Geometry of Stationary Sets for the Wave Equation in R^n, The Case of Finitely Supported Initial Data, An Announcement

Research Announcement: Finite-time Blow Up and Long-wave Unstable Thin Film Equations

We study short--time existence, long--time existence, finite speed of propagation, and finite--time blow--up of nonnegative solutions for long-wave unstable thin film equations $h_t = -a_0(h^n h_{xxx})_x - a_1(h^m h_x)_x$ with $n>0$, $a_0 > 0$, and $a_1 >0$. The existence and finite speed of propagation results extend those of [Comm Pure Appl Math 51:625--661, 1998]. For $0<n<2$ we prove the existence of a nonnegative, compactly--supported, strong solution on the line that blows up in finite time. The construction requires that the initial data be nonnegative, compactly supported in $\R^1$, be…

Characterization of three-dimensional Euler flows supported on finitely many Fourier modes

Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the physical side). On the other hand, in physics, it is observed that turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus our mathematical motivation in this study is to find another type of building blocks accompanied by vortex stretching and scale locality (possibly finitely many Fourier modes). In this paper, we give a complete l…

$\Z^3$ 上的精确离散Hardy不等式

The sharp discrete Hardy inequality on $\Z^3$

该研究确定了$\Z^3$上带欧几里得反平方权重的最近邻Hardy不等式的精确常数，通过显式倒易边场、逐边配方法及顶点权重凹性论证，证明系数$1/4$的最优性且非零有限支撑函数无法取等。

Wasserstein Barycenters over Riemannian manifolds

We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures $Ω\in P(P(M))$ on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier \cite{ac} to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger \cite{c-ems} for barycenters between two measures to the multi-marginal settin…

Quenching of Reaction by Cellular Flows

We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenario for the initial data that is compactly supported and the size of the support is large enough. If the flow cells are large compared to the reaction length scale, propagating fronts will always form. For the small cell size, any finitely supported initial data will be quenched by a sufficiently strong flow. We estimate that the flow amplitude required to quench the initial data of support $L_0$ is $A>CL_0^4\ln(L_0)$. The essence of the problem is the question abou…

Ionization of Coulomb systems in $\RR^3$ by time periodic forcings of arbitrary size

We analyze the long time behavior of solutions of the Schrödinger equation $iψ_t=(-Δ-b/r+V(t,x))ψ$, $x\in\RR^3$, $r=|x|$, describing a Coulomb system subjected to a spatially compactly supported time periodic potential $V(t,x)=V(t+2π/ω,x)$ with zero time average. We show that, for any $V(t,x)$ of the form $2Ω(r)\sin (ωt-θ)$, with $Ω(r)$ nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, {\em i.e.} $P(t,K)=\int_K|ψ(t,x)|^2dx\to 0$ as $t\to\infty$ for any compact set $K\subset\RR^3$. Furthermore, if the initial state is compactly supported and has on…