Mi-Ho Giga

Nonlinear Partial Differential Equations

Nonlinear partial differential equations (PDE’s) play a central role in the modelling of a great number of phenomena, ranging from Theoretical Physics, Astrophysics and Chemistry to Economy, Medicine and Population Dynamics. Among the phenomena encountered, the diffusion processes play a fundamental role. In the last 25 years a great deal of work has been devoted to semi-linear equations. In this type of equations, the interaction between a linear partial differential operator and the superlinear reaction term (source or absorption) can be understood, at least in part, thanks to the linear theory. One of the main observations, valid in the most interesting cases is the existence of critical exponents, for example the Fujita exponents for nonlinear heat equation, the Pohozaev exponent, the Sobolev exponent. In general, the first results (blow-up, global estimates, decay estimate), were proved up to a critical exponent by more or less easy applications of linear energy estimates, linked to ODE techniques. Then the study of what happens if the exponent is critical or even supercritical involves a very delicate analysis, often based upon very sharp linear estimates or even, in some cases a completely new approach. The main areas of research in the current proposal include the description of singular phenomenon: blow-up, singularities, problems with singular measure data in a large class of reaction diffusion equations, with quasi-linear or fully nonlinear diffusion and strong reaction.

SELF-SIMILAR EXPANDING SOLUTIONS IN A SECTOR FOR A CRYSTALLINE FLOW ∗

For a given sector a self-similar expanding solution to a crystalline flow is constructed. The solution is shown to be unique. Because of self-similarity the problem is reduced to solve a system of algebraic equations of degree two. The solution is constructed by a method of continuity and obtained by solving associated ordinary differential equations. The self-similar expanding solution is useful to construct a crystalline flow from an arbitrary polygon not necessarily admissible.

Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis

A numerical method for obtaining a crystalline flow starting from a general polygon is presented. A crystalline flow is a polygonal flow and can be regarded as a discrete version of a classical curvature flow. In some cases, new facets may be created instantaneously and their facet lengths are governed by a system of singular ordinary differential equations (ODEs). The proposed method solves the system of the ODEs numerically by using expanding selfsimilar solutions for newly created facets. The computation method is applied to a multi-scale analysis of a contour figure.

A Comparison Principle for Singular Diffusion Equations with Spatially Inhomogeneous Driving Force for Graphs

We introduce the notions of viscosity super- and subsolutions suitable for singular diffusion equations of non-divergence type with a general spatially inhomogeneous driving term. In particular, the viscosity super- and subsolutions support facets and allow a possible facet bending. We prove a comparison principle by a modified doubling variables technique. Finally, we present examples of viscosity solutions. Our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph-like curve.

On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force

A general anisotropic curvature flow equation with singular interfacial energy and spatially inhomogeneous driving force is considered for a curve given by the graph of a periodic function. We prove that the initial value problem admits a unique global-in-time viscosity solution for a general periodic continuous initial datum. The notion of a viscosity solution used here is the same as proposed by Giga, Giga and Rybka, who established a comparison principle. We construct the global-in-time solution by careful adaptation of Perron’s method.

A Computation of a Crystalline Flow Starting from Non-admissible Polygon Using Expanding Selfsimilar Solutions

A numerical method for obtaining a crystalline flow from a given polygon is presented. A crystalline flow is a discrete version of a classical curvature flow. In a crystalline flow, a given polygon evolves, and it remains polygonal through the evolving process. Each facet moves keeping its normal direction, and the normal velocity is determined by the length of the facet. In some cases, a set of new facets sprout out at the very beginning of the evolving process. The facet length is governed by a system of singular ordinary differential equations. The proposed method solves the system of ODEs, and obtain the length of each new facet, systematically. Experimental results show that the method obtains a crystalline flow from a given polygon successfully.

Self-Similar Solutions for Various Equations

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.

Convergence Theorems in the Theory of Integration

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.

Behavior Near Time Infinity of Solutions of the Vorticity Equations

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.

Behavior Near Time Infinity of Solutions of the Heat Equation

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.