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Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains
Author(s) -
Beneš Michal,
Kučera Petr
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400046
Subject(s) - mathematics , uniqueness , bounded function , mathematical analysis , banach space , operator (biology) , space (punctuation) , boundary (topology) , boundary value problem , bounded operator , navier–stokes equations , biochemistry , chemistry , linguistics , philosophy , repressor , transcription factor , compressibility , engineering , gene , aerospace engineering
In this paper we consider the system of the non‐steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y , respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator N : X → Y and formulate our problem in terms of operator equations. Let u ∈ X andG P u : X → Y be the Fréchet derivative of N at u . We prove thatG P u is one‐to‐one and onto Y . Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W 2, 2 ‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.