Renormalized entropy solutions for degenerate parabolic-hyperbolic equationswith time-space dependent coefficients

Zhigang Wang; Lei Wang; Yachun Li

doi:10.3934/cpaa.2013.12.1163

Article Contents

2013, Volume 12, Issue 3: 1163-1182. Doi: 10.3934/cpaa.2013.12.1163

This issue Previous Article Next Article The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows

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

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

Received: October 31, 2011

Revised: March 31, 2012

Published: April 2013

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Abstract

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.

Keywords:

Degenerate parabolic-hyperbolic equation,
entropy solutions,
renormalized entropy solutions,
device of doubling variables,
vanishing viscosity method.

Mathematics Subject Classification: Primary: 35B40, 35A05, 76Y05; Secondary: 35B35, 35L65, 85A05.

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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.poincaré Anal. Non Linéaire, 12 (1995), 727-761.
[2]	M. Bendahmane and K. H. Karlsen, Renormalized entropy solutions for quasilinear anisotropic degenerate parabolic equations, SIAM J. Math. Anal., 36 (2004), 405-422.doi: 10.1137/S0036141003428937.
[3]	M. C. Bustos, F. Concha, R. Bürger and E. M. Tory, "Sedimentation and Thicking: Phenomenological Foundation and Mathematical Theory," Kluwer Academic Publishers, Netherlands, 1999.
[4]	J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal., 147 (1999), 269-361.doi: 10.1007/s002050050152.
[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), 751-762.doi: 10.1137/S0036141001363597.
[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), 241-266.
[7]	G.-Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equation, Analyse non-lineaire, 20 (2003), 645-668.doi: 10.1016/S0294-1449(02)00014-8.
[8]	S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal., 37 (2000), 1838-1860.doi: 10.1137/S0036142998336138.
[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), 41-82.doi: 10.1007/s002110100342.
[10]	L. V. Juan, "The Porous Medium Equation: Mathematical Theory," The Clarendon Press, Oxford university press, Oxford, 2007.
[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), 439-458.doi: 10.1016/S0022-247X(02)00305-0.
[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), 239-269.doi: 10.1051/m2an:2001114.
[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), 1081-1104.doi: 10.3934/dcds.2003.9.1081.
[14]	S. N. Kružkov, First order quasilinear equations with several independent variables, Math. USSR. sb., 10 (1970), 217-243.doi: 10.1070/SM1970v010n02ABEH002156.
[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), 105-119.doi: 10.1016/0041-5553(76)90046-X.
[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), 355-387.
[17]	B. Perthame and P. E. Souganidis, Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws, Arch. Rational Mech Anal., 170 (2003), 359-370.doi: 10.1007/s00205-003-0282-5.
[18]	A. I. Volpert and S. I. Hudjaev, Cauchy's problem for degenerate second order quasilinear parabolic equation, Transl. Math. USSR Sb, 7 (1969), 365-387.doi: 10.1070/SM1969v007n03ABEH001095.
[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), 395-422.