Compressible Euler equations interacting with incompressible flow

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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

Abstract
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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.
[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. 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. doi: 10.1080/03605300500394389.
[5]	G.-Q. Chen, Euler equations and related hyperbolic conservation laws,, in Handbook of Differential Equations, (2005), 1.
[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. 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, (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. 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. 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. doi: 10.1142/S0218202515500050.
[12]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181. doi: 10.1007/BF00280740.
[13]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, (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. 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. 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. 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. 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. doi: 10.1080/03605300500394389.
[5]	G.-Q. Chen, Euler equations and related hyperbolic conservation laws,, in Handbook of Differential Equations, (2005), 1.
[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. 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, (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. 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. 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. doi: 10.1142/S0218202515500050.
[12]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181. doi: 10.1007/BF00280740.
[13]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, (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. 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. 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. doi: 10.1081/PDE-120020497.

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