Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, II
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arXiv
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2025
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| author | Hou, Fei Yin, Huicheng |
| author_facet | Hou, Fei Yin, Huicheng |
| contents | In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy problem of 2-D quadratically quasilinear wave equations with null conditions. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. When the 2-D quadratic nonlinearity admits a special $Q_0$ type null form, the global small solution is shown in our previous article [Hou Fei, Yin Huicheng, Yuan Meng, Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, arXiv:2411.06984]. In the present paper, we now solve this open problem through proving the global existence of small solutions to 2-D general quasilinear wave equations with null conditions in exterior domains. Our proof procedure is based on finding appropriate divergence structures of quasilinear wave equations under null conditions, introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity and deriving some new precise pointwise spacetime decay estimates of solutions and their derivatives. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_03348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, II Hou, Fei Yin, Huicheng Analysis of PDEs In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy problem of 2-D quadratically quasilinear wave equations with null conditions. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. When the 2-D quadratic nonlinearity admits a special $Q_0$ type null form, the global small solution is shown in our previous article [Hou Fei, Yin Huicheng, Yuan Meng, Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, arXiv:2411.06984]. In the present paper, we now solve this open problem through proving the global existence of small solutions to 2-D general quasilinear wave equations with null conditions in exterior domains. Our proof procedure is based on finding appropriate divergence structures of quasilinear wave equations under null conditions, introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity and deriving some new precise pointwise spacetime decay estimates of solutions and their derivatives. |
| title | Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, II |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.03348 |