Sharp well-posedness for a coupled system of mKdV-type equations

Author(s)

Carvajal, Xavier ^[1] ; Panthee, Mahendra ^[2]

Affiliation

^[1] Univ Fed Rio de Janeiro, Inst Matemat, BR-21945970 Rio De Janeiro, RJ - Brazil

^[2] Univ Estadual Campinas, Dept Math, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: JOURNAL OF EVOLUTION EQUATIONS; v. 19, n. 4, p. 1167-1197, DEC 2019.
• Web of Science Citations: 0

Abstract

We consider the initial value problem associated with a system consisting modified Korteweg-de Vries-type equations [partial derivative(t)v + partial derivative(3)(x)v + partial derivative(x)(vw(2)) = 0, u(x, 0) = phi(x), partial derivative(t)w + alpha partial derivative(3)(x)w + partial derivative(x)(v(2)w) = 0, v(x, 0) = psi(x), and prove the local well-posedness results for given data in low regularity Sobolev spaces H-s (R) x H-s (R), s > -1/2, for 0 < alpha < 1. Our result covers the whole scaling subcritical range of Sobolev regularity contrary to the case alpha = 1, where the local well-posedness holds only for s >= 1/4. We also prove that the local well-posedness result is sharp in two different ways; namely, for s < - 1/2 the key trilinear estimates used in the proof of the local well-posedness theorem fail to hold, and the flow map that takes initial data to the solution fails to be C-3 at the origin. These results hold for alpha > 1 as well. (AU)

• FAPESP's process: 16/25864-6 - Nonlinear Evolution Equations of Dispersive Type
• Grantee: Mahendra Prasad Panthee
• Support Opportunities: Regular Research Grants