American Institute of Mathematical Sciences

Advanced
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

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 & 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.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

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.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

[1]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[2]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
[3]

Young-Sam Kwon. Strong traces for degenerate parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1275-1286. doi: 10.3934/dcds.2009.25.1275
[4]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
[5]

M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827
[6]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-26. doi: 10.3934/dcds.2019226
[7]

Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019032
[8]

Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks & Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941
[9]

Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060
[10]

Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 833-849. doi: 10.3934/dcds.2016.36.833
[11]

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
[12]

Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767
[13]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
[14]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[15]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885
[16]

François Ledrappier, Seonhee Lim. Volume entropy of hyperbolic buildings. Journal of Modern Dynamics, 2010, 4 (1) : 139-165. doi: 10.3934/jmd.2010.4.139
[17]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[18]

Michiel Bertsch, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. Standing and travelling waves in a parabolic-hyperbolic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5603-5635. doi: 10.3934/dcds.2019246
[19]

Boris S. Kruglikov and Vladimir S. Matveev. Vanishing of the entropy pseudonorm for certain integrable systems. Electronic Research Announcements, 2006, 12: 19-28.

[20]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences