Analytical and Numerical Methods for Transport Equations

这个项目解决了输运方程中的一些问题，双曲型守恒律的Godunov型中心格式，经典和非线性逼近。贯穿拟议研究的共同主线是使用逼近理论、调和和泛函分析(Littlewood-Paley理论、小波分解、极大函数、插值法和K泛函)的技术来证明应用数学的其他领域的分析结果，并使用这些技术来开发数值方法。文中将着重讨论几个问题：将非线性问题线性化时所产生的几个空间维线性输运方程的满意理论的发展，求解多维守恒律组的Godunov型中心格式的发展，以及这些格式在各种问题和模型中的应用。平均引理从微观公式到宏观公式的扩展也将是主要关注的问题。这个项目的一部分将解决非线性逼近和多元立方体的基本问题。讨论的领域(非线性逼近、输运方程解的分析性质以及它们的计算的有效数值方法的发展)具有重要的实际意义。这些应用包括图像处理、统计估计、流体力学、地球物理、气象学、天体物理学、多组分流动、地下水流动、半导体和反应性流动。

0104112PetrovaThis project addresses some problems in transport equations, Godunov type central schemes for hyperbolic conservation laws, classical and nonlinear approximation. The common thread that runs through the proposed research is the use of techniques from approximation theory, harmonic and functional analysis (Littlewood-Paley theory, wavelet decompositions, maximal functions, interpolation and K-functionals) to prove analytic results in other areas of applied mathematics and to use these techniques to develop numerical methods. Particular emphases will be placed on several issues: development of a satisfactory theory for linear transport equations in several space dimensions which arise when linearizing the nonlinear problem, development of Godunov type central schemes for solving multidimensional systems of conservation laws, and application of these schemes to various problems and models. Extensions of averaging lemmas to go from microscopic to macroscopic formulations will also be of primary concern. A portion of this project will address fundamental questions in nonlinear approximation and multivariate cubature.The areas under discussion (nonlinear approximation, analytical properties of solutions to transport equations and development of effective numerical methods for their computation) are of significant practical interest. The applications include image processing, statistical estimation, fluid mechanics, geophysics, meteorology, astrophysics, multi-component flows, ground water flow, semiconductors, and reactive flows.