On the Cauchy Problem for the Generalized Korteweg-de Vries Equation

The Cauchy Problem of the Nonlinear Schrödinger Equations in ℝ1+1

“…We do not know if local existence holds for arbitrary large data iṅ B s k 2,∞ . The same situation occurs for the Navier-Stokes equations [2], the Korteweg-de Vries equations [17], the semilinear wave equations [23] and the 2D-cubic NLS equation [24].…”

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

“…(1.1) with small data in the Sobolev spacesḢ s κ for κ 4, s κ = 1/2 − 2/κ. Molinet and Ribaud [26] generalized their work to the case u 0 ∈Ḃ s κ 2,∞ . Due to M 2,1 ⊂Ḃ s κ 2,∞ if κ > 4, our Theorem 1.2 obtains new local and global well-posedness result for a class of rough Cauchy data.…”

The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

“…In [8], by constructing some special resolution spaces and using dyadic bilinear estimates as well as I-method, the authors proved that the KdV equation is globally well-posed for the initial data in H s (R) with s =− 3 4 . In [6], the authors consider the local and global Cauchy problem for the generalized Korteweg-de Vries equation…”