Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations
Résumé
We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by $ i |\xi|^\alpha \xi $ , with $1 \le \alpha\le 2$, and that the nonlinear term is of the form $\partial_x f(u) $ where $f$ is the sum of an entire series with infinite radius of convergence. Under these conditions, we prove the unconditional local well-posedness of the Cauchy problem in $H(\T)$ for $s ≥ 1 − \frac{\alpha}{2(\alpha+1)}$ . This leads to some global existence results in the energy space $H^{\alpha/2}(\T)$, for $\alpha\in [\sqrt{2},2] $.