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.**

非线性演化方程在描述各种模型方面有许多应用，例如反应扩散、激活剂-抑制剂、流体和量子力学以及群体生物学。许多方程涉及简并项或奇异项以及某些类型的爆炸特性，这在全局、爆炸和数值计算中引起许多具有挑战性的问题。该研究项目的目标是从理论上和数值上研究非线性演化方程的全局解和爆炸解的性质，包括收敛到稳态。预期结果将包括： 1. 引入一种新的函数方法来讨论可变阻尼系数的可压缩欧拉方程的全局解和爆炸解。还研究方程的渐近行为、爆炸速率和爆炸时间以及稳态。 2. 研究高阶非线性薛定谔方程解的存在性和爆炸性。 3. 修改现有的移动网格方法和其他自适应网格方法的算法，以数值求解可压缩欧拉方程、高阶非线性薛定谔方程和一些复杂方程，例如解在无限远空间爆炸的方程。还开发移动网格方案来模拟无界域中的渐近行为。 4. 处理一类更一般的准线性抛物线和双曲系统，以找到初始数据的充分条件，从而从理论上和数值上推论全局存在和爆炸特性。新的泛函方法是获得椭圆和抛物方程先验估计的非常强大的方法，并将被引入到双曲方程中。在泛函方法中，我们考虑多个解的 n 次方积分。对 t 求导并按部分积分，我们得到微分不等式。据我所知，如果超解法和子解法可以应用于系统，那么泛函方法也可以应用于系统。然而，泛函方法只需要较弱的条件。为了获得无界域中的数值解，我们首先将无界域映射到有界域，并用某种奇点改变方程。然后使用移动网格方法来减少奇点附近的误差。数值解还可以作为指导，显示解何时爆炸、全局存在或接近稳定状态。**