1. Introduction & Purposes

The main purpose of this lecture series is to demonstrate how to employ the maximal regularity for the parabolic-type linear equations together with semigroup theory to solve quasilinear equations.

2. Outline & Descriptions

The fundamental equations describing the motion of incompressible viscous fluids are the Navier-Stokes equations, which are semilinear equations. In Kato [2], the unique existence of strong solutions of the Navier-Stokes equations is proved by the Lp-Lq decay estimates of the Stokes semigroup. On the other hand, quasilinear equations also appear in fluid dynamics. For instance, free boundary problems for the Navier-Stokes equations and a Q-tensor model, which describes the nematic liquid crystal flows, are quasilinear equations. It is difficult to solve such quasilinear equations by semigroup theory only because of regularity-loss. One of the ideas to overcome this difficulty is to prove the maximal regularity for the parabolic-type linear equations based on the operator-valued Fourier multiplier theorem proved by Weis (2001). In order to apply Weis’s theorem, our main task is to prove the R-boundedness of the solution operators for the resolvent problem.

In this lecture, I first introduce the definitions of the maximal regularity and the R-boundedness, as well as Weis’s theorem. Furthermore, I explain that the maximal regularity and the resolvent estimates follow from the R-boundedness using the heat equation as an example. Next, we discuss the global solvability of the parabolic-type equations for small initialdata in the half-space. In particular, we consider the nonlinear heat equation as a toy model. The key issue is to obtain the weighted estimates of the higherorder terms for the linear equation by using the maximal regularity and the analytic semigroup theory.

3. Registration

4. Online Meeting Link

https://forms.gle/NCVDsxeZtfWsHL3HA