Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities

Abstract: We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local well-posedness theories. More precisely, assuming that a certain positive multiplier estimate holds at order $k_0$ and in dimension $d_0$, we prove a sharp local well-posedness result in $H^s(\mathbb{R}^d)$ for any $k\ge k_0$ and $d\ge d_0$. Moreover, we give an explicit bound on the gain of regularity observed in the difference between the linear and nonlinear solutions, confirming the conjecture made in CorreiaOliveiraSilva24. The result is then applied to generalized Korteweg-de Vries, Zakharov-Kuznetsov and nonlinear Schr\"odinger equations.