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April  2014, 34(4): 1285-1300. doi: 10.3934/dcds.2014.34.1285

Approximation of a simple Navier-Stokes model by monotonic rearrangement

Yann Brenier 1,

1. 

CMLS, Ecole Polytechnique, Palaiseau, FR-91128, France

Received  October 2012 Published  October 2013

We consider the very simple Navier-Stokes model for compressible fluids in one space dimension, where there is no temperature equation and both the pressure and the viscosity are proportional to the density. We show that the resolution of this Navier-Stokes system can be reduced, through the crucial intervention of a monotonic rearrangement operator, to the time discretization of a very elementary differential equation with noise. In addition, our result can be easily extended to a related Navier-Stokes-Poisson system.
Keywords: optimal transport, monotonic rearrangement, Fluid mechanics, systems of particles..
Mathematics Subject Classification: 35Q30, 35Q3.
Citation: Yann Brenier. Approximation of a simple Navier-Stokes model by monotonic rearrangement. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1285-1300. doi: 10.3934/dcds.2014.34.1285
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

A. Blanchet, E. Carlen and J. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal., 262 (2012), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar

[3]

F. Bolley, Y. Brenier and G. Loeper, Contractive metrics for scalar conservation laws, Journal of Hyperbolic Differential Equations, 2 (2005), 91-107. doi: 10.1142/S0219891605000397.  Google Scholar

[4]

Y. Brenier, Une application de la symétrisation de Steiner aux équations hyperboliques: La méthode de transport et écroulement, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 563-566.  Google Scholar

[5]

Y. Brenier, Résolution d'équations d'évolution quasilinéaires en dimension $N$ d'espace à l'aide d'équations linéaires en dimension $N+1$, J. Differential Equations, 50 (1983), 375-390. doi: 10.1016/0022-0396(83)90067-0.  Google Scholar

[6]

Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), 1013-1037. doi: 10.1137/0721063.  Google Scholar

[7]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.  Google Scholar

[8]

Y. Brenier, A particle method for nonlinear convection diffusion equations in dimension one, J. Comput. Appl. Math., 31 (1990), 35-56. doi: 10.1016/0377-0427(90)90334-V.  Google Scholar

[9]

Y. Brenier, Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics, Methods Appl. Anal., 11 (2004), 515-532.  Google Scholar

[10]

Y. Brenier, $L^2$ formulation of multidimensional scalar conservation laws, Arch. Rational Mech. Anal., 193 (2009), 1-19. doi: 10.1007/s00205-009-0214-0.  Google Scholar

[11]

Y. Brenier, Hilbertian approaches to some non-linear conservation laws, in "Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena," Contemporary Mathematics, 526, Amer. Math. Soc., Providence, RI, (2010), 19-35. doi: 10.1090/conm/526/10375.  Google Scholar

[12]

Y. Brenier, W. Gangbo, G. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, J. Math. Pures Appl. (9), 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013.  Google Scholar

[13]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," (French) North-Holland Mathematics Studies, No. 5, Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[14]

C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[15]

F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J., 33 (1984), 673-709. doi: 10.1512/iumj.1984.33.33036.  Google Scholar

[16]

A. Chorin, Numerical methods for use in combustion modeling, in "Computing Methods in Applied Sciences and Engineering" (Proc. Fourth Internat. Sympos., Versailles, 1979), North-Holland, Amsterdam-New York, (1980), 229-236.  Google Scholar

[17]

W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations, 34 (2009), 1041-1073. doi: 10.1080/03605300902892345.  Google Scholar

[18]

L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. Partial Differential Equations, 13 (2001), 377-403. doi: 10.1007/s005260000077.  Google Scholar

[19]

Y. Giga and T. Miyakawa, A kinetic construction of global solutions of first order quasilinear equations, Duke Math. J., 50 (1983), 505-515. doi: 10.1215/S0012-7094-83-05022-6.  Google Scholar

[20]

N. Gigli and S. Mosconi, A variational approach to the Navier-Stokes equations, Bull. Sci. Math., 136 (2012), 256-276. doi: 10.1016/j.bulsci.2012.01.001.  Google Scholar

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar

[22]

L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal., 41 (2009), 1340-1365. doi: 10.1137/090750809.  Google Scholar

[23]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

A. Blanchet, E. Carlen and J. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal., 262 (2012), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar

[3]

F. Bolley, Y. Brenier and G. Loeper, Contractive metrics for scalar conservation laws, Journal of Hyperbolic Differential Equations, 2 (2005), 91-107. doi: 10.1142/S0219891605000397.  Google Scholar

[4]

Y. Brenier, Une application de la symétrisation de Steiner aux équations hyperboliques: La méthode de transport et écroulement, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 563-566.  Google Scholar

[5]

Y. Brenier, Résolution d'équations d'évolution quasilinéaires en dimension $N$ d'espace à l'aide d'équations linéaires en dimension $N+1$, J. Differential Equations, 50 (1983), 375-390. doi: 10.1016/0022-0396(83)90067-0.  Google Scholar

[6]

Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), 1013-1037. doi: 10.1137/0721063.  Google Scholar

[7]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.  Google Scholar

[8]

Y. Brenier, A particle method for nonlinear convection diffusion equations in dimension one, J. Comput. Appl. Math., 31 (1990), 35-56. doi: 10.1016/0377-0427(90)90334-V.  Google Scholar

[9]

Y. Brenier, Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics, Methods Appl. Anal., 11 (2004), 515-532.  Google Scholar

[10]

Y. Brenier, $L^2$ formulation of multidimensional scalar conservation laws, Arch. Rational Mech. Anal., 193 (2009), 1-19. doi: 10.1007/s00205-009-0214-0.  Google Scholar

[11]

Y. Brenier, Hilbertian approaches to some non-linear conservation laws, in "Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena," Contemporary Mathematics, 526, Amer. Math. Soc., Providence, RI, (2010), 19-35. doi: 10.1090/conm/526/10375.  Google Scholar

[12]

Y. Brenier, W. Gangbo, G. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, J. Math. Pures Appl. (9), 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013.  Google Scholar

[13]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," (French) North-Holland Mathematics Studies, No. 5, Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[14]

C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[15]

F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J., 33 (1984), 673-709. doi: 10.1512/iumj.1984.33.33036.  Google Scholar

[16]

A. Chorin, Numerical methods for use in combustion modeling, in "Computing Methods in Applied Sciences and Engineering" (Proc. Fourth Internat. Sympos., Versailles, 1979), North-Holland, Amsterdam-New York, (1980), 229-236.  Google Scholar

[17]

W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations, 34 (2009), 1041-1073. doi: 10.1080/03605300902892345.  Google Scholar

[18]

L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. Partial Differential Equations, 13 (2001), 377-403. doi: 10.1007/s005260000077.  Google Scholar

[19]

Y. Giga and T. Miyakawa, A kinetic construction of global solutions of first order quasilinear equations, Duke Math. J., 50 (1983), 505-515. doi: 10.1215/S0012-7094-83-05022-6.  Google Scholar

[20]

N. Gigli and S. Mosconi, A variational approach to the Navier-Stokes equations, Bull. Sci. Math., 136 (2012), 256-276. doi: 10.1016/j.bulsci.2012.01.001.  Google Scholar

[21]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar

[22]

L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal., 41 (2009), 1340-1365. doi: 10.1137/090750809.  Google Scholar

[23]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

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