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

By Martin Hadac, Sebastian Herr, Herbert Koch. 2007

312 citations recorded by the source corpus.

Abstract The Cauchy problem for the Kadomtsev–Petviashvili-II equation ( u t + u x x x + u u x ) x + u y y = 0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space H ˙ − 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 space H ˙ − 1 2 , 0 ( R 2 ) and in the inhomogeneous space H − 1 2 , 0 ( R 2 ) , respectively.

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Citations in the corpus

Strichartz estimates for Schrödinger equations on irrational tori (2013, methodology): “Furthermore, by applying the well-posedness theory involving the U- and V -spaces developed by Tataru, Koch, and their collaborators [18, 21, 22, 30], we prove some critical local well-posedness.”
Almost sure well-posedness for the periodic 3D quintic nonlinear Schr\"odinger equation below the energy space (2013, background): “ following interpolation result at the end of Section 8 to obtain bounds in terms of the Xs spaces from those in U2 ∆H s and Up ∆H s just as in in [23] The proof relies solely on linear interpolation [22, 23]. Proposition 4.9 (Proposition 2.20 in [22] and Lemma 2.4 [23]). Let q1,...qm >2 where m=”
Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, methodology): “They have been successfully used for instance in [27], [26], [4], and [10].”
On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, methodology): “ssible pairs (q,r). 3. Function spaces and their properties In this section, we go over the basic deﬁnitions and properties of the Up- and Vp-spaces, developed byTataru, Koch, and their collaborators [41, 29, 30]. Thesespaces have been very eﬀective in establishing well-posedness of various dispersi”
Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb{T}^3)$ (2010, unknown): “kV p ∆H s = ke−it∆uk V p(R,Hs) (9) are ﬁnite. Spaces of this type have been succesfully used as replacements for Xs,b spaces which are still eﬀective at critical scaling, see for instance [12], [13], [9]. In the context of this article this would suggest that spaces of the form Up ∆H 1 and Vp ∆H 1 s”
WIENER RANDOMIZATION ON UNBOUNDED DOMAINS AND AN APPLICATION TO ALMOST SURE WELL-POSEDNESS OF NLS (2014, background): “5 can also be proven with the variants of the X-spaces adapted to the U- and V -spaces introduced by Koch, Tataru, and their collaborators [36, 29, 30].”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, unknown): “x2)ukDUp ∆ . (2.24) Finally, there is the duality relation (DUp ∆) ∗ = Vp ′ ∆ . (2.25) These spaces are also closed under truncation in time. χI : U p ∆ → U p ∆, χI : V p ∆ → V p ∆. (2.26) Proof: See [19]. Lemma 2.4 Suppose J = I1 ∪I2, I1 = [a,b], I2 = [b,c], a ≤ b ≤ c. kukp Up ∆(J×R) ≤ kukp Up ∆(I1”
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “See [46] for a general description of these spaces.”
Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, background): “(55) combined with Proposition 3.1 lead quickly to the frequency envelope control in (67) via a bootstrap argument. 1For further information on the Up and Vp spaces we refer the reader to [11], [12], [5] 8 JACOB STERBENZ AND DANIEL TATARU Section 11 contains the proof of the data truncation result i”