Related papers: On complex singularities of the 2D Euler equation …
We consider elliptic measure data problems of the type \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu \] in a bounded domain in $\mathbb{R}^n$, where $p<q$ and $a(\cdot) \ge 0$. We prove local Calder\'on--Zygmund estimates in the…
Michel proved that for a one-parameter family of elliptic curves over Q(T) with non-constant j(T) that the second moment of the number of solutions modulo p is p^2 + O(p^{3/2}). We show this bound is sharp by studying y^2 = x^3 + Tx^2 + 1.…
We continue the study of positive singular solutions of PDEs arising from double phase functionals started in [6]. In particular, we consider the case $p<q < 2$, and we relax the assumption on the capacity of the singular set using an…
We analyze the common four types of the finite-time singularities using a generic framework of the phase portrait geometric approach. This technique requires that the Friedmann system to be written as a one dimensional autonomous system. We…
We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $\epsilon>0$ proportional to the radius of the holes and a map $\phi$,…
A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to…
Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we…
We study the long-time asymptotics of solutions of the uniformly parabolic equation \[ u_t + F(D^2u) = 0 \quad {in} \R^n\times \R_+, \] for a positively homogeneous operator $F$, subject to the initial condition $u(x,0) = g(x)$, under the…
We prove the existence of a wide class of solutions to the isentropic relativistic Euler equations in 2 spacetime dimensions with an equation of state of the form $p=K\rho^2$ that have a fluid vacuum boundary. Near the fluid vacuum…
When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the two-dimensional Euler equations has been established only in some special cases, and the solutions to which these results…
Singularities arise in diverse disciplines and play a key role in both exploring fundamental laws of physics and making highly-sensitive sensors. Higher-order (>3) singularities, with further improved performance, however, usually require…
We study the existence and nonexistence of positive singular solutions to second-order non-divergence type elliptic inequalities with measurable coefficients. We prove the existence of a critical value $p^*$ that separates the existence…
In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)^\sigma u = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We…
We introduce some multiple integrals that are expected to have the same singularities as the singularities of the $ n$-particle contributions $\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find the Fuchsian linear…
To investigate the finite time singularity in three-dimensional (3D) Euler flows, the simplified model of 3D axisymmetric incompressible fluids (i.e., two-dimensional Boussinesq approximation equations) is studied numerically. The system…
We consider the 2D incompressible Euler equation on a bounded simply connected domain $\Omega$. We give sufficient conditions on the domain $\Omega$ so that for all initial vorticity $\omega_0 \in L^{\infty}(\Omega)$ the weak solutions are…
In this paper, we discuss the problem of derivation of kinetic equations from the theory of weak turbulence for the quintic Schr\"odinger equation. We study the quintic Schr\"odinger equation on $L\mathbb T$, with $L\gg 1$ and with a…
The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For some simplified formulation of the problem, a criterion for the…
We study the Euler equations describing the motion of an incompressible fluid on the cubic torus with real initial data. We construct solutions on the Fourier side which display a sudden loss of regularity within finite time even for highly…
Asymptotics of solutions to Schroedinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular…