The L p -Helmholtz projection in finite cylinders

Tobias Nau

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 1, page 119-134
  • ISSN: 0011-4642

Abstract

top
In this article we prove for 1 < p < ∞ the existence of the L p -Helmholtz projection in finite cylinders Ω . More precisely, Ω is considered to be given as the Cartesian product of a cube and a bounded domain V having C 1 -boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in Ω is solved, which implies existence and a representation of the L p -Helmholtz projection as a Fourier multiplier operator.

How to cite

top

Nau, Tobias. "The $L^p$-Helmholtz projection in finite cylinders." Czechoslovak Mathematical Journal 65.1 (2015): 119-134. <http://eudml.org/doc/270035>.

@article{Nau2015,
abstract = {In this article we prove for $1<p<\infty $ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega $. More precisely, $\Omega $ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega $ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as a Fourier multiplier operator.},
author = {Nau, Tobias},
journal = {Czechoslovak Mathematical Journal},
keywords = {Helmholtz projection; Helmholtz decomposition; weak Neumann problem; periodic boundary conditions; finite cylinder; cylindrical space domain; $L^p$-space; operator-valued Fourier multiplier; $\mathcal \{R\}$-boundedness; reflection technique; fluid dynamics; Helmholtz projection; Helmholtz decomposition; weak Neumann problem; periodic boundary conditions; finite cylinder; cylindrical space domain; -space; operator-valued Fourier multiplier; -boundedness; reflection technique; fluid dynamics},
language = {eng},
number = {1},
pages = {119-134},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The $L^p$-Helmholtz projection in finite cylinders},
url = {http://eudml.org/doc/270035},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Nau, Tobias
TI - The $L^p$-Helmholtz projection in finite cylinders
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 1
SP - 119
EP - 134
AB - In this article we prove for $1<p<\infty $ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega $. More precisely, $\Omega $ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega $ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as a Fourier multiplier operator.
LA - eng
KW - Helmholtz projection; Helmholtz decomposition; weak Neumann problem; periodic boundary conditions; finite cylinder; cylindrical space domain; $L^p$-space; operator-valued Fourier multiplier; $\mathcal {R}$-boundedness; reflection technique; fluid dynamics; Helmholtz projection; Helmholtz decomposition; weak Neumann problem; periodic boundary conditions; finite cylinder; cylindrical space domain; -space; operator-valued Fourier multiplier; -boundedness; reflection technique; fluid dynamics
UR - http://eudml.org/doc/270035
ER -

References

top
  1. Abels, H., 10.1007/s00021-004-0116-8, J. Math. Fluid Mech. 7 (2005), 201-222. (2005) Zbl1070.35020MR2177127DOI10.1007/s00021-004-0116-8
  2. Abels, H., 10.1007/s00021-004-0117-7, J. Math. Fluid Mech. 7 (2005), 223-260. (2005) Zbl1083.35085MR2177128DOI10.1007/s00021-004-0117-7
  3. Arendt, W., Bu, S., 10.1007/s002090100384, Math. Z. 240 (2002), 311-343. (2002) Zbl1018.47008MR1900314DOI10.1007/s002090100384
  4. Bernstein, S., Sur la convergence absolue des séries trigonométriques, C. R. Acad. Sci., Paris 158 French (1914), 1661-1663. (1914) 
  5. Bogovskiĭ, M. E., Decomposition of L p ( Ω ; ℝ n ) into a direct sum of subspaces of solenoidal and potential vector fields, Sov. Math., Dokl. 33 (1986), 161-165 translated from Russian original in Dokl. Akad. Nauk SSSR 286 (1986), 781-786. (1986) MR0828621
  6. Bu, S., 10.1007/s11425-006-0574-y, Sci. China, Ser. A 49 (2006), 574-576. (2006) Zbl1160.42305MR2250485DOI10.1007/s11425-006-0574-y
  7. Bu, S., Kim, J.-M., Operator-valued Fourier multiplier theorems on L p -spaces on 𝕋 d , Arch. Math. 82 (2004), 404-414. (2004) MR2061447
  8. Denk, R., Hieber, M., Prüss, J., ℛ -boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc. 788 (2003), 114. (2003) MR2006641
  9. Farwig, R., Weighted L q -Helmholtz decompositions in infinite cylinders and in infinite layers, Adv. Differ. Equ. 8 (2003), 357-384. (2003) Zbl1038.35068MR1948530
  10. Farwig, R., Kozono, H., Sohr, H., 10.1007/s00013-006-1910-8, Arch. Math. 88 (2007), 239-248. (2007) Zbl1121.35097MR2305602DOI10.1007/s00013-006-1910-8
  11. Farwig, R., Myong-Hwan, R., 10.1007/s00021-006-0235-5, J. Math. Fluid Mech. 10 (2008), 352-387. (2008) Zbl1162.76322MR2430805DOI10.1007/s00021-006-0235-5
  12. Farwig, R., Ri, M.-H., 10.4064/sm178-3-1, Stud. Math. 178 (2007), 197-216. (2007) Zbl1111.35034MR2289354DOI10.4064/sm178-3-1
  13. Farwig, R., Ri, M.-H., 10.1002/mana.200510536, Math. Nachr. 280 (2007), 1061-1082. (2007) Zbl1131.35055MR2334660DOI10.1002/mana.200510536
  14. Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems, Springer Tracts in Natural Philosophy 38 Springer, New York (1994). (1994) Zbl0949.35004MR1284205
  15. Kunstmann, P. C., Weis, L., Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus, Functional Analytic Methods for Evolution Equations Lecture Notes in Math. 1855 Springer, Berlin (2004), 65-311. (2004) Zbl1097.47041MR2108959
  16. Nau, T., L P -Theory of Cylindrical Boundary Value Problems, Springer Spektrum Wiesbaden (2012). (2012) Zbl1252.35003MR2987207
  17. Ruzhansky, M., Turunen, V., Pseudo-Differential Operators and Symmetries, Pseudo-Differential Operators. Theory and Applications 2: Background Analysis and Advanced Topics Birkhäuser, Basel (2010). (2010) Zbl1193.35261MR2567604
  18. Štrkalj, Ž., Weis, L., 10.1090/S0002-9947-07-04417-0, Trans. Am. Math. Soc. 359 (2007), 3529-3547. (2007) Zbl1209.42005MR2302504DOI10.1090/S0002-9947-07-04417-0

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.