Uniqueness Domains for ${\bf L}^\infty$ Solutions of $2 \times 2$ Hyperbolic Conservation Laws
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866908349125099520 |
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| author | Bressan, Alberto Marconi, Elio Vaidya, Ganesh |
| author_facet | Bressan, Alberto Marconi, Elio Vaidya, Ganesh |
| contents | For a genuinely nonlinear $2\times 2$ hyperbolic system of conservation laws, assuming that the initial data have small ${\bf L}^\infty$ norm but possibly unbounded total variation, the existence of global solutions was proved in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like $t^{-1}$. Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with faster decay rate: $\mathrm{Tot. Var. }\bigl\{u(t,\cdot)\bigr\}\leq C t^{α-1}$. For these solutions, a uniqueness theorem is proved. Indeed, as the initial data range over a domain of functions with $\|\bar u\|_{{\bf L}^\infty} \leq\varepsilon_1$ small enough, solutions with fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to $1$ by further shrinking the value of $\varepsilon_1>0$. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness Domains for ${\bf L}^\infty$ Solutions of $2 \times 2$ Hyperbolic Conservation Laws Bressan, Alberto Marconi, Elio Vaidya, Ganesh Analysis of PDEs 35L65 For a genuinely nonlinear $2\times 2$ hyperbolic system of conservation laws, assuming that the initial data have small ${\bf L}^\infty$ norm but possibly unbounded total variation, the existence of global solutions was proved in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like $t^{-1}$. Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with faster decay rate: $\mathrm{Tot. Var. }\bigl\{u(t,\cdot)\bigr\}\leq C t^{α-1}$. For these solutions, a uniqueness theorem is proved. Indeed, as the initial data range over a domain of functions with $\|\bar u\|_{{\bf L}^\infty} \leq\varepsilon_1$ small enough, solutions with fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to $1$ by further shrinking the value of $\varepsilon_1>0$. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation. |
| title | Uniqueness Domains for ${\bf L}^\infty$ Solutions of $2 \times 2$ Hyperbolic Conservation Laws |
| topic | Analysis of PDEs 35L65 |
| url | https://arxiv.org/abs/2505.00420 |