Stability and uniqueness for piecewise smooth solutions to Burgers-Hilbert among a large class of solutions
| Autor: | Krupa, Sam G., Vasseur, Alexis F. |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | SIAM J. Math. Anal., 52(3), 2491-2530, 2020 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1137/19M1257883 |
| Popis: | In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x=\mathbf{H}[u]$ where $\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]). Comment: 46 pages |
| Databáze: | arXiv |
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