The Modified Energy Method for Quasilinear Wave Equations of Kirchhoff Type

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.