Existence and blowup of solutions for nonlinear evolution equations and their numerical computations

非线性演化方程解的存在性、爆炸性及其数值计算

• 批准号: RGPIN-2019-05940
• 负责人: Chen, Shaohua
• 金额: $ 1.24万
• 依托单位: Cape Breton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675826
• 关键词: Existence; blowup; solutions; nonlinear; evolution

Nonlinear evolution equations have many applications in descriptions of various models, such as reaction-diffusion, activator-inhibitor, fluid and quantum mechanics and population biology. Many equations involve degenerate or singular terms and some kinds of blow-up properties which cause many challenging problems in global, blow-up and numerical computations. The objectives of this research program are to investigate the properties of global and blow-up solutions for nonlinear evolution equations both theoretically and numerically, including convergence to steady states. The expected results will include: 1. Introduce a new functional method to discuss the global and blow-up solutions for the compressible Euler equations with variable damping coefficient. Also study the asymptotic behaviors, blow-up rate and blow-up time and steady states to the equations. 2. Investigate the existence and blow-up of solutions to higher order nonlinear Schrodinger equations. 3. Modify the existing algorithms for moving mesh methods and other adaptive grid methods to numerically solve compressible Euler equations, higher order nonlinear Schrodinger equations and some complicated equations, such as the equations whose solutions blow up at space infinity. Also develop a moving mesh scheme to simulate asymptotic behaviours in an unbounded domain. 4. Deal with a class of more general quasi-linear parabolic and hyperbolic systems to find sufficient conditions on initial data to deduce global existence and blow-up properties both theoretically and numerically. The new functional method is a very powerful method to obtain a priori estimate for elliptic and parabolic equations and will be introduced to hyperbolic equations. In the functional method, we consider an integral of nth power of several solutions. Taking derivatives with respect to t and integrating by parts we obtain a differential inequality. To my knowledge, if super- and sub-solution methods can be applied to a system, then the functional method can also be applied to the system. However, the functional method only requires weaker conditions. To obtain a numerical solution in an unbounded domain, we first map the unbounded domain into a bounded domain and change equations with some kind of singularity. Then use moving mesh method to reduce errors near the singularity. The numerical solutions also serve as a guide showing when the solutions blow up, exist globally or approach a steady state.**

非线性进化方程在描述各种模型的描述中有许多应用，例如反应扩散，激活剂抑制剂，流体和量子力学和人口生物学。许多方程式涉及退化或奇异的术语以及某些爆炸特性，这些特性在全球，爆破和数值计算中引起许多具有挑战性的问题。该研究计划的目的是研究在理论上和数值上，包括融合到稳态，研究全球和爆破解决方案的特性。预期的结果将包括：1。引入一种新的功能方法，讨论具有可变阻尼系数的可压缩欧拉方程的全局和爆破解决方案。还研究方程式的渐近行为，爆炸率和爆炸时间以及稳态。 2。调查对高阶非线性Schrodinger方程的解决方案的存在和炸毁。 3。修改用于移动网格方法和其他自适应网格方法的现有算法，以数值求解可压缩的欧拉方程，高阶非线性schrodinger方程和一些复杂的方程，例如其解决方案在空间无限段吹来的方程。还要开发一个移动的网格方案，以模拟无限域中的渐近行为。 4.处理一类更一般的准线性抛物线和双曲线系统，以在初始数据上找到足够的条件，以在理论上和数字上推断出全球存在和爆炸属性。新的功能方法是一种非常强大的方法，可以获得椭圆和抛物线方程的先验估计，并将引入双曲线方程。在功能方法中，我们考虑了几种解决方案的第n个功率的积分。采用衍生物在t和零件整合的情况下，我们获得了差异不平等。据我所知，如果可以将超级和子解决方法应用于系统，则功能方法也可以应用于系统。但是，功能方法仅需要较弱的条件。为了在无界域中获得数值解决方案，我们首先将未绑定的域映射到有界域，并以某种奇异性更改方程。然后使用移动网格方法来减少奇异性附近的错误。数值解决方案还可以作为指导，以显示何时在全球范围内爆炸，存在稳定状态。**