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On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models
Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type
1. | Clermont Université, Université Blaise Pascal, 63000 Clermont-Ferrand, France |
2. | Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China |
References:
[1] |
S. Bianchini, Stability of $L^{\infty}$ solutions for hyperbolic systems with coinciding shocks and rarefactions, SIAM J. Math. Anal., 33 (2001), 959-981.
doi: 10.1137/S0036141000377900. |
[2] |
F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870.
doi: 10.1090/S0002-9947-98-02204-1. |
[3] |
Y. Brenier, Some geometric PDEs related to hydrodynamics and electrodynamics, Proceedings of the International Congress of Mathematicians, 3, Higher Education Press, Beijing, (2002), 761-772. |
[4] |
Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rat. Mech. Anal., 172 (2004), 65-91.
doi: 10.1007/s00205-003-0291-4. |
[5] |
A. Bressan, Hyperbolic Systems of Conservation Laws : The One Dimensional Cauchy Problem, Oxford Lecture Series in Math. and its Applications, 20. Oxford University Press, Oxford, 2000. |
[6] | |
[7] |
D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations, J. Math. Phys., 44 (2003), 6132-6139.
doi: 10.1063/1.1621057. |
[8] |
G. Q. Chen, The method of quasidecoupling for discontinuous solutions to conservation laws, Arch. Rational Mech. Anal., 121 (1992), 131-185.
doi: 10.1007/BF00375416. |
[9] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 695-715.
doi: 10.1002/cpa.3160180408. |
[10] |
J. Glimm and P. D. Lax, Decay of Solutions Of System of Nonlinear Hyperbolic Conservation Laws, Amer. Math. Soc., Providence, R.I. 1970. |
[11] |
F. John, Nonlinear Waves Equations, Formation of Singularities, Pitcher Lectures in Math. Sciences, Lehigh University, Amer. Math. Soc., 1990. |
[12] |
S. N. Kruzkov, First order quasilinear equations in several independent variables, Mat. Sbornik (N.S.), 81 (1970), 228-255. |
[13] |
D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $\mathbbR^{2+n}$, J. Math. Phys., 47 (2006), 013503, 16 pages.
doi: 10.1063/1.2158435. |
[14] |
D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Part. Diff. Eqs., 28 (2003), 1203-1220.
doi: 10.1081/PDE-120021192. |
[15] |
P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.
doi: 10.1002/cpa.3160100406. |
[16] |
T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Appl. Math., 32, Wiely/Masson, 1994. |
[17] |
T. T. Li, Y. J. Peng and J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl., 91 (2009), 553-568.
doi: 10.1016/j.matpur.2009.01.008. |
[18] |
T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Diff. Equations, 19 (1994), 1263-1317.
doi: 10.1080/03605309408821055. |
[19] |
H. Lindblad, A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102.
doi: 10.1090/S0002-9939-03-07246-0. |
[20] |
J. L. Liu and Y. Zhou, Asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500.
doi: 10.1002/mma.797. |
[21] |
T. P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796.
doi: 10.1002/cpa.3160300605. |
[22] |
Y. J. Peng, Explicit solutions for $2 \times 2$ linearly degenerate systems, Appl. Math. Letters, 11 (1998), 75-78.
doi: 10.1016/S0893-9659(98)00083-4. |
[23] |
Y. J. Peng, Euler-Lagrange change of variables in conservation laws and applications, Nonlinearity, 20 (2007), 1927-1953.
doi: 10.1088/0951-7715/20/8/007. |
[24] |
Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations, J. Math. Phys., 52 (2011), 053702 (23 pages).
doi: 10.1063/1.3591133. |
[25] |
D. Serre, Richness and the classification of quasilinear hyperbolic systems, in IMA Vol. Math. Appl., 29, Springer, New York, (1991), 315-333.
doi: 10.1007/978-1-4613-9121-0_24. |
[26] |
D. Serre, Systèmes de Lois de Conservation I-II, Diderot, Paris, 1996. |
[27] |
D. Serre, Hyperbolicity of the nonlinear models of Maxwell's equations, Arch. Rat. Mech. Anal., 172 (2004), 309-331.
doi: 10.1007/s00205-003-0303-4. |
[28] |
B. Sévennec, Géométrie des systèmes hyperboliques de lois de conservation, Mém. Soc. Math. France, 56 (1994), 125pp. |
[29] |
S. P. Tsarëv, On Poisson brackets and one-dimensional systems of hydrodynamic type, Dolk. Akad. Nauk SSSR, 282 (1985), 534-537. |
[30] |
S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419. |
[31] |
D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Eqs., 68 (1987), 118-136.
