American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation
  • DCDS-S Home
  • This Issue
  • Next Article
    Mean periodic solutions of a inhomogeneous heat equation with random coefficients
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

G. de Rham,, Variétés Différentiables, , Hermann, Paris, 1955.  Google Scholar

[5] K. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986.   Google Scholar
[6]

F. John, Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413.  doi: 10.1002/cpa.3160140316.  Google Scholar

[7]

T. Kato,, Perturbation Theory for Linear Operators, , Springer-Verlag, New York, 1966.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. Sohr,, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

G. de Rham,, Variétés Différentiables, , Hermann, Paris, 1955.  Google Scholar

[5] K. Falconer, The Geometry of Fractal Sets,, Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986.   Google Scholar
[6]

F. John, Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413.  doi: 10.1002/cpa.3160140316.  Google Scholar

[7]

T. Kato,, Perturbation Theory for Linear Operators, , Springer-Verlag, New York, 1966.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. Sohr,, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

Figure 1.  Koch snowflake
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The prefractal $ F_h $ at the step $ h = 2 $ and $ h = 5 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Surface $ S_3 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jorge Ferreira, Mauro De Lima Santos. Asymptotic behaviour for wave equations with memory in a noncylindrical domains. Communications on Pure & Applied Analysis, 2003, 2 (4) : 511-520. doi: 10.3934/cpaa.2003.2.511
[2]

Dorina Mitrea, Irina Mitrea, Marius Mitrea, Lixin Yan. Coercive energy estimates for differential forms in semi-convex domains. Communications on Pure & Applied Analysis, 2010, 9 (4) : 987-1010. doi: 10.3934/cpaa.2010.9.987
[3]

Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725
[4]

Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135
[5]

Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184
[6]

M. Chipot, A. Rougirel. On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 319-338. doi: 10.3934/dcdsb.2001.1.319
[7]

Gabriela Planas, Eduardo Hernández. Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1245-1258. doi: 10.3934/dcds.2008.21.1245
[8]

Raffaela Capitanelli, Maria Agostina Vivaldi. Uniform weighted estimates on pre-fractal domains. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1969-1985. doi: 10.3934/dcdsb.2014.19.1969
[9]

Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025
[10]

Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281
[11]

Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019
[12]

Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197
[13]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[14]

Simone Creo, Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. Approximation of a nonlinear fractal energy functional on varying Hilbert spaces. Communications on Pure & Applied Analysis, 2018, 17 (2) : 647-669. doi: 10.3934/cpaa.2018035
[15]

Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595
[16]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

[17]

V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117
[18]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
[19]

Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays. Electronic Research Archive, 2020, 28 (2) : 1037-1048. doi: 10.3934/era.2020056
[20]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (172)
  • HTML views (420)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences