The Modified Energy Method for Quasilinear Wave Equations of Kirchhoff Type

TL;DR

This paper advances the analysis of quasilinear Kirchhoff-type wave equations by developing an improved modified energy method, enabling better lifespan estimates and existence results for small initial data in lower regularity spaces.

Contribution

It introduces a novel modified energy approach tailored for Kirchhoff-type equations with nonlinearities depending on the solution's gradient norm.

Findings

Proves an improved quintic energy estimate for small initial data.

Establishes enhanced lifespan depending on lower regularity norms.

Demonstrates existence of weak solutions for initial data in $H^{5/4}_x \times H^{1/4}_x$.

Abstract

In this paper, we use the modified energy method of Hunter, Ifrim, Tataru, and Wongto prove an improved quintic energy estimate for initial data small in $\dot H^1_x \times L^2_x$ for a wide class of quasilinear wave equations of Kirchhoff type. This allows us to make the first steps towards small data $H^{5/4}_x \times H^{1/4}_x$ local well-posedness. In particular, we prove an enhanced lifespan for corresponding solutions depending only on the $\dot H^{5/4}_x \times \dot H^{1/4}_x$ norm of the initial data as well as the existence of weak solutions for $H^{5/4}_x \times H^{1/4}_x$ initial data, again small in $\dot H^1_x \times L^2_x$. In contrast to previous modified energy results, the nonlinearity in these models depends on an $\dot H^1_x$ norm of the solution. This means a modified energy cannot be deduced algebraically by analyzing resonant interactions between wave packets since all spatial dependence is integrated out in the nonlinearity. Instead, the modified energy is determined as a Taylor series of incremental leading order terms.