# American Institute of Mathematical Sciences

September  2014, 7(3): 477-492. doi: 10.3934/krm.2014.7.477

## A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations

 1 School of Mathematics and Information Sciences, Weifang University, Weifang 260240, China 2 Department of Mathematics and Key Lab of Scientific and Engineering Computing (MOE), Shanghai Jiao Tong University, Shanghai 200240, China 3 Department of Mathematics, Yunnan University, Kunming 650091, China

Received  November 2013 Revised  June 2014 Published  July 2014

This paper is devoted to the error estimate of approximate solutions for non-autonomous degenerate parabolic-hyperbolic equations, where the nonlinear convection flux, the diffusion matrix and source term depend on time $t$ explicitly. Our method is based on kinetic formulation and kinetic entropy formula. By developing the kinetic techniques, we obtain an error estimate of order $O(\sqrt{\mu})$, where $\mu$ is the artificial viscosity.
Citation: Xingwen Hao, Yachun Li, Qin Wang. A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations. Kinetic & Related Models, 2014, 7 (3) : 477-492. doi: 10.3934/krm.2014.7.477
##### References:
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Anal., 35 (2001), 239. doi: 10.1051/m2an:2001114. Google Scholar [15] K. Kobayasi and H. Ohwa, Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle,, J. Diff. Equa., 252 (2012), 137. doi: 10.1016/j.jde.2011.09.008. Google Scholar [16] S. N. Kružkov, First order quasilinear equation in several independent variables,, Math. USSR. sb., 81 (1970), 228. Google Scholar [17] Y. C. Li and Q. Wang, Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations,, J. Diff. Equa., 252 (2012), 4719. doi: 10.1016/j.jde.2012.01.027. Google Scholar [18] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations,, J. Amer. Math. Soc., 7 (1994), 169. doi: 10.1090/S0894-0347-1994-1201239-3. Google Scholar [19] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations,, Arch. Rat. Mech. Anal., 163 (2002), 87. doi: 10.1007/s002050200184. Google Scholar [20] A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods,, SIAM J. Numer. Anal., 41 (2003), 2262. doi: 10.1137/S0036142902406612. Google Scholar [21] M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations,, M2AN Math. Model. Numer. Anal., 35 (2001), 355. doi: 10.1051/m2an:2001119. Google Scholar [22] B. Perthame, Kinetic Formulation of Conservation Laws,, Oxford Lecture Series in Mathematics and its Applications, (2002). Google Scholar [23] B. Perthame and P. E. Souganidis, Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws,, Arch. Ration. Mech. Anal., 170 (2003), 359. doi: 10.1007/s00205-003-0282-5. Google Scholar [24] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory,, Oxford University Press, (2007). Google Scholar [25] A. I. Vol'pert and S. I. Hugjaev, Cauchy's problem for degenerate second order quasilinear parabolic equation,, Transl. Math. USSR Sb., 7 (1969), 365. doi: 10.1070/SM1969v007n03ABEH001095. Google Scholar [26] Z. G. Wang, L. Wang and Y. C. Li, Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients,, Comm. Pure Appl. Anal., 12 (2013), 1163. doi: 10.3934/cpaa.2013.12.1163. Google Scholar [27] Y. Q. Wu, Migrate Dynamic of Polluted Material in Porous Media,, Shanghai Jiao Tong University Press, (2007). Google Scholar

