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Well-posedness of the transport equation by stochastic perturbation

Abstract

We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.

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References

  1. 1.

    Albeverio, S., Flandoli, F., Sinai, Y.G.: In: Da Prato, G., Röckner, M. (eds.) SPDE in Hydrodynamic: Recent Progress and Prospects. LNM, vol. 1942. Springer, Berlin (2008). Fondazione C.I.M.E., Florence

    Chapter  Google Scholar 

  2. 2.

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

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    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. Lecture Notes of the Unione Matematica Italiana, vol. 5, pp. 1–5. Springer, Berlin (2008)

    Chapter  Google Scholar 

  4. 4.

    Brandt, A.: Interior Schauder estimates for parabolic differential (or difference) equations via the maximum principle. Israel J. Math. 7, 254–262 (1969)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    Brzezniak, Z., Flandoli, F.: Almost sure approximation of Wong-Zakai type for stochastic partial differential equations. Stoch. Process. Appl. 55, 329–358 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Crippa, G., De Lellis, C.: Oscillatory solutions to transport equations. Indiana Univ. Math. J. 55, 1–13 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  8. 8.

    Davie, A.M.: Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. 24, Article ID rnm124, 26 p. (2007)

  9. 9.

    Depauw, N.: Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan. C.R. Math. Acad. Sci. Paris 337(4), 249–252 (2003)

    MATH  MathSciNet  Google Scholar 

  10. 10.

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

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    Fang, S., Imkeller, P., Zhang, T.: Global flows for stochastic differential equations without global Lipschitz conditions. Ann. Probab. 35, 180–205 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254, 109–153 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Flandoli, F., Russo, F.: Generalized calculus and SDEs with non regular drift. Stoch. Stoch. Rep. 72(1–2), 11–54 (2002)

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Giaquinta, M., Modica, G., Soucek, J.: Cartesian Currents in the Calculus of Variations. I. Cartesian Currents. Springer, Berlin (1998)

    MATH  Google Scholar 

  15. 15.

    Gyöngy, I.: Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process. Appl. 73, 271–299 (1998)

    MATH  Article  Google Scholar 

  16. 16.

    Gyöngy, I., Krylov, N.V.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 105, 143–158 (1996)

    MATH  Article  Google Scholar 

  17. 17.

    Gyöngy, I., Shmatkov, A.: Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54, 315–341 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  19. 19.

    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, vol. 96. American Mathematical Society, Providence (2008)

    MATH  Google Scholar 

  20. 20.

    Krylov, N.V., Priola, E.: Elliptic and parabolic second-order PDEs with growing coefficients. To appear in Commun. Partial Differ. Equs. Preprint (2008)

  21. 21.

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

    MATH  Article  Google Scholar 

  22. 22.

    Krylov, N.V., Rozovskii, B.L.: Stochastic Evolution Equations (Russian). Current Problems in Mathematics, vol. 14 (Russian), pp. 71–147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979)

  23. 23.

    Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. In: Ecole d’été de Probabilités de Saint-Flour, XII—1982. Lecture Notes in Math., vol. 1097, pp. 143–303. Springer, Berlin (1984)

    Google Scholar 

  24. 24.

    Kunita, H.: First order stochastic partial differential equations. In: Stochastic Analysis, Katata/Kyoto, 1982. North-Holland Math. Library, vol. 32, pp. 249–269. North-Holland, Amsterdam (1984)

    Google Scholar 

  25. 25.

    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge Univ. Press, Cambridge (1990)

    MATH  Google Scholar 

  26. 26.

    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North Holland-Kodansha, Amsterdam (1981)

    MATH  Google Scholar 

  27. 27.

    Le Bris, C., Lions, P.L.: Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Commun. Partial Differ. Equs. 33, 1272–1317 (2008)

    MATH  Article  Google Scholar 

  28. 28.

    Le Jan, Y., Raimond, O.: Integration of Brownian vector fields. Ann. Probab. 30, 826–873 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  29. 29.

    Lions, P.L.: Mathematical Topics in Fluid Mechanics. Incompressible Models, vol. 1. Oxford Univ. Press, Oxford (1996)

    MATH  Google Scholar 

  30. 30.

    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Univ. Press, Cambridge (2002)

    MATH  Google Scholar 

  31. 31.

    Pardoux, E.: Equations aux dérivées partielles stochastiques non linéaires monotones. Etude de solutions fortes de type Itô. PhD Thesis, Université Paris Sud (1975)

  32. 32.

    Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004)

    MATH  Google Scholar 

  33. 33.

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)

    MATH  Google Scholar 

  34. 34.

    Rozovskii, B.L.: Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Mathematics and Its Applications (Soviet Series), vol. 35. Kluwer Academic, Dordrecht (1990). Translated from the Russian by A. Yarkho

    Google Scholar 

  35. 35.

    Tessitore, G., Zabczyk, J.: Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equs. 6, 621–655 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  36. 36.

    Veretennikov, Yu.A.: On strong solution and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb. 39, 387–403 (1981)

    MATH  Article  Google Scholar 

  37. 37.

    Zhang, X.: Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients. Stoch. Process. Appl. 115, 435–448 (2005)

    MATH  Article  Google Scholar 

  38. 38.

    Zvonkin, A.K.: A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. (NS) 93(135), 129–149 (1974) (Russian)

    MathSciNet  Google Scholar 

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Correspondence to M. Gubinelli.

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Flandoli, F., Gubinelli, M. & Priola, E. Well-posedness of the transport equation by stochastic perturbation. Invent. math. 180, 1–53 (2010). https://doi.org/10.1007/s00222-009-0224-4

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Keywords

  • Weak Solution
  • Transport Equation
  • Stochastic Differential Equation
  • Stochastic Partial Differential Equation
  • Stochastic Perturbation