outflow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278. [7] Y. P. Choi and J. Jung, Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain, preprint, arXiv: 1912.13134v2. [8] Y. P. Choi and J. Jung, Asymptotic analysis for Vlasov-Fokker-Planck/compressible Navier-Stokes equations with a density-dependent viscosity, arXiv: 1901.01221v1 [9] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. [10] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. [11] V. Girinon, Navier-Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain, J. Math. Fluid Mech., 13 (2011), 309-339. [12] S. Jiang and P. Zhang, Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pures Appl., 82 (2003), 949-973. [13] T. Karper, A. Mellet and K. Trivisa, Existence of weak solutions to the kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243. [14] F. Li and Y. Li, Global weak solutions and asymptotic analysis for a kinetic-fluid model with a heterogeneous friction force, preprint. [15] F. Li, Y. Mu and D. Wang, Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026. [16] Y. Li, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system with nonhomogeneous boundary data, Z. Angew. Math. Phys., 72 (2021), 29 pp. [17] P. L. Lions, Mathematical Topics in Fluid Mechanics-Volume 2: Compressible Models, Oxford Science Publications, Oxford, 1998. [18] A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. [19] A. Mellet and A. Vasseur, Asymptotic anslysis for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596. [20] P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations: theory and shape optimization, Springer-Verlag, New York, 2012. doi: 10.1007/978-3-0348-0367-0. [21] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-74.  doi: 10.1016/S1874-5792(02)80004-0.

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##### References:
 [1] S. Berres, R. Bürger, K. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80. [2] D. Bresch and P. E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. Math., 188 (2018), 577-684. [3] R. Bürger, W. L. Wendland and F. Concha, Model equations for gravitational sedimentation-consolidation processes, Z. Angew. Math. Mech., 80 (2000), 79-92. [4] J. A. Carrillo, Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system, Math. Methods Appl. Sci., 21 (1998), 907-938. [5] J. A. Carrillo, Y. P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, Ann. Inst. H. Poincaré Anal., 33 (2016), 273-307. [6] T. Chang, B. J. Jin and A. Novotný, Compressible Navier-Stokes system with general inflow-outflow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278. [7] Y. P. Choi and J. Jung, Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain, preprint, arXiv: 1912.13134v2. [8] Y. P. Choi and J. Jung, Asymptotic analysis for Vlasov-Fokker-Planck/compressible Navier-Stokes equations with a density-dependent viscosity, arXiv: 1901.01221v1 [9] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. [10] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. [11] V. Girinon, Navier-Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain, J. Math. Fluid Mech., 13 (2011), 309-339. [12] S. Jiang and P. Zhang, Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pures Appl., 82 (2003), 949-973. [13] T. Karper, A. Mellet and K. Trivisa, Existence of weak solutions to the kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243. [14] F. Li and Y. Li, Global weak solutions and asymptotic analysis for a kinetic-fluid model with a heterogeneous friction force, preprint. [15] F. Li, Y. Mu and D. Wang, Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026. [16] Y. Li, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system with nonhomogeneous boundary data, Z. Angew. Math. Phys., 72 (2021), 29 pp. [17] P. L. Lions, Mathematical Topics in Fluid Mechanics-Volume 2: Compressible Models, Oxford Science Publications, Oxford, 1998. [18] A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. [19] A. Mellet and A. Vasseur, Asymptotic anslysis for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596. [20] P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations: theory and shape optimization, Springer-Verlag, New York, 2012. doi: 10.1007/978-3-0348-0367-0. [21] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-74.  doi: 10.1016/S1874-5792(02)80004-0.
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