论文标题
球体上等等晶体方程的凸和传输
Convexity and transport for the isentropic Euler equations on the sphere
论文作者
论文摘要
该论文认为PDE的Euler系统在没有边界的正曲率的平滑紧凑型Riemannian歧管上，而Sphere $ {\ Mathbb {s}}}^2 $特别是。该论文将欧拉方程解释为流体密度下的流体密度的传输问题，该动力学由流体的内部能量梯度控制。该论文在切线捆绑包中提出了运输成本的概念，并将其特性与瓦瑟尔斯坦运输成本进行了比较。在Gangbo和Wesdickenberg（{\ sl comm。部分diff。方程} {\ bf 34}（2009），1041-1073）的样式中，有针对Euler方程的离散近似的应用，除了分析很大程度上依赖于下延义的弯曲曲率。假定内部能量满足凸状条件，该条件允许通过$φ$ - 内部熵产生的不平等分析，并且结果适用于功率法$ρ^γ$，其中$ 1 <γ<3/2 $，其中包括二离子气的情况。该论文证明了连续性方程的弱解决方案的存在，并提供了足够的条件，可以使加速度方程较弱的解决方案。
The paper considers the Euler system of PDE on a smooth compact Riemannian manifold of positive curvature without boundary, and the sphere ${\mathbb{S}}^2$ in particular. The paper interprets the Euler equations as a transport problem for the fluid density under dynamics governed by the gradient of the internal energy of the fluid. The paper develops the notion of transport cost in the tangent bundle, and compares its properties with the Wasserstein transportation cost on the manifold. There are applications to the discrete approximation to the Euler equations in the style of Gangbo and Wesdickenberg ({\sl Comm. Partial Diff. Equations} {\bf 34} (2009), 1041-1073), except that the analysis is heavily dependent upon the curvature of the underlying manifold. The internal energy is assumed to satisfy convexity conditions that allow analysis via $Φ$-entropy entropy-production inequalities, and the results apply to the power law $ρ^γ$ where $1<γ<3/2$, which includes the case of a diatomic gas. The paper proves existence of weak solutions of the continuity equation, and gives a sufficient condition for existence of weak solutions to the acceleration equation.
