American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 781-790. doi: 10.3934/proc.2013.2013.781

Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients

Hiroshi Watanabe 1,

1. 

Department of General Education, Salesian Polytechnic, 4-6-8 Oyamagaoka, Machida-city, Tokyo, 194-0215

Received  September 2012 Revised  February 2013 Published  November 2013

In this paper, we consider the initial value problem for strongly degenerate parabolic equations with discontinuous coefficients. This equation has the both properties of parabolic equation and hyperbolic equation. Therefore, we should choose entropy solutions as generalized solutions to the equation. Moreover, entropy solutions to the equation may not belong to $BV$ in our setting. These are difficult points for this type of equations.
    In particular, we consider the case that coefficients are the functions of bounded variation with respect to the space variable $x$. Then, we prove the existence of Kružkov type entropy solutions. Moreover, we prove the uniqueness of the solution under additional conditions.
Keywords: functions of bounded variation., discontinuous coefficients, Strongly degenerate parabolic.
Mathematics Subject Classification: Primary: 35K65, 35K55; Secondary: 35L65, 35R0.
Citation: 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
References:
[1]

J. Aleksić and D. Mitrovic, On the compactness for two dimensional scalar conservation law with discontinuous flux,, Comm. Math. Science, 4 (2009), 963.   Google Scholar

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems",, Oxford Science Publications, (2000).   Google Scholar

[3]

J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Rational. Anal., 147 (1999), 269.   Google Scholar

[4]

L. C. Evans and R. Gariepy, "Measure theory and fine properties of functions",, Studies in Advanced Math., (1992).   Google Scholar

[5]

K. H. Karlsen, M. Rascle and E. Tadmor, On the existence and compactness of a two-dimensional resonant system of conservation laws,, Commun. Math. Sci. 5, 5 (2007), 253.   Google Scholar

[6]

K. H. Karlsen, N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients,, Discrete Contin. Dyn., 9 (2003), 1081.   Google Scholar

[7]

K. H. Karlsen, N. H. Risebro and J. D. Towers, On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient,, Electron. J. Differential Equations, 28 (2002), 1.   Google Scholar

[8]

K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^{1}$ stability for entropy solutions of nonlinear degenerate parabolic convective-diffusion equations with discontinuous coefficients,, Skr. K. Vidensk. Selsk., (2003), 1.   Google Scholar

[9]

S. N. Kružkov, First order quasilinear equations in several independent variables,, Math. USSR Sbornik, 10 (1970), 217.   Google Scholar

[10]

C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations,, Arch. Rational Mech. Anal., 163 (2002), 87.   Google Scholar

[11]

E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Rational Mech. Anal., 195 (2010), 643.   Google Scholar

[12]

L. Tartar, Compensated compactness and applications to partial differential equations,, Nonlinear analysis and mechanics: Heriot-Watt Symposium, (1979), 136.   Google Scholar

[13]

H. Watanabe, Initial value problem for strongly degenerate parabolic equations with discontinuous coefficients,, Bulletin of Salesian Polytechnic 38 (2012), 38 (2012), 13.   Google Scholar

[14]

H. Watanabe, Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients,, Discrete Contin. Dyn. Syst. Ser. S, (2014), 177.   Google Scholar

[15]

H. Watanabe and S. Oharu, $BV$-entropy solutions to strongly degenerate parabolic equations,, Adv. Differential Equations 15 (2010), 15 (2010), 757.   Google Scholar

show all references

References:
[1]

J. Aleksić and D. Mitrovic, On the compactness for two dimensional scalar conservation law with discontinuous flux,, Comm. Math. Science, 4 (2009), 963.   Google Scholar

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems",, Oxford Science Publications, (2000).   Google Scholar

[3]

J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Rational. Anal., 147 (1999), 269.   Google Scholar

[4]

L. C. Evans and R. Gariepy, "Measure theory and fine properties of functions",, Studies in Advanced Math., (1992).   Google Scholar

