Harmonic Analysis Method for Nonlinear Evolution Equations, I

Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra

“…Proof. All these statements are well-known and the interested reader may find a proof in [11,10,21]. For the proof of statements (1), (2) and (3), see [11,Theorem 12.2.2], [18], and [7, Corollary 1.1] respectively.…”

The nonlinear Schrödinger equations with harmonic potential in modulation spaces

“…We refer to [23] for a classical foundation of these spaces and [43,34] for some recent developments for nonlinear dispersive equations and the references therein.…”

“…On the other hand, there is tremendous ongoing interest to use modulation spaces M p,q (R d ) (see Definition 2.1 below) as a regularity class for the Cauchy data for nonlinear dispersive equations, see e.g., [1,2,31,4,8,3,5,43]. Wang-Zhao-Guo [1] proved local well-posedness for nonlinear Schrödinger equations (NLS) and Navier-Stokes equations in M 2,1 (R d ).…”

“…The definition of the modulation space given above, is independent of the choice of the particular window function. See [23, Proposition 11.3.2(c)],[43]. (2) For a pair of functions f and the Gaussian Φ 0…”

Strongly singular integrals along curves on α-modulation spaces

“…It is the classical modulation space. Its related decomposition is called uniform decomposition; see [6, 7] and [14] for details. In order to define the Besov spaces, we introduce the dyadic decomposition.…”