Long time smooth solutions of 3D cubic quasilinear wave systems with small weakly decaying initial data

TL;DR

This paper proves long-time existence and scattering for small initial data solutions of 3D cubic quasilinear wave systems, extending known results to less restrictive decay conditions.

Contribution

It establishes almost global and global existence results for 3D cubic quasilinear wave systems with small initial data, using new weighted estimates and the strong Huygens' principle.

Findings

Existence of solutions up to exponential times for small data.

Global solutions with scattering for data with weighted decay.

Development of new weighted $L^ ext{infty}-L^2$ and Strichartz estimates.

Abstract

For the 3D cubic quasilinear wave system $\square_{c_i} u^i=G^i(u,\partial u,\partial^2u)=\displaystyle\sum_{\substack{0\le|\alpha|,|\beta|,|\gamma|\le1 \\ 1\le j,k,l \le m}}g_{\alpha\beta\gamma}^{ijkl}\partial^{\alpha}u^j\partial^{\beta}u^k\partial^{\gamma}u^l$, it is well known that global solution $u$ exists when the small smooth initial data $(u,\partial_tu)|_{t=0}$ $=(u_0(x), u_1(x))$ are compactly supported or decay rapidly at spatial infinity. However, when $(u_0, u_1)\in (H^{s+1}, H^s)$ with $s>\frac{5}{2}$ are small, it remains unknown whether $u$ exists globally or not. In this paper, we show that if $\|u_{0}\|_{H^{N+1}}+\|u_{1}\|_{H^N}\le\varepsilon$ ($N\ge 6$) is small, then the almost global solution $u$ exists in $[0, T_{\varepsilon}]$ with $T_{\varepsilon}\ge e^{C\varepsilon^{-1}}$ for the general $G(u,\partial u,\partial^2u)$ depending on $u$ and $T_{\varepsilon}\ge e^{C\varepsilon^{-2}}$ for the nonlinearity $G(\partial u,\partial^2u)$ independent of $u$, respectively. In addition, if $\displaystyle\sum_{|a|\le 5}\|\langle x\rangle^{\mu}\partial^a_x(u _0,u_1)\|_{L^2}\le\varepsilon$ holds for any fixed constant $\mu\in (0,1)$, then the solution $u$ exists globally and meanwhile the scattering property of $u$ is derived. Our main ingredients consist in establishing a series of new weighted $L^\infty-L^2$ estimates and Strichartz estimates based on the strong Huygens' principle for 3D linear wave equations.