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A general stability result in a memory-type Timoshenko system
Decay of solutions to fractal parabolic conservation laws with large initial data
1. | Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China, China |
References:
[1] |
P. Biler, T. Funaki and W. Woyczyński, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46.
doi: 10.1006/jdeq.1998.3458. |
[2] |
P. Biler, G. Karch and W. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré, 18 (2001), 613-637.
doi: 10.1016/S0294-1449(01)00080-4. |
[3] |
T. Cazenave, "Semilinear Schrödinger Equation," Courant Lecture Notes in Math. 10, Courant Ins. Math. Sci. and Amer. Math. Soc., 2003. |
[4] |
C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Math. 34, Springer-Verlag, New York, 2006. |
[5] |
P. Constantin, D. Córdoba and J. Wu, On the critical dissipative quasi-geostrophic equations, Indiana Univ. Math. J., 50 (2001), 97-107.
doi: 10.1512/iumj.2001.50.2153. |
[6] |
P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[7] |
A. Cródoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.
doi: 10.1007/s00220-004-1055-1. |
[8] |
J. Droniou, T. Gallouët and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499-521.
doi: 10.1007/s00028-003-0503-1. |
[9] |
J. Han and W. Wang, Decay estimates and the application in the stability for the solutions to the subcritical dissipative quasi-geostrophic equations, preprint, 2009. |
[10] |
E. Hopf, The partial differential equation $u_t+uu_x=\mu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230.
doi: 10.1002/cpa.3160030302. |
[11] |
N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[12] |
D. B. Kotlow, Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, J. Math. Anal. Appl., 35 (1971), 563-576.
doi: 10.1016/0022-247X(71)90204-6. |
[13] |
T.-T. Li and Y.-M. Chen, "Nonlinear Evolution Equations," in Chinese, Science Press, 1989. |
[14] |
S. Resnick, "Dynamical Problems in Non-linear Advective Partial Differential Equations," Ph.D. thesis, University of Chicago, 1995. |
[15] |
M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473.
doi: 10.1080/0360530800882145. |
[16] |
M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal., 10 (1986), 943-956.
doi: 10.1016/0362-546X(86)90080-5. |
[17] |
M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375.
doi: 10.1137/S0036141002409362. |
[18] |
W. Wang and L. Yu, Decay of solutions to parabolic conservation laws with large initial data, preprint, 2009. |
show all references
References:
[1] |
P. Biler, T. Funaki and W. Woyczyński, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46.
doi: 10.1006/jdeq.1998.3458. |
[2] |
P. Biler, G. Karch and W. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré, 18 (2001), 613-637.
doi: 10.1016/S0294-1449(01)00080-4. |
[3] |
T. Cazenave, "Semilinear Schrödinger Equation," Courant Lecture Notes in Math. 10, Courant Ins. Math. Sci. and Amer. Math. Soc., 2003. |
[4] |
C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Math. 34, Springer-Verlag, New York, 2006. |
[5] |
P. Constantin, D. Córdoba and J. Wu, On the critical dissipative quasi-geostrophic equations, Indiana Univ. Math. J., 50 (2001), 97-107.
doi: 10.1512/iumj.2001.50.2153. |
[6] |
P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[7] |
A. Cródoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.
doi: 10.1007/s00220-004-1055-1. |
[8] |
J. Droniou, T. Gallouët and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499-521.
doi: 10.1007/s00028-003-0503-1. |
[9] |
J. Han and W. Wang, Decay estimates and the application in the stability for the solutions to the subcritical dissipative quasi-geostrophic equations, preprint, 2009. |
[10] |
E. Hopf, The partial differential equation $u_t+uu_x=\mu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230.
doi: 10.1002/cpa.3160030302. |
[11] |
N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[12] |
D. B. Kotlow, Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, J. Math. Anal. Appl., 35 (1971), 563-576.
doi: 10.1016/0022-247X(71)90204-6. |
[13] |
T.-T. Li and Y.-M. Chen, "Nonlinear Evolution Equations," in Chinese, Science Press, 1989. |
[14] |
S. Resnick, "Dynamical Problems in Non-linear Advective Partial Differential Equations," Ph.D. thesis, University of Chicago, 1995. |
[15] |
M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473.
doi: 10.1080/0360530800882145. |
[16] |
M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal., 10 (1986), 943-956.
doi: 10.1016/0362-546X(86)90080-5. |
[17] |
M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375.
doi: 10.1137/S0036141002409362. |
[18] |
W. Wang and L. Yu, Decay of solutions to parabolic conservation laws with large initial data, preprint, 2009. |
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