Solutions to conservation laws are Hölder-stable in $L^2$ in the weak-BV setting
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908545757216768 |
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| author | Chen, Geng Faile, Cooper Krupa, Sam G. |
| author_facet | Chen, Geng Faile, Cooper Krupa, Sam G. |
| contents | We consider hyperbolic systems of conservation laws in one spatial dimension. For any limit of front tracking solutions $v$, and for a general weak solution $u\in L^\infty$ with no BV assumption, we prove the following Hölder-type stability estimate in $L^2$:
$$||u(\cdot,τ)-v(\cdot,τ)||_{L^2} \leq K \sqrt{||u( \cdot,0)-v( \cdot,0)||_{L^2}}$$
for all $τ$ without smallness and for a universal constant $K$. Our result holds for all limits of front tracking solutions $v$ with BV bound, either for general systems with small-BV data, or for special systems (isothermal Euler, Temple-class systems) with large-BV data. Our results apply to physical systems such as isentropic Euler. The stability estimate is completely independent of the BV norm of the potentially very wild solution $u$. We use the $L^2$ theory of shock stability modulo an artificial shift of position (Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]) but our stability results do not depend on an artificial shift. Moreover, we give the first result within this framework which can show uniqueness of some solutions with large $L^\infty$ and infinite BV initial data. We apply these techniques to isothermal Euler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions to conservation laws are Hölder-stable in $L^2$ in the weak-BV setting Chen, Geng Faile, Cooper Krupa, Sam G. Analysis of PDEs 35L65 (Primary) 76N15, 35L45, 35B35 (Secondary) We consider hyperbolic systems of conservation laws in one spatial dimension. For any limit of front tracking solutions $v$, and for a general weak solution $u\in L^\infty$ with no BV assumption, we prove the following Hölder-type stability estimate in $L^2$: $$||u(\cdot,τ)-v(\cdot,τ)||_{L^2} \leq K \sqrt{||u( \cdot,0)-v( \cdot,0)||_{L^2}}$$ for all $τ$ without smallness and for a universal constant $K$. Our result holds for all limits of front tracking solutions $v$ with BV bound, either for general systems with small-BV data, or for special systems (isothermal Euler, Temple-class systems) with large-BV data. Our results apply to physical systems such as isentropic Euler. The stability estimate is completely independent of the BV norm of the potentially very wild solution $u$. We use the $L^2$ theory of shock stability modulo an artificial shift of position (Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]) but our stability results do not depend on an artificial shift. Moreover, we give the first result within this framework which can show uniqueness of some solutions with large $L^\infty$ and infinite BV initial data. We apply these techniques to isothermal Euler. |
| title | Solutions to conservation laws are Hölder-stable in $L^2$ in the weak-BV setting |
| topic | Analysis of PDEs 35L65 (Primary) 76N15, 35L45, 35B35 (Secondary) |
| url | https://arxiv.org/abs/2507.23645 |