Electronic Communications in Probability

Uniqueness for an inviscid stochastic dyadic model on a tree

Luigi Bianchi

Full-text: Open access

Abstract

In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the   $q$-matrix associated to a continuous time Markov chain.

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 8, 12 pp.

Dates
Accepted: 31 January 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315547

Digital Object Identifier
doi:10.1214/ECP.v18-2382

Mathematical Reviews number (MathSciNet)
MR3019671

Zentralblatt MATH identifier
1329.60207

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35Q31: Euler equations [See also 76D05, 76D07, 76N10] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60J28: Applications of continuous-time Markov processes on discrete state spaces 76B03: Existence, uniqueness, and regularity theory [See also 35Q35]

Keywords
SPDE shell models dyadic model tree dyadic model q-matrix fluid dynamics Girsanov’s transform multiplicative noise

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Bianchi, Luigi. Uniqueness for an inviscid stochastic dyadic model on a tree. Electron. Commun. Probab. 18 (2013), paper no. 8, 12 pp. doi:10.1214/ECP.v18-2382. https://projecteuclid.org/euclid.ecp/1465315547


Export citation

References

  • Anderson, William J. Continuous-time Markov chains. An applications-oriented approach. Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York, 1991. xii+355 pp. ISBN: 0-387-97369-9
  • Attanasio, Stefano; Flandoli, Franco. Zero-noise solutions of linear transport equations without uniqueness: an example. C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 753–756.
  • Barbato, D.; Barsanti, M.; Bessaih, H.; Flandoli, F. Some rigorous results on a stochastic GOY model. J. Stat. Phys. 125 (2006), no. 3, 677–716.
  • Barbato, David; Bianchi, Luigi Amedeo; Flandoli, Franco; Morandin, Francesco. A dyadic model on a tree. (2012) To appear on Journal of Mathematical Physics
  • Barbato, D.; Flandoli, F.; Morandin, F. Uniqueness for a stochastic inviscid dyadic model. Proc. Amer. Math. Soc. 138 (2010), no. 7, 2607–2617.
  • Barbato, David; Flandoli, Franco; Morandin, Francesco. Anomalous dissipation in a stochastic inviscid dyadic model. Ann. Appl. Probab. 21 (2011), no. 6, 2424–2446.
  • Barbato, D.; Flandoli, F.; Morandin, F. Energy dissipation and self-similar solutions for an unforced inviscid dyadic model. Trans. Amer. Math. Soc. 363 (2011), no. 4, 1925–1946.
  • Barbato, David; Morandin, Francesco. Stochastic inviscid shell models: well-posedness and anomalous dissipation. (2012) Preprint.
  • Bernardin, Cédric. Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise. Stochastic Process. Appl. 117 (2007), no. 4, 487–513.
  • Bessaih, Hakima; Millet, Annie. Large deviation principle and inviscid shell models. Electron. J. Probab. 14 (2009), no. 89, 2551–2579.
  • Brzeźniak, Z.; Flandoli, F.; Neklyudov, M.; ZegarliÅ„ski, B. Conservative interacting particles system with anomalous rate of ergodicity. J. Stat. Phys. 144 (2011), no. 6, 1171–1185.
  • Constantin, Peter; Levant, Boris; Titi, Edriss S. Analytic study of shell models of turbulence. Phys. D 219 (2006), no. 2, 120–141.
  • Ferrario, Benedetta. Absolute continuity of laws for semilinear stochastic equations with additive noise. Commun. Stoch. Anal. 2 (2008), no. 2, 209–227.
  • Flandoli, Franco. Random perturbation of PDEs and fluid dynamic models. Lectures from the 40th Probability Summer School held in Saint-Flour, 2010. Lecture Notes in Mathematics, 2015. Springer, Heidelberg, 2011. x+176 pp. ISBN: 978-3-642-18230-3
  • Katz, Nets Hawk; Pavlović, NataÅ¡a. Finite time blow-up for a dyadic model of the Euler equations. Trans. Amer. Math. Soc. 357 (2005), no. 2, 695–708 (electronic).
  • Manna, Utpal; Mohan, Manil T. Shell model of turbulence perturbed by Lévy noise. NoDEA Nonlinear Differential Equations Appl. 18 (2011), no. 6, 615–648.
  • Da Prato, Giuseppe; Flandoli, Franco; Priola, Enrico; Röckner, Michael. Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. (2011) To appear on Annals of Probability.