Infinitely narrow soliton solutions to systems of conservation laws.
Kernel theorems in spaces of generalized functions
In analogy to the classical isomorphism between ((ℝⁿ), and (resp. and ), we show that a large class of moderate linear mappings acting between the space of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space of Colombeau rapidly decreasing generalized functions and the space of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of (resp. ). The main novelty is to use accelerated...
Linear and bilinear Hilbert transform.
Manifold-valued generalized functions in full Colombeau spaces
We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.
Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity.
On a nonlinear Peetre's theorem in full Colombeau algebras
We adapt a nonlinear version of Peetre's theorem on local operators in order to investigate representatives of nonlinear generalized functions occurring in the theory of full Colombeau algebras.
On Generalized Models and Singular Products of Distributions in Colombeau Algebra G(R)
MSC 2010: 46F30, 46F10Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of Colombeau that model such singularities. Moreover, we evaluate some products of singularity-modelling generalized functions whenever the result admits an associated distribution.
On generalized solutions to the wave equation in canonical form [Book]
On hypoellipticity in .
On hypoellipticity in g
On introduction of two diffeomorphism invariant Colombeau algebras
Equivalent definitions of two diffeomorphism invariant Colombeau algebras introduced in [7] and [5] (Grosser et al.) are listed and some new equivalent definitions are presented. The paper can be treated as tools for proving in [8] the equality of both algebras.
On systems of linear algebraic equations in the Colombeau algebra
From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra of generalized real numbers. It is worth mentioning that the algebra is not a field.
On the sign of Colombeau functions and applications to conservation laws
A generalized concept of sign is introduced in the context of Colombeau algebras. It extends the sign of the point-value in the case of sufficiently regular functions. This concept of generalized sign is then used to characterize the entropy condition for discontinuous solutions of scalar conservation laws.
On two-points boundary value problems for ordinary nonlinear differential equations of the fourth order in the Colombeau algebra
One-dimensional adhesion model for large scale structures.
Pseudodifferential Operators
Pseudodifferential operators with generalized symbols and regularity theory.
Recent progress in special Colombeau algebras: geometry, topology, and algebra
Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of modules over the ring of generalized numbers, and algebraic aspects of Colombeau theory. Some open problems are given and directions of further research are outlined.
Regular nonlinear generalized functions and applications
Regularized derivatives in a 2-dimensional model of self-interacting fields with singular data.