Skip to main content

Scalar conservation laws with rough flux and stochastic forcing

Abstract

In this paper, we study scalar conservation laws where the flux is driven by a geometric Hölder p-rough path for some \(p\in (2,3)\) and the forcing is given by an Itô stochastic integral driven by a Brownian motion. In particular, we derive the corresponding kinetic formulation and define an appropriate notion of kinetic solution. In this context, we are able to establish well-posedness, i.e., existence, uniqueness and the \(L^1\)-contraction property that leads to continuous dependence on initial condition. Our approach combines tools from rough path analysis, stochastic analysis and theory of kinetic solutions for conservation laws. As an application, this allows to cover the case of flux driven for instance by another (independent) Brownian motion enhanced with Lévy’s stochastic area.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    Here u is a function of \((\omega ,t,x)\) so \(F(\omega ,t,x,\xi )=\mathbf {1}_{u(\omega ,t,x)>\xi }\) is well-defined and regarded as a function of four variables \((\omega ,t,x,\xi )\).

  2. 2.

    For instance \((F,m)=(\mathbf {1}_{u>\xi },0)\) from the discussion above.

  3. 3.

    Here, \(|\cdot |_{\frac{1}{p}-\text {H}\ddot{o}\text {l};[0,T]}\) denotes the Hölder seminorm of a mapping taking values in \(\mathbb {R}^e\).

  4. 4.

    Note that we may assume without loss of generality that the \(\sigma \)-field \(\mathcal {F}\) is countably generated and hence, according to [6, Proposition3.4.5], the space \(L^1(\Omega )\) is separable.

  5. 5.

    By \(\langle \cdot ,\cdot \rangle _\sigma \) we denote the duality between the space of distributions on \(\mathbb {R}_\sigma \) and \(C^1_c(\mathbb {R}_\sigma )\).

  6. 6.

    By \(\mathbf {1}_{u^\varepsilon (t)>\xi }\circ \varphi _{0,t}\) we denote the composition of \((x,\xi )\mapsto \mathbf {1}_{u^\varepsilon (t,x)>\xi }\) with \((x,\xi )\mapsto \varphi _{0,t}(x,\xi )\).

  7. 7.

    Here \(\langle \cdot ,\cdot \rangle _\xi \) denotes the duality between the space of distributions on \(\mathbb {R}\) and \(C^1_c(\mathbb {R})\).

References

  1. 1.

    Bauzet, C., Vallet, G., Wittbold, P.: A degenerate parabolic–hyperbolic Cauchy problem with a stochastic force. J. Hyperbolic Differ. Equ. 12(3), 501–533 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bauzet, C., Vallet, G., Wittbold, P.: The Cauchy problem for conservation laws with a multiplicative noise. J. Hyperbolic Differ. Equ. 9(4), 661–709 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Berthelin, F., Vovelle, J.: A BGK approximation to scalar conservation laws with discontinuous flux. Proc. R. Soc. Edinb. A 140(5), 953–972 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Carrillo, J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147, 269–361 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Caruana, M., Friz, P.: Partial differential equations driven by rough paths. Proc. Lond. Math. Soc. (3) 100(1), 177–215 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cohn, D.L.: Measure Theory. Birkhäuser, Boston (1980)

    Book  MATH  Google Scholar 

  7. 7.

    Chen, C.Q., Ding, Q., Karlsen, K.H.: On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204, 707–743 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Chen, G.Q., Perthame, B.: Well-posedness for non-isotropic degenerate parabolic–hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(4), 645–668 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Crisan, D., Diehl, J., Friz, P.K., Oberhauser, H.: Robust filtering: correlated noise and multidimensional observation. Ann. Appl. Probab. 23(5), 2139–2160 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Debussche, A., Hofmanová, M., Vovelle, J.: Degenerate parabolic stochastic partial differential equations: quasilinear case, to appear in, Ann. Probab. arXiv:1309.5817

  11. 11.

    Debussche, A., Vovelle, J.: Invariant measure of scalar first-order conservation laws with stochastic forcing. Probab. Theory Relat. Fields. 163(3), 575–611 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259, 1014–1042 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing. Revised version. http://math.univ-lyon1.fr/~vovelle/DebusscheVovelleRevised

  14. 14.

    Diehl, J., Oberhauser, H., Riedel, S.: A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations. Stoch. Process. Appl. 125, 161–181 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

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

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Feng, J., Nualart, D.: Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Friz, P.K., Gess, B.: Stochastic scalar conservation laws driven by rough paths, to appear in Ann. Inst. H. Poincaré Analyse Non Linéaire. arXiv:1403.6785

  18. 18.

