On the Initial Value Problem for Hyperbolic Systems with Discontinuous Coefficients
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909554568069120 |
|---|---|
| author | Joseph, Kayyunnapara Divya |
| author_facet | Joseph, Kayyunnapara Divya |
| contents | Hyperbolic systems of the first and higher-order partial differential equations appear in many multiphysics problems. We will be dealing with a wave propagation problem in a piece-wise homogeneous medium. Mathematically, the problem is reduced to analyzing two systems of partial differential equations posed on two domains with a common boundary. The differential equations may not be satisfied on the boundary (or part of the boundary), but some interface conditions are satisfied. These interface conditions depend on a specific physical problem. We aim to prove the existence and regularity of the solution for the case of hyperbolic systems of first-order equations with different domains separated by a hyperplane, where we need to formulate the interface conditions. We do this for the initial value problem in 1D-space variable when the coefficient matrix has discontinuity on $m$ lines. More specifically, we find explicit solutions to the case when the coefficient matrix is piecewise constant with a discontinuity along $1$ line or $2$ lines. We also prove the existence of solution to the general initial value problem. We then formulate the weak solution of initial value problem for the corresponding symmetric hyperbolic system in $n $D-space variables with interface conditions, get the energy estimates for this system, and prove the existence of solution to the system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15100 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Initial Value Problem for Hyperbolic Systems with Discontinuous Coefficients Joseph, Kayyunnapara Divya Analysis of PDEs 2020 Mathematics Subject Classification. 35R05, 35L45, 35D30 Hyperbolic systems of the first and higher-order partial differential equations appear in many multiphysics problems. We will be dealing with a wave propagation problem in a piece-wise homogeneous medium. Mathematically, the problem is reduced to analyzing two systems of partial differential equations posed on two domains with a common boundary. The differential equations may not be satisfied on the boundary (or part of the boundary), but some interface conditions are satisfied. These interface conditions depend on a specific physical problem. We aim to prove the existence and regularity of the solution for the case of hyperbolic systems of first-order equations with different domains separated by a hyperplane, where we need to formulate the interface conditions. We do this for the initial value problem in 1D-space variable when the coefficient matrix has discontinuity on $m$ lines. More specifically, we find explicit solutions to the case when the coefficient matrix is piecewise constant with a discontinuity along $1$ line or $2$ lines. We also prove the existence of solution to the general initial value problem. We then formulate the weak solution of initial value problem for the corresponding symmetric hyperbolic system in $n $D-space variables with interface conditions, get the energy estimates for this system, and prove the existence of solution to the system. |
| title | On the Initial Value Problem for Hyperbolic Systems with Discontinuous Coefficients |
| topic | Analysis of PDEs 2020 Mathematics Subject Classification. 35R05, 35L45, 35D30 |
| url | https://arxiv.org/abs/2408.15100 |