On hyperbolic equations and systems with non-regular time dependent coefficients

In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than H\"older, namely bounded coefficients. As for second order equations in \cite{GR:14} we prove that such equations admit a `very weak solution' adapted to the type of solutions that exist for regular coefficients. The main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter. Classical solutions are recovered as limit of very weak solutions. Finally, by using a reduction to block Sylvester form we conclude that any first order hyperbolic system with non-regular coefficients is solvable in the very weak sense

Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity

We introduce different notions of wave front set for the functionals in the dual of the Colombeau algebra \Gc(\Om) providing a way to measure the \G and the \Ginf- regularity in \LL(\Gc(\Om),\wt{\C}). For the smaller family of functionals having a ``basic structure'' we obtain a Fourier transform-characterization for this type of generalized wave front sets and results of noncharacteristic \G and \Ginf-regularity

Fundamental solutions in the Colombeau framework: applications to solvability and regularity theory

In this article we introduce the notion of fundamental solution in the Colombeau context as an element of the dual \LL(\Gc(\R^n),\wt{\C}). After having proved the existence of a fundamental solution for a large class of partial differential operators with constant Colombeau coefficients, we investigate the relationships between fundamental solutions in \LL(\Gc(\R^n),\wt{\C}), Colombeau solvability and \G- and \Ginf-hypoellipticity respectively

LpL^p and Sobolev boundedness of pseudodifferential operators with non-regular symbol: a regularisation approach

In this paper we investigate $L^p$ and Sobolev boundedness of a certain class of pseudodifferential operators with non-regular symbols. We employ regularisation methods, namely convolution with a net of mollifiers (\rho_\eps)_\eps, and we study the corresponding net of pseudodifferential operators providing $L^p$ and Sobolev estimates which relate the parameter \eps with the non-regularity of the symbol

Pseudo-differential operators in algebras of generalized functions and global hypoellipticity

The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on $\mathbb{R}^n$

On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets

Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functionals and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis

Restricted Mobility Improves Delay-Throughput Trade-offs in Mobile Ad-Hoc Networks

In this paper we revisit two classes of mobility models which are widely used to repre-sent users ’ mobility in wireless networks: Random Waypoint (RWP) and Random Direction (RD). For both models we obtain systems of partial differential equations which describe the evolution of the users ’ distribution. For the RD model, we show how the equations can be solved analytically both in the stationary and transient regime adopting standard mathematical techniques. Our main contributions are i) simple expressions which relate the transient dura-tion to the model parameters; ii) the definition of a generalized random direction model whose stationary distribution of mobiles in the physical space corresponds to an assigned distribution