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Approximation of a simple Navier-Stokes model by monotonic rearrangement
1. | CMLS, Ecole Polytechnique, Palaiseau, FR-91128, France |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), 1013-1037.
doi: 10.1137/0721063. |
[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. |
[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. |
[9] |
Y. Brenier, Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics, Methods Appl. Anal., 11 (2004), 515-532. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003.
doi: 10.1007/b12016. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), 1013-1037.
doi: 10.1137/0721063. |
[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. |
[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. |
[9] |
Y. Brenier, Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics, Methods Appl. Anal., 11 (2004), 515-532. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, AMS, Providence, RI, 2003.
doi: 10.1007/b12016. |
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