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Oseen equation; weighted Sobolev space; anisotropic weight
This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in $L^{p}$ theory.
[1] C.  Amrouche, V.  Girault, J. Giroire: Weighted Sobolev spaces for Laplace’s equation in $\mathbb{R}^n$. J.  Math. Pures Appl., IX.  Sér. 73 (1994), 579–606. MR 1309165
[2] C.  Amrouche, U.  Razafison: Weighted Sobolev spaces for a scalar model of the stationary Oseen equation in  $\mathbb{R}^{3}$. J. Math. Fluids Mech (to appear). MR 2329264
[3] R.  Farwig: A variational approach in weighted Sobolev spaces to the operator $-\Delta + \partial /\partial x_{1}$ in exterior domains of  $\mathbb{R}^{3}$. Math.  Z. 210 (1992), 449–464. MR 1171183
[4] R.  Farwig: The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211 (1992), 409–448. DOI 10.1007/BF02571437 | MR 1190220
[5] R.  Farwig, H.  Sohr: Weighted estimates for the Oseen equations and the Navier-Stokes equations in exterior domains. Proc. 3rd International Conference on the Navier-Stokes Equations: Theory and Numerical Methods, Oberwolfach, Germany, June 5–11, 1994, J. G. Heywood (ed.), World Scientific, Ser. Adv. Math. Appl. Sci. Vol. 47, Singapore, 1998, pp. 11–30. MR 1643022
[6] R.  Finn: On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems. Arch. Ration. Mech. Anal. 19 (1965), 363–406. DOI 10.1007/BF00253485 | MR 0182816 | Zbl 0149.44606
[7] R.  Finn: Estimates at infinity for stationary solutions of the Navier-Stokes equations. Bull. Math. Soc. Sci. Math. Phys. R.  P.  R. 51 (1960), 387–418. MR 0166495 | Zbl 0106.39402
[8] G. P.  Galdi: An introduction to the mathematical theory of the Navier-Stokes equations. Vol.  I: Linearized steady problems. Springer Tracts in Natural Philosophy, Vol. 38, Springer, New York, 1994. MR 1284205
[9] B.  Hanouzet: Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46 (1972), 227–272. MR 0310417 | Zbl 0247.35041
[10] S.  Kračmar, A.  Novotný, M.  Pokorný: Estimates of Oseen kernels in weighted $L^p$ spaces. J.  Math. Soc. Japan 53 (2001), 59–111. DOI 10.2969/jmsj/05310059
[11] A.  Kufner: Weighted Sobolev spaces. Wiley-Interscience, New York, 1985. MR 0802206 | Zbl 0579.35021
[12] P. I.  Lizorkin: $(L^p,L^{q})$-multipliers of Fourier integrals. Dokl. Akad. Nauk SSSR 152 (1963), 808–811. MR 0154057 | Zbl 0199.44401
[13] C. W.  Oseen: Über die Stokessche Formel und über eine verwandte Aufgabe in der Hydrodynamik. Arkiv fór Mat. Astron. och Fys. 7 (1911), 1–36.
[14] C. W.  Oseen: Neuere Methoden und Ergebnisse in der Hydrodynamik. Akadem. Verlagsgesellschaft, Leipzig, 1927.
[15] C.  Pérez: Two weighted norm inequalities for Riesz potentials and uniform $L^{p}$- weighted Sobolev inequalities. Indiana Univ. Math. J. 39 (1990), 31–44. DOI 10.1512/iumj.1990.39.39004 | MR 1052009
[16] M.  Reed, B.  Simon: Methods of Modern Mathematical Physics. II.  Fourier Analysis, Self-adjointness. Academic Press, New York-San Francisco-London, 1975. MR 0493420
[17] E. M.  Stein: Singular Integrals and Differentiability Properties of Functions. University Press, Princeton, 1970. MR 0290095 | Zbl 0207.13501
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