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

Compressible Euler equations interacting with incompressible flow

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.
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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