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Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients
1.  Department of General Education, Salesian Polytechnic, 468 Oyamagaoka, Machidacity, Tokyo, 1940215 
We consider the type of equations under the zeroflux 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 KarlsenRisebroTowers [10].
References:
[1] 
J. Aleksić and D. Mitrovic, On the compactness for two dimensional scalar conservation law with discontinuous flux, Comm. Math. Science, 4 (2009), 963971. 
[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 wellposedness of entropy solutions to conservation laws with a zeroflux boundary condition, J. Math. Anal. Appl., 326 (2007), 108120. doi: 10.1016/j.jmaa.2006.02.072. 
[4] 
J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational. Anal., 147 (1999), 269361. 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), 337346. 
[7] 
J. Jimenez, Scalar conservation law with discontinuous flux in a bounded domain, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the $6^{th}$ AIMS International Conference, suppl., 520530. doi: 10.1007/s1066500791662. 
[8] 
K. H. Karlsen, M. Rascle and E. Tadmor, On the existence and compactness of a twodimensional resonant system of conservation laws, Commun. Math. Sci., 5 (2007), 253265. 
[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), 10811104. doi: 10.3934/dcds.2003.9.1081. 
[10] 
K. H. Karlsen, N. H. Risebro and J. D. Towers, On a nonlinear degenerate parabolic transportdiffusion equation with a discontinuous coefficient, Electron. J. Differential Equations, 28 (2002), 123 (electronic). 
[11] 
K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convectivediffusion equations with discontinuous coefficients, Skr. K. Vidensk. Selsk., 2003, 149. 
[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 parabolichyperbolic equations, Arch. Rational Mech. Anal., 163 (2002), 87124. doi: 10.1007/s002050200184. 
[14] 
E. Yu. Panov, Existence and strong precompactness properties for entropy solutions of a firstorder quasilinear equation with discontinuous flux, Arch. Rational Mech. Anal., 195 (2010), 643673. doi: 10.1007/s002050090217x. 
[15] 
L. Tartar, Compensated compactness and applications to partial differential equations, in "Nonlinear Analysis and Mechanics: HeriotWatt Symposium, Vol. IV," Res. Notes in Math., 39, Pitman, Boston, Mass.London, (1979), 136212. 
[16] 
A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181193. doi: 10.1007/s002050100157. 
[17] 
H. Watanabe and S. Oharu, $BV$entropy solutions to strongly degenerate parabolic equations, Adv. Differential Equations, 15 (2010), 757800. 
[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, SpringerVerlag, New York, 1989. doi: 10.1007/9781461210153. 
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), 963971. 
[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 wellposedness of entropy solutions to conservation laws with a zeroflux boundary condition, J. Math. Anal. Appl., 326 (2007), 108120. doi: 10.1016/j.jmaa.2006.02.072. 
[4] 
J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational. Anal., 147 (1999), 269361. 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), 337346. 
[7] 
J. Jimenez, Scalar conservation law with discontinuous flux in a bounded domain, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the $6^{th}$ AIMS International Conference, suppl., 520530. doi: 10.1007/s1066500791662. 
[8] 
K. H. Karlsen, M. Rascle and E. Tadmor, On the existence and compactness of a twodimensional resonant system of conservation laws, Commun. Math. Sci., 5 (2007), 253265. 
[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), 10811104. doi: 10.3934/dcds.2003.9.1081. 
[10] 
K. H. Karlsen, N. H. Risebro and J. D. Towers, On a nonlinear degenerate parabolic transportdiffusion equation with a discontinuous coefficient, Electron. J. Differential Equations, 28 (2002), 123 (electronic). 
[11] 
K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convectivediffusion equations with discontinuous coefficients, Skr. K. Vidensk. Selsk., 2003, 149. 
[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 parabolichyperbolic equations, Arch. Rational Mech. Anal., 163 (2002), 87124. doi: 10.1007/s002050200184. 
[14] 
E. Yu. Panov, Existence and strong precompactness properties for entropy solutions of a firstorder quasilinear equation with discontinuous flux, Arch. Rational Mech. Anal., 195 (2010), 643673. doi: 10.1007/s002050090217x. 
[15] 
L. Tartar, Compensated compactness and applications to partial differential equations, in "Nonlinear Analysis and Mechanics: HeriotWatt Symposium, Vol. IV," Res. Notes in Math., 39, Pitman, Boston, Mass.London, (1979), 136212. 
[16] 
A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181193. doi: 10.1007/s002050100157. 
[17] 
H. Watanabe and S. Oharu, $BV$entropy solutions to strongly degenerate parabolic equations, Adv. Differential Equations, 15 (2010), 757800. 
[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, SpringerVerlag, New York, 1989. doi: 10.1007/9781461210153. 
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