다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

In recent years two nonlinear dispersive partial differential equations have attracted a lot of attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations. In particular, they accomodate wave breaking phenomena.

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The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.

Solitons and the Inverse Scattering Transform

The course aims at presenting an introduction to the subject
of singularity formation in nonlinear evolution problems usually known as
blowup. In short, we are interested in the situation where, starting from a
smooth initial configuration, and after a first period of classical evolution, the
solution (or in some cases its derivatives) becomes infinite in finite time due
to the cumulative effect of the nonlinearities. We concentrate on problems involving
differential equations of parabolic type, or systems of such equations. 

A first part of the course introduces the subject and discusses the classical
questions addressed by the blow-up theory. We propose a list of main questions
that extends and hopefully updates on the existing literature. We also
introduce extinction problems as a parallel subject. 

In the main bulk of the paper we describe in some detail the developments
in which we have been involved in recent years, like rates of growth and pattern
formation before blow-up, the characterization of complete blow-up, the occurrence
of instantaneous blow-up (i.e., immediately after the initial moment)
and the construction of transient blow-up patterns (peaking solutions), as well
as similar questions for extinction. 

In a final part we have tried to give an idea of interesting lines of current
research. The survey concludes with an extensive list of references. Due to
the varied and intense activity in the field both aspects are partial, and reflect
necessarily the authors' tastes.

The problem of blow-up in nonlinear parabolic equations

This paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the Degasperis-Procesi equation on the line. Global strong solutions to the equation are determined for a class of initial profiles. On the other hand, it is shown that the first blow-up can occur only in the form of wave-breaking. A new wave-breaking mechanism for solutions is described in detail and two results of blow-up solutions with certain initial profiles are established.

Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation

Global weak solutions and blow-up structure for the Degasperis–Procesi equation

On the global existence and wave-breaking criteria for the two-component Camassa-Holm system

In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.

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Wave-Breaking and Peakons for a Modified Camassa-Holm Equation

In this paper we mainly study the problem of the development of singularities for solutions to the periodic Degasperis-Procesi equation. Firstly, we show that the first blow-up of strong solution to the equation must occur only in the form of wave breaking and shock waves possibly appear afterwards. Secondly, we established two new blow-up results. Thirdly, we investigate the blow-up rate for all non-global strong solutions and determine the blow-up set of blowing-up strong solutions to the equation for a large class of initial data. We finally give an explicit example of weak solutions to the equation, which may be considered as periodic shock waves.

Yue Liu

Papers

Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation

Global weak solutions and blow-up structure for the Degasperis–Procesi equation

On the global existence and wave-breaking criteria for the two-component Camassa-Holm system

Wave-Breaking and Peakons for a Modified Camassa-Holm Equation

Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation