Well-posedness and scattering for the KP-II equation in a critical space

Abstract

The Cauchy problem for the Kadomtsev-Petviashvili-II equation (ut + uxxx + uux)x + uyy = 0 is considered. A small data global wellposedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev spaceḢ − 1 2 ,0 (R 2 ) is derived. Additionally, it is proved that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous spaceḢ − 1 2 ,0 (R 2 ) and in the inhomogeneous space H − 1 2 ,0 (R 2 ), respectively.2000 Mathematics Subject Classification. 35Q55. Key words and phrases. Kadomtsev-Petviashvili-II equation, scale invariant space, wellposedness, scattering, bilinear estimates, bounded p-variation. 2 , the scaling symmetry suggests ill-posedness of the Cauchy problem (cp. also [10] Theorem 4.2), we will prove global well-posedness and scattering inḢ − 1 2 ,0 (R 2 ) for small initial data, see Theorem 1.1 and Corollary 1.3, and local well-posedness in H − 1 2 ,0 (R 2 ) andḢ − 1 2 ,0 (R 2 ) for arbitrarily large initial data, see Theorem 1.2.After J. Bourgain [2] established global well-posedness in L 2 (T 2 ) and L 2 (R 2 ) by the Fourier restriction norm method and opened up the way towards a low regularity well-posedness theory, there has been a lot of progress in this line of research. We will only mention the most recent results and also refer to the references therein. Local well-posedness in the full sub-critical range s > − 1 2 was obtained by H. Takaoka [19] in the homogeneous spaces and by the first author [7] in the inhomogeneous spaces. Global well-posedness for large, real valued data in H s,0 (R 2 ) has been pushed down to s > − 1 14 by P. Isaza -J. Mejía [8]. The first main result of this paper is concerned with small data global wellposedness inḢ − 1 2 ,0 (R 2 ). For δ > 0 we definė B δ := {u 0 ∈Ḣ − 1 2 ,0 (R 2 ) | u 0 Ḣ − 1 2 ,0 < δ}, and obtain the following:

Global well-posedness and scattering for small data for the 2D and 3D KP-II Cauchy problem

“…Then the following theorem holds. ) were also obtained in [4]. Some recent results on the two dimensional KP-II equation can be found in [9].…”

“…I report on work with M. Hadac and Sebastian Herr [5] on the two dimensional case and with junfeng Li on the three dimensional case.…”

“…We base our construction of the solution space on the space V 2 of functions of bounded 2 variation V 2 adapted to the dimensional KP-II equation. We refer to [4] for their properties. The x variable plays a prominent role and we use a Littlewood-Paley decomposition in the x variables: for λ ∈ 2 Z we define u λ by its Fourier transform with respect to x,…”

“…The integral on the right hand is a suggestive notation which takes some care to make sense of, see [5]. There are natural embeddings for p < q…”

“…It has often good regularity properties and it allows a transparent study of resonances. For critical problems one would want to use X [5] for the definition and explain here some properties. They 'interpolate' between the extremal Besov spaces:…”