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##### References:
 [1] M. Bendahmane and K. H. Karlsen, Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations,, SIAM J. Math. Anal., 36 (2004), 405. doi: 10.1137/S0036141003428937. Google Scholar [2] P. Bénilan, M. Carrillo and P. Wittbold, Renormalized entropy solutions of scalar conservation laws,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 313. Google Scholar [3] M. Bustos, Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory,, Springer, (1998). Google Scholar [4] J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Rational Mech. Anal., 147 (1999), 269. doi: 10.1007/s002050050152. Google Scholar [5] G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows Through Porous Media,, Access Online via Elsevier, (1986). Google Scholar [6] G. Q. Chen and E. DiBenedetto, Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations,, SIAM J. Math. Anal., 33 (2001), 751. doi: 10.1137/S0036141001363597. Google Scholar [7] G. Q. Chen and K. H. Karlsen, Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients,, Commun. Pure Appl. Anal., 4 (2005), 241. doi: 10.3934/cpaa.2005.4.241. Google Scholar [8] G. Q. Chen and K. H. Karlsen, $L^1$-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations,, Trans. Amer. Math. Soc., 358 (2006), 937. doi: 10.1090/S0002-9947-04-03689-X. Google Scholar [9] G. Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations,, Ann. Inst. H. PoincaréAnal. Non Linéaire, 20 (2003), 645. doi: 10.1016/S0294-1449(02)00014-8. Google Scholar [10] J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid,, Numer. Math., 105 (2006), 35. doi: 10.1007/s00211-006-0034-1. Google Scholar [11] J. Droniou, C. Imbert and J. Vovelle, An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 689. doi: 10.1016/j.anihpc.2003.11.001. Google Scholar [12] X. W. Hao, Well-posedness of the Second Order Quasilinear Anisotropic Degenerate Parabolic-Hyperbolic Equation,, Ph.D thesis, (2010). Google Scholar [13] C. Imbert and J. Vovelle, A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications,, SIAM J. Math. Anal., 36 (2004), 214. doi: 10.1137/S003614100342468X. Google Scholar [14] K. H. Karlsen and N. H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients,, M2AN Math. Model. Numer. Anal., 35 (2001), 239. doi: 10.1051/m2an:2001114. Google Scholar [15] K. Kobayasi and H. Ohwa, Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle,, J. Diff. Equa., 252 (2012), 137. doi: 10.1016/j.jde.2011.09.008. Google Scholar [16] S. N. Kružkov, First order quasilinear equation in several independent variables,, Math. USSR. sb., 81 (1970), 228. Google Scholar [17] Y. C. Li and Q. Wang, Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic-hyperbolic equations,, J. Diff. Equa., 252 (2012), 4719. doi: 10.1016/j.jde.2012.01.027. Google Scholar [18] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations,, J. Amer. Math. Soc., 7 (1994), 169. doi: 10.1090/S0894-0347-1994-1201239-3. Google Scholar [19] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations,, Arch. Rat. Mech. Anal., 163 (2002), 87. doi: 10.1007/s002050200184. Google Scholar [20] A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods,, SIAM J. Numer. Anal., 41 (2003), 2262. doi: 10.1137/S0036142902406612. Google Scholar [21] M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations,, M2AN Math. Model. Numer. Anal., 35 (2001), 355. doi: 10.1051/m2an:2001119. Google Scholar [22] B. Perthame, Kinetic Formulation of Conservation Laws,, Oxford Lecture Series in Mathematics and its Applications, (2002). Google Scholar [23] B. Perthame and P. E. Souganidis, Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws,, Arch. Ration. Mech. Anal., 170 (2003), 359. doi: 10.1007/s00205-003-0282-5. Google Scholar [24] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory,, Oxford University Press, (2007). Google Scholar [25] A. I. Vol'pert and S. I. Hugjaev, Cauchy's problem for degenerate second order quasilinear parabolic equation,, Transl. Math. USSR Sb., 7 (1969), 365. doi: 10.1070/SM1969v007n03ABEH001095. Google Scholar [26] Z. G. Wang, L. Wang and Y. C. Li, Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients,, Comm. Pure Appl. Anal., 12 (2013), 1163. doi: 10.3934/cpaa.2013.12.1163. Google Scholar [27] Y. Q. Wu, Migrate Dynamic of Polluted Material in Porous Media,, Shanghai Jiao Tong University Press, (2007). Google Scholar
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