Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations

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 $i |ξ|^αξ$, with $1\le α\le 2$, 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 $H^{s}(\mathbb{T})$ for $s\ge 1-\fracα{2(α+1)}$. This leads to some global existence results above the energy space $H^{α/2}(\mathbb{T})$, for $α\in [\sqrt{2},2]$.

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