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The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics,…
We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic H\"{o}lder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator…
We show that hyperbolicity is a necessary condition for the well posedness of the noncharacteristic Cauchy problem for nonlinear partial differential equations. We give conditions on the initial data which are necessary for the existence of…
For the paradigmatic case of the damped quantum harmonic oscillator we present two measurement-based feedback schemes to control the stability of its fixed point. The first scheme feeds back a Pyragas-like time-delayed reference signal and…
This paper focuses on investigating an inexact stochastic model-based optimization algorithm that integrates preconditioning techniques for solving stochastic composite optimization problems. The proposed framework unifies and extends the…
In the paper we investigate various inequalities for the one-dimensional Cauchy measure. We also consider analogous properties for one-dimensional sections of multidimensional isotropic Cauchy measure. The paper is a continuation of our…
We consider the Cauchy problem for a first-order evolution equation with memory in a finite-dimensional Hilbert space when the integral term is related to the time derivative of the solution. The main problems of the approximate solution of…
We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- \Delta)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$,…
We establish some quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field…
We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a…
We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence…
We investigate the initial-boundary value problem for linearized gravitational theory in harmonic coordinates. Rigorous techniques for hyperbolic systems are applied to establish well-posedness for various reductions of the system into a…
We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol…
In this paper we address the temporal energy growth associated with numerical approximations of the perfectly matched layer (PML) for Maxwell's equations in first order form. In the literature, several studies have shown that a numerical…
In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for 3D incompressible Navier-Stokes-Cahn-Hilliard equations. First, using Banach fixed point theorem, we establish the local well-posedness of…
We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative…
For problems of time-harmonic scattering by polygonal obstacles, embedding formulae provide a useful means of computing the far-field coefficient induced by any incident plane wave, given the far-field coefficient of a relatively small set…
The moment measure problem consists in finding a convex function $\psi$ whose moment measure, i.e., the pushforward by $\nabla \psi$ of the measure with density $e^{-\psi(\,\cdot\,)}$, is prescribed. It is highly non-linear and less…
Stability and error analysis remain challenging for problems that lack regularity properties near solutions, are subject to large perturbations, and might be infinite dimensional. We consider nonconvex optimization and generalized equations…
We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show…