Observability inequalities, unique continuation and applications to control problems for partial differential equations

数学科学：守恒定律的双曲系统-粘性守恒定律-应用摘要：本研究涉及双曲守恒定律和粘性守恒定律的一般领域。第一组问题涉及具有大初始数据的守恒定律双曲系统解的适定性，解的唯一性和正则性，以及一些守恒定律双曲系统在若干空间维度上的研究。第二组问题属于粘性守恒定律领域，涉及可压缩Navier-Stokes方程在流体动力学和燃烧理论中的应用研究。本研究项目的主要目的是对具有大初始数据的一般多维Navier-Stokes方程解的定性行为进行系统的研究。这是一项跨学科的研究，位于连续介质物理学和双曲（和粘性）守恒定律理论的界面上。研究者研究连续介质物理中出现的偏微分方程，期望潜在的物理结构将指导分析，而反过来，一些非线性偏微分方程的数学分析将进一步理解连续介质物理。

NSF Award Abstract - DMS-0072496Mathematical Sciences: Hyperbolic Systems of Conservation Laws - Viscous Conservation Laws - ApplicationsAbstract0072496 TrivisaThis research deals with the general areas of hyperbolic conservation laws and viscous conservation laws. The first set of questions concerns the well-posedness of solutions tohyperbolic systems of conservation laws with large initial data, the uniqueness and regularity of solutions, and the study of some hyperbolic systems of conservation laws in several space dimensions. The second set of questions lies in the area of viscous conservation laws and deals with the study of the compressible Navier-Stokes equations with applications in fluid dynamics and combustion theory. The main objective of this research project is to initiate a systematic investigation of the qualitative behavior of solutions to general multidimensional Navier-Stokes equations with large initial data.This is interdisciplinary research, lying on the interface between continuum physics and the theory of hyperbolic (and viscous) conservation laws. The investigator studies partial differential equations arising in continuum physics with the expectation that the underlying physical structure will direct the analysis, while in return the mathematical analysis of some nonlinear partial differential equations will further the understanding of continuum physics.