March 15, 2018

By Giovanni P. Galdi

ISBN-10: 0387096191

ISBN-13: 9780387096193

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Extra info for An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, 2nd Edition

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Besides the papers of Beir˜ao da Veiga (2004, 2005), which generalize and simplify the proof of the ˇcadilov, we refer the interested reader, for example, results of Solonnikov & Sˇ to Ebemeyer & Frehse (2001) for flow in bounded domains, Mucha (2003), Konieczny (2006), and Beir˜ ao da Veiga (2006) for flow in infinite channels and pipes, to Konieczny (2009) for flow in exterior domains, and to the literature cited therein. 2). 3). 1), with the objective of explaining the difference between the discharges in glass and copper tubes, as experimentally observed by Girard (1816).

0, ζ(z1 , 0, . . , 0)), z1 > 0 z (2) = (z1 , 0, . . , 0, ζ(z1 , 0, . . , 0)), z1 > 0 and so, at the same time, (1) tan α = z1 (1) ζ(z1 , 0, . . , 0) − yn (2) tan α = z1 (2) ζ(z1 , 0, . . , 0) − yn implying (1) (2) |ζ(z1 , 0, . . , 0) − ζ(z1 , 0, . . , 0)| (1) |z1 − (2) z1 | = 1 1 ≥ . tan α tan α Thus, if (say) 1 , 2κ ρ will cut ∂Ω ∩ Br (x0 ) at only one point. Next, denote by σ = σ(z) the intersection of Γ (y0 , α/2) with a plane orthogonal to xn-axis at a point z = (0, . . , zn ) with zn > yn , and set tan α ≤ R = R(z) ≡ dist (∂σ, z).

0) − yn (2) tan α = z1 (2) ζ(z1 , 0, . . , 0) − yn implying (1) (2) |ζ(z1 , 0, . . , 0) − ζ(z1 , 0, . . , 0)| (1) |z1 − (2) z1 | = 1 1 ≥ . tan α tan α Thus, if (say) 1 , 2κ ρ will cut ∂Ω ∩ Br (x0 ) at only one point. Next, denote by σ = σ(z) the intersection of Γ (y0 , α/2) with a plane orthogonal to xn-axis at a point z = (0, . . , zn ) with zn > yn , and set tan α ≤ R = R(z) ≡ dist (∂σ, z). Clearly, taking z sufficiently close to y0 (z = z, say), σ(z) will be entirely contained in Ω and, further, every ray starting from a point of σ(z) and lying within Γ (y0 , α/2) will form with the xn-axis an angle less than α and so, by what we have shown, it will cut ∂Ω ∩ Br (x0 ) at only one point.