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

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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

Zhigang Wang ^1, , Lei Wang ^2, and Yachun Li ^3,

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

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

Mathematics Subject Classification: Primary: 35B40, 35A05, 76Y05; Secondary: 35B35, 35L65, 85A0.

Citation: Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163

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.

show all references

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.

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