Fractional flows driven by subdifferentials
Prof. Goro Akagi, Mathematical Institute, Tohoku University
Sala conferenze IMATI-CNR, Pavia - Tuesday, March 6, 2018 h.15:00
Abstract. Nowadays, PDEs including various notions of fractional derivatives are vigorously studied. Indeed, in the last few decades, it turned out that fractional derivatives are related to some stochastic processes (e.g., Lévy process) and they are useful to describe non-classical dissipative phenomena (e.g., anomalous diffusion). In particular, PDEs including the fractional Laplacian have been energetically studied by many authors. On the other hand, frac- tional variants of time-derivative (e.g., Riemann-Liouville and Caputo derivatives) exhibit a different aspect from the fractional Laplacian and they have not yet been fully studied (particularly compared to classical derivative), although some notion was already discussed by Leibniz.
This talk is concerned with some extension of the so-called Brézis-K ̄omura theory to (nonlinear) evolution equations including fractional derivatives in time. More precisely, we shall consider
∂tα [u(t)−u0]+∂φ(u(t)) ∋ f(t), 0 < t < T,
where φ : H → [0, +∞] is a proper, lower semicontinuous, convex functional defined on a Hilbert space H, f : (0,T) → H and u0 ∈ H are prescribed data with T > 0 and ∂tαis the so-called Riemann-Liouville fractional derivative (in time) with 0 < α < 1 (i.e., subdiffusion case) and its precise definition will be given in this talk. The Riemann-Liouville derivative ∂tα is defined as a composition of classical derivative and convolution with a singular kernel. Hence we shall treat a more general setting. Evolution equations including nonlocal (time) derivatives have been studied in 1980s and 90s by several authors, and then, they have been recently studied by Zacher, Vergara and their collaborators. On the other hand, there seems no abstract theory for nonlinear evolution equations of gradient flow type, particularly, for strong solutions. In this talk, we shall present an abstract theory on fractional flows driven by subdifferential operators and also exhibit a couple of typical and simple applications to nonlinear PDEs.
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