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Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces
Alternative proof for the existence of Green's function
| 1. | Department of Mathematics Education, Gwangju National University of Education, 93 Pilmunlo Bugku, Gwangju 500-703, South Korea |
References:
| [1] |
A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble), 28 (1978), 169-213, x. See also for corrections: "Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien'' [Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 169-213; [MR 80d:31006], Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), Lecture Notes in Math., vol. 787, p. 28. (1980) Springer, Berlin. |
| [2] |
D. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694. |
| [3] |
P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc., 54 (1996), 284-296.
doi: 10.1112/jlms/54.2.284. |
| [4] |
P. Bauman, Equivalence of the Green's functions for diffusion operators in $R^n$: a counterexample, Proc. Amer. Math. Soc., 91 (1984), 64-68.
doi: 10.1090/S0002-9939-1984-0735565-4. |
| [5] |
P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat., 22 (1984), 153-173.
doi: 10.1007/BF02384378. |
| [6] |
S. Cho, Two-sided global estimates of the Green's function of parabolic equations, Potential Analysis, 25 (2006), 387-398.
doi: doi:10.1007/s11118-006-9026-0. |
| [7] |
R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. II., reprint of the 1962 original, John Wiley & Sons Inc., New York, 1989. |
| [8] |
E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Univ. Press, Cambridge, UK 1989.
doi: 10.1017/CBO9780511566158. |
| [9] |
S. Èĭdel'man, "Parabolicheskie sistemy," Izdat. "Nauka'', Moscow, 1964. Translation: "Parabolic systems," North-Holland Publishing Co., Amsterdam, 1964. |
| [10] |
L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form, Comm. Partial Differential Equations, 25 (2000), 821-845.
doi: 10.1080/03605300008821533. |
| [11] |
E. Fabes, N. Garofalo and S. Salsa, A control on the set where a Green's function vanishes, Colloq. Math., 60/61 (1990), 637-647. |
| [12] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, 1964. |
| [13] |
A. Il'in, A. Kalašnikov and O. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk, 17 (1962), 3-146.
doi: 10.1070/RM1962v017n03ABEH004115. |
| [14] |
H. Kalf, On E. E. Levi's method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo, 41 (1992), 251-294.
doi: 10.1007/BF02844669. |
| [15] |
O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, "Linear and Quasi-linear Equations of Parabolic Type," Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. |
| [16] |
E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali., Rend. Circ. Mat. Palermo, 24 (1907), 275-317. |
| [17] |
E. Levi, I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali., Memorie Mat. Fis. Soc. Ital. Scienze (detta dei XL) 16 (1909) 3-113 |
| [18] |
G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996. |
| [19] |
V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations, J. London Math. Soc., 62 (2000), 521-543.
doi: 10.1112/S0024610700001332. |
| [20] |
E. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005. |
| [21] |
F. Porper and S. Èĭdel'man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them, Uspekhi Math. Nauk, 39 (1984), 107-156.
doi: 10.1070/RM1984v039n03ABEH003164. |
| [22] |
L. Saloff-Coste, "Aspects of Sobolev-type Inequalities," London Mathematical Society Lecture Note Series 289, Cambridge University Press, 2002. |
| [23] |
P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory, Ark. Mat., 11 (1973), 153-165.
doi: 10.1007/BF02388513. |
| [24] |
W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen, Math. Z., 21 (1924), 286-311.
doi: 10.1007/BF01187471. |
| [25] |
Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians, Journal of Differential Equations, 182 (2002), 416-430.
doi: 10.1006/jdeq.2001.4112. |
show all references
References:
| [1] |
A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble), 28 (1978), 169-213, x. See also for corrections: "Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien'' [Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 169-213; [MR 80d:31006], Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), Lecture Notes in Math., vol. 787, p. 28. (1980) Springer, Berlin. |
| [2] |
D. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694. |
| [3] |
P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc., 54 (1996), 284-296.
doi: 10.1112/jlms/54.2.284. |
| [4] |
P. Bauman, Equivalence of the Green's functions for diffusion operators in $R^n$: a counterexample, Proc. Amer. Math. Soc., 91 (1984), 64-68.
doi: 10.1090/S0002-9939-1984-0735565-4. |
| [5] |
P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat., 22 (1984), 153-173.
doi: 10.1007/BF02384378. |
| [6] |
S. Cho, Two-sided global estimates of the Green's function of parabolic equations, Potential Analysis, 25 (2006), 387-398.
doi: doi:10.1007/s11118-006-9026-0. |
| [7] |
R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. II., reprint of the 1962 original, John Wiley & Sons Inc., New York, 1989. |
| [8] |
E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Univ. Press, Cambridge, UK 1989.
doi: 10.1017/CBO9780511566158. |
| [9] |
S. Èĭdel'man, "Parabolicheskie sistemy," Izdat. "Nauka'', Moscow, 1964. Translation: "Parabolic systems," North-Holland Publishing Co., Amsterdam, 1964. |
| [10] |
L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form, Comm. Partial Differential Equations, 25 (2000), 821-845.
doi: 10.1080/03605300008821533. |
| [11] |
E. Fabes, N. Garofalo and S. Salsa, A control on the set where a Green's function vanishes, Colloq. Math., 60/61 (1990), 637-647. |
| [12] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, 1964. |
| [13] |
A. Il'in, A. Kalašnikov and O. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk, 17 (1962), 3-146.
doi: 10.1070/RM1962v017n03ABEH004115. |
| [14] |
H. Kalf, On E. E. Levi's method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo, 41 (1992), 251-294.
doi: 10.1007/BF02844669. |
| [15] |
O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, "Linear and Quasi-linear Equations of Parabolic Type," Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. |
| [16] |
E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali., Rend. Circ. Mat. Palermo, 24 (1907), 275-317. |
| [17] |
E. Levi, I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali., Memorie Mat. Fis. Soc. Ital. Scienze (detta dei XL) 16 (1909) 3-113 |
| [18] |
G. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996. |
| [19] |
V. Liskevich and Y. Semenov, Estimates for fundamental solutions of second-order parabolic equations, J. London Math. Soc., 62 (2000), 521-543.
doi: 10.1112/S0024610700001332. |
| [20] |
E. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005. |
| [21] |
F. Porper and S. Èĭdel'man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them, Uspekhi Math. Nauk, 39 (1984), 107-156.
doi: 10.1070/RM1984v039n03ABEH003164. |
| [22] |
L. Saloff-Coste, "Aspects of Sobolev-type Inequalities," London Mathematical Society Lecture Note Series 289, Cambridge University Press, 2002. |
| [23] |
P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory, Ark. Mat., 11 (1973), 153-165.
doi: 10.1007/BF02388513. |
| [24] |
W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen, Math. Z., 21 (1924), 286-311.
doi: 10.1007/BF01187471. |
| [25] |
Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians, Journal of Differential Equations, 182 (2002), 416-430.
doi: 10.1006/jdeq.2001.4112. |
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