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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 2(356), Pages 105–120 (Mi umn719)  

This article is cited in 17 scientific papers (total in 17 papers)

Passive scalar equation in a turbulent incompressible Gaussian velocity field

S. V. Lototskii, B. L. Rozovskii

University of Southern California

Abstract: The time evolution of a passive scalar in a turbulent homogeneous incompressible Gaussian flow is considered. The turbulent nature of the flow results in non-smooth coefficients of the corresponding evolution equation. A strong solution (in the probabilistic sense) of the equation is constructed by using the Wiener Chaos expansion, and properties of the solution are studied. In particular, a certain $L_p$-regularity of the solution and a representation formula of Feynman–Kac type (or a Lagrangian formula) are among the results obtained. The results can be applied to both viscous and conservative flows.

DOI: https://doi.org/10.4213/rm719

Full text: PDF file (293 kB)
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English version:
Russian Mathematical Surveys, 2004, 59:2, 297–312

Bibliographic databases:

UDC: 519.218.1
MSC: Primary 60H15, 76F25; Secondary 35R60, 60G15, 76D99, 60G60
Received: 20.06.2003

Citation: S. V. Lototskii, B. L. Rozovskii, “Passive scalar equation in a turbulent incompressible Gaussian velocity field”, Uspekhi Mat. Nauk, 59:2(356) (2004), 105–120; Russian Math. Surveys, 59:2 (2004), 297–312

Citation in format AMSBIB
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\jour Uspekhi Mat. Nauk
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Lototsky S.V., Rozovskii B.L., “Wiener chaos solutions of linear stochastic evolution equations”, Ann. Probab., 34:2 (2006), 638–662  crossref  mathscinet  zmath  isi
    2. Fang Shizan, Luo Dejun, “Flow of homeomorphisms and stochastic transport equations”, Stoch. Anal. Appl., 25:5 (2007), 1079–1108  crossref  mathscinet  zmath  isi
    3. Luo Dejun, “Isotropic stochastic flow of homeomorphisms on $\mathbb R^d$ associated with the critical Sobolev exponent”, Stochastic Process. Appl., 118:8 (2008), 1463–1488  crossref  mathscinet  zmath  isi
    4. Lototsky S.V., Rozovskii B.L., “Stochastic partial differential equations driven by purely spatial noise”, SIAM J. Math. Anal., 41:4 (2009), 1295–1322  crossref  mathscinet  zmath  isi
    5. P. A. Razafimandimby, M. Sango, “Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids”, Bound Value Probl, 2010 (2010), 1  crossref  mathscinet  isi
    6. Hu Yaozhong, “A Random Transport-Diffusion Equation”, Acta Mathematica Scientia, 30:6 (2010), 2033–2050  crossref  mathscinet  zmath  isi
    7. Kim J.U., “On the Cauchy problem for the transport equation with random noise”, Journal of Functional Analysis, 259:12 (2010), 3328–3359  crossref  mathscinet  zmath  isi
    8. Sango M., “Magnetohydrodynamic turbulent flows: Existence results”, Physica D-Nonlinear Phenomena, 239:12 (2010), 912–923  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Barbato D., Flandoli F., Morandin F., “Anomalous Dissipation in a Stochastic Inviscid Dyadic Model”, Ann Appl Probab, 21:6 (2011), 2424–2446  crossref  mathscinet  zmath  isi
    10. Flandoli F., “Random Perturbation of PDEs and Fluid Dynamic Models”, Random Perturbation of Pdes and Fluid Dynamic Models, Lecture Notes in Mathematics, 2015, 2011, 1  crossref  mathscinet  isi
    11. Deugoue G., Sango M., “Weak solutions to stochastic 3D Navier–Stokes-alpha model of turbulence: alpha-Asymptotic behavior”, J Math Anal Appl, 384:1 (2011), 49–62  crossref  mathscinet  zmath  isi
    12. Zhang Z., Rozovskii B., Tretyakov M.V., Karniadakis G.E., “A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations”, SIAM J. Sci. Comput., 34:2 (2012), A914–A936  crossref  mathscinet  zmath  isi
    13. François Delarue, Franco Flandoli, Dario Vincenzi, “Noise Prevents Collapse of Vlasov–Poisson Point Charges”, Comm. Pure Appl. Math, 2013, n/a  crossref  mathscinet  isi
    14. Zhang Zh., Tretyakov M.V., Rozovskii B., Karniadakis G.E., “Wiener Chaos Versus Stochastic Collocation Methods For Linear Advection-Diffusion-Reaction Equations With Multiplicative White Noise”, 53, no. 1, 2015, 153–183  crossref  mathscinet  zmath  isi
    15. Lototsky S., Rozovsky B., “Stochastic Partial Differential Equations”, Stochastic Partial Differential Equations, Universitext, Springer, 2017, 1–508  crossref  mathscinet  isi
    16. Zhang Z., Karniadakis G., “Numerical Methods For Stochastic Partial Differential Equations With White Noise”, Numerical Methods For Stochastic Partial Differential Equations With White Noise, Applied Mathematical Sciences-Series, 196, Springer, 2017, 1–394  crossref  mathscinet  isi
    17. Chen T., Rozovskii B., Shu Ch.-W., “Numerical Solutions of Stochastic Pdes Driven By Arbitrary Type of Noise”, Stoch. Partial Differ. Equ.-Anal. Comput., 7:1 (2019), 1–39  crossref  mathscinet  isi  scopus
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