Global well-posedness of partially periodic KP-I equation in the energy space and application

Abstract

In this article, we address the Cauchy problem for the KP-I equationfor functions periodic in y. We prove global well-posedness of this problem for any data in the energy spaceWe then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.From the work of Benjamin [2], we know that these solutions are orbitally stable in H 1 (R) under the flow generated by the KdV equation (1.2), meaning that every solution of (1.2) with initial data close to Q c in H 1 (R) remains close in H 1 (R) to the Q c -orbit (under the action of translations) at any time t > 0.Looking at (1.1), we see that every solution of the KdV equation (1.2) is a solution of the KP equations (1.1), seen as a function independent of y. It is then a natural question to ask whether Q c is orbitally stable or unstable under the flow generated by (1.1). In order to do so, we first need a global well-posedness theory for (1.1) in a space containing Q c . In particular, this rules out any well-posedness result in an anisotropic Sobolev space H s1,s2 (R 2 ). A more suited space to look for is the energy space for functions periodic in y :where T = R/2πZ. Indeed, due to the Hamiltonian structure of (1.1), the mass M(u)(t) := R×T u 2 (t, x, y)dxdy (1.4) and the energy E(u)(t) := R×T (∂ x u) 2 (t, x, y) + (∂ −1 x ∂ y u) 2 (t, x, y) −

Well-posedness for a two-dimensional dispersive model arising from capillary-gravity flows

“…Now we collect some basic properties of the spaces X b N and F b N (T ). These results have been deduced in different contexts in [16,22,42,41,48] for instance.…”

On the Cauchy problem for the periodic fifth-order KP-I equation

“…These estimates are the analogous of those proved in [19, Subsections 2.1& 2.2] in the context of the bilinear estimate in standard Bourgain spaces. The proof is very similar to that one of [16,Proposition 5.5]. First, we split u 1 and u 2 depending on the value of m i on an M 3 scale, meaning…”

“…We collect here some basic properties of the spaces X M , F(T ) and N(T ). The proof of these results can be found e.g in [7,4,10,16]. First, for any f M ∈ X M , we have…”

“…The adaption [22] in the periodic setting revealed a logaritheoremeic divergence in the energy estimate due to a bad frequency interaction in the resonant set, establishing therefore a local well-posedness result in the Besov space B 1 2,1 (T 2 ) which is strictly larger than the natural energy space. To overcome this difficulty and recover a global well-posedness result in the energy space, one can look for a better dispersion effect by either removing the assumption of periodicity in one direction [16], or studying higher-order models.…”

Local wellposedness of the modified KP-I equations in periodic setting with small initial data

“…(3) = {φ ∈ L 2 : (1 + ξ 2 + n 2 ξ 2 ) φ(ξ, n) L 2 ξ,n < ∞} and Robert [27] proved global well-posedness in the first energy space Z 1 (3) = {φ ∈ L 2 : (1 + ξ + n ξ ) φ(ξ, n) L 2 ξ,n < ∞}. For our case of the third order partially periodic modified KP-I we prove the following theorem Theorem 1.1.…”