Related papers: Transition operators of diffusions reduce zero-cro…

We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for…

This article presents the finite size analysis of two consecutive crossovers leading laminar-turbulent bands to uniform wall turbulence in transitional plane Couette flow. Direct numerical simulations and low order modeling simulations of…

We show that bounded divergence-free vector fields $u : [0,\infty) \times \mathbb{R}^d \to\mathbb{R}^d$ decrease the ''concentration'', quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to…

Dynamical transitions, such as a change from bound to unbound motion, often occur as post-adiabatic crossings of a time-dependent separatrix. Whether or not any given orbit will include such a crossing transition typically depends…

In the present work, we demonstrate how the pseudoinverse concept from linear algebra can be used to represent and analyze the boundary conditions of linear systems of partial differential equations. This approach has theoretical and…

We consider a cross-diffusion system for which the diffusion of each species is governed solely by the aggregate density through a pressure law of logarithmic or fast diffusion type. The model is set over a one dimensional bounded interval,…

This work concerns the exponential stabilization of underactuated linear homogeneous systems of m parabolic partial differential equations (PDEs) in cascade (reaction-diffusion systems), where only the first state is controlled either…

The generalization of neural networks is a central challenge in machine learning, especially concerning the performance under distributions that differ from training ones. Current methods, mainly based on the data-driven paradigm such as…

In this article, we consider the space-time Fractional (nonlocal) diffusion equation $$\partial_t^\beta u(t,x)={\mathtt{L}_D^{\alpha_1,\alpha_2}} u(t,x), \ \ t\geq 0, \ x\in D, $$ where $\partial_t^\beta$ is the Caputo fractional derivative…

Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained…

We consider time-changed Poisson processes, and derive the governing difference-differential equations (DDE) these processes. In particular, we consider the time-changed Poisson processes where the the time-change is inverse Gaussian, or…

We investigate the transition probability rate of a Unruh-DeWitt (UD) detector interacting with massless scalar field for a finite duration of proper time, $T$, of the detector. For a UD detector moving at a uniform acceleration, $a$, we…

We prove that the only global, strong, spatially periodic solution to the Degasperis-Procesi equation, vanishing at some point (t0, x0), is the identically zero solution. We also establish the analogue of such Liouville-type theorem for the…

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original…

We prove the Vollhardt and Wolfle hypothesis that the irreducible vertex U_{kk'}(q) appearing in the Bethe--Salpeter equation contains a diffusion pole (with the observable diffusion coefficient D(\omega,q)) in the limit k+k'\to 0. The…

The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace…

In this paper, we study the fully fractional master equation \begin{equation}\label{pdeq1} (\partial_t-\Delta)^s u(x,t) =f(x,t,u(x,t)),\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}. \end{equation} First we prove a Liouville type theorem for…

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties.…

Consider the stochastic PDE, $\partial_tu = \partial^2_x u + \sigma(u) \dot{W}$ on $\mathbb{R}_+\times\mathbb{R}$, subject to $u(0)\equiv1$, where $\dot{W}$ denotes space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and…

Let $L$ be a linear differential operator acting on functions defined over an open set $\mathcal{D}\subset \mathbb{R}^d$. In this article, we characterize the measurable second order random fields $U = (U(x))_{x\in\mathcal{D}}$ whose sample…