Local well-posedness of nonlinear dispersive equations on modulation spaces

Abstract

By using tools of time‐frequency analysis, we obtain some improved local well‐posedness results for the nonlinear Schrödinger, nonlinear wave and nonlinear Klein–Gordon equations with Cauchy data in modulation spaces ℳ0,sp,1.

The inclusion relation between Sobolev and modulation spaces

“…We remark that Theorem B with α = 2 is established in a more general form by [26]. We also remark that Bényi and Okoudjou [3] extends Theorem B to the case 1 p ∞, 0 < q ∞, 0 α 2 and the case n/(n + 1) p < 1, 0 < q ∞, α = 1, 2. For α > 2, we have a different type of boundedness: Miyachi,et al [16].)…”

Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation

“…The estimate above is attained by applying Minkowski's Inequality and the Hardy-LittlewoodSobolev inequality (5) to the estimate (48). The dual homogeneous estimates (44) follow by duality.…”

“…We record that hybrid spaces like the Wiener amalgam ones had appeared before as a technical tool in PDEs (see, e.g., Tao [37]). Notice that fixed-time estimates between modulation spaces in the case H = were first considered in [1] and, independently, in [3,4], and they were used to obtain well-posedness results on such spaces [2,5].…”

Remarks on Fourier multipliers and applications to the wave equation

“…The integral version of the problem (1) has the form u(t, ·) = K (t)u 0 + K (t)u 1 + BF (u), (2) where…”