Abstract
The strong existence and the pathwise uniqueness of solutions with \({L^{\infty}}\)-vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved.
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Ambrosio L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004)
Ambrosio, L., Crippa, G.: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws. Lect. Notes of the Unione Matematica Italiana, Vol. 5, I, 3–57, 2008
Aubin T.: Some nonlinear problems in Riemannian geometry. Springer-Verlag, Berlin (1998)
Batt J., Rein G.: A rigorous stability result for the Vlasov-Poisson system in three dimensions. Ann. Mat. Pura Appl. (4) 164, 133–154 (1993)
Baxendale P., Harris T.E.: Isotropic stochastic flows. Ann. Probab. 14(4), 1155–1179 (1986)
Bernard D., Gawedzki K., Kupiainen A.: Anomalous scaling in the N-point functions of a passive scalar. Phys. Rev. E (3) 54(3), 2564–2572 (1996)
Bessaih H.: Martingale solutions for stochastic Euler equations. Stoch. Anal. Appl. 17(5), 713–725 (1999)
Bessaih H.: Stochastic weak attractor for a dissipative Euler equation. Electron. J. Probab. 5(3), 16 (2000)
Bessaih, H.: Stationary solutions for the 2D stochastic dissipative Euler equation. Seminar on Stochastic Analysis, Random Fields and Applications V, 23–36, Progr. Probab., 59, Birkhäuser, Basel, 2008
Bessaih H., Flandoli F.: 2-D Euler equation perturbed by noise. NoDEA Nonlinear Differ. Equ. Appl. 6(1), 35–54 (1999)
Brzeźniak, Z., Goldys, B., Ondreját, M.: Stochastic geometric partial differential equations, In: New Trends in Stochastic Analysis and Related Topics, 1–32, Interdiscip. Math. Sci., 12, World Sci. Publ., Hackensack, NJ, 2012
Brzeźniak Z., Peszat S.: Stochastic two dimensional Euler equations. Ann. Probab. 29(4), 1796–1832 (2001)
Capiński M., Cutland N.J.: Stochastic Euler equations on the torus. Ann. Appl. Probab. 9(3), 688–705 (1999)
Celani A., Vincenzi D.: Intermittency in passive scalar decay. Phys. D 172(1-4), 103–110 (2002)
Chemin, J.Y.: Èquations d’Euler d’un fluide incompressible. Facettes mathèmatiques de la mècanique des fluides, 9-30, Ed. Éc. Polytech., Palaiseau, 2010
De Lellis C., Szèkelyhidi L. Jr.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)
DiPerna R.J., Lions P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Doss H.: Liens entre éuations différentielles stochastiques et ordinaires. Ann. Inst. H. PoincarÃl’ Sect. B (N.S.) 13(2), 99–125 (1977)
Falkovich G., Gawedzki K.: Vergassola M., Particles and fields in fluid turbulence. Rev. Mod. Phys. 73(4), 913–975 (2001)
Flandoli, F.: Random Perturbation of PDEs and Fluid Dynamic Models, Saint Flour Summer School Lectures 2010. Lecture Notes in Math. Vol. 2015, Springer, Berlin, 2011
Flandoli F., Gubinelli M., Priola E.: Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations. Stoch. Process. Appl. 121(7), 1445–1463 (2011)
Gawedzki, K.: Stochastic processes in turbulent transport. arXiv:0806.1949v2
Glatt-Holtz, N., Šverák, V., Vicol, V.: On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models. arXiv:1302.0542
Glatt-Holtz N., Vicol V.C.: Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise. Ann. Probab. 42(1), 80–145 (2014)
Kim J.U.: On the stochastic Euler equations in a two-dimensional domain. SIAM J. Math. Anal. 33(5), 1211–1227 (2002)
Kim J.U.: Existence of a local smooth solution in probability to the stochastic Euler equations in \({{\mathbb{R}}^{3}}\). J. Funct. Anal. 256(11), 3660–3687 (2009)
Kraichnan R.H.: Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–963 (1968)
Kraichnan R.H.: Anomalous scaling of a randomly advected passive scalar. Phys. Rev. Lett. 72, 1016–1019 (1994)
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms, Ecole d’été de probabilités de Saint-Flour, XII—1982, 143–303, Lecture Notes in Math. 1097, Springer, Berlin, 1984
Kunita H.: Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Math. 24. Cambridge University Press, Cambridge (1997)
Kupiainen A., Muratore-Ginanneschi P.: Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection. J. Stat. Phys. 126(3), 669–724 (2007)
Lamperti J.: A simple construction of certain diffusion porcesses. J. Math. Kyoto Univ. 4, 161–170 (1964)
Le Jan Y., Raimond O.: Integration of Brownian vector fields. Ann. Probab. 30(2), 826–873 (2002)
Majda A.J., Bertozzi A.L.: Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
Marchioro C., Pulvirenti M.: Mathematical Theory of Incompressible Nonviscous Fluids. Springer, Berlin (1994)
Mikulevicius R., Rozovskii B.L.: Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250–1310 (2004)
Mikulevicius, R., Valiukevicius, G.: On stochastic Euler equation. Liet. Mat. Rink. 38 (2), 234–247 (1998); translation in Lithuanian Math. J. 38 (1998), no. 2, 181–192 (1999)
Mikulevicius R., Valiukevicius G.: On stochastic Euler equation in \({{\mathbb{R}}^{d}}\). Electron. J. Probab. 5(6), 20 (2000)
Rozovskii B.L.: Stochastic Evolution Equations. Linear Theory and Applications to Non-linear Filtering. Kluwer, Dordrecht (1990)
Sussmann H.J.: On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6(1), 19–41 (1978)
Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983
Wolibner W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698–726 (1933)
Yokoyama S.: Construction of weak solutions of a certain stochastic Navier–Stokes equation. Stochastics 86(4), 573–593 (2014)
Yudovich V.I.: Non-stationary flows of an ideal incompressible fluid (Russian). Z̆. Vyc̆isl. Mat. i Mat. Fiz. 3, 1032–1066 (1963)
Yudovich V.I.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2(1), 27–38 (1995)
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Brzeźniak, Z., Flandoli, F. & Maurelli, M. Existence and Uniqueness for Stochastic 2D Euler Flows with Bounded Vorticity. Arch Rational Mech Anal 221, 107–142 (2016). https://doi.org/10.1007/s00205-015-0957-8
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DOI: https://doi.org/10.1007/s00205-015-0957-8