doi: 10.1016/0022-0396(87)90188-4. |
show all references
References:
[1] |
S. Bianchini, Stability of $L^{\infty}$ solutions for hyperbolic systems with coinciding shocks and rarefactions, SIAM J. Math. Anal., 33 (2001), 959-981.
doi: 10.1137/S0036141000377900. |
[2] |
F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870.
doi: 10.1090/S0002-9947-98-02204-1. |
[3] |
Y. Brenier, Some geometric PDEs related to hydrodynamics and electrodynamics, Proceedings of the International Congress of Mathematicians, 3, Higher Education Press, Beijing, (2002), 761-772. |
[4] |
Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rat. Mech. Anal., 172 (2004), 65-91.
doi: 10.1007/s00205-003-0291-4. |
[5] |
A. Bressan, Hyperbolic Systems of Conservation Laws : The One Dimensional Cauchy Problem, Oxford Lecture Series in Math. and its Applications, 20. Oxford University Press, Oxford, 2000. |
[6] | |
[7] |
D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations, J. Math. Phys., 44 (2003), 6132-6139.
doi: 10.1063/1.1621057. |
[8] |
G. Q. Chen, The method of quasidecoupling for discontinuous solutions to conservation laws, Arch. Rational Mech. Anal., 121 (1992), 131-185.
doi: 10.1007/BF00375416. |
[9] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 695-715.
doi: 10.1002/cpa.3160180408. |
[10] |
J. Glimm and P. D. Lax, Decay of Solutions Of System of Nonlinear Hyperbolic Conservation Laws, Amer. Math. Soc., Providence, R.I. 1970. |
[11] |
F. John, Nonlinear Waves Equations, Formation of Singularities, Pitcher Lectures in Math. Sciences, Lehigh University, Amer. Math. Soc., 1990. |
[12] |
S. N. Kruzkov, First order quasilinear equations in several independent variables, Mat. Sbornik (N.S.), 81 (1970), 228-255. |
[13] |
D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $\mathbbR^{2+n}$, J. Math. Phys., 47 (2006), 013503, 16 pages.
doi: 10.1063/1.2158435. |
[14] |
D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Part. Diff. Eqs., 28 (2003), 1203-1220.
doi: 10.1081/PDE-120021192. |
[15] |
P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.
doi: 10.1002/cpa.3160100406. |
[16] |
T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Appl. Math., 32, Wiely/Masson, 1994. |
[17] |
T. T. Li, Y. J. Peng and J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl., 91 (2009), 553-568.
doi: 10.1016/j.matpur.2009.01.008. |
[18] |
T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Diff. Equations, 19 (1994), 1263-1317.
doi: 10.1080/03605309408821055. |
[19] |
H. Lindblad, A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102.
doi: 10.1090/S0002-9939-03-07246-0. |
[20] |
J. L. Liu and Y. Zhou, Asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500.
doi: 10.1002/mma.797. |
[21] |
T. P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796.
doi: 10.1002/cpa.3160300605. |
[22] |
Y. J. Peng, Explicit solutions for $2 \times 2$ linearly degenerate systems, Appl. Math. Letters, 11 (1998), 75-78.
doi: 10.1016/S0893-9659(98)00083-4. |
[23] |
Y. J. Peng, Euler-Lagrange change of variables in conservation laws and applications, Nonlinearity, 20 (2007), 1927-1953.
doi: 10.1088/0951-7715/20/8/007. |
[24] |
Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations, J. Math. Phys., 52 (2011), 053702 (23 pages).
doi: 10.1063/1.3591133. |
[25] |
D. Serre, Richness and the classification of quasilinear hyperbolic systems, in IMA Vol. Math. Appl., 29, Springer, New York, (1991), 315-333.
doi: 10.1007/978-1-4613-9121-0_24. |
[26] |
D. Serre, Systèmes de Lois de Conservation I-II, Diderot, Paris, 1996. |
[27] |
D. Serre, Hyperbolicity of the nonlinear models of Maxwell's equations, Arch. Rat. Mech. Anal., 172 (2004), 309-331.
doi: 10.1007/s00205-003-0303-4. |
[28] |
B. Sévennec, Géométrie des systèmes hyperboliques de lois de conservation, Mém. Soc. Math. France, 56 (1994), 125pp. |
[29] |
S. P. Tsarëv, On Poisson brackets and one-dimensional systems of hydrodynamic type, Dolk. Akad. Nauk SSSR, 282 (1985), 534-537. |
[30] |
S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419. |
[31] |
D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Eqs., 68 (1987), 118-136.
doi: 10.1016/0022-0396(87)90188-4. |
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