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Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain
Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain
Department of Mathematics, Nanjing University, Nanjing, 210093, China |
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.
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. |
show all references
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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