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| Main Authors: | , |
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| Format: | Preprint |
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2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1912.04692 |
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| _version_ | 1866909072321675264 |
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| author | Abbrescia, Leonardo Enrique Wong, Willie Wai Yeung |
| author_facet | Abbrescia, Leonardo Enrique Wong, Willie Wai Yeung |
| contents | We prove global well-posedness of the initial value problem for a class of variational quasilinear wave equations, in one spatial dimension, with initial data that is not-necessarily small. Key to our argument is a form of quasilinear null condition (a "nilpotent structure") that persists for our class of equations even in the large data setting. This in particular allows us to prove global well-posedness for $C^2$ initial data of moderate decrease, provided the data is sufficiently close to that which generates a simple traveling wave.
We take here a geometric approach inspired by works in mathematical relativity and recent works on shock formation for fluid systems. First we recast the equations of motion in terms of a dynamical double-null coordinate system; we show that this formulation semilinearizes our system and decouples the wave variables from the null structure equations. After solving for the wave variables in the double-null coordinate system, we next analyze the null structure equations, using the wave variables as input, to show that the dynamical coordinates are $C^1$ regular and covers the entire space-time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_04692 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Geometric analysis of 1+1 dimensional quasilinear wave equations Abbrescia, Leonardo Enrique Wong, Willie Wai Yeung Analysis of PDEs 35L72, 35B35, 35A01, 35A30 We prove global well-posedness of the initial value problem for a class of variational quasilinear wave equations, in one spatial dimension, with initial data that is not-necessarily small. Key to our argument is a form of quasilinear null condition (a "nilpotent structure") that persists for our class of equations even in the large data setting. This in particular allows us to prove global well-posedness for $C^2$ initial data of moderate decrease, provided the data is sufficiently close to that which generates a simple traveling wave. We take here a geometric approach inspired by works in mathematical relativity and recent works on shock formation for fluid systems. First we recast the equations of motion in terms of a dynamical double-null coordinate system; we show that this formulation semilinearizes our system and decouples the wave variables from the null structure equations. After solving for the wave variables in the double-null coordinate system, we next analyze the null structure equations, using the wave variables as input, to show that the dynamical coordinates are $C^1$ regular and covers the entire space-time. |
| title | Geometric analysis of 1+1 dimensional quasilinear wave equations |
| topic | Analysis of PDEs 35L72, 35B35, 35A01, 35A30 |
| url | https://arxiv.org/abs/1912.04692 |