The Westervelt--Pennes--Cattaneo model: local well-posedness and singular limit for vanishing relaxation time
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916234717560832 |
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| author | Benabbas, Imen Said-Houari, Belkacem |
| author_facet | Benabbas, Imen Said-Houari, Belkacem |
| contents | In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). Using the energy method together with a fixed point argument, we prove that our model is locally well-posed and does not degenerate under a smallness assumption on the pressure data in the Westervelt equation. In addition, we perform a singular limit analysis and show that the Westervelt--Pennes--Fourier model can be seen as an approximation of the Westervelt--Pennes--Cattaneo model as the relaxation parameter tends to zero. This is done by deriving uniform bounds of the solution with respect to the relaxation parameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_02881 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Westervelt--Pennes--Cattaneo model: local well-posedness and singular limit for vanishing relaxation time Benabbas, Imen Said-Houari, Belkacem Analysis of PDEs In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). Using the energy method together with a fixed point argument, we prove that our model is locally well-posed and does not degenerate under a smallness assumption on the pressure data in the Westervelt equation. In addition, we perform a singular limit analysis and show that the Westervelt--Pennes--Fourier model can be seen as an approximation of the Westervelt--Pennes--Cattaneo model as the relaxation parameter tends to zero. This is done by deriving uniform bounds of the solution with respect to the relaxation parameter. |
| title | The Westervelt--Pennes--Cattaneo model: local well-posedness and singular limit for vanishing relaxation time |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2304.02881 |