论文标题
流体动力学中自由表面流的功能不平等和强大的Lyapunov功能
Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics
论文作者
论文摘要
本文的动机是由四个方程式的Lyapunov功能进行的，这些方程描述了流体动力学中的自由表面流：Hele-Shaw和Mullins-Sekerka方程及其润滑近似，Boussinesq和薄膜方程。我们确定了新的Lyapunov功能，包括一些以凸的方式衰减的功能（这些称为强Lyapunov功能）。对于Hele-Shaw方程和Mullins-Sekerka方程，我们证明了自由表面高程的$ L^2 $ - norm和自由表面的面积是Lyapunov功能，以及薄膜和BousSinesQ方程的并行结果。这些证明将耗散率的确切身份与功能不平等相结合。对于薄膜和BousSinesQ方程，我们引入了独立利益的Sobolev不平等，重新审视了一些已知的结果并表现出强大的Lyapunov功能。对于Hele-Shaw和Mullins-sekerka方程，我们引入了一个功能，该功能控制了$ l^2 $ - 三分之二的空间衍生物。在初始数据的轻度小假设下，我们表明后一个数量也是Hele-shaw方程的Lyapunov功能，这意味着该区域功能是强大的Lyapunov功能。建立了耗散率的精确下限，表明这些Lyapunov功能实际上是熵。还研究了其他数量，例如Lebesgue Norms或Boltzmann的熵。
This paper is motivated by the study of Lyapunov functionals for four equations describing free surface flows in fluid dynamics: the Hele-Shaw and Mullins-Sekerka equations together with their lubrication approximations, the Boussinesq and thin-film equations. We identify new Lyapunov functionals, including some which decay in a convex manner (these are called strong Lyapunov functionals). For the Hele-Shaw equation and the Mullins-Sekerka equation, we prove that the $L^2$-norm of the free surface elevation and the area of the free surface are Lyapunov functionals, together with parallel results for the thin-film and Boussinesq equations. The proofs combine exact identities for the dissipation rates with functional inequalities. For the thin-film and Boussinesq equations, we introduce a Sobolev inequality of independent interest which revisits some known results and exhibits strong Lyapunov functionals. For the Hele-Shaw and Mullins-Sekerka equations, we introduce a functional which controls the $L^2$-norm of three-half spatial derivative. Under a mild smallness assumption on the initial data, we show that the latter quantity is also a Lyapunov functional for the Hele-Shaw equation, implying that the area functional is a strong Lyapunov functional. Precise lower bounds for the dissipation rates are established, showing that these Lyapunov functionals are in fact entropies. Other quantities are also studied such as Lebesgue norms or the Boltzmann's entropy.
