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Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. (English) Zbl 1275.35129
Summary: 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 E. Yu. Panov [Arch. Ration. Mech. Anal. 195, No. 2, 643–673 (2010; Zbl 1191.35102)] and the estimate of degenerate diffusion term derived by Karlsen-Risebro-Towers [K. H. Karlsen et al., Electron. J. Differ. Equ. 2002, Paper No. 93, 23 p. (2002; Zbl 1015.35049)].
MSC:
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35L65 Hyperbolic conservation laws
35R05 PDEs with low regular coefficients and/or low regular data
35K20 Initial-boundary value problems for second-order parabolic equations
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[1] J. Aleksić, <em>On the compactness for two dimensional scalar conservation law with discontinuous flux</em>,, Comm. Math. Science, 4, 963, (2009) · Zbl 1190.35148
[2] L. Ambrosio, “Functions of Bounded Variation and Free Discontinuity Problems,”, Oxford Science Publications, (2000) · Zbl 0957.49001
[3] R. Bürger, <em>On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition</em>,, J. Math. Anal. Appl., 326, 108, (2007) · Zbl 1160.35048
[4] J. Carrillo, <em>Entropy solutions for nonlinear degenerate problems</em>,, Arch. Rational. Anal., 147, 269, (1999) · Zbl 0935.35056
[5] L. C. Evans, “Measure Theory and Fine Properties of Functions,”, Studies in Advanced Math., (1992) · Zbl 0804.28001
[6] S. Evje, <em>A continuous dependence result for nonlinear degenerate parabolic equations with spatially dependent flux function</em>,, in , 140, 141, 337, (2001)
[7] J. Jimenez, <em>Scalar conservation law with discontinuous flux in a bounded domain</em>,, Discrete Contin. Dyn. Syst., 2007, 520 · Zbl 1133.74030
[8] K. H. Karlsen, <em>On the existence and compactness of a two-dimensional resonant system of conservation laws</em>,, Commun. Math. Sci., 5, 253, (2007) · Zbl 1165.35412
[9] K. H. Karlsen, <em>On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients</em>,, Discrete Contin. Dyn., 9, 1081, (2003) · Zbl 1027.35057
[10] K. H. Karlsen, <em>On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient</em>,, Electron. J. Differential Equations, 28, 1, (2002) · Zbl 1015.35049
[11] K. H. Karlsen, <em>\(L^1\) stability for entropy solutions of nonlinear degenerate parabolic convective-diffusion equations with discontinuous coefficients,</em>, Skr. K. Vidensk. Selsk., 2003, 1 · Zbl 1036.35104
[12] O. A. Ladyženskaja, “Linear and Quasilinear Equations of Parabolic Type,”, Translations of Mathematical Monographs, (1968)
[13] C. Mascia, <em>Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations</em>,, Arch. Rational Mech. Anal., 163, 87, (2002) · Zbl 1027.35081
[14] E. Yu. Panov, <em>Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux</em>,, Arch. Rational Mech. Anal., 195, 643, (2010) · Zbl 1191.35102
[15] L. Tartar, <em>Compensated compactness and applications to partial differential equations</em>,, in , 39, 136, (1979) · Zbl 0437.35004
[16] A. Vasseur, <em>Strong traces for solutions of multidimensional scalar conservation laws</em>,, Arch. Ration. Mech. Anal., 160, 181, (2001) · Zbl 0999.35018
[17] H. Watanabe, <em>\(BV\)-entropy solutions to strongly degenerate parabolic equations</em>,, Adv. Differential Equations, 15, 757, (2010) · Zbl 1215.35094
[18] H. Watanabe, <em>Strongly degenerate parabolic equations with nonlocal convective terms</em>,, preprint.
[19] W. P. Ziemer, “Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,”, Graduate Texts in Mathematics, 120, (1989) · Zbl 0692.46022
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