Shigeaki Koike - Nonlinear Partial Differential Equations for Future Applications: Sendai, Japan, July 10–28 and October 2–6, 2017

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The proceedings contain several original surveys by invited speakers in a series of workshops entitled Partial Differential Equation and Future Applications, which was organized by Tohoku Forum for Creativity (TFC for short) supported in Tohoku University from July 2017 to October 2017, and also some research papers in related fields. The TFC program has started since 2013 and is evolving over various research fields on natural and human sciences. The above title Nonlinear Partial Differential Equations for Future Applications is one of the thematic programs in TFC. In our program, we focussed on nonlinear partial differential equations arising in fluid mechanics, reaction diffusion, optimal control, modern physics, material sciences, and geometry. Furthermore, in order to search for new applications, we invited experts from other areas.

In this note, we give an introduction to the concept of maximal -regularity as a method to solve nonlinear partial differential equations. We first define maximal regularity for autonomous and non-autonomous problems and describe the connection to Fourier multipliers and -boundedness. The abstract results are applied to a large class of parabolic systems in the whole space and to general parabolic boundary value problems. For this, both the construction of solution operators for boundary value problems and a characterization of trace spaces of Sobolev spaces are discussed. For the nonlinear equation, we obtain local in time well-posedness in appropriately chosen Sobolev spaces. This manuscript is based on known results and consists of an extended version of lecture notes on this topic.