February  2014, 7(1): 177-189. doi: 10.3934/dcdss.2014.7.177

Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients

1. 

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

Received  February 2012 Revised  August 2012 Published  July 2013

In this paper we consider the initial boundary value problem for strongly degenerate parabolic equations with discontinuous coefficients. This equation has the both properties of parabolic equation and hyperbolic equation. Moreover, approximate solutions for this equation may not belong to $BV$. These are difficult points for this type of equations.
    We consider the type of equations under the zero-flux boundary conditions. In particular, we prove the existence and partial uniqueness of weak solutions to such problems. Our proof use the compactness theorem derived by Panov [14] and the estimate of degenerate diffusion term derived by Karlsen-Risebro-Towers [10].
Citation: Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
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.

[2]

L. Ambrosio, N. Fusco and Paliara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Science Publications, 2000.

[3]

R. Bürger, H. Frid and K. H. Karlsen, On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition, J. Math. Anal. Appl., 326 (2007), 108-120. doi: 10.1016/j.jmaa.2006.02.072.

[4]

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.

[5]

L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Math., CRC Press, London, 1992.

[6]

S. Evje, K. H. Karlsen and N. H. Risebro, A continuous dependence result for nonlinear degenerate parabolic equations with spatially dependent flux function, in "Hyperbolic Problems: Theory, Numerics, Applications, Vol. I, II" (Magdeburg, 2000), Internat. Ser. Numer. Math., 140, 141, Birkhäuser, Basel, (2001), 337-346.

[7]

J. Jimenez, Scalar conservation law with discontinuous flux in a bounded domain,, Discrete Contin. Dyn. Syst., 2007 (): 520.  doi: 10.1007/s10665-007-9166-2.

[8]

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.

[9]

K. H. Karlsen and 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. doi: 10.3934/dcds.2003.9.1081.

[10]

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).

[11]

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. 

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[13]

C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Rational Mech. Anal., 163 (2002), 87-124. doi: 10.1007/s002050200184.

[14]

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. doi: 10.1007/s00205-009-0217-x.

[15]

L. Tartar, Compensated compactness and applications to partial differential equations, in "Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV," Res. Notes in Math., 39, Pitman, Boston, Mass.-London, (1979), 136-212.

[16]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.

[17]

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

[18]

H. Watanabe and S. Oharu, Strongly degenerate parabolic equations with nonlocal convective terms,, preprint., (). 

[19]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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.

[2]

L. Ambrosio, N. Fusco and Paliara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Science Publications, 2000.

[3]

R. Bürger, H. Frid and K. H. Karlsen, On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition, J. Math. Anal. Appl., 326 (2007), 108-120. doi: 10.1016/j.jmaa.2006.02.072.

[4]

J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.

[5]

L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Math., CRC Press, London, 1992.

[6]

S. Evje, K. H. Karlsen and N. H. Risebro, A continuous dependence result for nonlinear degenerate parabolic equations with spatially dependent flux function, in "Hyperbolic Problems: Theory, Numerics, Applications, Vol. I, II" (Magdeburg, 2000), Internat. Ser. Numer. Math., 140, 141, Birkhäuser, Basel, (2001), 337-346.

[7]

J. Jimenez, Scalar conservation law with discontinuous flux in a bounded domain,, Discrete Contin. Dyn. Syst., 2007 (): 520.  doi: 10.1007/s10665-007-9166-2.

[8]

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.

[9]

K. H. Karlsen and 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. doi: 10.3934/dcds.2003.9.1081.

[10]

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).

[11]

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. 

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[13]

C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Rational Mech. Anal., 163 (2002), 87-124. doi: 10.1007/s002050200184.

[14]

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. doi: 10.1007/s00205-009-0217-x.

[15]

L. Tartar, Compensated compactness and applications to partial differential equations, in "Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV," Res. Notes in Math., 39, Pitman, Boston, Mass.-London, (1979), 136-212.

[16]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.

[17]

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

[18]

H. Watanabe and S. Oharu, Strongly degenerate parabolic equations with nonlocal convective terms,, preprint., (). 

[19]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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