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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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