Nonlinear Partial Differential Equations for Future Applications

Inhaltsverzeichnis

1. Frontmatter

2. An Introduction to Maximal Regularity for Parabolic Evolution Equations

   Robert Denk

   Abstract

   In this note, we give an introduction to the concept of maximal \(L^p\)-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 \(\mathcal {R}\)-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.

3. On Stability and Bifurcation in Parallel Flows of Compressible Navier-Stokes Equations

   Yoshiyuki Kagei

   Abstract

   The stability analysis of parallel flows of the compressible Navier-Stokes equations is overviewed. The asymptotic behaviour of solutions is firstly considered for small Reynolds and Mach numbers. An instability result of the plane Poiseuille flow is then given for a certain range of Reynolds and Mach numbers, together with a result of the bifurcation of wave trains from the plane Poiseuille flow.

4. Uniform Regularity for a Compressible Gross-Pitaevskii-Navier-Stokes System

   Jishan Fan, Tohru Ozawa

   Abstract

   Uniform regularity estimates are proved for a compressible Gross-Pitaevskii-Navier-Stokes system in \(\mathbb {T}^n\) with \(n\ge 3\).

5. Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity

   Takayoshi Ogawa

   Abstract

   We consider singular limit problems of the Cauchy problem for the Patlak-Keller-Segel equation and related problems appeared in the theory of medical and biochemical dynamics. It is shown that the solution to the Patlak-Keller-Segel equation in a scaling critical function class converges strongly to a solution of the drift-diffusion system of parabolic-elliptic equations as the relaxation time parameter \(\tau \rightarrow \infty \). Analogous problem related to the Chaplain-Anderson model for cancer growth model is also presented as well as Arzhimer’s model that involves the multi-component drift-diffusion system. For the proof, we use generalized maximal regularity for the heat equations and systematically apply embeddings between the interpolation spaces shown in [40, 41]. The argument requires generalized version of maximal regularity developed in [40, 61], for the Cauchy problem of the heat equation.

6. HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control

   Andrzej Święch

   Abstract

   The paper is an extended version of lecture notes from a mini-course given by the author in the workshop Optimal Control and PDE in Tohoku University in 2017. The main objective of the lecture notes is to give a short but rigorous introduction to the dynamic programming approach to stochastic optimal control problems. The manuscript discusses, among other things, the classical necessary and sufficient conditions for optimality, properties of the value function, and it contains a proof of the dynamic programming principle, and a proof that the value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

7. Regularity of Solutions of Obstacle Problems –Old & New–

   Shigeaki Koike

   Abstract

   Two kinds of machinery to show regularity of solutions of bilateral/unilateral obstacle problems are presented. Some generalizations of known results in the lit- erature are included. Several important open problems in the topics are given.

8. High-Energy Eigenfunctions of the Laplacian on the Torus and the Sphere with Nodal Sets of Complicated Topology

   A. Enciso, D. Peralta-Salas, F. Torres de Lizaur

   Abstract

   Let \(\Sigma \) be an oriented compact hypersurface in the round sphere \(\mathbb {S}^n\) or in the flat torus \(\mathbb {T}^n\), \(n\ge 3\). In the case of the torus, \(\Sigma \) is further assumed to be contained in a contractible subset of \(\mathbb {T}^n\). We show that for any sufficiently large enough odd integer N there exists an eigenfunctions \(\psi \) of the Laplacian on \(\mathbb {S}^n\) or \(\mathbb {T}^n\) satisfying \(\Delta \psi =-\lambda \psi \) (with \(\lambda =N(N+n-1)\) or \(N^2\) on \(\mathbb {S}^n\) or \(\mathbb {T}^n\), respectively), and with a connected component of the nodal set of \(\psi \) given by \(\Sigma \), up to an ambient diffeomorphism.