Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations

Abstract

Sufficient and necessary conditions for the embeddings between Besov spaces B s 1 p,q and modulation spaces M s 2 p,q are obtained. Moreover, using the frequency-uniform decomposition method, we study the Cauchy problem for the generalized BO, KdV and NLS equations, for which the global well-posedness of solutions with the small rough data in certain modulation spaces M s 2,1 is shown.

Embedding relations between local Hardy and modulation spaces

“…It will be interesting to compare our result with the following result of Wang-Huang [19 is the Besov space. Let 0 < p ≤ 1.…”

Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces

“…The modulation space M s 2,1 can be equivalently defined in the following way (cf. [40,41,42]): 6) where k = 1 + |k|. For any Banach function spaces X defined in R × R n , we will use the function spaces 1,s (X) ,…”

“…To answer this question, modulation spaces seem very useful tools. Recalling the sharp embedding (see [35,38,40]) M sc 2,1 ⊂ H sc , M s 2,1 ⊂ H sc if s < s c , it will be interesting if we can establish the global well-posedness for the initial data in M s 2,1 with s < s c . In fact, there are some recent works which have been devoted to the study of the well-posedness for a class of nonlinear evolution equation in modulation spaces; cf.…”

“…In fact, there are some recent works which have been devoted to the study of the well-posedness for a class of nonlinear evolution equation in modulation spaces; cf. [2,3,5,7,8,9,20,24,25,26,31,39,40,41,42]. Our main goal of this paper is to study the global well-posedness of 4NLS in modulation spaces M 3+1/2 2,1…”

Estimates for unimodular Fourier multipliers on modulation spaces

“…Recall that supp ψ ⊂ (−1, 1) n . Therefore, from (27), (28), and Lemma 4.1 with Ω = (−1, 1) n , we obtain…”