Acta Mathematica Sinica, English Series

, Volume 32, Issue 9, pp 1027–1034 | Cite as

A stochastic variational approach to viscous Burgers equations

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Abstract

We consider viscous Burgers equations in one dimension of space and derive their solutions from stochastic variational principles on the corresponding group of homeomorphisms. The metrics considered on this group are L p metrics. The velocity corresponds to the drift of some stochastic Lagrangian processes. Existence of minima is proved in some cases by direct methods. We also give a representation of the solutions of viscous Burgers equations in terms of stochastic forward-backward systems.

Keywords

Burgers equations stochastic variational principles Lagrangian diffusions 

MR(2010) Subject Classification

60H30 35Q40 
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References

  1. [1]
    Arnaudon, M., Chen, X., Cruzeiro, A. B.: Stochastic Euler–Poincaré reduction. J. Math. Physics., 55, 081507 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Arnold, V. I.: Sur la gémétrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier, 16, 316–361 (1966)CrossRefGoogle Scholar
  3. [3]
    Arnold, V. I., Khesin, B. A.: Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1988MATHGoogle Scholar
  4. [4]
    Cipriano, F., Cruzeiro, A. B.: Navier–Stokes equation and diffusions on the group of homeomorphisms of the torus. Comm. Math. Phys., 275(1), 255–269 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Cruzeiro, A. B., Liu, G. P.: A stochastic variational approach to the viscous Camassa–Holm and related equations. Stochastic Process. Appl., to appear, arxiv:1508.04064Google Scholar
  6. [6]
    Cruzeiro, A. B., Qian, Z. M.: Backward stochastic differential equations associated with the vorticity equations. J. Funct. Anal., 267(3), 660–677 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Cruzeiro, A. B., Shamarova, E.: Navier–Stokes equations and forward-backward SDEs on the group of diffeomorphisms of the torus. Stochastic Process. Appl., 119, 4034–4060 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Krylov, N. V., Rockner, M.: Strong solutions of stochastic equations with singular time dependent drift. Prob. Th. Related Fields, 131(2), 154–196 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Lepeltier, J. P., San Martin, J.: Existence for BSDE with superlinear-quadratic coefficient. Stochastic and Stochastic Reports, 63, 227–240 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Miao, C. X., Zhang, B.: The Cauchy problem for semilinear parabolic equations in Besov spaces. Houston Journal of Mathematics, 30(3), 829–878 (2004)MathSciNetMATHGoogle Scholar
  11. [11]
    Zhang, X. C.: Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electr. J. Probab., 16(38), 1096–1116 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.GFMUL and Dep. de Matemática Instituto Superior Técnico (Universidade de Lisboa)LisboaPortugal
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingP. R. China

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