Energy conservation for the Euler equations on $\mathbb{T}^2\times \mathbb{R}_+$ for weak solutions defined without reference to the pressure
| Autor: | Robinson, James C., Rodrigo, José L., Skipper, Jack W. D. |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | We study weak solutions of the incompressible Euler equations on $\mathbb{T}^2\times \mathbb{R}_+$; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the pressure. We prove energy conservation under the assumptions that $u\in L^3(0,T;L^3(\mathbb{T}^2\times \mathbb{R}_+))$, $$ \lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{T}^2}\int^\infty_{x_3>|y|} |u(x+y)-u(x)|^3\mathrm{d} x\, \mathrm{d} t=0, $$ and an additional continuity condition near the boundary: for some $\delta>0$ we require $u\in L^3(0,T;C^0(\mathbb{T}^2\times [0,\delta])))$. We note that all our conditions are satisfied whenever $u(x,t)\in C^\alpha$, for some $\alpha>1/3$, with H\"older constant $C(x,t)\in L^3(\mathbb{T}^2\times\mathbb{R}^+\times(0,T))$. Comment: 21 pages |
| Databáze: | arXiv |
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