American Institute of Mathematical Sciences

Advanced
June  2015, 8(2): 335-358. doi: 10.3934/krm.2015.8.335

Compressible Euler equations interacting with incompressible flow

Young-Pil Choi 1,

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ

Received  December 2014 Revised  January 2015 Published  March 2015

We investigate the global existence and large-time behavior of classical solutions to the compressible Euler equations coupled to the incompressible Navier-Stokes equations. The coupled hydrodynamic equations are rigorously derived in [1] as the hydrodynamic limit of the Vlasov/incompressible Navier-Stokes system with strong noise and local alignment. We prove the existence and uniqueness of global classical solutions of the coupled system under suitable assumptions. As a direct consequence of our result, we can conclude that the estimates of hydrodynamic limit studied in [1] hold for all time. For the large-time behavior of the classical solutions, we show that two fluid velocities will be aligned with each other exponentially fast as time evolves.
Keywords: hydrodynamic limit, kinetic-fluid equations, coupled hydrodynamic equations., large-time behavior, Global existence of classical solutions.
Mathematics Subject Classification: Primary: 35Q30, 35Q70; Secondary: 76N10, 76N1.
Citation: Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic and Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335
References:
[1]

J. A. Carrillo, Y.-P. Choi and T. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, to appear in Annales de lIHP-ANL. doi: 10.1016/j.anihpc.2014.10.002.

[2]

J. A. Carrillo, Y.-P. Choi, E. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with nonlocal forces, preprint, 2014.

[3]

J. A. Carrillo, R. Duan and A. Moussa, Global classical solutions close to the equilibrium to the Vlasov-Fokker-Planck-Euler system, Kinetic and Related Models, 4 (2011), 227-258. doi: 10.3934/krm.2011.4.227.

[4]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Comm. Partial Differential Equations,, 31 (2006), 1349-1379. doi: 10.1080/03605300500394389.

[5]

G.-Q. Chen, Euler equations and related hyperbolic conservation laws, in Handbook of Differential Equations, Vol. 2 (eds. C. M. Dafermos and E. Feireisl), Elsevier Science, Amsterdam, 2005, 1-104.

[6]

Y.-P. Choi, A revisit to the large-time behavior of the Vlasov/compressible Navier-Stokes equations, preprint, 2014.

[7]

R. Duan and S. Liu, Cauchy problem on the Vlaosv-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinetic and Related Models, 6 (2013), 687-700. doi: 10.3934/krm.2013.6.687.

[8]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[9]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508.

[10]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.

[11]

T. Karper, A. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Mod. Meth. Appl. Sci., 25 (2015), 131-163. doi: 10.1142/S0218202515500050.

[12]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.

[13]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[14]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Mod. Meth. Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.

[15]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/Compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 573-596. doi: 10.1007/s00220-008-0523-4.

[16]

T. C. Sideris, B. Thomases and D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations, 28 (2003), 795-816. doi: 10.1081/PDE-120020497.

show all references

References:
[1]

J. A. Carrillo, Y.-P. Choi and T. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, to appear in Annales de lIHP-ANL. doi: 10.1016/j.anihpc.2014.10.002.

[2]

J. A. Carrillo, Y.-P. Choi, E. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with nonlocal forces, preprint, 2014.

[3]

J. A. Carrillo, R. Duan and A. Moussa, Global classical solutions close to the equilibrium to the Vlasov-Fokker-Planck-Euler system, Kinetic and Related Models, 4 (2011), 227-258. doi: 10.3934/krm.2011.4.227.

[4]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Comm. Partial Differential Equations,, 31 (2006), 1349-1379. doi: 10.1080/03605300500394389.

[5]

G.-Q. Chen, Euler equations and related hyperbolic conservation laws, in Handbook of Differential Equations, Vol. 2 (eds. C. M. Dafermos and E. Feireisl), Elsevier Science, Amsterdam, 2005, 1-104.

[6]

Y.-P. Choi, A revisit to the large-time behavior of the Vlasov/compressible Navier-Stokes equations, preprint, 2014.

[7]

R. Duan and S. Liu, Cauchy problem on the Vlaosv-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinetic and Related Models, 6 (2013), 687-700. doi: 10.3934/krm.2013.6.687.

[8]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[9]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508.

[10]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.

[11]

T. Karper, A. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Mod. Meth. Appl. Sci., 25 (2015), 131-163. doi: 10.1142/S0218202515500050.

[12]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.

[13]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[14]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Mod. Meth. Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.

[15]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/Compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 573-596. doi: 10.1007/s00220-008-0523-4.

[16]

T. C. Sideris, B. Thomases and D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations, 28 (2003), 795-816. doi: 10.1081/PDE-120020497.

[1]

Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039
[2]

Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic and Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165
[3]

Feimin Huang, Yeping Li. Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 455-470. doi: 10.3934/dcds.2009.24.455
[4]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic and Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
[5]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure and Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003
[6]

Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic and Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013
[7]

Yachun Li, Shengguo Zhu. Existence results for compressible radiation hydrodynamic equations with vacuum. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1023-1052. doi: 10.3934/cpaa.2015.14.1023
[8]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[9]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic and Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016
[10]

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
[11]

Jens Lorenz, Wilberclay G. Melo, Natã Firmino Rocha. The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3819-3841. doi: 10.3934/dcdsb.2018332
[12]

Colette Guillopé, Samer Israwi, Raafat Talhouk. Large-time existence for one-dimensional Green-Naghdi equations with vorticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2947-2974. doi: 10.3934/dcdss.2021040
[13]

Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087
[14]

Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2271-2297. doi: 10.3934/dcds.2013.33.2271
[15]

Adrian Constantin, Joachim Escher. Introduction to the special issue on hydrodynamic model equations. Communications on Pure and Applied Analysis, 2012, 11 (4) : i-iii. doi: 10.3934/cpaa.2012.11.4i
[16]

Alexander V. Bobylev, Sergey V. Meleshko. On group symmetries of the hydrodynamic equations for rarefied gas. Kinetic and Related Models, 2021, 14 (3) : 469-482. doi: 10.3934/krm.2021012
[17]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[18]

Ming Mei, Bruno Rubino, Rosella Sampalmieri. Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain. Kinetic and Related Models, 2012, 5 (3) : 537-550. doi: 10.3934/krm.2012.5.537
[19]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[20]

Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381

2021 Impact Factor: 1.398

Tools

Metrics

  • PDF downloads (142)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy