One-sided H\"older regularity of global weak solutions of negative order dispersive equations

TL;DR

This paper establishes the global existence, uniqueness, and stability of entropy solutions for negative order dispersive equations with initial data in certain function spaces, and introduces one-sided Hölder regularity that decays over time.

Contribution

It extends the solution concept to all L^2 initial data and demonstrates decay properties of solutions, providing bounds on classical solution lifespan.

Findings

Global existence and uniqueness of entropy solutions

Extension of solutions to all L^2 initial data

Decay of Hölder coefficients over time

Abstract

We prove global existence, uniqueness and stability of entropy solutions with $L^2\cap L^\infty$ initial data for a general family of negative order dispersive equations. It is further demonstrated that this solution concept extends in a unique continuous manner to all $L^2$ initial data. These weak solutions are found to satisfy one sided H\"older conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.