1:00 pm Wednesday, March 30, 2005
Analysis seminar: Optimal convergence rates for the fastest conservative nonlinear diffusions by Robert McCann (University of Toronto) in RLM 10.176
In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features --- such as size and location --- will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Joint work with Yong Jung Kim (University of California at Riverside and KAIST) addresses this issue for the fastest conservative nonlinearities in a model problem known as the fast diffusion equation $ u_t = \Delta(u^m), \qquad (n-2)_+/n < m \le n/(n+2), \quad u,t \ge 0, \quad x \in &ob;\bf R&cb;^n, $ which governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. For other values of the parameter , this equation has been used to model heat transport, population spreading, fluid seepage, curvature flows, and avalanches in sandpiles. For the fastest conservative nonlinearities, we develop a potential theoretic comparison technique which establishes the sharp conjectured power law rate of decay $1/t$ uniformly in relative error, and in weaker norms such as $L^1(&ob;\bf R&cb;^n)$. Submitted by
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