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May  2020, 13(5): 1553-1565. doi: 10.3934/dcdss.2020088

The Stokes problem in fractal domains: Asymptotic behaviour of the solutions

Maria Rosaria Lancia , and  Paola Vernole

Dipartimento di Scienze di Base e Applicate per I'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy

* Corresponding author: Maria Rosaria Lancia

Received  January 2018 Revised  June 2018 Published  June 2019

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.

Keywords: Asymptotic behaviour, energy forms, fractal domains, semigroups, Stokes problem.
Mathematics Subject Classification: Primary: 35Q35, 47D06, 47D06, 28A80.
Citation: Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1553-1565. doi: 10.3934/dcdss.2020088
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. 

Figure 1.  Koch snowflake
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Figure 2.  The prefractal $ F_h $ at the step $ h = 2 $ and $ h = 5 $
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Figure 3.  Surface $ S_3 $
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