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On a class of reversible elliptic systems
Entropy solutions of forward-backward parabolic equations with Devonshire free energy
1. | Department of Mathematics "G. Castelnuovo", University of Rome "La Sapienza", P.le A. Moro 5, I-00185 Rome, Italy, Italy |
References:
[1] |
G. I. Barenblatt, M. Bertsch, R. Dal Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439.
doi: 10.1137/0524082. |
[2] |
M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci. 121, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[3] |
L. T. T. Bui, F. Smarrazzo, and A. Tesei,, forthcoming., ().
|
[4] |
L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620.
doi: 10.1142/S0218202504003763. |
[5] |
O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1991. |
[6] |
C. Mascia, A. Terracina and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in: "Asymptotic Analysis and Singularities" (edited by H. Kozono, T. Ogawa, K. Tanaka, Y. Tsutsumi and E. Yanagida), 451-478, Advanced Studies in Pure Mathematics, 47-2 (Math. Soc. Japan, 2007). |
[7] |
C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Rational Mech. Anal., 194 (2009), 887-925.
doi: 10.1007/s00205-008-0185-6. |
[8] |
A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.
doi: 10.2307/2001511. |
[9] |
V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Differential Equations, 23 (1998), 457-486.
doi: 10.1080/03605309808821353. |
[10] |
P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. |
[11] |
P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Equ., 30 (1994), 614-622. |
[12] |
D. Serre, "Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves," Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511612374. |
[13] |
F. Smarrazzo and A. Terracina, Local existence and uniqueness of two-phase entropy solutions,, Discrete Contin. Dyn. Syst. (to appear)., ().
|
[14] |
A. Terracina, Qualitative behaviour of the two-phase entropy solution of a forward-backward parabolic equation, SIAM J. Math. Anal., 43 (2011), 228-252.
doi: 10.1137/090778833. |
[15] |
M. Valadier, A course on Young measures, Rend. Ist. Mat. Univ. Trieste, 26 (1995), 349-394. |
show all references
References:
[1] |
G. I. Barenblatt, M. Bertsch, R. Dal Passo and M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439.
doi: 10.1137/0524082. |
[2] |
M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci. 121, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[3] |
L. T. T. Bui, F. Smarrazzo, and A. Tesei,, forthcoming., ().
|
[4] |
L. C. Evans and M. Portilheiro, Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620.
doi: 10.1142/S0218202504003763. |
[5] |
O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type," Amer. Math. Soc., Providence, 1991. |
[6] |
C. Mascia, A. Terracina and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in: "Asymptotic Analysis and Singularities" (edited by H. Kozono, T. Ogawa, K. Tanaka, Y. Tsutsumi and E. Yanagida), 451-478, Advanced Studies in Pure Mathematics, 47-2 (Math. Soc. Japan, 2007). |
[7] |
C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Rational Mech. Anal., 194 (2009), 887-925.
doi: 10.1007/s00205-008-0185-6. |
[8] |
A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.
doi: 10.2307/2001511. |
[9] |
V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Differential Equations, 23 (1998), 457-486.
doi: 10.1080/03605309808821353. |
[10] |
P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. |
[11] |
P. I. Plotnikov, Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Equ., 30 (1994), 614-622. |
[12] |
D. Serre, "Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves," Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511612374. |
[13] |
F. Smarrazzo and A. Terracina, Local existence and uniqueness of two-phase entropy solutions,, Discrete Contin. Dyn. Syst. (to appear)., ().
|
[14] |
A. Terracina, Qualitative behaviour of the two-phase entropy solution of a forward-backward parabolic equation, SIAM J. Math. Anal., 43 (2011), 228-252.
doi: 10.1137/090778833. |
[15] |
M. Valadier, A course on Young measures, Rend. Ist. Mat. Univ. Trieste, 26 (1995), 349-394. |
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