Abstract
Function spaces with weighted norms and detached asymptotics naturally appear in the treatment of boundary value problems when linear and nonlinear terms have got same asymptotic behavior either at a singularity point of the boundary, or at infinity. The characteristic feature of these spaces is that their norms are composed from both, norms of angular parts in the detached terms and norms of asymptotic remainders. The developed approach is described for the Navier-Stokes problems in domains with conical (angular) outlets to infinity while the 3-D exterior and 2-D aperture problems imply representative examples. With a view towards compressible and non-Newtonian fluids, the described technique is applied to the transport equation as well.
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References
H. Beirao da Veiga: On a stationary transport equation, Ann. Univ. Ferrara, Nuova Ser., Sez. VII, 32 (1986), 79–91.
W. Borchers, T. Miyakawa: On stability of exterior stationary Navier-Stokes flows, Acta Mathematica, 174 (1995), 311–382.
W. Borchers, K. Pileckas: Existence, uniqueness and asymptotics of steady jets, Arch. Rat. Mech. Analysis, 120 (1992), 1–49.
G. P. Galdi, M. Padula, V.A. Solonnikov: Existence, uniqueness and asymptotic behavior of solutions of steady-state Navier-Stokes equations in a plane aperture domain, Indiana Univ. Math. J., 45: 4 (1996), 961–995.
I. VKamotskiï, S.A. Nazarov: Spectral problems in singularly perturbed domains and self-adjoint extensions of differential operators, Trudy Sankt-Peterburg. Mat. Obshch., 6 (1998), 151–212 (Russian).
V. A. Kondrat’ev: Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obshch., 16 (1967), 209–292; English transl. in Trans. Moscow Math. Soc, 16 (1967), 227–313.
O. A. Ladyzhenskaya: The mathematical theory of viscous incompressible flow, Gordon and Breach, New-York, London, Paris, 1969.
V. G. Maz’ya, B.A. Plamenevskii: On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points, Math. Nachr., 76 (1977), 29–60; English transl. in Amer. Math. Soc. Transl., 123 (2) (1984), 57–88.
V. G. Maz’ya, B.A. Plamenevskii: Estimates in L p and Hölder classes and Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Math. Nachr., 81 (1978), 25–82; English transl. in Amer. Math. Soc. Transl., 123 (2) (1984), 1–56.
V. G. Maz’ya, B.A. Plamenevskii, L.I. Stupyalis: The three dimensional problem of steady-state motion of a fluid with a free surface, Differentsial’nye Uravneniya i Primenen. Trudy Sem. Protsessy Upravleniya. I Sektsiya, 23 (1979), 1–155; English transl. in Amer. Math. Soc. Transl. 123 (2) (1984), 171–268.
I. Mogilevskii, V.A. Solonnikov: Problem on stationary flow of second grade fluid in Hölder spaces of functions, Zapiski nauchn. seminar POMI, 243 (1997), 154–165 (Russian).
S. A. Nazarov: Self-adjoint extensions of the operator of the Dirichlet problem in weighted function spaces, Mat. sbornik, 137 (1988), 224–241; English transl. in Math. USSR Sbornik. 65 (1990), 229–247.
S. A. Nazarov, B.A. Plamenevskii: Selfadjoint elliptic problems with radiation conditions on the edges of the boundary, Algebra i Analiz, 4:3 (1992), 196–225; English transl. in St.Petersburg Math. J. 4 (1993), 569–594.
S. A. Nazarov, B.A. Plamenevskii: Elliptic Problems in domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin, 1994.
S. A. Nazarov: Asymptotic conditions at a point, selfadjoint extensions of operators and the method of matched asymptotic expansions, Trudy Sankt-Peterburg. Mat. Obshch., 5 (1996), 112–183; English transl. in Amer. Math. Soc. Transl., 193 (2) (1999), 77–126.
S. A. Nazarov: On the two-dimensional aperture problem for Navier-Stokes equations, C.R. Acad. Sei. Paris. Sér. 1, 323 (1996), 699–703.
S. A. Nazarov: The operator of a boundary value problem with Chaplygin-Zhukovskii-Kutta type conditions on an edge of the boundary has the Fredhom property, Funkt. Analiz i Ego Prilozheniya, 31:3 (1997), 44–56; English transl. in Functional Analysis and Its Applications, 31:3 (1997), 183–192.
S. A. Nazarov: The Navier-Stokes problem in a two-dimensional domain with angular outlets to infinity, Zapiski nauchn. seminar. POMI, 257 (1999), 207–227 (Russian).
S. A. Nazarov: The polynomial property of self-adjoint elliptic problems and an algebraic description of their attributes, Uspekhi Mat. Nauk, 54:5 (1999) 77–142 (Russian).
S. A. Nazarov, K. Pileckas: Asymptotics of solutions to the Navier-Stokes equations in the exterior of a bounded body, Doklady RAN, 367:4 (1999), 461–463; English tranl. in Doklady Math, 60:1 (1999), 133–135.
S. A. Nazarov, K. Pileckas: On steady Stokes and Navier-Stokes problems with zero velocity at infinity in a three-dimensional exterior domain, Kyoto University Math. J. (to appear).
S. A. Nazarov, O.R. Polyakova: Rupture criteria, asymptotic conditions at crack tips, and selfadjoint extensions of the Lame operator, Trudy Moskov. Mat. Obshch., 57 (1996), 16–75; English transl. in Trans. Moscow Math. Soc, 57 (1996), 13–66.
S. A. Nazarov, A. Sequeira, J.H. Videman: Asymptotic behavior at infinity of three dimensional steady viscoelastic flows, Pacific Journal (submitted).
S. A. Nazarov, A. Sequeira, J.H. Videman: Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. 1. Linearization on an asymmetric solution (in preparation).
S. A. Nazarov, A. Sequeira, J.H. Videman: Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. 2. Linearization on a symmetric solution (in preparation).
S. A. Nazarov, M. Specovius-Neugebauer, G. Thäter: Quiet flows for Stokes and Navier-Stokes problems in domains with cylindrical outlets to infinity, Kyshu J. Math, 53:2 (1999), 369–394.
A. Novotny: About steady transport equation. II — Shauder estimates in domains with smooth boundaries, Portugaliae Mathematica, 54:3 (1997), 317–333.
A. Novotny, M. Padula: L p-approach to steady flows of viscous compressible fluids in exterior domains, Arch. Rat. Mech. Analysis, 126 (1994), 243–297.
A. Novotny, M. Padula: Note on decay of solutions of steady Navier-Stokes equations in 3-D exterior domains, Differential Integral Equations, 8 (1995), 1833–1844.
K. Pileckas, A. Sequeira, J.H. Videman: A note on steady flows of non-Newtonian fluids in channels and pipes, in: L. Magalhäes, L. Sanchez, C. Rocha (eds), EQUADIFF-95, World Scientific (1998), 458–467.
B.-W. Schulze: Boundary Value Problems and Singular Pseudo-differential Operators, John Wiley &, Sons, Chichester, New-York, 1999.
G. Thäter: Quiet flows for the steady Navier-Stokes problem in domains with quasicylindrical outlets to infinity, in: H. Amann, G.P. Galdi et all. (eds.) Navier-Stokes Equations and Related Nonlinear problems, TEV/VSP. Vilnius/Utrecht (1998), 412–438.
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Nazarov, S.A. (2000). Weighted Spaces with Detached Asymptotics in Application to the Navier-Stokes Equations. In: Málek, J., Nečas, J., Rokyta, M. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57308-8_5
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DOI: https://doi.org/10.1007/978-3-642-57308-8_5
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