# American Institute of Mathematical Sciences

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

 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.
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 and 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. [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. [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. [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. [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. [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. [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. [8] S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651. doi: 10.1002/cpa.3160350503. [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. [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. [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. [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.

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