Abstract
This work is devoted to examining the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator and a multiplicative noise. Our proof for uniqueness is based upon the analysis on double variables method and the existence is enabled by a parabolic approximation.
Acknowledgements
This research was partly supported by the NSF of China Grants 11501577, 11771123 and 11531006, and the Startup Foundation for Introducing Talent of NUIST.
References
[1] N. Alibaud, Entropy formulation for fractal conservation laws. J. Evol. Equ. 7, No 1 (2007), 145–175.10.1007/s00028-006-0253-zSearch in Google Scholar
[2] C. Bauzet, G. Vallet and P. Wittbold, The Cauchy problem for a conservation law with a multiplicative stochastic perturbation. J. Hyperbolic Differential Equations 9, No 4 (2012), 661–709.10.1142/S0219891612500221Search in Google Scholar
[3] C. Bauzet, G. Vallet and P. Wittbold, The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation. J. Funct. Anal. 266, No 4 (2014), 2503–2545.10.1016/j.jfa.2013.06.022Search in Google Scholar
[4] I. Biswas and A. Majee, Stochastic conservation laws: weak-in-time formulation and strong entropy condition. J. Funct. Anal. 267, No 7 (2014), 2199–2252.10.1016/j.jfa.2014.07.008Search in Google Scholar
[5] G.Q. Chen, Q. Ding and K.H. Karlsen, On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204, No 3 (2012), 707–743.10.1007/s00205-011-0489-9Search in Google Scholar
[6] P. Chen, Y. Li, X. Zhang, On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Commun. Pure Appl. Anal. 14, No 5 (2015), 1817–1840.10.3934/cpaa.2015.14.1817Search in Google Scholar
[7] P. Chen, X. Zhang and Y. Li, Nonlocal problem for fractional stochastic evolution equations with solution operators. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1507–1526; 10.1515/fca-2016-0078; https://www.degruyter.com/journal/key/FCA/19/6/html.Search in Google Scholar
[8] P. Chen, X. Zhang and Y. Li, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators. Fract. Calc. Appl. Anal. 23, No 1 (2020), 268–291; 10.1515/fca-2020-0011; https://www.degruyter.com/journal/key/FCA/23/1/html.Search in Google Scholar
[9] P. Chen, X. Zhang, Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete Contin. Dyn. Syst. Ser. B (2020); 10.3934/dcdsb.2020290.Search in Google Scholar
[10] A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259, No 4 (2010), 1014–1042.10.1016/j.jfa.2010.02.016Search in Google Scholar
[11] A. Debussche, M. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: quasilinear case. Ann. Probab. 44, No 3 (2016), 1916–1955.10.1214/15-AOP1013Search in Google Scholar
[12] J. Droniou, Intégration et Espaces de Sobolev à Valeurs Vectorielles (2001); http://www-gm3.univ-mrs.fr/polys/.Search in Google Scholar
[13] J. Droniou and C. Imbert, Fractal first-order partial differential equations. Arch. Ration. Mech. Anal. 182, No 2 (2006), 299–331.10.1007/s00205-006-0429-2Search in Google Scholar
[14] J. Duan, An Introduction to Stochastic Dynamics. Cambridge University Press, New York, 2015.Search in Google Scholar
[15] J. Feng and D. Nualart, Stochastic scalar conservation laws. J. Funct. Anal. 255, No 2 (2008), 313–373.10.1016/j.jfa.2008.02.004Search in Google Scholar
[16] J.U. Kim, On a stochastic scalar conservation law. Indiana Univ. Math. J. 52, No 1 (2003), 227–256.10.1512/iumj.2003.52.2310Search in Google Scholar
[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, No 1 (1994), 169–191.10.1090/S0894-0347-1994-1201239-3Search in Google Scholar
[18] G. Lv, J. Duan and H. Gao, Stochastic nonlocal conservation laws on whole space. Comput. Math. Appl. 77, No 7 (2019), 1945–1962.10.1016/j.camwa.2018.11.027Search in Google Scholar
[19] G. Lv, J. Duan, H. Gao and J. Wu, On a stochastic nonlocal conservation law in a bounded domain. Bull. Sci. Math. 140, No 6 (2016), 718–746.10.1016/j.bulsci.2016.03.003Search in Google Scholar
[20] C. Olivera, Well-posedness of the non-local conservation law by stochastic perturbation. Manuscripta Math. 162, No 3-4 (2020), 367–387.10.1007/s00229-019-01129-6Search in Google Scholar
[21] J. Sanchez-Ortiz, F. Ariza-Hernandez, M. Arciga-Alejandre and E. Garcia-Murcia, Stochastic diffusion equation with fractional Laplacian on the first quadrant. Fract. Calc. Appl. Anal. 22, No 3 (2019), 795–806; 10.1515/fca-2019-0043; https://www.degruyter.com/journal/key/FCA/22/3/html.Search in Google Scholar
[22] M. Simon and C. Olivera, Non-local conservation law from stochastic particle systems. J. Dynam. Differential Equations 30, No 4 (2018), 1661–1682.10.1007/s10884-017-9620-4Search in Google Scholar
[23] D.W. Stroock, Diffusion processes associated with Lévy generators. Z. Wahr. Verw. Geb. 32, No 3 (1975), 209–244.10.1007/BF00532614Search in Google Scholar
[24] M.F. Shlesinger, G.M. Zaslavsky and U. Frisch, Lévy Flights and Related Topics in Physics. Lecture Notes in Phys. 450, Springer-Verlag, Berlin, 1995.10.1007/3-540-59222-9Search in Google Scholar
[25] G. Vallet and P. Wittbold, On a stochastic first-order hyperbolic equation in a bounded domain. Infin. Dimens. Anal. Quantum Probab. 12, No 4 (2009), 1–39.10.1142/S0219025709003872Search in Google Scholar
[26] J. Wei, J. Duan and G. Lv, Kinetic solutions for nonlocal scalar conservation laws. SIAM J. Math. Anal. 50, No 2 (2018), 1521–1543.10.1137/16M108687XSearch in Google Scholar
© 2021 Diogenes Co., Sofia
