doi: 10.3934/dcdsb.2022059
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The isothermal limit for the compressible Euler equations with damping

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received  September 2021 Revised  February 2022 Early access March 2022

We consider the isothermal Euler system with damping. We explicitly compute the propagation and the behavior of Gaussian initial data, then we show the weak $ L^1 $ convergence of the density as well as the asymptotic behavior of its first and second moments. We also rigorously show the convergence of Barenblatt solutions towards a limit Gaussian profile in the isothermal limit $ \gamma \rightarrow 1 $.

Citation: Quentin Chauleur. The isothermal limit for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022059
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, No. 55. U. S. Government Printing Office, Washington, D. C., 1964. For sale by the Superintendent of Documents.

[2]

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur Les inéGalités De Sobolev Logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2000. With a preface by Dominique Bakry and Michel Ledoux.

[3]

A. ArnoldP. Markowich and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.

[4]

D. G. Aronson, The porous medium equation, In Nonlinear Diffusion Problems (Montecatini Terme, 1985), 1224 (1986), 1–46. doi: 10.1007/BFb0072687.

[5]

G. I. Barenblatt, On a class of exact solutions of the plane one-dimensional problem of unsteady filtration of a gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh., 17 (1953), 739-742. 

[6]

R. CarlesK. Carrapatoso and M. Hillairet, Rigidity results in generalized isothermal fluids, Ann. H. Lebesgue, 1 (2018), 47-85.  doi: 10.5802/ahl.2.

[7]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[8]

Q. Chauleur, Dynamics of the Schrödinger-Langevin equation, Nonlinearity, 34 (2021), 1943-1974.  doi: 10.1088/1361-6544/abd528.

[9]

C. Dellacherie and P.-A. Meyer, Probabilités Et Potentiel, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV. Hermann, Paris, 1975.

[10]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958.

[11]

S. Geng and F. Huang, $L^1$-convergence rates to the Barenblatt solution for the damped compressible Euler equations, J. Differential Equations, 266 (2019), 7890-7908.  doi: 10.1016/j.jde.2018.12.016.

[12]

F. Huang, Large time behavior for compressible Euler equations with damping and vacuum, Mathematical Analysis in Fluid and Gas Dynamics (Japanese) (Kyoto, 2001), (2002), 57–66.

[13]

F. HuangP. Marcati and R. Pan, Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24.  doi: 10.1007/s00205-004-0349-y.

[14]

F. Huang and R. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differential Equations, 220 (2006), 207-233.  doi: 10.1016/j.jde.2005.03.012.

[15]

F. HuangR. Pan and Z. Wang, $L^1$ convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal., 200 (2011), 665-689.  doi: 10.1007/s00205-010-0355-1.

[16]

P. G. LeFloch and V. Shelukhin, Symmetries and global solvability of the isothermal gas dynamics equations, Arch. Ration. Mech. Anal., 175 (2005), 389-430.  doi: 10.1007/s00205-004-0344-3.

[17]

T. Li and D. Wang, Blowup phenomena of solutions to the Euler equations for compressible fluid flow, J. Differential Equations, 221 (2006), 91-101.  doi: 10.1016/j.jde.2004.12.004.

[18]

T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32.  doi: 10.1007/BF03167296.

[19]

P. Marcati and R. Pan, Cauchy problem for compressible Euler equations with damping, In International Conference on Differential Equations, (Berlin, 1999), World Sci. Publ., River Edge, NJ, 1/2 (2000), 315–317.

[20]

T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Mathématiques d'Orsay, No. 78-02. Département de Mathématique, Université de Paris-Sud, Orsay, 1978.

[21]

K. Zhao, On the isothermal compressible Euler equations with frictional damping, Commun. Math. Anal., 9 (2010), 77-97. 

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, No. 55. U. S. Government Printing Office, Washington, D. C., 1964. For sale by the Superintendent of Documents.

[2]

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur Les inéGalités De Sobolev Logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2000. With a preface by Dominique Bakry and Michel Ledoux.

[3]

A. ArnoldP. Markowich and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.

[4]

D. G. Aronson, The porous medium equation, In Nonlinear Diffusion Problems (Montecatini Terme, 1985), 1224 (1986), 1–46. doi: 10.1007/BFb0072687.

[5]

G. I. Barenblatt, On a class of exact solutions of the plane one-dimensional problem of unsteady filtration of a gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh., 17 (1953), 739-742. 

[6]

R. CarlesK. Carrapatoso and M. Hillairet, Rigidity results in generalized isothermal fluids, Ann. H. Lebesgue, 1 (2018), 47-85.  doi: 10.5802/ahl.2.

[7]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[8]

Q. Chauleur, Dynamics of the Schrödinger-Langevin equation, Nonlinearity, 34 (2021), 1943-1974.  doi: 10.1088/1361-6544/abd528.

[9]

C. Dellacherie and P.-A. Meyer, Probabilités Et Potentiel, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV. Hermann, Paris, 1975.

[10]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958.

[11]

S. Geng and F. Huang, $L^1$-convergence rates to the Barenblatt solution for the damped compressible Euler equations, J. Differential Equations, 266 (2019), 7890-7908.  doi: 10.1016/j.jde.2018.12.016.

[12]

F. Huang, Large time behavior for compressible Euler equations with damping and vacuum, Mathematical Analysis in Fluid and Gas Dynamics (Japanese) (Kyoto, 2001), (2002), 57–66.

[13]

F. HuangP. Marcati and R. Pan, Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24.  doi: 10.1007/s00205-004-0349-y.

[14]

F. Huang and R. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differential Equations, 220 (2006), 207-233.  doi: 10.1016/j.jde.2005.03.012.

[15]

F. HuangR. Pan and Z. Wang, $L^1$ convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal., 200 (2011), 665-689.  doi: 10.1007/s00205-010-0355-1.

[16]

P. G. LeFloch and V. Shelukhin, Symmetries and global solvability of the isothermal gas dynamics equations, Arch. Ration. Mech. Anal., 175 (2005), 389-430.  doi: 10.1007/s00205-004-0344-3.

[17]

T. Li and D. Wang, Blowup phenomena of solutions to the Euler equations for compressible fluid flow, J. Differential Equations, 221 (2006), 91-101.  doi: 10.1016/j.jde.2004.12.004.

[18]

T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32.  doi: 10.1007/BF03167296.

[19]

P. Marcati and R. Pan, Cauchy problem for compressible Euler equations with damping, In International Conference on Differential Equations, (Berlin, 1999), World Sci. Publ., River Edge, NJ, 1/2 (2000), 315–317.

[20]

T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Mathématiques d'Orsay, No. 78-02. Département de Mathématique, Université de Paris-Sud, Orsay, 1978.

[21]

K. Zhao, On the isothermal compressible Euler equations with frictional damping, Commun. Math. Anal., 9 (2010), 77-97. 

Figure 1.  Convergence of $ \mathcal{B}_{\gamma} $ towards its limit Gaussian profile
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