American Institute of Mathematical Sciences

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 & 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. Google Scholar [2] L. Ambrosio, N. Fusco and Paliara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Science Publications, (2000). Google Scholar [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. doi: 10.1016/j.jmaa.2006.02.072. Google Scholar [4] J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Rational. Anal., 147 (1999), 269. doi: 10.1007/s002050050152. Google Scholar [5] L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Math., (1992). Google Scholar [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, 140, 141 (2001), 337. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.3934/dcds.2003.9.1081. Google Scholar [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. Google Scholar [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. Google Scholar [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968). Google Scholar [13] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations,, Arch. Rational Mech. Anal., 163 (2002), 87. doi: 10.1007/s002050200184. Google Scholar [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. doi: 10.1007/s00205-009-0217-x. Google Scholar [15] L. Tartar, Compensated compactness and applications to partial differential equations,, in, 39 (1979), 136. Google Scholar [16] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181. doi: 10.1007/s002050100157. Google Scholar [17] H. Watanabe and S. Oharu, $BV$-entropy solutions to strongly degenerate parabolic equations,, Adv. Differential Equations, 15 (2010), 757. Google Scholar [18] H. Watanabe and S. Oharu, Strongly degenerate parabolic equations with nonlocal convective terms,, preprint., (). Google Scholar [19] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,", Graduate Texts in Mathematics, 120 (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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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 Paliara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Science Publications, (2000). Google Scholar [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. doi: 10.1016/j.jmaa.2006.02.072. Google Scholar [4] J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Rational. Anal., 147 (1999), 269. doi: 10.1007/s002050050152. Google Scholar [5] L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Math., (1992). Google Scholar [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, 140, 141 (2001), 337. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.3934/dcds.2003.9.1081. Google Scholar [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. Google Scholar [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. Google Scholar [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968). Google Scholar [13] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations,, Arch. Rational Mech. Anal., 163 (2002), 87. doi: 10.1007/s002050200184. Google Scholar [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. doi: 10.1007/s00205-009-0217-x. Google Scholar [15] L. Tartar, Compensated compactness and applications to partial differential equations,, in, 39 (1979), 136. Google Scholar [16] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181. doi: 10.1007/s002050100157. Google Scholar [17] H. Watanabe and S. Oharu, $BV$-entropy solutions to strongly degenerate parabolic equations,, Adv. Differential Equations, 15 (2010), 757. Google Scholar [18] H. Watanabe and S. Oharu, Strongly degenerate parabolic equations with nonlocal convective terms,, preprint., (). Google Scholar [19] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,", Graduate Texts in Mathematics, 120 (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar
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