71 research outputs found
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
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
and Sobolev boundedness of pseudodifferential operators with non-regular symbol: a regularisation approach
In this paper we investigate 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 and Sobolev estimates which relate the parameter
\eps with the non-regularity of the symbol
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
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