Speaker:  

Garrett Rea

Date of Talk:  

03/14/13

Affiliation:  

Department of Mathematics, Missouri Southern State University

Title:  

The Evolution of the Heat Equation from the 1800's Till Now

Starting in the 1800's much has been known about the diffusion of heat. We give a tour through historical developments in our understanding of the heat equation from the perspective of how the diffusion is modeled. This culminates in the current work.

In 1964, Jürgen Moser proved a Harnack inequality and Hölder continuity for weak solutions to certain parabolic operators in ℝⁿ. Since then, his approach has been adapted to many situations. We employ his iteration scheme to show a parabolic Harnack inequality and Hölder continuity of weak solutions of nonlinear parabolic operators of the form

Here u ∈ S^1,2_loc(ℝⁿ⁺¹), {X_i}^m_i=1 are smooth vector fields satisfying Hömander's condition, X_j* is the formal adjoint of X_j and A : ℝ^m → ℝ^m is a measurable function.

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