An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations
An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations
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DOI：
10.1007/s00220-017-3078-4
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发表时间：
2018-04-01
影响因子：
2.4
通讯作者：
Eyink, Gregory L.
中科院分区：
文献类型：
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作者：
Drivas, Theodore D.;Eyink, Gregory L.
我们证明了可压缩欧拉方程的有界弱解将守恒热力学熵，除非解场具有足够低的时空Besov正则性。对于这样的欧拉解，测量动能级联的量也将消失，除非满足相同的奇性条件。进一步证明了具有反常耗散的有界可压缩N-S方程的解的强极限是弱Euler解。这些无粘极限解具有非负的反常熵产生和动能耗散，当解超过Besov正则性的临界度时，两者都消失。欧氏空间中具有理想气体状态方程的定常平面激波提供了满足我们定理条件的简单例子，并证明了我们基于L(3)的条件的敏锐性。这些条件涉及时空Besov正则性，但我们证明了时间一致地具有相似空间正则性的欧拉解满足这些条件。
We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our L (3)-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.