    Friz, P., Victoir, N.B.: Multidimensional Stochastic Processes as Rough Paths, Theory and Applications, Cambridge Studies in Advanced Mathematics, vol. 120. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  19. 19.

    Gess, B., Souganidis, P.E.: Long-time behavior, invariant measures and regularizing effects for stochastic scalar conservation laws. arXiv:1411.3939

  20. 20.

    Gess, B., Souganidis, P.E.: Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci. 13(6), 1569–1597 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Hofmanová, M.: A Bhatnagar-Gross-Krook approximation to stochastic scalar conservation laws. Ann. Inst. H. Poincaré Probab. Stat. 51(4), 1500–1528 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Hofmanová, M.: Degenerate parabolic stochastic partial differential equations. Stoch. Process. Appl. 123(12), 4294–4336 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Holden, H., Risebro, N.H.: Conservation laws with a random source. Appl. Math. Optim. 36(2), 229–241 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Imbert, C., Vovelle, J.: A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications. SIAM J. Math. Anal. 36(1), 214–232 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Jakubowski, A.: The almost sure Skorokhod representation for subsequences in nonmetric spaces. Teor. Veroyatnost. i Primenen 42(1), 209–216 (1997); translation in Theory Probab. Appl. 42(1), 167–174 (1998)

  26. 26.

    Kavian, O.: Introduction à la théorie des points critiques. Springer, Berlin (1993)

    MATH  Google Scholar 

  27. 27.

    Kim, J.U.: On a stochastic scalar conservation law. Indiana Univ. Math. J. 52(1), 227–256 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970)

    MathSciNet  Google Scholar 

  29. 29.

    Lieb, E.H., Loss, M.: Analysis, 2nd edn. AMS, Providence (2001)

    MATH  Google Scholar 

  30. 30.

    Lions, P.L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ. 1(4), 664–686 (2013)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Lions, P.L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case. Stoch. Partial Diff. Equ. Anal. Comp. 2(4), 517–538 (2014)

  32. 32.

    Lions, P.L., Perthame, B., Tadmor, E.: Formulation cinétique des lois de conservation scalaires multidimensionnelles. C. R. Acad. Sci. Paris. Série I 312(1), 97–102 (1991)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Lions, P.L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7(1), 169–191 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Lyons, T.J., Caruana, M., Lévy, T.: Differential equations driven by rough paths. In: Lectures from the 34th Summer School on Probability Theory held in Saint Flour, July 6–24, 2004. Lecture Notes in Mathematics, vol. 1908. Springer, Berlin (2007)

  35. 35.

    Lyons, T.J., Qian, Z.: System Control and Rough Paths, Oxford Mathematical Monographs. Oxford University Press, Oxford (2002)

    Book  MATH  Google Scholar 

  36. 36.

    Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996)

    Book  MATH  Google Scholar 

  37. 37.

    Perthame, B.: Kinetic Formulation of Conservation Laws. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  38. 38.

    Perthame, B., Tadmor, E.: A kinetic equation with kinetic entropy functions for scalar conservation laws. Commun. Math. Phys. 136(3), 501–517 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

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

    MATH  Google Scholar 

  40. 40.

    Revuz, D., Yor, M.: Continuous martingales and Brownian motion, third ed., Grundlehren der Mathematischen Wissenschaften, vol. 293, Springer, Berlin (1999)

  41. 41.

    Saussereau, B., Stoica, I.L.: Scalar conservation laws with fractional stochastic forcing: existence, uniqueness and invariant measure. Stoch. Process. Appl. 122, 1456–1486 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Vallet, G., Wittbold, P.: On a stochastic first order hyperbolic equation in a bounded domain, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12(4), 613–651 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Young, L.C.: Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, Philadelphia (1969)

    MATH  Google Scholar 

Download references

Acknowledgments

The author wishes to thank the anonymous referees for providing many useful suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Martina Hofmanová.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hofmanová, M. Scalar conservation laws with rough flux and stochastic forcing. Stoch PDE: Anal Comp 4, 635–690 (2016). https://doi.org/10.1007/s40072-016-0072-3

Download citation

Keywords

  • Scalar conservation laws
  • Rough paths
  • Kinetic formulation
  • Kinetic solution
  • BGK approximation
  • Method of characteristics

Mathematics Subject Classification

  • 60H15
  • 35R60
  • 35L65