流体动力学中的守恒量和规律性
Conserved quantities and regularity in fluid dynamics
摘要：
守恒量或耗散量，如能量或熵，是与流体力学有关的许多类时变偏微分方程研究的核心。例如，欧拉方程和纳维尔-斯托克斯方程、守恒律系统和输运方程就是这种情况。在所有这些情况下，形式守恒量对于低正则性的弱解可能不再是时间常数。规则性和守恒量之间微妙的相互作用与DiPerna-Lions理论中的输运和连续性方程的重整化有关，也与Onsager在理想不可压缩流体领域的猜想有关。我们将回顾DiPerna-Lions和Constantin-E-Titi的经典换位子方法，然后继续讨论最近的结果。
英文摘要：
Conserved or dissipated quantities, like energy or entropy, are at the heart of the study of many classes of time-dependent PDEs in connection with fluid mechanics. This is the case, for instance, for the Euler and Navier-Stokes equations, for systems of conservation laws, and for transport equations. In all these cases, a formally conserved quantity may no longer be constant in time for a weak solution at low regularity. The delicate interplay between regularity and conservation of the respective quantity relates to renormalisation in the DiPerna-Lions theory of transport and continuity equations, and to Onsager's conjecture in the realm of ideal incompressible fluids. We will review the classical commutator methods of DiPerna-Lions and Constantin-E-Titi, and then proceed to more recent results.
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