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On the uniqueness of blowup solutions of fully nonlinear elliptic equations
Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients
1.  Department of General Education, Salesian Polytechnic, 468 Oyamagaoka, Machidacity, Tokyo, 1940215 
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.
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 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. Google Scholar 
[4] 
L. C. Evans and R. Gariepy, "Measure theory and fine properties of functions",, Studies in Advanced Math., (1992). Google Scholar 
[5] 
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, 5 (2007), 253. 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. Google Scholar 
[7] 
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), 1. Google Scholar 
[8] 
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), 1. Google Scholar 
[9] 
S. N. Kružkov, First order quasilinear equations in several independent variables,, Math. USSR Sbornik, 10 (1970), 217. Google Scholar 
[10] 
C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic equations,, Arch. Rational Mech. Anal., 163 (2002), 87. Google Scholar 
[11] 
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), 643. Google Scholar 
[12] 
L. Tartar, Compensated compactness and applications to partial differential equations,, Nonlinear analysis and mechanics: HeriotWatt Symposium, (1979), 136. Google Scholar 
[13] 
H. Watanabe, Initial value problem for strongly degenerate parabolic equations with discontinuous coefficients,, Bulletin of Salesian Polytechnic 38 (2012), 38 (2012), 13. Google Scholar 
[14] 
H. Watanabe, Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients,, Discrete Contin. Dyn. Syst. Ser. S, (2014), 177. Google Scholar 
[15] 
H. Watanabe and S. Oharu, $BV$entropy solutions to strongly degenerate parabolic equations,, Adv. Differential Equations 15 (2010), 15 (2010), 757. 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. 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. Google Scholar 
[4] 
L. C. Evans and R. Gariepy, "Measure theory and fine properties of functions",, Studies in Advanced Math., (1992). Google Scholar 
[5] 
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, 5 (2007), 253. 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. Google Scholar 
[7] 
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), 1. Google Scholar 
[8] 
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), 1. Google Scholar 
[9] 
S. N. Kružkov, First order quasilinear equations in several independent variables,, Math. USSR Sbornik, 10 (1970), 217. Google Scholar 
[10] 
C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic equations,, Arch. Rational Mech. Anal., 163 (2002), 87. Google Scholar 
[11] 
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), 643. Google Scholar 
[12] 
L. Tartar, Compensated compactness and applications to partial differential equations,, Nonlinear analysis and mechanics: HeriotWatt Symposium, (1979), 136. Google Scholar 
[13] 
H. Watanabe, Initial value problem for strongly degenerate parabolic equations with discontinuous coefficients,, Bulletin of Salesian Polytechnic 38 (2012), 38 (2012), 13. Google Scholar 
[14] 
H. Watanabe, Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients,, Discrete Contin. Dyn. Syst. Ser. S, (2014), 177. Google Scholar 
[15] 
H. Watanabe and S. Oharu, $BV$entropy solutions to strongly degenerate parabolic equations,, Adv. Differential Equations 15 (2010), 15 (2010), 757. Google Scholar 
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