Well-posedness and numerical approximation of nonlinear conservation laws with hysteresis
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| Format: | Preprint |
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2026
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| _version_ | 1866912845407453184 |
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| author | Goatin, Paola Moreti, Stefan |
| author_facet | Goatin, Paola Moreti, Stefan |
| contents | This article studies the Cauchy problem for the scalar conservation law \[ \partial_t u + \partial_t w + \partial_x f(u) = 0, \] where $w(x,t) = [\mathcal{F}(u)(x,t)]$ is the output of a specific hysteresis operator, namely the Play hysteresis operator, and $f$ is a $\mathbf{C}^2$ convex flux function. The hysteresis operator models a rate-independent memory effect, introducing a specific non-local feature into the partial differential equation. We define a suitable notion of entropy weak solution and analyse in detail the Riemann problem. Furthermore, a Godunov-type finite volume numerical scheme is developed to compute approximate solutions. The convergence of the scheme for $\mathrm{BV}$ initial data provides the existence of an entropy weak solution. Finally, a stability estimate is established, implying the uniqueness and overall well-posedness of the entropy weak solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17403 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Well-posedness and numerical approximation of nonlinear conservation laws with hysteresis Goatin, Paola Moreti, Stefan Analysis of PDEs Numerical Analysis 35L65, 47J40, 65M08 This article studies the Cauchy problem for the scalar conservation law \[ \partial_t u + \partial_t w + \partial_x f(u) = 0, \] where $w(x,t) = [\mathcal{F}(u)(x,t)]$ is the output of a specific hysteresis operator, namely the Play hysteresis operator, and $f$ is a $\mathbf{C}^2$ convex flux function. The hysteresis operator models a rate-independent memory effect, introducing a specific non-local feature into the partial differential equation. We define a suitable notion of entropy weak solution and analyse in detail the Riemann problem. Furthermore, a Godunov-type finite volume numerical scheme is developed to compute approximate solutions. The convergence of the scheme for $\mathrm{BV}$ initial data provides the existence of an entropy weak solution. Finally, a stability estimate is established, implying the uniqueness and overall well-posedness of the entropy weak solution. |
| title | Well-posedness and numerical approximation of nonlinear conservation laws with hysteresis |
| topic | Analysis of PDEs Numerical Analysis 35L65, 47J40, 65M08 |
| url | https://arxiv.org/abs/2601.17403 |