Abstract
This paper is concerned with the problem of energy conservation for the two dimensional inhomogeneous Euler equations, with an emphasis on the vorticity \(\omega =\mathrm{curl}u\) of the flow. In particular, two types of sufficient conditions are obtained. The first one assumes \(L^p\)-regularity on the spatial gradient of the density \(\nabla \rho \) and the vorticity \(\omega \). The second one removes the regularity condition on \(\nabla \rho \) while requires certain time regularity of \(\omega \). Furthermore, we phrase the energy spectrum in terms of the Littlewood–Paley decomposition and show that the energy flux \(\Pi _q\) vanishes as the dyadic exponent \(q\rightarrow \infty \).
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This work is supported in part by National Natural Science Foundation of China-NSAF (No. 11301439) and Natural Science Foundation of Fujian Province, China (No. 2018J01430).
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Chen, Q. Energy Conservation in 2-D Density-Dependent Euler Equations with Regularity Assumptions on the Vorticity. J. Math. Fluid Mech. 22, 6 (2020). https://doi.org/10.1007/s00021-019-0470-1
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DOI: https://doi.org/10.1007/s00021-019-0470-1