二维Euler和Vlasov方程的不变测度
Invariant measures of the 2D Euler and Vlasov equations
摘要：
我们讨论的不变措施的偏微分方程，如二维欧拉或弗拉索夫方程。对于二维欧拉方程，从Liouville定理出发，有效的N维近似的动力学，我们定义的微正则测度作为一个极限测度，其中N走向无穷大。当只考虑能量和拟能不变量时，我们给出了一个显式计算，证明了以下结果：微正则测度实际上是一个与平均场熵最大化相对应的Young测度.我们解释了为什么这个结果仍然是真实的更一般的微正则措施，当所有的动力学不变量被考虑在内。我们给出了一个明确的证明，这些微正则措施是不变的措施的动力学的二维欧拉方程。我们描述了一组更一般的不变测度，并简要讨论了它们的稳定性和它们对二维欧拉方程遍历性的影响。将这些结果推广到Vlasov方程，并证明了具有排斥凸势的Vlasov方程的统计平衡点的唯一性。即使我们认为，在本文中，不变的措施只为哈密顿方程，没有通量的保守量，我们认为这项工作是一个重要的一步，对非平衡不变的措施与通量的描述。
英文摘要：
We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures, and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials. Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.
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