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Explore the mathematical foundations of compressible fluid mechanics in this 57-minute lecture from the Institute for Advanced Study's Joint IAS/PU Analysis and Mathematical Physics seminar. Delve into the complex relationship between the compressible Euler equation and shock discontinuities that emerge in finite time, such as those observed behind supersonic aircraft. Examine the mathematical challenges of studying solutions as inviscid limits of Navier-Stokes solutions with vanishing viscosities, particularly the destabilization effects caused by viscosities. Learn about Bianchini and Bressan's 2004 proof of the inviscid limit to small BV solutions using artificial viscosities, and discover why achieving this limit with physical viscosities remained an open problem until recently. Understand the fundamental concepts of classical mathematical theories in compressible fluid mechanics and explore the innovative a-contraction with shifts method. See how this method successfully describes the physical inviscid limit in the context of the barotropic Euler equation and resolves the Bianchini and Bressan conjecture in this specialized case, based on collaborative research with Geng Chen and Moon-Jin Kang.
