23 research outputs found
First-order hyperbolic pseudodifferential equations with generalized symbols
We consider the Cauchy problem for a hyperbolic pseudodifferential operator
whose symbol is generalized, resembling a representative of a Colombeau
generalized function. Such equations arise, for example, after a
reduction-decoupling of second-order model systems of differential equations in
seismology. We prove existence of a unique generalized solution under log-type
growth conditions on the symbol, thereby extending known results for the case
of differential operators with generalized functions as coefficients
Wave breaking of periodic solutions to the Fornberg-Whitham equation
Based on recent well-posedness results in Sobolev (or Besov spaces) for
periodic solutions to the Fornberg-Whitham equations we investigate here the
questions of wave breaking and blow-up for these solutions. We show first that
finite maximal life time of a solution necessarily leads to wave breaking.
Second, we prove that for a certain class of initial wave profiles the
corresponding solutions do indeed blow-up in finite time
Microlocal properties of basic operations in Colombeau algebras
The Colombeau algebra of generalized functions allows to unrestrictedly carry
out products of distributions. We analyze this operation from a microlocal
point of view, deriving a general inclusion relation for wave front sets of
products in the algebra. Furthermore, we give explicit examples showing that
the given result is optimal, i.e. its assumptions cannot be weakened. Finally,
we discuss the interrelation of these results with the concept of pullback
under smooth maps.Comment: LaTeX, 18 page
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
Elliptic regularity and solvability for partial differential equations with Colombeau coefficients
The paper addresses questions of existence and regularity of solutions to
linear partial differential equations whose coefficients are generalized
functions or generalized constants in the sense of Colombeau. We introduce
various new notions of ellipticity and hypoellipticity, study their
interrelation, and give a number of new examples and counterexamples. Using the
concept of \G^\infty-regularity of generalized functions, we derive a general
global regularity result in the case of operators with constant generalized
coefficients, a more specialized result for second order operators, and a
microlocal regularity result for certain first order operators with variable
generalized coefficients. We also prove a global solvability result for
operators with constant generalized coefficients and compactly supported
Colombeau generalized functions as right hand sides
Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities
We characterize microlocal regularity of Colombeau generalized functions by
an appropriate extension of the classical notion of micro-ellipticity to
pseudodifferential operators with slow scale generalized symbols. Thus we
obtain an alternative, yet equivalent, way to determine generalized wave front
sets, which is analogous to the original definition of the wave front set of
distributions via intersections over characteristic sets. The new methods are
then applied to regularity theory of generalized solutions of
(pseudo-)differential equations, where we extend the general noncharacteristic
regularity result for distributional solutions and consider propagation of
generalized singularities for homogeneous first-order hyperbolic equations
Distributional solution concepts for the Euler-Bernoulli beam equation with discontinuous coefficients
We study existence and uniqueness of distributional solutions to the
differential equation of the Euler-Bernoulli rod with discontinuous
coefficients and right-hand side. Upon checking the validity of a solution the
occurring products of singular coefficients with the distributional solution
have no obvious meaning. When interpreted on the most general level of the
so-called hierarchy of distributional products, it turns out that existence of
a solution forces a minimum regularity to hold. Curiously, the choice of the
distributional product concept is thus incompatible with the possibility of
having a discontinuous displacement function as a solution. We also give
conditions for unique solvability.Comment: 3 figure