Related papers: Regularization for fractional integral. Applicatio…

We consider fractional Schr\"odinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragm\'en-Lindel\"of theorem for polynomially bounded…

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only…

The purpose of this article is to address the issues of dimensional consistency that arise in the process of replacing the ordinary time derivative operator by a fractional derivative operator in order to write a fractional differential…

In this article, we explore the fractional Kirchhoff-Choquard system given by $$ \left\{ \begin{array}{lr} (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} u|^2\;dx)(-\Delta)^su=\lambda_1u+(I_{\mu}*|v|^{{2^*_{\mu,s}}})|u|^{{2^*_{\mu,s}}-2}u…

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form \begin{align*} \begin{aligned} - \Delta u &= f(|x|,u,v)\quad\text{in}\R^n, - \Delta v &=…

In this paper, we establish refined Strichartz estimates for higher-order Schr\"odinger equations with initial data exhibiting partial regularity. By partial regularity, we mean that the initial data are not required to have full Sobolev…

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the $1$-dimensional semi-linear fractional equations: \begin{align*} \vert D \vert^\alpha u + u -f(u)=0, \end{align*} with $\alpha\in (0,2)$, a…

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schr\"odinger equations with subcritical exponent. For some smooth bounded domain $\Omega\subset \mathbf R^n$, our boundary condition is given…

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schr\"odinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS…

This paper considers the numerical analysis of a semilinear fractional diffusion equation with nonsmooth initial data. A new Gr\"onwall's inequality and its discrete version are proposed. By the two inequalities, error estimates in three…

In this note we consider the 1-D cubic Schr\"odinger equation with data given as small perturbations of a Dirac-$\delta$ function and some other related equations. We first recall that although the problem for this type of data is ill-posed…

The goal of this work is to discuss how should we impose initial values in fractional problems to ensure that they have exactly one smooth unique solution, where smooth simply means that the solution lies in a certain suitable space of…

We consider families of schemes over arbitrary fields resp. analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular families of generically reduced curves. We define a modified delta…

We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces,…

We study the notion of regular singularities for parameterized complex ordinary linear differential systems, prove an analogue of the Schlesinger theorem for systems with regular singularities and solve both a parameterized version of the…

In this paper, we study the following nonlinear Dirac equations \begin{align*} \begin{cases} -i\sum\limits_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u, \displaystyle \int_{\mathbb{R}^3} |u|^2dx=a^2, \end{cases} \end{align*}…

Solutions to fractional models inherently exhibit non-smooth behavior, which significantly deteriorates the accuracy and therefore efficiency of existing numerical methods. We develop a two-stage data-infused computational framework for…

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schr\"odinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a…

Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of ${\mathcal G}^\infty$-regularity, which reduces to $C^\infty$-regularity when restricted to Schwartz distributions. Surprisingly, in the…

We study the maximal regularity problem for abstract time-fractional Schr\"odinger equations $\partial_t^\alpha(u-u_0) -\mathrm{i} A u=f$, with a fractional derivative $\partial_t^\alpha$ of order $\alpha \in (0,1)$. We assume that $A$ is a…