American Institute of Mathematical Sciences

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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), 963-971.  Google Scholar

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

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J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational. Anal., 147 (1999), 269-361.  Google Scholar

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

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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, (2007), 253-265.  Google Scholar

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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), 1081-1104.  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), 1-23 (electronic).  Google Scholar

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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., (3) (2003), 1-49.  Google Scholar

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S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10 (1970), 217-243. Google Scholar

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C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Rational Mech. Anal., 163 (2002), 87-124.  Google Scholar

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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), 643-673.  Google Scholar

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L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Pitman, Boston, Mass. London, (1979), 136-212.  Google Scholar

[13]

H. Watanabe, Initial value problem for strongly degenerate parabolic equations with discontinuous coefficients, Bulletin of Salesian Polytechnic 38 (2012), 13-20. Google Scholar

[14]

H. Watanabe, Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients, Discrete Contin. Dyn. Syst. Ser. S, 7(2014), no.1, 177-189. Google Scholar

[15]

H. Watanabe and S. Oharu, $BV$-entropy solutions to strongly degenerate parabolic equations, Adv. Differential Equations 15 (2010), 757-800.  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), 963-971.  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), 269-361.  Google Scholar

[4]

L. C. Evans and R. Gariepy, "Measure theory and fine properties of functions", Studies in Advanced Math., CRC Press, London, (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, (2007), 253-265.  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), 1081-1104.  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), 1-23 (electronic).  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., (3) (2003), 1-49.  Google Scholar

[9]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10 (1970), 217-243. Google Scholar

[10]

C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Rational Mech. Anal., 163 (2002), 87-124.  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), 643-673.  Google Scholar

[12]

L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Pitman, Boston, Mass. London, (1979), 136-212.  Google Scholar

[13]

H. Watanabe, Initial value problem for strongly degenerate parabolic equations with discontinuous coefficients, Bulletin of Salesian Polytechnic 38 (2012), 13-20. Google Scholar

[14]

H. Watanabe, Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients, Discrete Contin. Dyn. Syst. Ser. S, 7(2014), no.1, 177-189. Google Scholar

[15]

H. Watanabe and S. Oharu, $BV$-entropy solutions to strongly degenerate parabolic equations, Adv. Differential Equations 15 (2010), 757-800.  Google Scholar

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