A Bhatnagar–Gross–Krook approximation to stochastic scalar conservation laws
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 4, pp. 1500-1528.

We study a BGK-like approximation to hyperbolic conservation laws forced by a multiplicative noise. First, we make use of the stochastic characteristics method and establish the existence of a solution for any fixed parameter ε. In the next step, we investigate the limit as ε tends to 0 and show the convergence to the kinetic solution of the limit problem.

Dans ce papier, nous étudions une approximation de type BGK pour des lois de conservations hyperboliques soumises à un bruit multiplicatif. Dans un premier temps, nous utilisons la méthode des caractéristiques dans le cadre stochastique et établissons l’existence d’une solution pour tout paramètre ε fixé. Nous nous intéressons ensuite à la limite quand ε tend vers 0 et prouvons la convergence vers la solution cinétique du problème limite.

DOI: 10.1214/14-AIHP610
Keywords: stochastic conservation laws, kinetic solution, BGK model, hydrodynamic limit, stochastic characteristics method
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Hofmanová, Martina. A Bhatnagar–Gross–Krook approximation to stochastic scalar conservation laws. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 4, pp. 1500-1528. doi : 10.1214/14-AIHP610. http://www.numdam.org/articles/10.1214/14-AIHP610/

[1] F. Berthelin and J. Vovelle. A BGK approximation to scalar conservation laws with discontinuous flux. Proc. Roy. Soc. Edinburgh Sect. A 140 (2010) 953–972. | MR | Zbl

[2] C. Bauzet, G. Vallet and P. Wittbolt. The Cauchy problem for conservation laws with a multiplicative noise. J. Hyperbolic Differ. Equ. 9 (4) (2012) 661–709. | MR | Zbl

[3] C. Q. Chen, Q. Ding and K. H. Karlsen. On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204 (2012) 707–743. | MR | Zbl

[4] G. Q. Chen and B. Perthame. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (4) (2003) 645–668. | Numdam | MR | Zbl

[5] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia Math. Appl. 44. Cambridge Univ. Press, Cambridge, 1992. | MR | Zbl

[6] A. Debussche and J. Vovelle. Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259 (2010) 1014–1042. | MR | Zbl

[7] R. J. Diperna and P. L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–547. | MR | Zbl

[8] J. Feng and D. Nualart. Stochastic scalar conservation laws. J. Funct. Anal. 255 (2) (2008) 313–373. | MR | Zbl

[9] F. Flandoli, M. Gubinelli and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (2010) 1–53. | MR | Zbl

[10] M. Hofmanová. Degenerate parabolic stochastic partial differential equations. Stochastic Process. Appl. 123 (2013) 4294–4336. | MR | Zbl

[11] H. Holden and N. H. Risebro. Conservation laws with a random source. Appl. Math. Optim. 36 (2) (1997) 229–241. | MR | Zbl

[12] C. Imbert and J. Vovelle. A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications. SIAM J. Math. Anal. 36 (1) (2004) 214–232. | MR | Zbl

[13] J. U. Kim. On a stochastic scalar conservation law. Indiana Univ. Math. J. 52 (1) (2003) 227–256. | MR | Zbl

[14] H. Kunita. Stochastic differential equations and stochastic flows of diffeomorphisms. In École d’Été de Probabilités de Saint-Flour XII – 1982 143–303. Lecture Notes in Math. 1097. Springer, Berlin, 1984. | MR | Zbl

[15] H. Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press, Cambridge, 1990. | MR | Zbl

[16] P. L. Lions, B. Perthame and E. Tadmor. Formulation cinétique des lois de conservation scalaires multidimensionnelles. C. R. Acad. Sci. Paris Sér. I Math. 312 (1991) 97–102. | MR | Zbl

[17] P. L. Lions, B. Perthame and E. Tadmor. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1) (1994) 169–191. | MR | Zbl

[18] A. Nouri, A. Omrane and J. P. Vila. Boundary conditions for scalar conservation laws from a kinetic point of view. J. Stat. Phys. 94 (5-6) (1999) 779–804. | MR | Zbl

[19] A. Nouri, A. Omrane and J. P. Vila. Erratum to “Boundary conditions for scalar conservation laws from a kinetic point of view.” J. Stat. Phys. 115 (2004) 1755–1756. | MR | Zbl

[20] B. Perthame and E. Tadmor. A kinetic equation with kinetic entropy functions for scalar conservation laws. Comm. Math. Phys. 136 (3) (1991) 501–517. | MR | Zbl

[21] B. Perthame. Kinetic Formulation of Conservation Laws. Oxford Lecture Ser. Math. Appl. 21. Oxford Univ. Press, Oxford, 2002. | MR | Zbl

[22] P. E. Protter. Stochastic Integration and Differential Equations. Springer, Berlin, 2004. | MR | Zbl

[23] B. Saussereau and I. L. Stoica. Scalar conservation laws with fractional stochastic forcing: Existence, uniqueness and invariant measure. Stochastic Process. Appl. 122 (2012) 1456–1486. | MR | Zbl

[24] G. Vallet and P. Wittbolt. On a stochastic first order hyperbolic equation in a bounded domain. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (4) (2009) 613–651. | MR | Zbl

[25] E. Weinan, K. Khanin, A. Mazel and Ya. Sinai. Invariant measures for Burgers equation with stochastic forcing. Ann. of Math. (2) 151 (2000) 877–960. | MR | Zbl

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