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We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index \sigma\in [3/2, 2], we obtain the local in time well-posedness in the space of Gevrey class \sigma\in [3/2, 2], in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. Grard-Varet and N. Masmoudi [{\it Ann. Sci. c. Norm. Supr}.(4) 48 (2015), no. 6, 1273-1325], in which the case \sigma\in [3/2, 2], is solved.

By introducing a new approximate Green function, we obtain the pointwise estimates on the solutions of Euler equations with linear frictional damping, from which we can deduce the optimal L p (1 p+) convergence rates to the nonlinear diffusion waves. The pointwise estimates and L p (1 p< 2) convergence rates given in this paper are new.

We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.