The Modified Energy Method for Quasilinear Wave Equations of Kirchhoff Type

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 ${\dot{H}}_x^1\times L_x^2$ for a wide class of quasilinear wave equations of Kirchhoff type. This allows us to make the first steps towards small data $H_x^{5\mathrm{/}4}\times H_x^{1\mathrm{/}4}$ local well-posedness. In particular, we prove an enhanced lifespan for corresponding solutions depending only on the ${\dot{H}}_x^{5\mathrm{/}4}\times {\dot{H}}_x^{1\mathrm{/}4}$ norm of the initial data as well as the existence of weak solutions for $H_x^{5\mathrm{/}4}\times H_x^{1\mathrm{/}4}$ initial data, again small in ${\dot{H}}_x^1\times L_x^2$. In contrast to previous modified energy results, the nonlinearity in these models depends on an ${\dot{H}}_x^1$ 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.