
Regularity of solutions to partial differential equations
(PDEs) is one of the central research issues in classical as well as modern
analysis of partial differential operators (PDOs). By now, a far reaching
regularity theory with powerful analytic methods is established in case of
operators with (relatively) smooth coefficients. Furthermore, proofs of
existence and uniqueness of solutions often rely substantially on such
smoothness assumptions.
One of the "key technologies" of modern regularity theory of PDEs is
microlocal analysis. It combines information on the spatial concentration of
singularities of a distributional solution with their spectral properties in
the sense of Fourier analysis, leading to the notion of wave front set. In the
context of bicharacteristics of PDOs microlocal analysis has become the
crucial method in the study of propagation of singularities, which occur in a
solution due to singularities in the given data (i.e., righthand side or the
initial values). Refined analytical techniques in determining microlocal
regularity are based on the calculus of pseudodifferential operators and
Fourier integral operators and their mapping properties with respect to wave
front sets.
Singularities of the given data for a PDE with smooth coefficients are
typically reflected by a corresponding loss of regularity in the solution.
This may limit the applicability of nonlinear operations, such as
multiplications, involving the solution and distributional objects. In
particular, this difficulty arises already when discussing solution concepts
if we remove the smoothness constraints on the coefficients of the PDO.
Situations of this kind appear naturally, for example, in models from the
physics of wave propagation in complex media, although even in very simple
cases the mere question of existence of solutions lies beyond the reach of
distribution theory. One may then employ theories of algebras of generalized
functions, in which such PDE problems become welldefined and are subject to a
consistent extension of solvability theory. Specifically, with the type of
partial (or pseudo) differential operators appearing in models of prominent
applications, e.g., mathematical geophysics, a rigorous analysis has to
concurrently allow for
 a wide range of singular coefficients (or symbols),
 highly singular initial values and source terms,
 generalized solutions,
 nonlinear differentialalgebraic operations thereon.
Within Colombeau algebras an intrinsic notion of regularity exists, which is
compatible with the concept of smoothness of (embedded) distributions. By
allowing for more general objects as solutions to PDEs, a natural further step
is the extension of microlocal analysis to the context of generalized function
algebras. In Colombeau theory this has been started in recent years, with
successful case studies in applications to hyperbolic PDEs with discontinuous
coefficients, theories of microhypoellipticity for linear PDOs with
generalized function coefficients, propagation of singularities, and
microlocal properties of basic nonlinear operations. On a refined level of
quantitative regularity analysis, the influence of coefficient singularities
on the solutions of PDEs is attempted to be measured in terms of generalized
HölderZygmund continuity. Here, the socalled HölderZygmund classes of
temperate distributions can be embedded into the broader nonlinear context of
Colombeau algebras with an intrinsic, yet compatible, extension of their
regularity scales.
Primary new challenges in this whole analytical program are due to the
interaction of the coefficient singularities with propagating singularities in
the solution. A qualitatively new phenomenon to be taken into account here is
the, classically unexpected, sensitivity of regularity properties to the lower
order terms in the operators. A future comprehensive theory and the systematic
development of refined tools in this direction has high potential for
immediate applications in many models of mathematical physics (e.g. wave
equations on singular curved space times, electrodynamics with interaction of
singular charge distributions in complex media), and all problems involving
wave propagation through highly irregular media (such as (non)linear
elasticity, ocean acoustics, and seismology).

