A first order in time wave equation modeling nonlinear acoustics
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910413562576896 |
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| author | Kaltenbacher, Barbara Lehner, Pascal |
| author_facet | Kaltenbacher, Barbara Lehner, Pascal |
| contents | In this paper we focus on a small amplitude approximation of a Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third and higher order terms with respect to certain small parameters, we obtain a first order in time system containing linear and quadratic pressure and velocity terms.
Subsequently, the well-posedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the well-posedness of a linearized equation using energy estimates and then the well-posedness of the nonlinear system using a Newton-Kantorovich type argument. Based on this, we also obtain global in time well-posedness for small enough data and exponential decay. This is in line with semigroup results for a linear part of the system that we provide as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A first order in time wave equation modeling nonlinear acoustics Kaltenbacher, Barbara Lehner, Pascal Analysis of PDEs 35L60, 35L50 In this paper we focus on a small amplitude approximation of a Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third and higher order terms with respect to certain small parameters, we obtain a first order in time system containing linear and quadratic pressure and velocity terms. Subsequently, the well-posedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the well-posedness of a linearized equation using energy estimates and then the well-posedness of the nonlinear system using a Newton-Kantorovich type argument. Based on this, we also obtain global in time well-posedness for small enough data and exponential decay. This is in line with semigroup results for a linear part of the system that we provide as well. |
| title | A first order in time wave equation modeling nonlinear acoustics |
| topic | Analysis of PDEs 35L60, 35L50 |
| url | https://arxiv.org/abs/2404.11250 |