On scalar nonlinear balance laws with singular nonlocal sources
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913136886415360 |
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| author | Ftaka, Evangelia Nguyen, Khai T. |
| author_facet | Ftaka, Evangelia Nguyen, Khai T. |
| contents | We investigate one-dimensional scalar balance laws with singular convolution-type source terms. Under appropriate convexity and kernel assumptions, we establish the global existence of entropy weak solutions in ${\bf L}^2(\mathbb{R})$, together with two partial uniqueness results, in the ${\bf L}^2$-periodic setting and non-periodic setting with ${\bf L}^1(\mathbb{R})$ kernel. In the ${\bf L}^1$-kernel case, the characteristic speed satisfies an Oleinik-type estimate, and entropy weak solutions possess locally bounded fractional variation for all positive times. Furthermore, we derive a simple criterion characterizing local smoothness and wave breaking of solutions, which, in particular, includes both the Burger-Poisson and the Burgers-Hilbert equation as special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17422 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On scalar nonlinear balance laws with singular nonlocal sources Ftaka, Evangelia Nguyen, Khai T. Analysis of PDEs 35B65, , 35L60, 35L03, 76B15 We investigate one-dimensional scalar balance laws with singular convolution-type source terms. Under appropriate convexity and kernel assumptions, we establish the global existence of entropy weak solutions in ${\bf L}^2(\mathbb{R})$, together with two partial uniqueness results, in the ${\bf L}^2$-periodic setting and non-periodic setting with ${\bf L}^1(\mathbb{R})$ kernel. In the ${\bf L}^1$-kernel case, the characteristic speed satisfies an Oleinik-type estimate, and entropy weak solutions possess locally bounded fractional variation for all positive times. Furthermore, we derive a simple criterion characterizing local smoothness and wave breaking of solutions, which, in particular, includes both the Burger-Poisson and the Burgers-Hilbert equation as special cases. |
| title | On scalar nonlinear balance laws with singular nonlocal sources |
| topic | Analysis of PDEs 35B65, , 35L60, 35L03, 76B15 |
| url | https://arxiv.org/abs/2605.17422 |