相关论文: On complex singularities of the 2D Euler equation …

We consider a spacetime singularity at $t = 0$ arising in a Kasner-type metric that solves the gravitational equations modified by quantum effects of a conformal field theory (CFT). The resulting constraints can be solved efficiently when…

In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a…

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

We consider the simplest instabilities involving multiple unstable electrostatic plasma waves corresponding to four-dimensional systems of mode amplitude equations. In each case the coupled amplitude equations are derived up to third order…

We analyze the wave equation in families of pp-wave geometries developing strong localized scale-invariant singularities in certain limits. For both cases of well-localized pp-waves and the so-called null-cosmologies, we observe an…

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^\alpha$…

In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar…

We present a self-contained interior quadrupole mechanism for finite-time singularity formation in the axisymmetric three-dimensional incompressible Euler equations with swirl in the whole space. The construction is localized away from the…

We provide a complete description of the critical threshold phenomena for the two-dimensional localized Euler-Poisson equations, introduced by the authors in [Liu & Tadmor, Comm. Math Phys., To appear]. Here, the questions of global…

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time "splash" (or "splat") singularity first introduced in [9], wherein the…

The search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by…

The quasi-geostrophic two-layer (QS2L) system models the dynamic evolution of two interconnected potential vorticities, each is governed by an active scalar equation. These vorticities are linked through a distinctive combination of their…

We determine the position and the type of spontaneous singularities of solutions of generic analytic nonlinear differential systems in the complex plane, arising along antistokes directions towards irregular singular points of the system.…

We consider weak solutions of the adjoint equation for an elliptic operator in nondivergent form, and their asymptotic properties at an interior point. We assume that the coefficients a_{ij} are bounded, measurable, complex-valued functions…

We provide a complete classification of the asymptotic behavior of isolated singularities for solutions satisfying \[ 0\le-\Delta_{p}u(x)\le \tau u^{\frac{n(p-1)}{n-p}} (x),\,\,u(x)\ge0,\,\,1<p<n,\,\,n\ge2, \]where $u(x)\in…

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in…

We study the coupled Einstein-Klein-Gordon equations for a complex scalar field with and without a quartic self-interaction in a curvatureless Friedman-Lema\^{\i}\-tre Universe. The equations can be written as a set of four coupled first…

We first introduce a family of binary $pq^2$-periodic sequences based on the Euler quotients modulo $pq$, where $p$ and $q$ are two distinct odd primes and $p$ divides $q-1$. The minimal polynomials and linear complexities are determined…

In the first part of the paper, we discuss eigenvalue problems of the form -w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on two rays in the complex plane. We give sufficient conditions for continuity of the…

The aim of this work is to study the existence of a periodic solutions of third order differential equations $z'''(t) = Az(t) + f(t)$ with the periodic condition $x(0) = x(2\pi), x'(0) = x'(2\pi)$ and $x''(0) = x''(2\pi)$. Our approach is…