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]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[2]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297
[3]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[4]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
[5]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078
[6]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117
[7]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120
[8]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015
[9]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[10]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

 Impact Factor: 

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences