On the Cauchy problem for gravity water waves

By Thomas Alazard, Nicolas Burq, Claude Zuily. 2012

235 citations recorded by the source corpus.

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of $$C^{3/2+\epsilon }$$C3/2+ϵ-class for some $$\epsilon >0$$ϵ>0 and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.

Subfields: Applied Mathematics, Mathematical Physics, Oceanography

View Paper: https://www.semanticscholar.org/paper/35f217eed2713548533b20d02c7aedd6e60081e7

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In Applied Mathematics

In Mathematical Physics

In Oceanography

Citations in the corpus

Two dimensional water waves in holomorphic coordinates II: global solutions (2014, unknown): “…au numéro : 43 e ($ 64) Abonnement Europe : 178 e, hors Europe : 194 e ($ 291) Des conditions spéciales sont accordées aux membres de la SMF. Secrétariat : Nathalie Christiaën Bulletin de la Société Mathématique de France Société Mathématique de France Institut Henri Poincaré,…”
Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, methodology): “For the sake of shortness, this is the only point that we do not prove in details in this paper (referring to [2] for the proof of a similar result in a case with similar difficulties).”
Global solutions and asymptotic behavior for two dimensional gravity water waves (2013, methodology): “For 0 ≤ ℓ ≤ s2 +N0 and 0 ≤ k ≤ s2 +N0 − ℓ we define (6.1) W k,(ℓ) = ( Zk ′ w(ℓ) ) 0≤k′≤k where w(ℓ) = Oph ( 〈ξ〉ℓ ) w, as in the preceding section.”
Time quasi-periodic gravity water waves in finite depth (2017, background): “Note that the profile η ∈ Hs0− 14 (Tx) is less regular than the velocity potential ψ ∈ H0 14 (Tx), as it happens for pure gravity waves, see [2].”
A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, background): “0]) and kr x;zvk X˙ 1 ([z 0;0]) F(kk Hs) kr xfk H ˙ 1 + kF 0k Y z 1;0]) + kr x;zvk X 1 2 ([z 0;0]) for some F: R+ !R+ depending only on (˙;s;h;z 0;z 1). REMARK 3.13.In fact, Proposition 3.16 in [3] assumes f2H˙(Rd). This comes from estimating vsolving @ zv+ T Av= w z2[z 0;0]; vj z=0 = f where w2X˙”