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KSU Math Department
Previous Prairie Seminar

Location and Dates:
Department of Mathematics
Kansas State University
Manhattan, Kansas
October 23, 2009
List of
Participants
Principal Lecturer:
Vanderbilt University
Title: Measure Theoretical Methods in Degenerate
and/or Singular Parabolic Partial Differential Equations Part 1 Part
2
Abstract: Degenerate
and/or singular parabolic equations arise in a number of physical
models, including porous media, nonnewtonianfluids, phase transition
etc. Their mathematical interestis in that their solutions exhibit a
behavior considerably different than solutions of the heat equation
and/or itsquasilinear counterparts.
Harnack and local
Hölder estimates are a suitable tool to explore the local behavior
of
their solutions as opposed to that of nondegenerate and/or
nonsingular equations. The theory, begun in the 80's, has recently
experienced a considerable growth due to the discovery of some novel
measure theoretical methods, of independent interest on their own
right.
We present a spectrum of these new techniques and apply them
to establish Harnack estimates in the intrinsic geometry generated by
both the PDEs and their solutions. Degenerate equations and singular
equations afford different forms of Harnack estimates. We discuss their
differences, their significance and their implications and point to
some open problems for subcritically singular equations.
Invited Speakers:
University of Pavia
Title: Continuity of the Saturation in the Flow of
Two Immiscible Fluids in a Porous Medium
Abstract: The weakly coupled system
v_t = div[A(v)grad v + B(v)] = V Â·
grad(C(v))
in E_T ,
div V = 0
where V = K(v)[grad u + e(v)] consists of an elliptic equation, and a
degenerate parabolic equation, and it arises in the theory of flows of
immiscible fluids in a porous medium.
The unknown functions u and v and the equations they satisfy, represent
the pressure and the saturation respectively, subject to Darcy's law
and the BuckelyLeverett coupling. Due to the empirical nature of these
laws no determination is possible on the structure of the degeneracy
exhibited by the system.
It is established that the saturation is a locally continuous function
in its space time domain of definition, irrespective of the nature of
the degeneracy of the principal part of the system.
Vincenzo Vespri
University of Florence
Title: A new regularity approach for weak solutions
of degenerate parabolic equations
Abstract: In order to prove the Hoelder regularity of
weak solutions to quasilinear degenerate (singular) parabolic equations
we use the same
approach originally introduced in recent papers by
DiBenedettoGianazzaV to obtain Harnack inequalities for
nonnegative solutions to theseequations. The new approach gives a
more
geometric and intuitive proof to the regularity and avoids covering and
alternative arguments.
Contributed Talks:
There will be time allocated for short contributed talks by
participants. Priority will be given to graduate students and those
in early stages of their carriers.
Participants interested in given a contributed talk and/or receiving
financial support should email Marianne Korten at
marianne@math.ksu.edu.
Organizers:
Estela
A. Gavosto, KU
Marianne Korten, KSU
Charles
Moore, KSU
Rodolfo
H. Torres, KU
Contact Information:
marianne@math.ksu.edu
