Well-posedness of the non-local conservation law by stochastic perturbation

  • Christian OliveraEmail author


Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an \(L^{1}\cap L^{2}\) setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove existence and uniqueness of weak solutions. The solution is strong solution in the probabilistic sense. The proofs are constructive and based on the method of characteristics (in the presence of noise), Itô–Wentzell–Kunita formula and commutators. Our results are new , to the best of our knowledge, and are the first nonlinear extension of the seminar paper (Flandoli et al. in Invent Math 180:1–53, 2010) where the linear case was addressed.

Mathematics Subject Classification

60H15 35R60 35F10 60H30 



Christian Olivera is partially supported by FAPESP by the Grants 2017/17670-0 and 2015/07278-0 and by CNPq by the Grant 426747/2018-6.


  1. 1.
    Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158, 227–260 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Crippa, G.: Continuity equations and ODE flows with non-smooth velocity. Proc. R. Soc. Edinb. Sect. A Math. 144, 1191–1244 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Attanasio, S., Flandoli, F.: Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative noise. Commun. Partial Differ. Equ. 36, 1455–1474 (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Beck, L., Flandoli, F., Gubinelli, M., Maurelli, M.: Stochastic ODEs and Stochastic Linear PDEs with Critical Drift: Regularity, Duality and Uniqueness. (2014) Preprint available on arXiv:1401.1530
  5. 5.
    Carrillo, J.A., Colombo, R.M., Gwiazda, P., Ulikowska, A.: Structured populations, cell growth and measure valued balance laws. J. Differ. Equ. 252(4), 3245–3277 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Colombo, R.M., Herty, M., Mercier, M.: Control of the continuity equation with a non local flow. ESAIM Control Optim. Calc. Var. 17, 353–379 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coviello, R., Russo, F.: Nonsemimartingales: stochastic differential equations and weak Dirichlet processes. Ann. Probab. 35, 255–308 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, G.Q., Perthame, B.: Well-posedness for non-isotropic degenerate parabolic–hyperbolic equations. Ann. Inst. H. Poincar e Anal. Non Lineaire 20, 645–668 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, G.-Q., Ding, Q., Karlsen, K.H.: On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204, 707–743 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chow, P.L.: Stochastic Partial Differential Equations. Chapman Hall/CRC, London (2007)zbMATHGoogle Scholar
  11. 11.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 3rd edn. Springer, Berlin (2010)Google Scholar
  12. 12.
    De Lellis, C.: Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio. Semin. Bourbaki, Preprint, 1–26 (2007)Google Scholar
  13. 13.
    Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259, 1014–1042 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    DiPerna, R., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duboscq, R., Reveillac, A.: Stochastic regularization effects of semi-martingales on random functions. J. Math. Pures Appl. 106, 1141–1173 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fedrizzi, E., Flandoli, F.: Noise prevents singularities in linear transport equations. J. Funct. Anal. 264, 1329–1354 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fedrizzi, E., Neves, W., Olivera, C.: On a class of stochastic transport equations for \(L_{loc}^{2}\) vector fields. Ann. Sc. Norm. Super. Pisa Cl. Sci. 18(397–419), 2018 (2014)Google Scholar
  18. 18.
    Feng, J., Nualart, D.: Stochastic scalar conservation laws. J. Funct. Anal. 255, 313–373 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Flandoli, F.: Random Perturbation of PDEs and Fluid Dynamic Model. Lectures from the 40th Probability Summer School held in Saint-Flour, 2010. Lecture Notes in Mathematics, 2015. Springer, Heidelberg (2011)Google Scholar
  20. 20.
    Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180, 1–53 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Flandoli, F., Russo, F.: Generalized integration and stochastic ODEs. Ann. Probab. 30, 270–292 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Friedman, A.: Conservation laws in mathematical biology. Discrete Contin. Dyn. Syst. 32, 3081–3097 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gess, B., Maurelli, M.: Well-Posedness by Noise for Scalar Conservation Laws (2017). arXiv:1701.05393
  24. 24.
    Gess, B., Souganidis, P.E.: Long? Time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws. Commun. Pure Appl. Math. 70(8), 1562–1597 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hofmanova, M.: Scalar conservation laws with rough flux and stochastic forcing. Stoch. Partial Differ. Equ. Anal. Comput. 4, 635–690 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kruzhkov, S.: First-order quasilinear equations with several space variables. Mat. Sb. 123, 228–255. English transl. in Math. USSR Sb. 10, 217–273 (1970)Google Scholar
  27. 27.
    Kunita, H.: Stochastic Differential Equations and Stochastic Flows of Diffeomorphisms, Lectures Notes in Mathematics, vol. 1097, pp. 143–303. Springer, Berlin (1982)Google Scholar
  28. 28.
    Kunita, H.: First order stochastic partial differential equations. In: Stochastic Analysis, Katata Kyoto, North-Holland Math. Library, vol. 32, pp. 249–269 (1984)Google Scholar
  29. 29.
    Lions, P.L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7, 169–191 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lions, P.L., Benoit, P., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ. Anal. Comput. 1(4), 664–686 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Lions, P.L., Benoit, P., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case. Stoch. Partial Differ. Equ. Anal. Comput. 2, 517–538 (2014)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Lions, P.L., Benoit, P., Souganidis, P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun. Pure Appl. Math. 49(6), 599–638 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mohammed, S.A., Nilssen, T.K., Proske, F.N.: Sobolev differentiable stochastic flows for SDE’s with singular coefficients: applications to the transport equation. Ann. Probab. 43, 1535–1576 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mollinedo, D.A.C., Olivera, C.: Stochastic continuity equation with non-smooth velocity. Ann. Mat. Pura Appl. 196, 1669–1684 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Neves, W., Olivera, C.: Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition. Nonlinear Differ. Equ. Appl. 22, 1247–1258 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Olivera, C.: Regularization by noise in one-dimensional continuity equation. Potential Anal. (2018). Google Scholar
  37. 37.
    Perthame, B.: Kinetic Formulation of Conservation Laws, Oxford Lecture Series in Mathematics and Its Applications, vol. 21. Oxford University Press, Oxford (2002)Google Scholar
  38. 38.
    Piccoli, B., Tosin, A.: Time-evolving measures and macroscopic modeling of pedestrian flow. Arch. Ration. Mech. Anal. 199, 707–738 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Russo, F., Vallois, P.: Elements of stochastic calculus via regularizations. Séminaire de Probabilités XL, Lecture Notes in Math. 1899, pp. 147–186 (2007)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Estadual de CampinasCampinasBrazil

Personalised recommendations