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## Über dieses Buch

The purpose of this book is to present typical methods (including rescaling methods) for the examination of the behavior of solutions of nonlinear partial di?erential equations of di?usion type. For instance, we examine such eq- tions by analyzing special so-called self-similar solutions. We are in particular interested in equations describing various phenomena such as the Navier– Stokesequations.Therescalingmethod describedherecanalsobeinterpreted as a renormalization group method, which represents a strong tool in the asymptotic analysis of solutions of nonlinear partial di?erential equations. Although such asymptotic analysis is used formally in various disciplines, not seldom there is a lack of a rigorous mathematical treatment. The intention of this monograph is to ?ll this gap. We intend to develop a rigorous mat- matical foundation of such a formalasymptotic analysis related to self-similar solutions. A self-similar solution is, roughly speaking, a solution invariant under a scaling transformationthat does not change the equation. For several typical equations we shall give mathematical proofs that certain self-similar solutions asymptotically approximate the typical behavior of a wide class of solutions. Since nonlinear partial di?erential equations are used not only in mat- matics but also in various ?elds of science and technology, there is a huge variety of approaches. Moreover,even the attempt to cover only a few typical ?elds and methods requires many pages of explanations and collateral tools so that the approaches are self-contained and accessible to a large audience.

## Inhaltsverzeichnis

### 1. Behavior Near Time Infinity of Solutions of the Heat Equation

Abstract
Partial differential equations that include time derivatives of unknown functions are often called evolution equations. One important problem about evolution equations is to analyze the behavior of solutions at sufficiently large time. Such problems have been studied extensively from various points of view. Here, we are concerned with the initial value problem of the heat equation, which is a linear partial differential equation. It is not difficult to determine the asymptotic behavior of solutions of the heat equation near time infinity, and we introduce two methods to analyze its behavior. The first method is based on a representation formula of the solution of the equation directly; here we shall give a proof, which is short and easy. This method is sufficient to obtain the result for the heat equation; however, it may not apply to nonlinear problems in general, since we do not expect that solutions for nonlinear problems usually have a representation formula. The second method is based on a scaling transformation of the solution using the structure of the heat equation. By this method we shall give a proof of the behavior of solutions again. The proof by the second method is longer and it seems to be inefficient, but its idea can apply to nonlinear problems, which we study in Chapter 2 and in several parts of Chapter 3. To be familiar with the method, we give the proof for the heat equation, which is easier and more transparent to handle than nonlinear problems.
Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

### 2. Behavior Near Time Infinity of Solutions of the Vorticity Equations

Abstract
The Navier–Stokes equations are famous as fundamental equations of fluid mechanics and have been well studied as typical nonlinear partial differential equations in mathematics. It is not too much to say that various mathematical methods for analyzing nonlinear partial differential equations have been developed through studies of the Navier–Stokes equations. There have been many studies of the behavior of solutions of the Navier–Stokes equations near time infinity. In this chapter, as an application of the previous section, we study the behavior of the vorticity of a two dimensional flow near time infinity. In particular, we study whether or not the vorticity converges to a self-similar solution.
Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

### 3. Self-Similar Solutions for Various Equations

Abstract
We first present for the porous medium equation, a typical nonlinear degenerate diffusion equation, that its (forward) self-similar solution well describes asymptotic behavior of solutions, as is observed for the heat equation, without proof. We next explain that it is important to classify backward self-similar solutions in order to analyze behavior of solutions near singularities for the axisymmetric mean curvature flow equation as an example. In what follows, a self-similar solution is regarded as a stationary solution of the equation written with similarity variables. Convergence behavior of a solution of the equation to its stationary corresponds to the asymptotic behavior of the solution of the original equation near singularities. We give an outline of the proof of convergence and mention that a monotonicity formula plays a key role. Moreover, we give a simple proof of uniqueness of the stationary solutions, i.e., the backward self-similar solutions of the original equation. The proof is simpler and easier than that in the literature. We remark that the method using similarity variables is applicable, to some extent, to other diffusion equations such as semilinear heat equations and harmonic map flow equations. Finally, we note that the existence of forward self-similar solutions has also been proved for nonlinear equations of nondiffusion type.
Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

### 4. Various Properties of Solutions of the Heat Equation

Abstract
Here we establish the tools used in Chapter 1 in order to analyze the asymptotic behavior of solutions for the heat equation. We start by deriving L p -L q estimates for solutions and their derivatives and the uniqueness theorem for weak solutions. For this purpose, we prepare the Young inequality for convolution, which has a wide range of applications. Furthermore, algebraic and commutativity properties, in particular concerning differentiation of convolutions, are stated. These properties turn out to be helpful in the proof of smoothness for t > 0 for the solution of the heat equation in Chapter 1. Next, we consider the continuity of the solution at time t= 0, in the case that the initial value is continuous. Continuity is proved by a fairly general method that applies to a large class of equations.
Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

### 5. Compactness Theorems

Abstract
In this section we prove the Ascoli–Arzelà-type compactness theorem introduced in §1.3.2. The theorem is fundamental since a variety of compactness results on various function spaces follows. Here we give a detailed proof, since the case that the domain of definition of functions is not compact is usually not contained in elementary course books.
Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

### 6. Calculus Inequalities

Abstract
In this section we introduce the Nash inequality and its generalized version, the Gagliardo–Nirenberg inequality. Roughly speaking, these inequalities provide estimates for an integral of a function by its derivatives, a tool that is very helpful not only in the analysis of the vorticity equations as demonstrated in Chapter 2, but in the analysis of nonlinear PDE in general.
Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

### 7. Convergence Theorems in the Theory of Integration

Abstract
This section gives a summary of some elementary facts used frequently throughout this book, and can be regarded as an appendix. In particular, we consider sufficient conditions for the interchange of integration and limit operations. In detail, we discuss a result on uniform convergence, the dominated convergence theorem, the bounded convergence theorem, Fatou’s lemma, and the monotone convergence theorem from the points of view of both Lebesgue integration theory and Riemann integration theory. Note that these are well-known results; hence we will be brief in details. For the proof of the monotone convergence theorem and Fubini’s theorem we merely refer to the appropriate literature.
Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

### Backmatter

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