A modified least action principle allowing mass concentrations for the early universe reconstruction problem
Confluentes Mathematici, Volume 3 (2011) no. 3, pp. 361-385.

We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors in [26]), and the related Zeldovich approximate model [46]. By substituting the fully nonlinear Monge–Ampère equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model ("Monge-Ampère gravitation/MAG"), for which the Zeldovich approximation becomes exact. The MAG model enjoys a least action principle in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developed by Ghoussoub [29]. A fully discrete algorithm is introduced for the EUR problem in one space dimension.

Published online:
DOI: 10.1142/S1793744211000400
Yann Brenier 1

1
@article{CML_2011__3_3_361_0,
     author = {Yann Brenier},
     title = {A modified least action principle allowing mass concentrations for the early universe reconstruction problem},
     journal = {Confluentes Mathematici},
     pages = {361--385},
     publisher = {World Scientific Publishing Co Pte Ltd},
     volume = {3},
     number = {3},
     year = {2011},
     doi = {10.1142/S1793744211000400},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000400/}
}
TY  - JOUR
AU  - Yann Brenier
TI  - A modified least action principle allowing mass concentrations for the early universe reconstruction problem
JO  - Confluentes Mathematici
PY  - 2011
SP  - 361
EP  - 385
VL  - 3
IS  - 3
PB  - World Scientific Publishing Co Pte Ltd
UR  - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000400/
DO  - 10.1142/S1793744211000400
LA  - en
ID  - CML_2011__3_3_361_0
ER  - 
%0 Journal Article
%A Yann Brenier
%T A modified least action principle allowing mass concentrations for the early universe reconstruction problem
%J Confluentes Mathematici
%D 2011
%P 361-385
%V 3
%N 3
%I World Scientific Publishing Co Pte Ltd
%U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000400/
%R 10.1142/S1793744211000400
%G en
%F CML_2011__3_3_361_0
Yann Brenier. A modified least action principle allowing mass concentrations for the early universe reconstruction problem. Confluentes Mathematici, Volume 3 (2011) no. 3, pp. 361-385. doi : 10.1142/S1793744211000400. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000400/

[1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004) 227–260.

[2] L. Ambrosio and W. Gangbo, Hamiltonian ODE in the Wasserstein spaces of proba- bility measures, Comm. Pure Appl. Math. 61 (2008) 18–53.

[3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and the Wasser- stein Spaces of Probability Measures, Lectures in Mathematics (Birkhäuser, 2005).

[4] J.-P. Aubin, Mathematical Methods of Game and Economic Theory, Studies in Math- ematics and Its Applications, Vol. 7 (North-Holland, 1979).

[5] M. Avellaneda and W. E, Statistical properties of shocks in Burgers turbulence, Comm. Math. Phys. 172 (1995) 13–38.

[6] E. Aurell, U. Frisch, J. Lutsko and M. Vergassola, On the multifractal properties of the energy dissipation derived from turbulence data, J. Fluid Mech. 238 (1992) 467–486.

[7] M. Blaser and T. Rivière, A minimality property for entropic solutions to scalar conservation laws in 1 + 1 dimensions, arXiv:0907.4215v1.

[8] F. Bouchut, Advances in Kinetic Theory and Computing, Ser. Adv. Math. Appl. Sci., Vol. 22 (World Scientific, 1994).

[9] F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rational Mech. Anal. 157 (2001) 75–90.

[10] L. Boudin, A solution with bounded expansion rate to the model of viscous pressure- less gases, SIAM J. Math. Anal. 32 (2000) 172–193.

[11] Y. Brenier, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal. 21 (1984) 1013–1037.

[12] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris I Math. 305 (1987) 805–808.

[13] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued func- tions, Comm. Pure Appl. Math. 44 (1991) 375–417.

[14] Y. Brenier, Order preserving vibrating strings and applications to electrodynamics and magnetohydrodynamics, Methods Appl. Anal. 11 (2004) 515–532.

[15] Y. Brenier, L2 formulation of multidimensional scalar conservation laws, Arch. Ratio- nal Mech. Anal. 193 (2009) 1–19.

