Geodesic
The solution of a spectral problem for the curl and the Stokes operators with periodic boundary
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 246-276 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper, the relations between eigenvalues and eigenfunctions of the curl operator and the Stokes operator (with periodic boundary condition) are considered. These relations show that the curl operator is a square root of the Stokes operator with $\nu=1$. The multiplicity of zero eigenvalue of the curl operator is infinite. The space $\mathbf{L}_2(Q,2\pi)$ is decomposed into a directe sum of the eigensubspaces of the operator curl. For any complex number $\lambda$, the equation $\operatorname{rot}\mathbf{u}+\lambda\mathbf{u}=\mathbf{f}$ and the Stokes equation $-\nu(\Delta v+\lambda^2v)+\nabla p=\mathbf{f}$, $\operatorname{div}v=0$, are solved. 
@article{ZNSL_2004_318_a12,
     author = {R. S. Saks},
     title = {The solution of a~spectral problem for the curl and the {Stokes} operators with periodic boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {246--276},
     year = {2004},
     volume = {318},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a12/}
}
TY  - JOUR
AU  - R. S. Saks
TI  - The solution of a spectral problem for the curl and the Stokes operators with periodic boundary
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2004
SP  - 246
EP  - 276
VL  - 318
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a12/
LA  - ru
ID  - ZNSL_2004_318_a12
ER  - 
%0 Journal Article
%A R. S. Saks
%T The solution of a spectral problem for the curl and the Stokes operators with periodic boundary
%J Zapiski Nauchnykh Seminarov POMI
%D 2004
%P 246-276
%V 318
%U http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a12/
%G ru
%F ZNSL_2004_318_a12
R. S. Saks. The solution of a spectral problem for the curl and the Stokes operators with periodic boundary. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 246-276. http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a12/

[1] V. A. Arnold, Izbrannoe – 60, FAZIS, M., 1997 | MR

[2] V. I. Arnold, “Zamechaniya o povedenii techenii trekhmernoi idealnoi zhidkosti”, Izbrannoe – 60, FAZIS, M., 1997, 203–212 | MR

[3] V. I. Arnold, E. I. Korkina, Izbrannoe – 60, FAZIS, M., 1997 | MR

[4] B. R. Vainberg, V. V. Grushin, “O ravnomerno neellipticheskikh zadachakh, I”, Mat. sb., 72:4 (1967), 126–154 | Zbl

[5] V. S. Vladimirov, Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR

[6] V. A. Zorich, Matematicheskii analiz, Ch. II, Nauka, M., 1984 | MR

[7] M. Henon, “Sur la topologie des lignes de courant dans un cas particulier”, C. R. Acad. Sci. Paris, 262 (1966), 312–314

[8] A. I. Maltsev, Osnovy lineinoi algebry, Nauka, M., 1975

[9] O. A. Ladyzhenskaya, Matematicheskie voprosy dmnamiki vyazkoi neszhimaemoi zhidkosti, 2-oe izd., Nauka, M., 1970

[10] R. S. Saks, “O kraevykh zadachakh dlya sistemy $\operatorname{rot}u+\lambda u=h$”, DAN SSSR, 199:5 (1971), 1022–1025 | MR | Zbl

[11] R. S. Saks, “O kraevykh zadachakh dlya sistemy $\operatorname{rot}u+\lambda u=h$”, Diff. uravn., 8:1 (1972), 126–133 | MR | Zbl

[12] R. S. Saks, “Normalno razreshimye i neterovye kraevye zadachi dlya nekotorykh sistem uravnenii matematicheskoi fiziki. Primenenie funktsionalnogo analiza k uravneniyam s chastnymi proizvodnymi”, Trudy seminara akad. S. L. Soboleva, IM SO AN SSSR, 2, Novosibirsk, 1983, 129–158 | MR | Zbl

[13] R. S. Saks, “Spektr operatora vikhrya v share pri uslovii neprotekaniya i sobstvennye znacheniya kolebanii uprugogo shara, zakreplennogo na granitse”, Trudy mezhdunarodnoi konferentsii “Kompleksnyi analiz, differentsialnye uravneniya i smezhnye voprosy”, IV, Prikladnaya matematika, IMVTs UNTs RAN, Ufa, 2000, 61–68

[14] R. S. Saks, Yu. N. Polyakov, “O spektre operatora vikhrya v klasse periodicheskikh vektor-funktsii”, Metody funktsionalnogo analiza i teorii funktsii v razlichnykh zadachakh matematicheskoi fiziki, materialy mezhdunarodnogo nauchnogo seminara-soveschaniya s odnoimennym nazvaniem, BashGU, IMVTs UNTs RAN, Ufa, 2000, 52–55

[15] B. V. Shabat, Vvedenie v kompleksnyi analiz, Nauka, M., 1976 | MR