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The Stokes problem in fractal domains: Asymptotic behaviour of the solutions
Dipartimento di Scienze di Base e Applicate per I'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy |
We study a Stokes problem in a three dimensional fractal domain of Koch type and in the corresponding prefractal approximating domains. We prove that the prefractal solutions do converge to the limit fractal one in a suitable sense. Namely the approximating velocity vector fields as well as the approximating associated pressures converge to the limit fractal ones respectively.
References:
[1] |
G. Acosta, R. G. Durán and M. A. Muschietti,
Solutions of the divergence operator on John domains, Adv. Math., 206 (2006), 373-401.
doi: 10.1016/j.aim.2005.09.004. |
[2] |
M. Cefalo, G. Dell'acqua and M. R. Lancia,
Numerical approximation of transmission problems across Koch-type highly conductive layers,, AMC, 218 (2012), 5453-5473.
doi: 10.1016/j.amc.2011.11.033. |
[3] |
M. Cefalo, M. R. Lancia and H. Liang,, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential and Integral Equations, 26 (2013), 1027–1054. |
[4] |
G. de Rham,, Variétés Différentiables, , Hermann, Paris, 1955. |
[5] |
K. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986.
![]() ![]() |
[6] |
F. John,
Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413.
doi: 10.1002/cpa.3160140316. |
[7] |
T. Kato,, Perturbation Theory for Linear Operators, , Springer-Verlag, New York, 1966. |
[8] |
M. R. Lancia and P. Vernole,
Convergence results for parabolic transmission problems across higly conductive layers with small capacity, Adv. Math. Sci. Appl., 16 (2006), 411-445.
|
[9] |
S. Monniaux,
Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme,, Math. Res. Lett., 13 (2006), 455-461.
doi: 10.4310/MRL.2006.v13.n3.a9. |
[10] |
U. Mosco,
Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.
doi: 10.1006/jfan.1994.1093. |
[11] |
U. Mosco,
Convergence of convex sets and solutions of variational inequalities,, Adv. in Math., 3 (1969), 510-585.
doi: 10.1016/0001-8708(69)90009-7. |
[12] |
S. Shen, J. Xu, J. Zhou and Y. Chen,
Flow and heat transfer in microchannels with rough wall surface, Energy Convers. Manage., 47 (2006), 1311-1325.
doi: 10.1016/j.enconman.2005.09.001. |
[13] |
H. Sohr,, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhuser Verlag, Basel, 2001.
doi: 10.1007/978-3-0348-8255-2. |
[14] |
B. Taylor, A. L. Carrano and S. G. Kandlikar,
Characterization of the effect of surface roughness and texture on fluid flow past, present, and future, Int. J. Thermal Sci., 45 (2006), 962-968.
doi: 10.1016/j.ijthermalsci.2006.01.004. |
[15] |
R. Temam, Roger Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979. |
[16] |
S. S. Yang, B. Yu, M. Zou and M. Liang,
A fractal analysis of laminar flow resistance in roughened microchannels, Int. J. Heat Mass Transf., 77 (2014), 208-217.
|
show all references
References:
[1] |
G. Acosta, R. G. Durán and M. A. Muschietti,
Solutions of the divergence operator on John domains, Adv. Math., 206 (2006), 373-401.
doi: 10.1016/j.aim.2005.09.004. |
[2] |
M. Cefalo, G. Dell'acqua and M. R. Lancia,
Numerical approximation of transmission problems across Koch-type highly conductive layers,, AMC, 218 (2012), 5453-5473.
doi: 10.1016/j.amc.2011.11.033. |
[3] |
M. Cefalo, M. R. Lancia and H. Liang,, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential and Integral Equations, 26 (2013), 1027–1054. |
[4] |
G. de Rham,, Variétés Différentiables, , Hermann, Paris, 1955. |
[5] |
K. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986.
![]() ![]() |
[6] |
F. John,
Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413.
doi: 10.1002/cpa.3160140316. |
[7] |
T. Kato,, Perturbation Theory for Linear Operators, , Springer-Verlag, New York, 1966. |
[8] |
M. R. Lancia and P. Vernole,
Convergence results for parabolic transmission problems across higly conductive layers with small capacity, Adv. Math. Sci. Appl., 16 (2006), 411-445.
|
[9] |
S. Monniaux,
Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme,, Math. Res. Lett., 13 (2006), 455-461.
doi: 10.4310/MRL.2006.v13.n3.a9. |
[10] |
U. Mosco,
Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.
doi: 10.1006/jfan.1994.1093. |
[11] |
U. Mosco,
Convergence of convex sets and solutions of variational inequalities,, Adv. in Math., 3 (1969), 510-585.
doi: 10.1016/0001-8708(69)90009-7. |
[12] |
S. Shen, J. Xu, J. Zhou and Y. Chen,
Flow and heat transfer in microchannels with rough wall surface, Energy Convers. Manage., 47 (2006), 1311-1325.
doi: 10.1016/j.enconman.2005.09.001. |
[13] |
H. Sohr,, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhuser Verlag, Basel, 2001.
doi: 10.1007/978-3-0348-8255-2. |
[14] |
B. Taylor, A. L. Carrano and S. G. Kandlikar,
Characterization of the effect of surface roughness and texture on fluid flow past, present, and future, Int. J. Thermal Sci., 45 (2006), 962-968.
doi: 10.1016/j.ijthermalsci.2006.01.004. |
[15] |
R. Temam, Roger Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979. |
[16] |
S. S. Yang, B. Yu, M. Zou and M. Liang,
A fractal analysis of laminar flow resistance in roughened microchannels, Int. J. Heat Mass Transf., 77 (2014), 208-217.
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