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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.
|
[2] |
D. Aronson, Non-negative solutions of linear parabolic equations,, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607.
|
[3] |
P. Auscher, Regularity theorems and heat kernel for elliptic operators,, J. London Math. Soc., 54 (1996), 284.
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.
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.
doi: 10.1007/BF02384378. |
[6] |
S. Cho, Two-sided global estimates of the Green's function of parabolic equations,, Potential Analysis, 25 (2006), 387.
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, (1962).
|
[8] |
E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Univ. Press, (1989).
doi: 10.1017/CBO9780511566158. |
[9] |
S. Èĭdel'man, "Parabolicheskie sistemy,", Izdat., (1964).
|
[10] |
L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form,, Comm. Partial Differential Equations, 25 (2000), 821.
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.
|
[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.
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.
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, (1967).
|
[16] |
E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali.,, Rend. Circ. Mat. Palermo, 24 (1907), 275. Google Scholar |
[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) \textbf{16} (1909) 3-113, 16 (1909), 3. Google Scholar |
[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.
doi: 10.1112/S0024610700001332. |
[20] |
E. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series \textbf{31}, 31 (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.
doi: 10.1070/RM1984v039n03ABEH003164. |
[22] |
L. Saloff-Coste, "Aspects of Sobolev-type Inequalities,", London Mathematical Society Lecture Note Series \textbf{289}, 289 (2002).
|
[23] |
P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory,, Ark. Mat., 11 (1973), 153.
doi: 10.1007/BF02388513. |
[24] |
W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen,, Math. Z., 21 (1924), 286.
doi: 10.1007/BF01187471. |
[25] |
Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians,, Journal of Differential Equations, 182 (2002), 416.
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.
|
[2] |
D. Aronson, Non-negative solutions of linear parabolic equations,, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607.
|
[3] |
P. Auscher, Regularity theorems and heat kernel for elliptic operators,, J. London Math. Soc., 54 (1996), 284.
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.
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.
doi: 10.1007/BF02384378. |
[6] |
S. Cho, Two-sided global estimates of the Green's function of parabolic equations,, Potential Analysis, 25 (2006), 387.
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, (1962).
|
[8] |
E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Univ. Press, (1989).
doi: 10.1017/CBO9780511566158. |
[9] |
S. Èĭdel'man, "Parabolicheskie sistemy,", Izdat., (1964).
|
[10] |
L. Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form,, Comm. Partial Differential Equations, 25 (2000), 821.
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.
|
[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.
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.
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, (1967).
|
[16] |
E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali.,, Rend. Circ. Mat. Palermo, 24 (1907), 275. Google Scholar |
[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) \textbf{16} (1909) 3-113, 16 (1909), 3. Google Scholar |
[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.
doi: 10.1112/S0024610700001332. |
[20] |
E. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs Series \textbf{31}, 31 (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.
doi: 10.1070/RM1984v039n03ABEH003164. |
[22] |
L. Saloff-Coste, "Aspects of Sobolev-type Inequalities,", London Mathematical Society Lecture Note Series \textbf{289}, 289 (2002).
|
[23] |
P. Sjögren, On the adjoint of an elliptic linear differential operator and its potential theory,, Ark. Mat., 11 (1973), 153.
doi: 10.1007/BF02388513. |
[24] |
W. Sternberg, Über die lineare elliptische Differentialgleichung zweiter Ordnung mit drei unabhängigen Veränderlichen,, Math. Z., 21 (1924), 286.
doi: 10.1007/BF01187471. |
[25] |
Q. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians,, Journal of Differential Equations, 182 (2002), 416.
doi: 10.1006/jdeq.2001.4112. |
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