具有竞争型 Van Der Waals 型势的 Schrödinger 方程的指定质量驻波

Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials

\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation \begin{equation*}\label{1} -Δu=λu+μ(|x|^{-α}\ast|u|^{2})u+(|x|^{-β}\ast|u|^{2})u,\ \ x\in \R^{N} \end{equation*} under the normalized constraint \[\int_{{\mathbb{R}^N}} {{u}^2}=c>0,\] where $N\geq 3$, $μ\!>\!0$, $α$, $β\in (0,N)$, and the frequency $λ\in \mathbb{R}$ is unknown and appears as Lagrange multiplier. Compared with the…

On steady states of van der Waals force driven thin film equations

Let $Ω\subset\mathbb{R}^{N}$, $N\geq2$ be a bounded smooth domain and $α>1$. We are interested in the singular elliptic equation% \[ \triangle h=\frac{1}αh^{-α}-p\quad\text{in}Ω\] with Neumann boundary conditions. In this paper, we gave a complete description of all continuous radially symmetric solutions. In particular, we constructed nontrivial smooth solutions as well as rupture solutions. Here a continuous solution is said to be a rupture solution if its zero set is nonempty. When N=2 and $α=3$, the equation has been used to model steady states of van der Waals force driven thin films of v…

基于范德瓦尔斯状态方程的二维直管喷管中的声速-超声速射流

Sonic-supersonic jet flows from a straight two-dimensional nozzle with van der Waals equation of state

研究基于范德瓦尔斯状态方程的二维直管喷管声速-超声速射流，通过特征分解和先验估计，建立了向真空中扩展的射流全局存在性，以及在特定静态大气条件下喷管出口附近射流的局部存在性。

Global smooth solutions of Euler equations for Van der Waals gases

We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in $Hm$, for $m$ large enough. To do so, we introduce a specific symmetrisation allowing areas of null density. Next, we make estimates in $Hm$, using for some terms the estimates done by M. Grassin, who proved the same theorem in the easier case of a perfect polytropic gas. We treat the remaining terms separately, due to their non-linearity.

Analytical and numerical study of weakly nonlinear hyperbolic waves in a van der Waals gas

In this paper, we characterized resonant interaction of weakly nonlinear hyperbolic waves in gas dynamics with a real gas background. An asymptotic approach is used to study the interaction between waves, governed by the Euler equations of gas dynamics, supplemented by a van der Waal equation of state; the evolution equation is an integro-differential equation composed of a Burgers type nonlinear term and a convolution term with a known kernel. A one parameter family of traveling wave solutions for various values of van der Waals parameter is studied analytically and numerically. Effect of the…

Van der Waals shock polars with multiple or supersonic critical points

It is shown that the $γ$-van der Waals equation of state (eos) permits shock polars with supersonic critical points, corresponding to critical or strong-type shock reflections that are supersonic, which is not possible for ideal gas. It is also shown that general van der Waals eos permit polars with multiple critical points, corresponding to four or more reflected shocks for same deflection angle. Of these reflected shocks at least two are weak-type, i.e.\ deflection angle increasing with increasing shock strength, so that standard literature has no criteria to select one of the two. Both phen…