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

2024/12/16 by Simão Correia, Correia, Simão, Pedro Leite +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2412.11808

Abstract

We consider a general nonlinear dispersive equation with monomial nonlinearity of order k over 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 k₀ and in dimension d₀, we prove a sharp local well-posedness result in H^s(ℝ^d) for any k≥ k₀ and d≥ d₀. 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] (doi.org/10.1137/23M156923X). The result is then applied to generalized Korteweg-de Vries, Zakharov-Kuznetsov and nonlinear Schrödinger equations.

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