Regularity and energy conservation for the compressible Euler equations.(English)Zbl 1365.35113

In this paper the authors consider both the inhomogeneous incompressible Euler equations and the compressible isentropic Euler equations. They derive sufficient conditions on the regularity of density $$\rho$$ and velocity $$u$$ to ensure the conservation of energy. Their approach is based on the idea of P. Constantin et al. [Commun. Math. Phys. 165, No. 1, 207–209 (1994; Zbl 0818.35085)] to use suitable commutator estimates. Therefore, the corresponding regularity assumptions are stated in terms of Besov spaces $$B^{\alpha, \infty}_{p}$$ similar to those in [loc. cit.].
The bibliography contains 21 items. The authors give an appropriate overview on the problems. The paper is self-contained and reads good.

MSC:

 35Q31 Euler equations 35B65 Smoothness and regularity of solutions to PDEs 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Zbl 0818.35085
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References:

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