[16] Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, J. Nonlinear Sci. 19 (2009) 547–570.

[17] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal. 35 (1998) 2317–2328.

[18] Y. Brenier, U. Frisch, M. Hénon, G. Loeper, S. Matarrese, R. Mohayaee and A. Sobolevskii, Reconstruction of the early universe as a convex optimization problem, Mon. Not. R. Astron. Soc. 2002.

[19] Y. Brenier and G. Loeper, A geometric approximation to the Euler equations: The Vlasov–Monge–Ampère equation, Geom. Funct. Anal. 14 (2004) 1182–1218.

[20] H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, No. 5 (North-Holland, 1973).

[21] J. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, to appear in Duke Math. J., CVGMT preprint 2010.

[22] D. Christodoulou, The Action Principle and PDEs, Annals of Maths Studies, Vol. 146 (Princeton Univ. Press, 2000).

[23] M. Cullen and J. Purser, An extended Lagrangian theory of semigeostrophic fronto- genesis, J. Atmospheric Sci. 41 (1984) 1477–1497.

[24] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer, Berlin, 2000).

[25] W. E, Y. Rykov and Y. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys. 177 (1996) 349–380.

[26] U. Frisch, S. Matarrese, R. Mohayaee and A. Sobolevski, A reconstruction of the initial conditions of the universe by optimal mass transportation, Nature 417 (2002) 260–262.

[27] W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton–Jacobi equations in the Wasser- stein space, Methods Appl. Anal. 15 (2008) 155–183.

[28] W. Gangbo, T. Nguyen and A. Tudorascu, Euler–Poisson systems as action- minimizing paths in the Wasserstein space, Arch. Rational Mech. Anal. 192 (2009) 419–452.

[29] N. Ghoussoub, Self-Dual Partial Differential Systems and Their Variational Principles (Springer, 2009).

[30] G. Gibbons and S. Hawking, The Very Early Universe (Cambridge Univ. Press, 1983).

[31] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992) 819–824.

[32] E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971) 225–234.

[33] G. Loeper, The reconstruction problem for the Euler–Poisson system in cosmology, Arch. Rational Mech. Anal. 179 (2006) 153–216.

[34] L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal. 41 (2009) 1340–1365.

[35] T. Nguyen and A. Tudorascu, Pressureless Euler/Euler–Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. Math. Anal. 40 (2008) 754–775.

[36] J. Nieto, F. Poupaud and J. Soler, High-field limit for the Vlasov–Poisson–Fokker– Planck system, Arch. Rational Mech. Anal. 158 (2001) 29–59.

[37] F. Otto, The geometry of dissipative evolution equations, CPDE 26 (2001) 873–915.

[38] P. J. E. Peebles, Astrophys. J. 344 (1989) L53–L56.

[39] M. Sever, An existence theorem in the large for zero-pressure gas dynamics, Diff. Integral Eq. 14 (2001) 1077–1092.

[40] S. Shandarin and Y. Zeldovich, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Mod. Phys. 61 (1989) 185–220.

[41] A. Shnirelman, On the principle of the shortest way in the dynamics of systems with constraints, in Global Analysis Studies and Applications, Vol. II, Lecture Notes in Math., Vol. 1214 (Springer, 1986), pp. 117–130.

[42] A. Sobolevskii, The small viscosity method for a one-dimensional system of equations of gas dynamic type without pressure, Dokl. Akad. Nauk 356 (1997) 310–312.

[43] M. Vergassola, B. Dubrulle, U. Frisch and A. Noullez, Burgers’ equation, Devil’s staircases and the mass distribution for large-scale structures, Astron. Astrophys. 289 (1994) 325–356.

[44] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58 (Amer. Math. Soc., 2003).

[45] G. Wolansky, On time reversible description of the process of coagulation and frag- mentation, Arch. Rational Mech. Anal. 193 (2009) 57–115.

[46] Y. Zeldovich, Gravitational instability: An approximate theory for large density perturbations, Astron. Astrophys. 5 (1970) 84–89.

Cited by Sources: