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:

A refined trilinear Strichartz estimate for solutions to the Schrödinger equation on the flat rational torus T 3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic Nonlinear Schrödinger Equation in H s (T 3 ) for all s ≥ 1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L 2 × H −1/2 × H −3/2 . This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L 2 -based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schrödinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data -a result which is false for the cubic nonlinear Schrödinger equation in dimension two -and it is optimal because Glangetas-Merle's solutions blow up at that time. AMS classification scheme numbers: 35Q551 2 × L 2 × H −1 , one-half a derivative away from the result in Theorem 1.1.One motivation for considering the space L 2 × H −1/2 × H −3/2 is the connection to the cubic nonlinear Schrödinger equation in two spatial dimensions i∂ t u + ∆u + |u| 2 u = 0.(1.2) Consider the Zakharov system with wave speed λ > 0:On the 2d Zakharov system 3 Then formally (1.3) converges to (1.2) as λ → ∞ in the sense that for fixed initial data u λ → u, where (u λ , n λ ) solves (1.3) and u solves (1.2) with the same initial data. Rigorous results of this type in a high regularity setting were obtained by Schochet-Weinstein [21], Added-Added [1], Ozawa-Tsutsumi [20], see also the recent work by Masmoudi-Nakanishi [18] on this issue in 3d.Local well-posedness in L 2 of (1.2) was obtained by Cazenave-Weissler [7]. However, in this version of well-posedness, the time interval of existence depends directly upon the initial data, not just on the L 2 norm of the initial data. Indeed, via the pseudoconformal transformation, it can be shown that a result giving the maximal time of existence in terms of the L 2 norm alone is not possible ‡.Remark 3. Our result gives local well-posedness of (1.3) with a time of existence depending on the L 2 norm of u 0 , but also on the H −1/2 × H −3/2 norm of the wave data (n 0 , n 1 ) as well as the wave speed. Indeed, this claim follows by combining the rescalingand Theorem 1.1. However, note that the lower bound on the maximal time of existence obtain by this method tends to zero as the wave speed goes to infinity.Global well-posedness of (1.1) is known for initial data in the energy space [6,13]; see also [10] regarding bounds on higher order Sobolev norms. Recently, the imposed regularity assumption has been slightly weakened in [11]. Here, Q is the ground state solution for (1.2), i.e. Q is the unique solution to − Q + ∆Q + |Q| 2 Q = 0, Q > 0, Q(x) = Q(|x|), Q ∈ S(R 2 ) (1.4) of minimal L 2 mass. This gives rise to a blow-up solution of (1.2) by the pseudoconformal transformation. This idea is exploited in [14], where Glangetas-Merle construct a family of blow-up solutions for (1.1) of the formfor parameters θ ∈ S 1 , T > 0, and ω ≫ 1, such that P ω ∈ H 1 is smooth and radially symmetric, N ω ∈ L 2 is a radially symmetric Schwartz function, and (P ω , ...

The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for data u 0 in the space b H s r (T), defined by the normsis shown in the parameter range s ≥ 1 2 , 2 > r > 4 3 . The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method.2000 Mathematics Subject Classification. 35Q55.

Uniform L 2 -estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.