We provide a unified viewpoint on two ill-posedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi-Mizohata condition. Our approach is based on a robust energy-and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydrodynamics. Concretely, the main results in this paper concern strong ill-posedness of the Cauchy problem (e.g., nonexistence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schr & ouml;dinger-and KdV-type equations, including the Hunter-Smothers equation, K (m, n) models of Rosenau-Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of the combination of two effects: degenerate dispersion-which is a property of the principal term in the presence of degenerating coefficients - and the evolution of the amplitude governed by the Takeuchi-Mizohata condition - which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical L2-ill-posedness result by Mizohata for linear variable-coefficient Schr & ouml;dinger equations with failed Takeuchi-Mizohata condition.
We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in R-3, which converges to a plane with multiplicity 2 as time approaches infinity.
We identify a new way to divide the delta-neighborhood of surfaces M C DI83 into a finitely overlapping collection of rectangular boxes S. We obtain a sharp (& ell;2, Lp) decoupling estimate using this decomposition for the sharp range of exponents 2 <= p <= 4. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line.
In this paper, we prove that the first eigenfunction of the Laplacian for a horo-convex domain Ω⊂ℍ^n is super log-concave when diam(Ω) is not large. Our result is optimal in the sense that there are counterexamples
We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. This estimate is crucial in the integrability analysis mean-field interactions induced by a singular kernel. The tensorial operator arises from a reformulation of integrability. Additionally, for smooth periodic solutions of the reformulated system, we derive a stability result.
