Abstract
We prove the existence of weak solutions for the forward and backward statistical transport equations associated with the 2D Euler equations. Such solutions can be interpreted, respectively, as a statistical Lagrangian and a statistical Eulerian description of the motion of the fluid.
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Albeverio S., Hoegh-Krohn R., Merlini D. (1985). Some remarks on Euler flow, associated generalized random fields and Coulomb systems. In: Albeverio S. (eds) Infinite dimensional analysis and stochastic processes. Pitman, London, pp. 216–244
Albeverio S., Ribeiro de Faria M., Hoegh-Krohn R. (1979). Stationary measures for the periodic Euler flow in two dimensions. J. Stat. Phys. 20:585–595
Albeverio S., Cruzeiro A.B. (1990). Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids. Commun. Math. Phys. 129:431–444
Boldrighini C., Frigio S. (1978). Equilibrium states for the two-dimensional incompressible Euler fluid. Atti Sem. Mat. Fis. Univ. Moderna XXVII:106–125
Cipriano F. (1999). The two dimensional Euler equation: a statistical study. Commun. Math. Phys. 201:139–154
Chemetov N.V., Starovoitov V.N. (2002). On a motion of perfect fluid in a domain with sources and sinks. J. Math. Fluid Mech. 4:128–144
Cruzeiro A.B. (1983). Équations différentielles sur l’espace de Wiener et formules de Cameron- Martin non linéaires. J. Funct. Anal. 54:206–227
Caprino S., De Gregorio S. (1985). On the statistical solutions of the two-dimensional periodic Euler equations. Math. Methods Appl. Sci. 7:55–73
Delort J.M. (1991). Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4:553–586
DiPerna R.J., Lions P.L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98:511–547
Diperna R.J., Majda A.J. (1987). Concentrations in regularizations for 2D incompressible flow. Commun. Pure Appl. Math. 40:301–345
Foias C., Manley O., Rosa R., Temam R. (2001). Navier-Stokes equations and turbulence. Cambridge University Press, Cambridge
Gihman I.I., Skorohod A.V. (1974). The theory of stochastic processes. Springer Monographs in Mathematics. Springer Verlag, New York
Kato T. (1967). On classical solutions of the two-dimensional nonstationary Euler equations. Arch. Rat. Mech. Anal. 25(3):188–200
Krasny, R.: Computing vortex sheet motion. In: Proceedings of Int. Congress of Math. Kyoto 1990, New York: Springer-Verlag, 2, 1991, pp. 1573–1583
Malliavin P. (1982). Introduction au cours d’Analyse. Masson, Paris
Peters G. (1996). Anticipating flows on the Wiener space generated by vector fields of low regularity. J. Funct. Anal. 142:129–192
Ustunel A.S., Zakai M. (2000). Transformation of measure on the Wiener space. Springer Monographs in Mathematics, Springer Verlag, Berlin
Yudovic V. (1963). Non-stationary flow of an incompressible liquid. Zh. Vychysl. Mat. Mat. Fiz. 3:1032–1066
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Chemetov, N.V., Cipriano, F. The 2D Euler Equations and the Statistical Transport Equations. Commun. Math. Phys. 267, 543–558 (2006). https://doi.org/10.1007/s00220-006-0078-1
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DOI: https://doi.org/10.1007/s00220-006-0078-1