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Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation
Localization, smoothness, and convergence to equilibrium for a thin film equation
1. | Department of Mathematics, Hill Center, Rutgers University, Piscataway, NJ 08854, United States |
2. | Department of Mathematics, Faculty of Education, Zirve University, Gaziantep, Turkey |
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the Wasserstein space of probability measures, Birkhäuser Verlag, Basel, 2005. |
[2] |
L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equation in one space dimension, Arch. Ration. Mech. Anal., 173 (2004), 89-131.
doi: 10.1007/s00205-004-0313-x. |
[3] |
J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter, 17 (2005), 291-307.
doi: 10.1088/0953-8984/17/9/002. |
[4] |
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Eqns., 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[5] |
A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices AMS, 45 (1998), 689-697. |
[6] |
A. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123.
doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. |
[7] |
M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Diff. Eqns., 3 (1998), 417-440. |
[8] |
M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal. TMA, 69 (2008), 1268-1286.
doi: 10.1016/j.na.2007.06.028. |
[9] |
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417.
doi: 10.1002/cpa.3160440402. |
[10] |
E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, J. Diff. Eqns., 241 (2007), 279-292.
doi: 10.1016/j.jde.2007.07.005. |
[11] |
J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin-film equation, Comm. Math. Phys., 225 (2002), 551-571.
doi: 10.1007/s002200100591. |
[12] |
M. Chugunova, M. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. Math. Anal., 42 (2010), 1826-1853.
doi: 10.1137/090777062. |
[13] |
R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.
doi: 10.1137/S0036141096306170. |
[14] |
L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. Part. Diff. Eq., 13 (2001), 377-403.
doi: 10.1007/s005260000077. |
[15] |
G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. PDEs, 29 (2004), 1697-1744.
doi: 10.1081/PDE-200040193. |
[16] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[17] |
R. Laugesen, New dissipated energies for the thin fluid film equation, Comm. Pure Appl. Analysis, 4 (2005), 613-634.
doi: 10.3934/cpaa.2005.4.613. |
[18] |
D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. PDEs, 34 (2009), 1352-1397.
doi: 10.1080/03605300903296256. |
[19] |
R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.
doi: 10.1006/aima.1997.1634. |
[20] |
T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462.
doi: 10.1137/S003614459529284X. |
[21] |
F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. PDEs, 23 (1998), 2077-2164.
doi: 10.1080/03605309808821411. |
[22] |
F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. PDEs, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[23] |
J. E. Rakotoson, J. M. Rakotoson and C. Verbeke, Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data, J. Diff. Eqns., 244 (2008), 2693-2740.
doi: 10.1016/j.jde.2008.03.009. |
[24] |
N. F. Smyth and J. M. Hill, High-order nonlinear diffusion, IMA J. Appl. Math., 40 (1988), 73-86.
doi: 10.1093/imamat/40.2.73. |
[25] |
A. Tudorascu, Lubrication approximation for viscous flows: asyptotic behavior of nonnegative solutions, Comm. PDEs, 32 (2007), 1147-1172.
doi: 10.1080/03605300600987272. |
[26] |
S. Ulusoy, A new family of higher order nonlinear degenerate parabolic equations, Nonlinearity, 20 (2007), 685-712.
doi: 10.1088/0951-7715/20/3/007. |
[27] |
S. Ulusoy, On a new family of higher order nonlinear degenerate parabolic equations, Appl. Math. Res. eXpress, 2007 (2007), Article ID abm010, 28 pages.
doi: 10.1088/0951-7715/20/3/007. |
[28] |
S. Ulusoy, The Mathematical Theory of Thin Film Evolution, Ph.D thesis, Georgia Institute of Technology, 2007. |
[29] |
C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., 58, AMS, Providence, RI, 2003.
doi: 10.1007/b12016. |
[30] |
J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the Wasserstein space of probability measures, Birkhäuser Verlag, Basel, 2005. |
[2] |
L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equation in one space dimension, Arch. Ration. Mech. Anal., 173 (2004), 89-131.
doi: 10.1007/s00205-004-0313-x. |
[3] |
J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter, 17 (2005), 291-307.
doi: 10.1088/0953-8984/17/9/002. |
[4] |
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Eqns., 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[5] |
A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices AMS, 45 (1998), 689-697. |
[6] |
A. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123.
doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. |
[7] |
M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Diff. Eqns., 3 (1998), 417-440. |
[8] |
M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal. TMA, 69 (2008), 1268-1286.
doi: 10.1016/j.na.2007.06.028. |
[9] |
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417.
doi: 10.1002/cpa.3160440402. |
[10] |
E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, J. Diff. Eqns., 241 (2007), 279-292.
doi: 10.1016/j.jde.2007.07.005. |
[11] |
J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin-film equation, Comm. Math. Phys., 225 (2002), 551-571.
doi: 10.1007/s002200100591. |
[12] |
M. Chugunova, M. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. Math. Anal., 42 (2010), 1826-1853.
doi: 10.1137/090777062. |
[13] |
R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.
doi: 10.1137/S0036141096306170. |
[14] |
L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. Part. Diff. Eq., 13 (2001), 377-403.
doi: 10.1007/s005260000077. |
[15] |
G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. PDEs, 29 (2004), 1697-1744.
doi: 10.1081/PDE-200040193. |
[16] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[17] |
R. Laugesen, New dissipated energies for the thin fluid film equation, Comm. Pure Appl. Analysis, 4 (2005), 613-634.
doi: 10.3934/cpaa.2005.4.613. |
[18] |
D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. PDEs, 34 (2009), 1352-1397.
doi: 10.1080/03605300903296256. |
[19] |
R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.
doi: 10.1006/aima.1997.1634. |
[20] |
T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462.
doi: 10.1137/S003614459529284X. |
[21] |
F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. PDEs, 23 (1998), 2077-2164.
doi: 10.1080/03605309808821411. |
[22] |
F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. PDEs, 26 (2001), 101-174.
doi: 10.1081/PDE-100002243. |
[23] |
J. E. Rakotoson, J. M. Rakotoson and C. Verbeke, Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data, J. Diff. Eqns., 244 (2008), 2693-2740.
doi: 10.1016/j.jde.2008.03.009. |
[24] |
N. F. Smyth and J. M. Hill, High-order nonlinear diffusion, IMA J. Appl. Math., 40 (1988), 73-86.
doi: 10.1093/imamat/40.2.73. |
[25] |
A. Tudorascu, Lubrication approximation for viscous flows: asyptotic behavior of nonnegative solutions, Comm. PDEs, 32 (2007), 1147-1172.
doi: 10.1080/03605300600987272. |
[26] |
S. Ulusoy, A new family of higher order nonlinear degenerate parabolic equations, Nonlinearity, 20 (2007), 685-712.
doi: 10.1088/0951-7715/20/3/007. |
[27] |
S. Ulusoy, On a new family of higher order nonlinear degenerate parabolic equations, Appl. Math. Res. eXpress, 2007 (2007), Article ID abm010, 28 pages.
doi: 10.1088/0951-7715/20/3/007. |
[28] |
S. Ulusoy, The Mathematical Theory of Thin Film Evolution, Ph.D thesis, Georgia Institute of Technology, 2007. |
[29] |
C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., 58, AMS, Providence, RI, 2003.
doi: 10.1007/b12016. |
[30] |
J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. |
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