Analytical and Numerical Methods for Transport Equations

0104112 Petrova这个项目解决了迁移方程中的一些问题，双曲守恒律的Goddom型中心格式，经典和非线性逼近。贯穿拟议的研究的共同点是使用近似理论，谐波和功能分析（Littlewood-Paley理论，小波分解，极大函数，插值和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。特别强调将放在几个问题：发展一个令人满意的理论，线性运输方程在几个空间维度时出现的线性化的非线性问题，发展Goddom型中央计划解决多维系统的守恒律，以及应用这些计划的各种问题和模型。从微观到宏观的平均引理的扩展也将是主要关注的问题。 本专题的一部分将讨论非线性近似和多元体积中的基本问题。所讨论的领域（非线性近似、输运方程解的分析性质和有效的数值计算方法的发展）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、地球物理学、气象学、天体物理学、多组分流、地下水流、半导体和反应流。

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.