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Analysis and PDE Seminar (Master's Exam)

Date:
-
Location:
745 Patterson Office Tower
Speaker(s) / Presenter(s):
Joseph Lindgren, University of Kentucky

Title:  Continuity of a Right Inverse of the Divergence Operator

Abstract:  The divergence of a vector field u = (u1, ... , un), often written as div u = L,j=1  or V · u, is·a well-known  quantity in vector  calculus, rneasuring  'sinks' and 'sources' Of u. In fluid dynamics, this quantity manifests  itself  in the compression  and rarefaction  of a fluid whose velocity is given by u. The incompressibility  condition on _such a fluid, formulated as div u = 0, is well-known. A more general case, div u = J, is naturally a PDE of interest.

Given J E L§(D), a right inverse of divergence can be constructed from a singular integral kernel and used to solve div u = f. In my talk, I present a proof (due to Ricardo G.Duran) that this right inverse is bounded from (fl) to HJ (l1r using the Fourier transform and elementary techniques.