Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems

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Time quasi-periodic gravity water waves in finite depth (2017, unknown): “(2.36) We now recall the pseudo-differential norm introduced in Definition 2.11 in [21] (inspired by Métivier [49], chapter 5), which controls the regularity in (ϕ, x), and the decay in ξ, of the symbol a(ϕ, x, ξ) ∈ Sm, together with its derivatives ∂βξ a ∈ Sm−β, 0 ≤ β ≤ α, in the Sobolev norm ‖ ‖s”
Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, methodology): “Classical results from paradifferential calculus (see [6, 16, 39]) would allow us to estimate them provided that we work in nonhomogeneous Sobolev spaces.”
The Cauchy problem on large time for surface waves Boussinesq systems (2012, methodology): “8 in [22], one could prove that {U ν}ν>0 converges to a unique U in C([0, T ′];Xs ε3 (R2)) as ν → 0 and such a U ∈ C([0, T ′];Xs ε3 (R2)) ∩ C1([0, T ′];Xs−1 ε3 (R2)) is the unique solution of (2.”
On the Cauchy problem for gravity water waves (2012, methodology): “We shall use quantitative results from [39] about operator norms estimates in symbolic calculus.”
A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, unknown): “[12, 45, 52, 3]).”