[5]

K. H. Karlsen, M. Rascle and E. Tadmor, On the existence and compactness of a two-dimensional resonant system of conservation laws,, Commun. Math. Sci. 5, 5 (2007), 253.   Google Scholar

[6]

K. H. Karlsen, N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients,, Discrete Contin. Dyn., 9 (2003), 1081.   Google Scholar

[7]

K. H. Karlsen, N. H. Risebro and J. D. Towers, On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient,, Electron. J. Differential Equations, 28 (2002), 1.   Google Scholar

[8]

K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^{1}$ stability for entropy solutions of nonlinear degenerate parabolic convective-diffusion equations with discontinuous coefficients,, Skr. K. Vidensk. Selsk., (2003), 1.   Google Scholar

[9]

S. N. Kružkov, First order quasilinear equations in several independent variables,, Math. USSR Sbornik, 10 (1970), 217.   Google Scholar

[10]

C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations,, Arch. Rational Mech. Anal., 163 (2002), 87.   Google Scholar

[11]

E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Rational Mech. Anal., 195 (2010), 643.   Google Scholar

[12]

L. Tartar, Compensated compactness and applications to partial differential equations,, Nonlinear analysis and mechanics: Heriot-Watt Symposium, (1979), 136.   Google Scholar

[13]

H. Watanabe, Initial value problem for strongly degenerate parabolic equations with discontinuous coefficients,, Bulletin of Salesian Polytechnic 38 (2012), 38 (2012), 13.   Google Scholar

[14]

H. Watanabe, Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients,, Discrete Contin. Dyn. Syst. Ser. S, (2014), 177.   Google Scholar

[15]

H. Watanabe and S. Oharu, $BV$-entropy solutions to strongly degenerate parabolic equations,, Adv. Differential Equations 15 (2010), 15 (2010), 757.   Google Scholar

[1]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[2]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[3]

Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43
[4]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213
[5]

Dung Le. Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems. Conference Publications, 2005, 2005 (Special) : 576-586. doi: 10.3934/proc.2005.2005.576
[6]

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
[7]

Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051
[8]

Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure & Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241
[9]

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
[10]

Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129
[11]

Roberto Alicandro, Andrea Braides, Marco Cicalese. $L^\infty$ jenergies on discontinuous functions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 905-928. doi: 10.3934/dcds.2005.12.905
[12]

Cai-Ping Liu. Some characterizations and applications on strongly $\alpha$-preinvex and strongly $\alpha$-invex functions. Journal of Industrial & Management Optimization, 2008, 4 (4) : 727-738. doi: 10.3934/jimo.2008.4.727
[13]

Jian Liu, Sihem Mesnager, Lusheng Chen. Variation on correlation immune Boolean and vectorial functions. Advances in Mathematics of Communications, 2016, 10 (4) : 895-919. doi: 10.3934/amc.2016048
[14]

Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520
[15]

Renato C. Calleja, Alessandra Celletti, Rafael de la Llave. Construction of response functions in forced strongly dissipative systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4411-4433. doi: 10.3934/dcds.2013.33.4411
[16]

Ciprian Preda, Petre Preda, Adriana Petre. On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1637-1645. doi: 10.3934/cpaa.2009.8.1637
[17]

Karim Boulabiar, Gerard Buskes and Gleb Sirotkin. A strongly diagonal power of algebraic order bounded disjointness preserving operators. Electronic Research Announcements, 2003, 9: 94-98.

[18]

G. P. Trachanas, Nikolaos B. Zographopoulos. A strongly singular parabolic problem on an unbounded domain. Communications on Pure & Applied Analysis, 2014, 13 (2) : 789-809. doi: 10.3934/cpaa.2014.13.789
[19]

Maria Alessandra Ragusa. Parabolic systems with non continuous coefficients. Conference Publications, 2003, 2003 (Special) : 727-733. doi: 10.3934/proc.2003.2003.727
[20]

Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems & Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565

 Impact Factor: 

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences