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Ergodicity for Stochastic Conservation Laws with Multiplicative Noise

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Abstract

We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing property is established. This is done in the setting of kinetic solutions taking values in an \(L^1\)-weighted space.

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The authors declare that all data supporting the findings of this study are available within the article.

References

  1. Ammar, K., Willbold, P., Carrillo, J.: Scalar conservation laws with general boundary condition and continuous flux function. J. Differ. Equ. 228(1), 111–139 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bardos, C., Le Roux, A.Y., Nédélec, J.-C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauzet, C., Vallet, G., Wittbold, P.: The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation. J. Funct. Anal. 266(4), 2503–2545 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, G.-Q., Pang, P.H.C.: Invariant measures for nonlinear conservation laws driven by stochastic forcing. Chin. Ann. Math. Ser. B 40(6), 967–1004 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, G.-Q., Perthame, B.: Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(4), 645–68 (2003)

  6. Chung, K.L., Williams, R.J.: Introduction to Stochastic Integration. Probability and its Applications, 2nd edn. Birkhäuser Boston Inc, Boston (1990)

    Google Scholar 

  7. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 2nd edn. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  8. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 152, 2nd edn. Cambridge University Press, Cambridge (2014)

    Book  MATH  Google Scholar 

  9. Dareiotis, K., Gess, B., Tsatsoulis, P.: Ergodicity for stochastic porous media equations. arXiv:1907.04605, to appear in SIAM J. Math. Anal. (2020)

  10. Debussche, A., Hofmanová, M., Vovelle, J.: Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab. 44(3), 1916–1955 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing (revised version). J. Funct. Anal. 259(4), 1014–1042 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Debussche, A., Vovelle, J.: Invariant measure of scalar first-order conservation laws with Stochastic forcing. Probab. Theory Related Fields 163(3–4), 575–611 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dong, Z., Wu, J.-L., Zhang, R., Zhang, T.: Large deviation principles for first-order scalar conservation laws with stochastic forcing. Ann. Appl. Probab. 30(1), 324–367 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. E, W., Khanin, K., Mazel, A., Ya, S.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. (2) 151(3), 877-960 (2000)

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Gyöngy, I., Rovira, C.: On \(L^p\) solutions of semilinear stochastic partial differential equations. Stochastic Process. Appl. 90(1), 83–108 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Kobayasi, K.: A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions. J. Differ. Equ. 230(2), 682–701 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Kobayasi, K., Noboriguchi, D.: A stochastic conservation law with nonhomogeneous Dirichlet boundary conditions. Acta Math. Vietnam 41(4), 607–632 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kobayasi, K., Noboriguchi, D.: Well-posedness for stochastic scalar conservation laws with the initial-boundary condition. J. Math. Anal. Appl. 461(2), 1416–1458 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Noboriguchi, D.: An \(L^1-\)theory for scalar conservation laws with multiplicative noise on a periodic domain. Nihonkai Math. J. 28(1), 43–53 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Otto, F.: Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729-734 (1996)

  25. Porretta, A., Vovelle, J.: \(L^1\) solutions to first order hyperbolic equations in bounded domains. Commun. Partial Differ. Equ. 28(1–2), 381–408 (2003)

    Article  MATH  Google Scholar 

  26. Villani, C.: Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer (2008)

  27. 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)

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Acknowledgements

This work is partly supported by National Key R and D Program of China (No. 2020YFA0712700), Beijing Natural Science Foundation (No. 1212008), National Natural Science Foundation of China (Nos. 12131019, 12171032, 12090014, 11931004, 11721101, 11971227), Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, CAS (No. 2008DP173182), Beijing Institute of Technology Research Fund Program for Young Scholars and MIIT Key Laboratory of Mathematical Theory and Computation in Information Security.

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Authors and Affiliations

  1. RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhao Dong

  2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Rangrang Zhang

  3. School of Mathematics, University of Science and Technology of China, Hefei, 230026, China

    Tusheng Zhang

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  1. Zhao Dong
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  2. Rangrang Zhang
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Correspondence to Rangrang Zhang.

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Dong, Z., Zhang, R. & Zhang, T. Ergodicity for Stochastic Conservation Laws with Multiplicative Noise. Commun. Math. Phys. 400, 1739–1789 (2023). https://doi.org/10.1007/s00220-022-04629-x

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