Related papers: L^2-concentration phenomenon for Zakharov system b…
We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.
We apply the recently generalized Levinson theorem for potentials with inverse square singularities [Sheka et al, Phys.Rev.A, v.68, 012707 (2003)] to Aharonov-Bohm systems in two-dimensions. By this theorem, the number of bound states in a…
We calculate the maximal Lyapunov exponent, the generalized entropies, the asymptotic distance between nearby trajectories and the fractal dimensions for a finite two dimensional system at different initial excitation energies. We show that…
We prove there exist solutions to the focusing cubic nonlinear Schr\"odinger equation in three dimensions that blowup on a circle, in the sense of L^2 concentration on a ring, bounded H^1 norm outside any surrounding toroid, and growth of…
We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in $H^s, s>\frac{d}{2}+1$, the…
We study the theory of scattering for the Zakharov system in space dimension 3. We prove in particular the existence of wave operators for that system with no size restriction on the data in larger spaces and for more general asymptotic…
Motivated by an open problem posed by J.P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of…
In this paper we mainly investigate the initial value problem of the periodic Euler-Poincar\'e equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant…
In this paper, we develop the well-posedness theory and uncover the noise-regularization effect on scattering for the stochastic Zakharov system in dimensions $d \geq 4$ and beyond the energy space. Our focus is particularly directed at the…
We consider the focusing inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +\Delta u + |x|^{-b} |u|^{2\sigma}u = 0,$$ where $N\geq 2$ and $\sigma$, $b>0$. We first obtain a small data global result in…
In this paper, we prove the soliton resolution conjecture for general type II solutions to the focusing energy critical wave equation, in space dimension 3,4 or 5, along a sequence of times. This is an important step towards the full…
In the previous work, we classified the solutions to a family of systems of Klein-Gordon equations with non-negative energy below the ground state into two parts: one blows up in finite time while the other extends to a global solution. In…
We consider the magnetic nonlinear inhomogeneous Schr\"odinger equation $$i\partial_t u -\left(-i\nabla+\frac{\alpha}{|x|^2}(-x_2,x_1)\right)^2 u =\pm|x|^{-\varrho}|u|^{p-1}u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^2,$$ where…
We prove a blow-up criterion for the solutions to the $\nu$-dimensional Patlak-Keller-Segel equation in the whole space. The condition is new in dimension three and higher. In dimension two it is exactly Dolbeault's and Perthame's blow-up…
We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space…
This paper studies the regularity of solutions to the Zakharov and Klein-Gordon-Schr\"{o}dinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial…
We consider the 1D cubic NLS on $\mathbb R$ and prove a blow-up result for functions that are of borderline regularity, i.e. $H^s$ for any $s<-\frac 12$ for the Sobolev scale and $\mathcal F L^\infty$ for the Fourier-Lebesgue scale. This is…
Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field blows up in the L-infinity norm as the distance between the conductors tends to zero. We give here…
In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in…
We study the nonlinear Schr\"odinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing…