Nonlinear partial differential equations of mixed elliptic-hyperbolic type in geometry and related areas
几何及相关领域混合椭圆双曲型非线性偏微分方程
基本信息
- 批准号:2271985
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:英国
- 项目类别:Studentship
- 财政年份:2019
- 资助国家:英国
- 起止时间:2019 至 无数据
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Whilst the theory for purely elliptic or purely hyperbolic partial differential equations is relatively well understood, the theory for nonlinear equations of mixed elliptic-hyperbolic type is much less developed. However, equations of mixed elliptic-hyperbolic type have important applications in geometry, as well as a wide range of related areas such as the mathematical study of fluid mechanics, solid mechanics, and elasticity. Developing a greater understanding of equations of mixed type is therefore critical to progress in a number of fields. We illustrate the role of these equations by now giving two examples of problems which can be reduced to the study of equations of mixed elliptic-hyperbolic type, each of which contains a number of open problems and is the subject of current research.Given a two-dimensional Riemannian manifold, an important problem in differential geometry concerns whether the metric can be realised as an isometric immersion into R-3. This requires us to solve the Gauss-Codazzi equations, which relate the coefficients of the second fundamental form to those of the metric. For surfaces of positive Gauss curvature these can be formulated as an elliptic boundary value problem, whilst for surfaces of negative Gauss curvature we instead have a hyperbolic initial or initial-boundary value problem. In a surface for which the Gauss curvature changes signs, we therefore have to solve an initial-boundary value problem of mixed elliptic-hyperbolic type, for which the available results are much weaker than in the preceding cases. The equivalent problems for higher dimensional manifolds or pseudo-Riemannian metrics can also be studied similarly, and have applications to areas of physics such as relativity.We also give an example outside geometry, arising from the study of transonic potential flows. We consider a plane shock wave hitting an angled wedge head-on and wish to study the resulting reflection-diffraction pattern, modelled mathematically as a global entropy solution of the two-dimensional Riemann problem for hyperbolic conservation laws. The equations are hyperbolic in the far field and elliptic near the wedge vertex, so again we have a nonlinear PDE of mixed type. We are then interested in the potentially quite complicated structure of the reflection-diffraction patterns, and in how these patterns depend on the wedge angle as well as various physical parameters.The aim of the research will therefore be to study mixed elliptic-hyperbolic nonlinear PDEs, either in the context of the immersion problem from differential geometry or in a related area of geometry, mechanics or other areas of mathematics in which they arise.This project falls within the EPSRC Mathematical Analysis research area.
虽然纯椭圆或纯双曲型偏微分方程的理论相对较好理解,但混合椭圆-双曲型非线性方程的理论发展得较少。然而,椭圆-双曲混合型方程在几何学中有重要的应用,以及广泛的相关领域,如流体力学、固体力学和弹性力学的数学研究。因此,发展对混合型方程的更好理解对于许多领域的进展至关重要。我们说明了这些方程的作用,现在给出两个例子的问题,可以减少到研究方程的混合椭圆双曲型,其中每个包含一些开放的问题,是目前的研究主题。鉴于一个二维黎曼流形,一个重要的问题,在微分几何的度量是否可以实现为等距浸入R-3。这就要求我们求解高斯-科达齐方程,该方程将第二基本形式的系数与度规的系数联系起来。对于正高斯曲率的曲面,这些可以用椭圆边值问题来表示,而对于负高斯曲率的曲面,我们则有一个双曲初值或初边值问题。在高斯曲率改变符号的曲面中,我们因此必须求解混合椭圆-双曲型的初边值问题,对于这种情况,现有的结果比前面的情况弱得多。高维流形或伪黎曼度量的等价问题也可以类似地研究,并应用于物理学领域,如相对论。我们还给出了一个几何学之外的例子,来自跨音速势流的研究。我们考虑一个平面冲击波击中一个角度的楔形头上,并希望研究由此产生的反射衍射图案,数学建模为一个全局熵解的二维黎曼问题的双曲守恒律。方程在远场是双曲型的,而在楔形顶点附近是椭圆型的,所以我们又得到了一个混合型的非线性偏微分方程。然后,我们感兴趣的是潜在的相当复杂的结构的反射-衍射图案,以及在这些图案如何依赖于楔角以及各种物理参数,因此,研究的目的将是研究混合椭圆-双曲非线性偏微分方程,无论是在浸入问题的背景下,从微分几何或在相关领域的几何,力学或它们出现的其他数学领域。该项目属于EPSRC数学分析研究领域。福尔斯。
项目成果
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其他文献
吉治仁志 他: "トランスジェニックマウスによるTIMP-1の線維化促進機序"最新医学. 55. 1781-1787 (2000)
Hitoshi Yoshiji 等:“转基因小鼠中 TIMP-1 的促纤维化机制”现代医学 55. 1781-1787 (2000)。
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LiDAR Implementations for Autonomous Vehicle Applications
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2021 - 期刊:
- 影响因子:0
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吉治仁志 他: "イラスト医学&サイエンスシリーズ血管の分子医学"羊土社(渋谷正史編). 125 (2000)
Hitoshi Yoshiji 等人:“血管医学与科学系列分子医学图解”Yodosha(涉谷正志编辑)125(2000)。
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Effect of manidipine hydrochloride,a calcium antagonist,on isoproterenol-induced left ventricular hypertrophy: "Yoshiyama,M.,Takeuchi,K.,Kim,S.,Hanatani,A.,Omura,T.,Toda,I.,Akioka,K.,Teragaki,M.,Iwao,H.and Yoshikawa,J." Jpn Circ J. 62(1). 47-52 (1998)
钙拮抗剂盐酸马尼地平对异丙肾上腺素引起的左心室肥厚的影响:“Yoshiyama,M.,Takeuchi,K.,Kim,S.,Hanatani,A.,Omura,T.,Toda,I.,Akioka,
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