When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The dissipative counterpart of such a system coincides with the porous medium equation, which can be cast in the form of a gradient flow for the same internal energy. Motivated by these related variational structures, we propose a particle method for both problems in which the internal energy is replaced by its Moreau-Yosida regularization in the L2 sense, which can be efficiently computed as a semi-discrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the problem in Eulerian variables, we prove quantitative convergence estimates towards smooth solutions. We verify such estimates by means of several numerical tests.

翻译：当用拉格朗日变量表达时,可压缩(巴罗罗亚)流体运动的方程式具有典型的汉密尔顿系统的结构,其中潜在的能量由流体的内部能量提供。这种系统的消散对应物与多孔的中位方程式相吻合,该方程式可以以同一种内部能量的梯度流的形式投射。受这些相关变异结构的驱动,我们为这两个问题提出了一个粒子方法,其中内部能量被其L2级莫罗-约斯达正规化所取代,这可以有效地计算成半分位最佳运输问题。我们利用利用尤利安变量中问题共性的经调整的能源参数,证明量化趋同估计是通畅的解决办法。我们通过若干数字测试来核查这种估计。