The global solvability of the Kirchhoff equation with Sobolev data

TL;DR

This paper proves the long-standing open problem of global existence of solutions to the Kirchhoff equation with Sobolev initial data, using new estimates and fixed point methods, also extending results to Gevrey spaces.

Contribution

It establishes the first proof of global solvability for the Kirchhoff equation with Sobolev initial data, a problem unsolved for over eighty years.

Findings

Global existence of solutions with Sobolev initial data

New uniform estimate for linear equations with time-dependent coefficients

Extension of results to Gevrey spaces

Abstract

We consider linear and non-linear Cauchy equations in the context of Sobolev spaces. In particular, we show the global existence of solutions to the Kirchhoff equation with initial data in the Sobolev spaces, a problem that has been open for more than eighty years. Our proof is based on a new uniform estimate for solutions to the linear equation with time-dependent coefficient and a fixed point argument. As an immediate consequence of our result, the global solvability of the Kirchhoff equation with initial data in the Gevrey spaces is also obtained.