The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in L r -framework

Tomáš Neustupa

Applications of Mathematics (2023)

  • Volume: 68, Issue: 2, page 171-190
  • ISSN: 0862-7940

Abstract

top
We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain Ω , which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves Γ - and Γ + (lower and upper parts of ∂ Ω ), the Dirichlet boundary conditions on Γ in (the inflow) and Γ 0 (boundary of the profile) and an artificial “do nothing”-type boundary condition on Γ out (the outflow). We show that the considered problem has a strong solution with the L r -maximum regularity property for appropriately integrable given data. From this we deduce a series of properties of the corresponding strong Stokes operator.

How to cite

top

Neustupa, Tomáš. "The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework." Applications of Mathematics 68.2 (2023): 171-190. <http://eudml.org/doc/299513>.

@article{Neustupa2023,
abstract = {We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain $\Omega $, which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves $\Gamma _\{-\}$ and $\Gamma _\{+\}$ (lower and upper parts of $\partial \Omega $), the Dirichlet boundary conditions on $\Gamma _\{\rm in\}$ (the inflow) and $\Gamma _\{0\}$ (boundary of the profile) and an artificial “do nothing”-type boundary condition on $\Gamma _\{\rm out\}$ (the outflow). We show that the considered problem has a strong solution with the $L^r$-maximum regularity property for appropriately integrable given data. From this we deduce a series of properties of the corresponding strong Stokes operator.},
author = {Neustupa, Tomáš},
journal = {Applications of Mathematics},
keywords = {Stokes problem; artificial boundary condition; maximum regularity property},
language = {eng},
number = {2},
pages = {171-190},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework},
url = {http://eudml.org/doc/299513},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Neustupa, Tomáš
TI - The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 2
SP - 171
EP - 190
AB - We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain $\Omega $, which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves $\Gamma _{-}$ and $\Gamma _{+}$ (lower and upper parts of $\partial \Omega $), the Dirichlet boundary conditions on $\Gamma _{\rm in}$ (the inflow) and $\Gamma _{0}$ (boundary of the profile) and an artificial “do nothing”-type boundary condition on $\Gamma _{\rm out}$ (the outflow). We show that the considered problem has a strong solution with the $L^r$-maximum regularity property for appropriately integrable given data. From this we deduce a series of properties of the corresponding strong Stokes operator.
LA - eng
KW - Stokes problem; artificial boundary condition; maximum regularity property
UR - http://eudml.org/doc/299513
ER -

