Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$

Abstract

We prove the local well-posedness of a 1-D quadratic nonlinear Schrödinger equationiut + uxx = u 2 in H s (R) for s ≥ −1 and ill-posedness below H −1 . The same result for another quadratic nonlinearity u 2 was given by I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal. 233 (2006), but the function space of solutions depended heavily on the special property of the nonlinearity u 2 . We construct the solution space suitable for the nonlinearity u 2 .1123 1124 NOBU KISHIMOTO of the equation with this nonlinearity, since the L 2 (R) norm is conserved by this invariance. Thus it would be possible to show the well-posedness under some negative Sobolev regularity if we choose the nonlinearity such that the equation is not Galilean invariant, and that the critical scaling regularityIndeed, in [9] Kenig, Ponce and Vega obtained the local well-posedness of (1) with the quadratic nonlinearities N (u) = u 2 , u 2 and |u| 2 for the regularity s > −3/4, s > −3/4 and s > −1/4, respectively. To show such low-regularity wellposedness results, the useful function space of solutions is the space X s,b introduced by Bourgain [3], defined for s, b ∈ R as the completion of the Schwartz class S x,t (R 2 ) with respect to the norm v X s,b := ξ s τ − ξ 2 b v L 2 ξ,τ

Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation

“…Of course the second modification with the same weight factor will not work when we consider other nonlinearities. Let us consider another nonlinearity c 2 u 2 in 1D, which was treated in [7] to establish the same well-posedness and ill-posedness results. It turns out that in this situation the Fourier transform of a solution has high density near the parabola {τ = − 1 2 ξ 2 } as well as the characteristic curve of free Schrödinger equation {τ = ξ 2 }.…”

“…Fortunately, this type of estimate will not be indispensable if we take advantage of the scaling invariance to establish the well-posedness. In this situation, however, the uniqueness result will be obtained only in a weak sense; the solution is the unique strong C 0 t (H s x )-limit of smooth solutions and unique only in a small ball of the function space (see [1,2,7]). In this article we first establish the well-posedness by the scaling and the iteration, and then extend the uniqueness results to the whole function space in which we work (this is the same uniqueness assertion obtained in [6,4]) using the argument of Muramatu and Taoka [8].…”

“…One clue to this problem is the weight with "variable exponent," which is the main novelty of the present paper. Note that each of the weight factor in [2,7] has a constant power of ξ near the reflected parabola {τ = −ξ 2 }; w ∼ ξ 20 for u 2 , w ∼ ξ 5/8 for u 2 (see Fig. 3).…”

“…The weight w(ξ, ·) in the cases of u 2[2] and u 2[7] for a fixed ξ 1. The vertical scale is logarithmic, while the horizontal is normal.…”

Improved local well-posedness for the periodic “good” Boussinesq equation

“…With this assignment for h ε , we will certainly not get the nice exact identity (12). However, we get something similar (up to an error term), which is good enough for our purposes.…”