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The Interaction Between Noise and Transport Mechanisms in PDEs

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Abstract

The understanding of transport mechanisms in PDEs is at the core of some of the main open problems in fluid dynamics. The introduction of random perturbations in these equations may have an influence on the properties of transport. We report here some recent progresses on the interaction between noise and transport. The Lagrangian viewpoint is used to investigate the problem. In some example we observe that noise improves the properties of the Lagrangian motion and restores uniqueness or prevents blow-up of the associated PDE; in other examples this does not happen.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Attanasio, F. Flandoli, Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative noise, arXiv:1007.4102, to appear on Comm. Partial Diff. Eq.

  3. Da Prato G., Debussche A.: Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures Appl. (9) 82(8), 877–947 (2003)

    MathSciNet  MATH  Google Scholar 

  4. F. Delarue, F. Flandoli, D. Vincenzi, Noise prevents coalescence in the Vlasov-Poisson motion of n particles, work in progress.

  5. DiPerna R.J., Lions P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Fedrizzi, F. Flandoli, Hölder flow and differentiability for SDEs with nonregular drift, Stoch. Anal. Appl., to appear.

  7. F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Lecture Notes in Mathematics 2015, Springer 2011.

  8. Flandoli F., Gubinelli M., Priola E.: Well posedness of the transport equation by stochastic perturbation. Invent. Math. 180, 1–53 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Flandoli, M. Gubinelli, E. Priola, Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations, to appear on Stoch. Proc. Appl.

  10. Flandoli F., Romito M.: Markov selections for the 3D stochastic Navier-Stokes equations, Probab. Theory Related Fields 140(3-4), 407–458 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, 12. American Mathematical Society, Providence, RI, 1996.

  12. Krylov N.V., Röckner M.: Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131, 154–196 (2005)

    Article  MATH  Google Scholar 

  13. Majda A.J., Bertozzi A.L.: Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  14. Marchioro C., Pulvirenti M.: Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, 96. Springer-Verlag, New York (1994)

    Book  Google Scholar 

  15. Poupaud F.: Diagonal defect measures, adhesion dynamics and Euler equation. Methods Appl. Anal. 9(4), 533–561 (2002)

    MathSciNet  MATH  Google Scholar 

  16. Romito R.: Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise. J. Stat. Phys. 131(3), 415–444 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università di Pisa, Via F. Buonarroti 1/c, I-56127, Pisa, Italy

    Franco Flandoli

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  1. Franco Flandoli
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Correspondence to Franco Flandoli.

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Lecture held in the Seminario Matematico e Fisico di Milano on November 8, 2010

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Flandoli, F. The Interaction Between Noise and Transport Mechanisms in PDEs. Milan J. Math. 79, 543–560 (2011). https://doi.org/10.1007/s00032-011-0164-5

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  • DOI: https://doi.org/10.1007/s00032-011-0164-5

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