Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

Abstract

For both localized and periodic initial data, we prove local existence in classical energy space $H^s,s>\frac{3}{2}$, for a class of dispersive equations $u_t+(n(u){)}_x+L{u}_x=0$ with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators $L$ whose symbol is of temperate growth, and $n(\cdot )$ in local Sobolev space $H_{\mathrm{loc}}^{s+2}(\mathbb{R})$. In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semi-group methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on $\mathbb{R}$ to the periodic setting by using the difference-derivative characterization of Besov spaces.

References in corpus (3)

• Sharp Global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\T$
• On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type
• Global well-posedness of the Benjamin-Ono equation in H^1(R)

Cited by in corpus (2)

• A comparative study of bi-directional Whitham systems
• Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity