A bilinear estimate with applications to the KdV equation

Strichartz Estimates for a Schrödinger Operator With Nonsmooth Coefficients

“…Before we go to the details of the presentation of the problem we considerin this paper, we would like to recall that the Scrödinger operator, with smooth variable coefficients, appears in many papers concerning questions on well-posedness and smoothing effect (see Doi [5] [4], Creg, Kappler and Strauss [3], Kapitanski and Safanov [9] [8], Kenig, Ponce and Vega [11] and Rolvung [14]). As mentioned above, in our work the coefficients are instead relatively rough.…”

Discrete Fourier restriction associated with KdV equations

“…It was proved by Colliander, Keel, Staffilani, Takaoka, and Tao [Colliander et al 2004], but our method is different. The method used by those authors is based on a rescaling argument and the bilinear estimates proved by Kenig, Ponce and Vega [Kenig et al 1996]. Our method is more straightforward and does not need the rescaling argument, the bilinear estimates, or the multilinear estimates in the earlier papers.…”

On one dimensional quantum Zakharov system

“…The key step is to derive non-linear estimates by extensive use of Strichartz inequalities which has its origin from Bourgain [5]. For one dimensional Zakharov system, Ginibre, Tsutsumi and Velo [8] established the local well-posedness result in low regularity spaces by adapting a method first proposed by Kenig, Ponce and Vega [14] to treat the Korteweg-de Vries equation which is a variant of Bourgain's method. Their method does not use Strichartz inequalities in the derivation of non-linear estimates, and relies instead on using Schwarz inequality cleverly followed by a direct estimation.…”

“…The conservation of mass comes from the imaginary part of (1a) and hence contains no contribution of n. The momentum J is not conservative due to the coupling of the density |E| 2 and ∂n ∂x . The O( 2 ) term of (14) and (15) shows the quantum effect of the Langmuir wave. Formally letting ε → 0 in (14) and (15) we have the hydrodynamical equations ∂ρ ∂t…”