Local and Global Cauchy Problems for the Kadomtsev–petviashvili (Kp–ii) Equation in Sobolev Spaces of Negative Indices

Abstract. We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H s 1 ,s 2 (R 2 ) withand s 2 ≥ 0. On the H s 1 ,0 (R 2 ) scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation:α,, s 2 ≥ 0 and u 0 ∈ H s 1 ,s 2 (R 2 ). We deduce global well-posedness for s 1 ≥ 0, s 2 = 0 and real valued initial data.

Section: Introductionmentioning

confidence: 93%

“…We have the following refined bilinear Strichartz estimate which for the case α = 2 was already implicitly used in [18,15,19,16,17,6]. …”

Section: Now We Havementioning

confidence: 99%

Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations

Hadac

2008

“…In [8], Bourgain proved the local (and therefore global due to the L 2 conservation law) well-posedness of the KP-II equation with L 2 initial data. Local and global well-posedness for the KP-II equation with data below L 2 were obtained in [20,33,34,37]. Their results were generalized in [17] to the sharp results in the critical space.…”

Section: Introductionmentioning

confidence: 98%

Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation

Chen

Liu

Zhang

2012

Abstract. This paper is mainly concerned with the local low regularity of solutions and decay estimates of solitary waves to the rotation-modified Kadomtsev-Petviashvili (rmKP) equation. It is shown that with negative dispersion, the rmKP equation is locally well-posed for data in H s 1 ,s 2 (R 2 ) forand s 2 0, and hence globally well-posed in the space L 2 . Moreover, an improved result on the decay property of the solitary waves is established, which shows that all solitary waves of the rmKP equation decay exponentially at infinity.

“…We also refer to [24] and [19] for related well-posedness results for v 0 in negative exponent Sobolev spaces and simpler proofs.…”

Section: Proposition 6 For Anymentioning

confidence: 99%

“…We refer to Section 4 for a definition of these spaces and a precise well-posedness result (see Theorem 7). Following Bourgain, several authors have studied local and global well-posedness for initial data in negative exponent Sobolev spaces; see for example Takaoka and Tzvetkov [24] and Isaza and Mejía [19].…”

Section: Well-posedness For Mkp II 2449mentioning

confidence: 99%

Global well-posedness in the energy space for a modified KP II equation via the Miura transform

Kenig

Martel

2006

Abstract. We prove global well-posedness of the initial value problem for a modified Kadomtsev-Petviashvili II (mKP II) equation in the energy space. The proof proceeds in three main steps and involves several different techniques.In the first step, we make use of several linear estimates to solve a fourthorder parabolic regularization of the mKP II equation by a fixed point argument, for regular initial data (one estimate is similar to the sharp Kato smoothing effect proved for the KdV equation by Kenig, Ponce, and Vega, 1991).Then, compactness arguments (the energy method performed through the Miura transform) give the existence of a local solution of the mKP II equation for regular data.Finally, we approximate any data in the energy space by a sequence of smooth initial data. Using Bourgain's result concerning the global well-posedness of the KP II equation in L 2 and the Miura transformation, we obtain convergence of the sequence of smooth solutions to a solution of mKP II in the energy space.