Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation

Abstract

In this article we establish the bilinear estimates corresponding to the 1D and 2D NLS with a quadratic nonlinearity c 1 u 2 + c 2 u 2 , which imply the local well-posedness of the Cauchy problem in H s for s −1 in the 1D case and for s > −1 in the 2D case. This is a continuation of our study [N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity u 2 , Commun. Pure Appl. Anal. 7 (2008) 1123-1143] on the 1D NLS with nonlinearity cu 2 . Previous papers by Kenig, Ponce and Vega, and Colliander, Delort, Kenig and Staffilani established local well-posedness for s > −3/4 in 1D and in 2D, respectively, and when the nonlinearity is restricted to cu 2 , papers by Bejenaru and Tao, and Bejenaru and De Silva improved these results to s −1 in 1D and s > −1 in 2D. The bilinear estimate for 2D also yields an improvement on the growth rate of Sobolev norms of finite energy global-in-time solutions to the 2D cubic NLS.

A remark on norm inflation for nonlinear Schrödinger equations

“…Their argument is based on the power series expansion of the solution, and they proved ill-posedness by observing that high-to-low frequency cascades break the continuity of the first nonlinear term in the series. A similar dichotomy was shown for other quadratic nonlinearitiesū 2 , uū in [25,26] by employing the idea of [2].…”

“…Their argument is based on the power series expansion of the solution, and they proved ill-posedness by observing that high-to-low frequency cascades break the continuity of the first nonlinear term in the series. A similar dichotomy was shown for other quadratic nonlinearitiesū 2 , uū in [25,26] by employing the idea of [2].Later, Iwabuchi and Ogawa [20] considered the nonlinearity u 2 ,ū 2 in R, R 2 and refined the idea of [2] to prove ill-posedness in the sense of NI s for s < −1 in R and s ≤ −1 in R 2 . In particular, in the two-dimensional case they could complement the local well-posedness result in H s (R 2 ), s > −1, which had been obtained in [25].…”

“…A similar dichotomy was shown for other quadratic nonlinearitiesū 2 , uū in [25,26] by employing the idea of [2].Later, Iwabuchi and Ogawa [20] considered the nonlinearity u 2 ,ū 2 in R, R 2 and refined the idea of [2] to prove ill-posedness in the sense of NI s for s < −1 in R and s ≤ −1 in R 2 . In particular, in the two-dimensional case they could complement the local well-posedness result in H s (R 2 ), s > −1, which had been obtained in [25]. Note that the original argument of [2] is not likely to yield norm inflation phenomena nor discontinuity of the solution map at the threshold regularity such as s = −1 in the above R 2 case.…”

Some non-existence results for the semilinear Schrödinger equation without gauge invariance

“…in H s (R d ) with s > − 1 8 was obtained in [21]. On the other hand, ill-posedness to (NLS) has not been less understood, though in the case F (z) = c 1 z 2 + c 2z 2 for some c 1 , c 2 ∈ C\{0}, ill-posedness results were studied in [1,9,37]. For more information about (NLS) with F (z) = |z| p , see [15,[23][24][25]42,43].…”

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

“…where F is a bilinear form, which contains expressions in the form u 2 , uū,ū 2 . Recall that Kenig, Ponce and Vega, [11] have established the local well-posedness in H −1/4+ (R 1 ) for (2), while later Kishimoto-Tsugava, [14] (see also [12], [13]) have established the sharpness of this result on the line (when the nonlinearity is uū).…”