Recent progress on very high order schemes for compressible flows

We will talk about necessary and sufficient conditions for an entire solution $u$ of a biharmonic equation with exponential nonlinearity $e^u$ to be a radially symmetric solution. The standard tool to obtain the radial symmetry for a system of equations is the Moving-Plane-Method (MPM). In order to apply the MPM, we need to know the asymptotic expansions of $u$ and $-\Delta u$ at $\infty$. The difficulties come from the fact that $e^u$ is supercritical for $N\geq5$, $e^u \not \in L^{\frac{N}{4}} (\mathbb{R}^N)$. We need to get the right decay rate of $u$ and $-\Delta u$ at $\infty$ in order to start the moving plane procedure. This is a joint work with Z.M.Guo and X.