Long time smooth solutions of 3D cubic quasilinear wave systems with small weakly decaying initial data
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2026
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| _version_ | 1866913045824929792 |
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| author | Gao, Mu Li, Jun Yin, Huicheng |
| author_facet | Gao, Mu Li, Jun Yin, Huicheng |
| contents | For the 3D cubic quasilinear wave system $\square_{c_i} u^i=G^i(u,\partial u,\partial^2u)=\displaystyle\sum_{\substack{0\le|α|,|β|,|γ|\le1 \\ 1\le j,k,l \le m}}g_{αβγ}^{ijkl}\partial^αu^j\partial^βu^k\partial^γu^l$, 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 $\displaystyle\sum_{|a|\le 5}\|\langle x\rangle^μ\partial^a_x(u _0,u_1)\|_{L^2}\le\varepsilon$ holds for any fixed constant $μ\in (0,1)$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17683 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long time smooth solutions of 3D cubic quasilinear wave systems with small weakly decaying initial data Gao, Mu Li, Jun Yin, Huicheng Analysis of PDEs For the 3D cubic quasilinear wave system $\square_{c_i} u^i=G^i(u,\partial u,\partial^2u)=\displaystyle\sum_{\substack{0\le|α|,|β|,|γ|\le1 \\ 1\le j,k,l \le m}}g_{αβγ}^{ijkl}\partial^αu^j\partial^βu^k\partial^γu^l$, 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 $\displaystyle\sum_{|a|\le 5}\|\langle x\rangle^μ\partial^a_x(u _0,u_1)\|_{L^2}\le\varepsilon$ holds for any fixed constant $μ\in (0,1)$, 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. |
| title | Long time smooth solutions of 3D cubic quasilinear wave systems with small weakly decaying initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.17683 |