References

top
  1. Abe, T., 10.1002/mma.483, Math. Methods Appl. Sci. 27 (2004), 1007-1048. (2004) Zbl1050.35065MR2063094DOI10.1002/mma.483
  2. Acevedo, P., Amrouche, C., Conca, C., Ghosh, A., 10.1016/j.crma.2018.12.002, C. R., Math., Acad. Sci. Paris 357 (2019), 115-119. (2019) Zbl1412.35212MR3927018DOI10.1016/j.crma.2018.12.002
  3. Agmon, S., Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies 2. D. Van Nostrand, New York (1965). (1965) Zbl0142.37401MR0178246
  4. Amrouche, C., Seloula, N. El Houda, 10.7153/dea-03-36, Differ. Equ. Appl. 3 (2011), 581-607. (2011) Zbl1259.35092MR2918930DOI10.7153/dea-03-36
  5. Amrouche, C., Escobedo, M., Ghosh, A., Semigroup theory for the Stokes operator with Navier boundary condition in L p spaces, Available at https://arxiv.org/abs/1808.02001 (2018), 45 pages. (2018) MR4299328
  6. Amrouche, C., Girault, V., 10.21136/CMJ.1994.128452, Czech. Math. J. 44 (1994), 109-140. (1994) Zbl0823.35140MR1257940DOI10.21136/CMJ.1994.128452
  7. Beneš, M., Kučera, P., 10.1002/mana.201400046, Math. Nachr. 289 (2016), 194-212. (2016) Zbl1381.35116MR3458302DOI10.1002/mana.201400046
  8. Bruneau, C.-H., Fabrie, P., 10.1051/m2an/1996300708151, RAIRO, Modélisation Math. Anal. Numér. 30 (1996), 815-840. (1996) Zbl0865.76016MR1423081DOI10.1051/m2an/1996300708151
  9. Chen, G., Osborne, D., Qian, Z., 10.1016/S0252-9602(09)60078-3, Acta Math. Sci., Ser. B, Engl. Ed. 29 (2009), 919-948. (2009) Zbl1212.35346MR2509999DOI10.1016/S0252-9602(09)60078-3
  10. Chen, G.-Q., Qian, Z., 10.1512/iumj.2010.59.3898, Indiana Univ. Math. J. 59 (2010), 721-760. (2010) Zbl1206.35193MR2648084DOI10.1512/iumj.2010.59.3898
  11. Dauge, M., 10.1137/0520006, SIAM J. Math. Anal. 20 (1989), 74-97. (1989) Zbl0681.35071MR0977489DOI10.1137/0520006
  12. Feistauer, M., Felcman, J., Dolejší, V., 10.1002/zamm.19960761422, Z. Angew. Math. Mech. 76 (1996), 297-300. (1996) Zbl0925.76443DOI10.1002/zamm.19960761422
  13. Feistauer, M., Neustupa, T., 10.1007/978-3-0348-7926-2_29, Operator Theoretical Methods and Applications to Mathematical Physics Operator Theory: Advances and Applications 147. Birkhäuser, Basel (2004), 257-276. (2004) Zbl1054.35051MR2053693DOI10.1007/978-3-0348-7926-2_29
  14. Feistauer, M., Neustupa, T., 10.1002/mma.755, Math. Methods Appl. Sci. 29 (2006), 1907-1941. (2006) Zbl1124.35054MR2259990DOI10.1002/mma.755
  15. Feistauer, M., Neustupa, T., 10.1007/s00021-013-0135-4, J. Math. Fluid Mech. 15 (2013), 701-715. (2013) Zbl1293.35204MR3127015DOI10.1007/s00021-013-0135-4
  16. Galdi, G. P., 10.1007/978-0-387-09620-9, Springer Monographs in Mathematics. Springer, New York (2011). (2011) Zbl1245.35002MR2808162DOI10.1007/978-0-387-09620-9
  17. Glowinski, R., 10.1007/978-3-662-12613-4, Springer Series in Computational Physics. Springer, New York (1984). (1984) Zbl0536.65054MR0737005DOI10.1007/978-3-662-12613-4
  18. Grisvard, P., Singularités des solutions du probléme de Stokes dans un polygone, Université de Nice, Nice (1979), preprint French. 
  19. Grisvard, P., 10.1137/1.9781611972030, Monographs and Studies in Mathematics 24. Pitman, Boston (1985). (1985) Zbl0695.35060MR0775683DOI10.1137/1.9781611972030
  20. Heywood, J. G., Rannacher, R., Turek, S., 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y, Int. J. Numer. Methods Fluids 22 (1996), 325-352. (1996) Zbl0863.76016MR1380844DOI10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y
  21. Kato, T., Mitrea, M., Ponce, G., Taylor, M., 10.4310/MRL.2000.v7.n5.a10, Math. Res. Lett. 7 (2000), 643-650. (2000) Zbl0980.53022MR1809290DOI10.4310/MRL.2000.v7.n5.a10
  22. Kellogg, R. B., Osborn, J. E., 10.1016/0022-1236(76)90035-5, J. Func. Anal. 21 (1976), 397-431. (1976) Zbl0317.35037MR0404849DOI10.1016/0022-1236(76)90035-5
  23. Kozel, K., Louda, P., Příhoda, J., 10.1002/pamm.200610352, Proc. Appl. Math. Mech. 6 (2006), 743-744. (2006) DOI10.1002/pamm.200610352
  24. Kračmar, S., Neustupa, J., Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities, Z. Angew. Math. Mech. 74 (1994), T637--T639. (1994) Zbl0836.35121
  25. Kračmar, S., Neustupa, J., 10.1016/S0362-546X(01)00534-X, Nonlinear Anal., Theory Methods Appl. 47 (2001), 4169-4180. (2001) Zbl1042.35605MR1972357DOI10.1016/S0362-546X(01)00534-X
  26. Kračmar, S., Neustupa, J., 10.1002/mana.201700228, Math. Nachr. 291 (2018), 1801-1814. (2018) Zbl1401.35239MR3844807DOI10.1002/mana.201700228
  27. Kučera, P., 10.1007/s11565-009-0082-4, Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 55 (2009), 289-308. (2009) Zbl1205.35198MR2563661DOI10.1007/s11565-009-0082-4
  28. Kučera, P., Skalák, Z., 10.1023/A:1006185601807, Acta Appl. Math. 54 (1998), 275-288. (1998) Zbl0924.35097MR1671783DOI10.1023/A:1006185601807
  29. Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York (1969). (1969) Zbl0184.52603MR0254401
  30. Medková, D., 10.1016/j.jde.2016.08.007, J. Differ. Equations 261 (2016), 5670-5689. (2016) Zbl1356.35181MR3548266DOI10.1016/j.jde.2016.08.007
  31. Medková, D., 10.1515/anly-2018-0035, Analysis, München 40 (2020), 1-17. (2020) Zbl1437.35586MR4069875DOI10.1515/anly-2018-0035
  32. Neustupa, T., 10.1007/978-3-642-00464-3_49, Numerical Analysis and Its Applications Lecture Notes in Computer Science 5434. Springer, Berlin (2009), 431-438. (2009) Zbl1233.35159DOI10.1007/978-3-642-00464-3_49
  33. Neustupa, T., 10.1016/j.amc.2011.05.020, Appl. Math. Comput. 219 (2012), 3316-3322. (2012) Zbl1309.76054MR2993903DOI10.1016/j.amc.2011.05.020
  34. Neustupa, T., 10.1016/j.amc.2015.05.066, Appl. Math. Comput. 272 (2016), 687-691. (2016) Zbl1410.35098MR3423376DOI10.1016/j.amc.2015.05.066
  35. Neustupa, T., The weak Stokes problem associated with a flow through a profile cascade in L r -framework, Available at https://arxiv.org/abs/2009.08234v2 (2020), 20 pages. (2020) MR4553623
  36. Neustupa, T., 10.1007/s10440-021-00396-4, Acta Appl. Math. 172 (2021), Article ID 3, 23 pages. (2021) Zbl1471.35220MR4220789DOI10.1007/s10440-021-00396-4
  37. Sohr, H., 10.1007/978-3-0348-8255-2, Birkhäuser Advanced Texts. Birkhäuser, Basel (2001). (2001) Zbl0983.35004MR1928881DOI10.1007/978-3-0348-8255-2
  38. Straka, P., Příhoda, J., Kožíšek, M., Fürst, J., 10.1051/epjconf/201714302118, EPJ Web Conf. 143 (2017), Article ID 02118, 6 pages. (2017) DOI10.1051/epjconf/201714302118
  39. Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications 2. North-Holland, Amsterdam (1977). (1977) Zbl0383.35057MR0609732

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.