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June  2016, 11(2): 313-330. doi: 10.3934/nhm.2016.11.313

A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case

Rinaldo M. Colombo 1, and  Graziano Guerra 2,

1. 

INdAM Unit c/o DII, Università degli Studi di Brescia, Via Branze 38, 25123 Brescia, Italy
2. 

Dipartimento di Matematica e Applicazioni Via Roberto Cozzi, 55 - 20125 Milano, Italy

Received  April 2015 Published  March 2016

We consider two compressible immiscible fluids in one space dimension and in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the sound speed of the first fluid diverges to infinity, we present the proof of rigorous convergence for the fully non--linear compressible to incompressible limit of the coupled dynamics of the two fluids. A linear example is considered in detail, where fully explicit computations are possible.
Keywords: hyperbolic conservation laws., Incompressible limit, compressible Euler equations.
Mathematics Subject Classification: Primary: 35L65, 35Q35; Secondary: 35Q3.
Citation: Rinaldo M. Colombo, Graziano Guerra. A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case. Networks & Heterogeneous Media, 2016, 11 (2) : 313-330. doi: 10.3934/nhm.2016.11.313
References:
[1]

D. Amadori and G. Guerra, Global BV solutions and relaxation limit for a system of conservation laws, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1-26. doi: 10.1017/S0308210500000767.  Google Scholar

[2]

R. Borsche, R. M. Colombo and M. Garavello, Mixed systems: ODEs - balance laws, J. Differential Equations, 252 (2012), 2311-2338. doi: 10.1016/j.jde.2011.08.051.  Google Scholar

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem.  Google Scholar

[4]

R. M. Colombo and G. Guerra, Bv solutions to 1d isentropic euler equations in the zero mach number limit, J. Hyperbolic Differ. Equ., 2016, to appear. Google Scholar

[5]

R. M. Colombo, G. Guerra and V. Schleper, The compressible to incompressible limit of one dimensional euler equations: The non smooth case, Arch. Ration. Mech. Anal., 219 (2016), 701-718. doi: 10.1007/s00205-015-0904-8.  Google Scholar

[6]

R. M. Colombo and V. Schleper, Two-phase flows: Non-smooth well posedness and the compressible to incompressible limit, Nonlinear Anal. Real World Appl., 13 (2012), 2195-2213. doi: 10.1016/j.nonrwa.2012.01.015.  Google Scholar

[7]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.  Google Scholar

[8]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.  Google Scholar

[9]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90. doi: 10.1007/PL00004241.  Google Scholar

[10]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.  Google Scholar

[11]

S. Schochet, The mathematical theory of low Mach number flows, M2AN Math. Model. Numer. Anal., 39 (2005), 441-458. doi: 10.1051/m2an:2005017.  Google Scholar

[12]

J. Xu and W.-A. Yong, A note on incompressible limit for compressible Euler equations, Math. Methods Appl. Sci., 34 (2011), 831-838. doi: 10.1002/mma.1405.  Google Scholar

show all references

References:
[1]

D. Amadori and G. Guerra, Global BV solutions and relaxation limit for a system of conservation laws, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1-26. doi: 10.1017/S0308210500000767.  Google Scholar

[2]

R. Borsche, R. M. Colombo and M. Garavello, Mixed systems: ODEs - balance laws, J. Differential Equations, 252 (2012), 2311-2338. doi: 10.1016/j.jde.2011.08.051.  Google Scholar

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem.  Google Scholar

[4]

R. M. Colombo and G. Guerra, Bv solutions to 1d isentropic euler equations in the zero mach number limit, J. Hyperbolic Differ. Equ., 2016, to appear. Google Scholar

[5]

R. M. Colombo, G. Guerra and V. Schleper, The compressible to incompressible limit of one dimensional euler equations: The non smooth case, Arch. Ration. Mech. Anal., 219 (2016), 701-718. doi: 10.1007/s00205-015-0904-8.  Google Scholar

[6]

R. M. Colombo and V. Schleper, Two-phase flows: Non-smooth well posedness and the compressible to incompressible limit, Nonlinear Anal. Real World Appl., 13 (2012), 2195-2213. doi: 10.1016/j.nonrwa.2012.01.015.  Google Scholar

[7]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.  Google Scholar

[8]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. doi: 10.1002/cpa.3160350503.  Google Scholar

[9]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90. doi: 10.1007/PL00004241.  Google Scholar

[10]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.  Google Scholar

[11]

S. Schochet, The mathematical theory of low Mach number flows, M2AN Math. Model. Numer. Anal., 39 (2005), 441-458. doi: 10.1051/m2an:2005017.  Google Scholar

[12]

J. Xu and W.-A. Yong, A note on incompressible limit for compressible Euler equations, Math. Methods Appl. Sci., 34 (2011), 831-838. doi: 10.1002/mma.1405.  Google Scholar

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