By Matthias Köhne

Navier-Stokes Equations.- strength keeping Boundary Condition.- Weakly Singular Domain.- Maximal Lp-Regularity

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**Extra info for Lp-theory for incompressible newtonian flows : energy preserving boundary conditions, weakly**

**Sample text**

Thus, ω +Aα,β is sectorial with spectral angle φω+Aα,β < π/2 and −Aα,β generates an analytic semigroup in Xpβ (Ω). 45 To treat the non-linear Navier-Stokes equations (N | a, Ω)α,β f,g,h,u0 it is convenient to recast them into an operator equation. 31 the Navier-Stokes equations are equivalent to α,β Lα,β a,γ (u, p) = Na,γ (u, 0) + (f, g, h), u(0) = u0 in Ω, where the non-linear operator α,β : Xβγ (a, Ω) −→ Yf (a, Ω) × Yg (a, Ω) × Yα,β Na,γ h,γ (a, Γ) is given as α,β Na,γ (v, q) = (−(v · ∇)v, 0, 0), (v, q) ∈ Xβγ (a, Ω).

30 with some constant M > 0. 45 it remains to uniquely solve (¯ u, p¯) = Ka¯α,β u, p¯), (¯ u, p¯) ∈ 0 Xβγ (¯ a, Ω) ,γ (¯ in order to obtain the unique maximal regular solution to the Navier-Stokes equations (N | a ¯, Ω)α,β f,g,h,u0 . Note that from this point on, we may assume γ = 1/2 − 1/2p, if β = ±1. Therefore, all function spaces, which occur in the next paragraphs, constitute Banach spaces. 50 First observe that ∗ Na¯α,β a, Ω) −→ 0 Yα,β a, Ω) ,γ ( · + u , 0) : 0 Xu (¯ γ (¯ is Fr´echet diﬀerentiable with u + u∗ , 0)¯ v = (−((¯ u + u∗ ) · ∇)¯ v − (¯ v · ∇)(¯ u + u∗ ), 0, 0), DNa¯α,β ,γ (¯ u¯, v¯ ∈ 0 Xu (¯ a, Ω).

Also note that the tangential part of the boundary operator Bα,β neither depends on β nor on the pressure. Therefore, we will in slight abuse of notation make frequent use of the abbreviation PΓ B α u = PΓ B α,β (u, p), α, β ∈ { −1, 0, +1 }. Analogously, the normal part of the boundary operator B α,β does not depend on α, which motivates the abbreviation QΓ B β (u, p) = QΓ B α,β (u, p), α, β ∈ { −1, 0, +1 }. This abbreviations allow the boundary condition to be split into two equations on the boundary, each of which is deﬁned by one of the parameters, which will sometimes be convenient.