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 Prairie Analysis Seminar 2009

  


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Previous Prairie  Seminar

Location and Dates:

Department of Mathematics
Kansas State University
Manhattan, Kansas
October 2-3, 2009

List of Participants


Principal Lecturer:

Emmanuele DiBenedetto

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, non-newtonianfluids, phase transition etc. Their mathematical interestis in that their solutions exhibit a behavior considerably different than solutions of the heat equation and/or itsquasi-linear counterparts.

Harnack and local Hölder estimates are a suitable tool to explore the local behavior of their solutions as opposed to that of non-degenerate and/or non-singular 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 sub-critically singular equations.


Invited Speakers:

Ugo Gianazza

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 · 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 Buckely-Leverett 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 DiBenedetto-Gianazza-V 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 e-mail 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

This conference is supported
by the National Science Foundation (NSF) and by the KSU Mathematics Department through the Isidore and Hilda Dressler Endowment for the Enrichment of Mathematics. 
Thanks to our new NSF grant we will be able to assist all speakers who need travel and housing expenses.


The Prairie Analysis Seminar is a joint project of the Department of Mathematics of  Kansas State University  and the Department of Mathematics of the University of Kansas.

 
  


The picture of the Kansas Prairie is a courtesy of the Kansas Geological Survey.