Analytical and Numerical Methods for Transport Equations

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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.