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

2021/05/18 by Molinet, Luc, Tanaka, Tomoyuki · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.08731

Abstract

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≤ α≤ 2 , and that the nonlinear term is of the form ∂_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(\mathbbT) for s≥ 1-\fracα2(α+1). This leads to some global existence results above the energy space H^α/2(\mathbbT) , for α∈ [√(2),2].

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