Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity
Mingwen Fei Huicheng Yin
We will focus on the existence and concentration of nodal solutions to the following critical nonlinear Schrödinger equations in $\Bbb R^2$ $$ -\epsilon^2\triangle u_{\epsilon}+V(x)u_{\epsilon}=K(x) |u_{\epsilon}|^{p-2}u_{\epsilon}e^{\alpha_{0}|u_{\epsilon}| ^{2}},\quad u_{\epsilon}\in H^1(\Bbb R^2), $$ where $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, and $\epsilon>0$ is a small constant. For the positive potential $V(x)$ which decays at infinity like $(1+|x|)^{-\alpha}$ with $0 < \alpha \le 2$, we will show that a nodal solution with one positive and one negative peaks exists, and concentrates around local minimum points of the related ground energy function $G(\xi)$ of the Schrödinger equation $ -\triangle u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{2}}$.
keywords: multi-peak critical exponent concentration-compactness. Nonlinear Schrödinger equation nodal solution
The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations
Huicheng Yin Lin Zhang

We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In this paper, as the second part of our three papers, we will confirm this physical phenomenon for the compressible viscid fluids by obtaining the exact lower and upper bound on the density function.

keywords: Compressible Navier-Stokes equations expanding ball weighted energy estimates global existence vacuum state
Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations
Bingbing Ding Ingo Witt Huicheng Yin

For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in [1,2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (u,\nabla u)\partial _{ij}^2u = 0$, whose coefficients depend on $u$ and $\nabla u$ simultaneously. When the first weak null condition is not fulfilled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution $u$ blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.

keywords: Weak null condition lifespan blowup blowup system Nash-Moser-Hörmander iteration
The global stability of 2-D viscous axisymmetric circulatory flows
Huicheng Yin Lin Zhang

In this paper, we study the global existence and stability problem of a perturbed viscous circulatory flow around a disc. This flow is described by two-dimensional Navier-Stokes equations. By introducing some suitable weighted energy space and establishing a priori estimates, we show that the 2-D circulatory flow is globally stable in time when the corresponding initial-boundary values are perturbed sufficiently small.

keywords: Compressible Navier-Stokes equations circulatory flow weighted energy space global existence

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