In this paper, we construct stationary classical solutions of the shallow water equation with vanishing Froude number $Fr$ in the so-called lake model. To this end we need to study solutions to the following semilinear elliptic problem

for small $ε$ > 0, where $p$ > 1, div($\frac{∇q}{b}$) = 0 and $Ω$ ⊂ $\mathbb{R}$² is a smooth bounded domain.
We show that if $\frac{q^2}{b}$ has $m$ strictly local minimum (maximum) points $\widetilde{z}_i$, $i$ = 1, · · · , $m$, then there is a stationary classical solution approximating stationary $m$ points vortex solution of shallow water equations with vorticity $\sum\limits_{i=1}^m$ $\frac{2πq(\widetilde{z}_i)}{b(\widetilde{z}_i)}$. Moreover, strictly local minimum points of $\frac{q^2}{b}$ on the boundary can also give vortex solutions for the shallow water equation.

The usual heat flow moves along the direction from high temperature place to the low one, as often observed in the daily life. However, when the gas is very rarefied, the gas may move along a different way, that is, the so-called thermal creep flow moves along the direction from the low temperature place to the high one. In this note, we will survey our recent mathematical works on this topic, mainly based on [27] and [25].

In this paper, we shall prove that any minimizer of Ginzburg-Landau functional from an Alexandrov space with curvature bounded below into a nonpositively curved metric cone must be locally Lipschitz continuous.

This is to comment on the well-posedness of weak solutions for the initial value problem for partial differential equations. In recent decades, and particularly in recent years, there have been substantial progresses on construction by convex integration for the study of non-uniqueness of solutions for incompressible Euler equations, and even for compressible Euler equations. This prompts the question of whether it is possible to give a sense of well-posedness, which is narrower than the canonical Hadamard sense, so that the evolutionary equations are well-posed. We give a brief and partial review of the related results and offer some thoughts on this fundamental topic.

The paper is concerned with time-asymptotic behavior of solution near a local Maxwellian with rarefaction wave to a fluid-particle model described by the Vlasov-Fokker-Planck equation coupled with the compressible and inviscid fluid by Euler-Poisson equations through the relaxation drag frictions, Vlasov forces between the macroscopic and microscopic momentums and the electrostatic potential forces. Precisely, based on a new micro-macro decomposition around the local Maxwellian to the kinetic part of the fluid-particle coupled system, which was first developed in [16], we show the time-asymptotically nonlinear stability of rarefaction wave to the one-dimensional compressible inviscid Euler equations coupled with both the Vlasov-Fokker-Planck equation and Poisson equation.