Index of notations

General notations

• d refers to the surface dimension (and thus, d = 1 or 2).

• |α| =

∑d

j=1

αj for all multi-index α ∈

Nd.

• Cst is a generic notation for a constant whose exact value is of no impor-

tance.

• A B means A ≤ Cst B.

• s (s ∈ R) denotes the smallest integer larger or equal to s.

• A + B

sr

is equal to A if s ≤ r and equal to A + B if s r.

• A ∨ B stands for max{A, B}.

• c.c. means “complex conjugate”.

• Ω denotes the fluid domain.

• S is the flat strip S = {(X, z) ∈

Rd × R, −1 z 0}.

• Γ is the boundary of Ω. We sometimes write Γ = (ζ, b), where ζ and b are

the surface and bottom parametrizations.

• Γ ⊂ Ht0+2(Rd)2 is the set of all ζ and b such that infRd (1+ εζ − βb) 0.

• ψ† is an explicit smoothing extension of ψ on S (see Notation 2.28).

• ψh is the “harmonic extension” of ψ (see Notation 2.33).

Matrices and vectors

• ez denotes the unit (upward) vector in the vertical direction.

• n always denotes the unit upward normal vector evaluated at the bound-

ary of the fluid domain.

• I always stands for the identity matrix (its size is usually obvious from

the context).

• Iμ,γ is the (d + 1) × (d + 1) diagonal matrix with diagonal (

√

μ, γ

√

μ, 1)

when d = 2 and (

√

μ, 1) when d = 1.

• Pμ,γ(Σ): see Equation (2.13).

• |v|, where v is a vector of

Rd+1

stands for the euclidean norm of v.

• |M|, where M is a (d + 1) × (d + 1) matrix is the matricial norm subor-

dinated to the euclidean norm.

Variables and standard operators

• X ∈

Rd

always refers to the horizontal variables; we sometimes write

X = (x, y) or X = (x1,x2) when d = 2 and X = x or X = x1 when d = 1.

• z always refers to the vertical variable (also sometimes denoted xd+1)

• ∇ =

(∂x,∂y)T

when d = 2 and ∇ = ∂x when d = 1.

• Δ = ∂x

2

+ ∂y

2

when d = 2 and Δ = ∂x

2

when d = 1.

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