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Abstract

Two famous developments in nonlinear partial differential equations in the past century continue to inspire modern mathematicians: (1) Solvability theory of incompressible three dimensional fluid dynamics initiated by. J. Leray in the 1930’s and further crystallized by E. Hopf and O. A. Ladyzhenskaya in the 1950’s which is also one of the six remaining Clay Institute Millennium Prize Problems; (2) Solvability theory of compressible three dimensional fluid dynamics initiated by James Serrin (1959) and John Nash (1962), with the most decisive results coming from P. L. Lions in the 1990’s. Both of these subjects are important in engineering applications as they address mathematical aspects of low-speed and high-speed aerodynamics. In particular, stochastic analysis, control theory, filtering/estimation, turbulence closure modeling, and numerical analysis of viscous aerodynamics essentially depend on our understanding of the solvability theory of these physical problems which add to the importance of such analysis. In this talk we will discuss the recent progress and open problems in the rigorous theory of compressible viscous and inviscid fluid dynamics in multi-dimensions. Both weak and strong solutions have famous developments and open problems. We will also discuss the stochastic counterpart and control theoretic aspects.