# American Institute of Mathematical Sciences

May  2013, 12(3): 1163-1182. doi: 10.3934/cpaa.2013.12.1163

## Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients

 1 Department of Mathematics, Fuyang Teachers College, Anhui 236037, China 2 Wuxi Teachers' College, Jiangsu 214153, China 3 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030

Received  November 2011 Revised  April 2012 Published  September 2012

We study the well-posedness of renormalized entropy solutions to the Cauchy problem for general degenerate parabolic-hyperbolic equations of the form \begin{eqnarray*} \partial_{t}u+ \sum_{i=1}^{d}\partial_{x_{i}f_{i}(u,t,x)}= \sum_{i,j=1}^{d}\partial_{x_j}(a_{ij}(u,t,x)\partial_{x_i}u)+\gamma(t,x) \end{eqnarray*} with initial data $u(0,x)=u_{0}(x)$, where the convection flux function $f$, the diffusion function $a$, and the source term $\gamma$ depend explicitly on the independent variables $t$ and $x$. We prove the uniqueness by using Kružkov's device of doubling variables and the existence by using vanishing viscosity method.
Citation: Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163
##### References:
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##### References:
 [1] P. Bénilan and H. Touré, Sur l'équation générale $u_t=a(\cdot,u,\phi(\cdot,u)_x)_x+v$ dans L1. II. Le probléme d'évolutions,, Ann. Inst. H.poincar$\acutee$ Anal. Non Lin$\acutee$aire, 12 (1995), 727. Google Scholar [2] M. Bendahmane and K. H. Karlsen, Renormalized entropy solutions for quasilinear anisotropic degenerate parabolic equations,, SIAM J. Math. Anal., 36 (2004), 405. doi: 10.1137/S0036141003428937. Google Scholar [3] M. C. Bustos, F. Concha, R. Bürger and E. M. Tory, "Sedimentation and Thicking: Phenomenological Foundation and Mathematical Theory,", Kluwer Academic Publishers, (1999). 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.-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 [6] 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. Google Scholar [7] G.-Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equation,, Analyse non-lineaire, 20 (2003), 645. doi: 10.1016/S0294-1449(02)00014-8. Google Scholar [8] S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations,, SIAM J. Numer. Anal., 37 (2000), 1838. doi: 10.1137/S0036142998336138. Google Scholar [9] R. Eymard, T. Gallou¨et, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations,, Numer. Math., 92 (2002), 41. doi: 10.1007/s002110100342. Google Scholar [10] L. V. Juan, "The Porous Medium Equation: Mathematical Theory,", The Clarendon Press, (2007). Google Scholar [11] K. H. Karlsen and M. Ohlberger, A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations,, J. Math. Anal. Appl., 275 (2002), 439. doi: 10.1016/S0022-247X(02)00305-0. Google Scholar [12] 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 [13] K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coeffcients,, Discrete Contin. Dyn. Syst., 9 (2003), 1081. doi: 10.3934/dcds.2003.9.1081. Google Scholar [14] S. N. Kružkov, First order quasilinear equations with several independent variables,, Math. USSR. sb., 10 (1970), 217. doi: 10.1070/SM1970v010n02ABEH002156. Google Scholar [15] N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solution of first-order quasi-linear equation,, USSR comput. Math. and Math. Phys., 16 (1976), 105. doi: 10.1016/0041-5553(76)90046-X. Google Scholar [16] 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. Google Scholar [17] B. Perthame and P. E. Souganidis, Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws,, Arch. Rational Mech Anal., 170 (2003), 359. doi: 10.1007/s00205-003-0282-5. Google Scholar [18] A. I. Volpert and S. I. Hudjaev, Cauchy's problem for degenerate second order quasilinear parabolic equation,, Transl. Math. USSR Sb, 7 (1969), 365. doi: 10.1070/SM1969v007n03ABEH001095. Google Scholar [19] Z. Wu and J. Yin, Some properties of functions in $BV_x$ and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations,, Northeastern Math. J., 5 (1989), 395. Google Scholar
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