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Green's functions and Well-posedness of Compressible Navier-Stokes equation
[2018 KAIST-HKUST-NUS Joint Workshop in Mathematics : Analysis, PDE and Probability]
Date: 2018-11-17
Speaker : Shih-Hsien Yu (National University of Singapore)
Abstract : A class of decomposition of Green's functions for the compressilbe Navier-Stokes linearized around a constant state is introduced. The singular structures of the Green's functions are developed as essential devices to use the nonlinearity directly to covert the 2nd order quasi-linear PDE into a system of zero-th order integral equation with regular integral kernels. The system of integrable equations allows a wider class of functions such as BV solutions. We have shown global existence and well-posedness of the compressible Navier-Stokes equation for isentropic gas with the gas constant $\gamma \in (0,e)$ in the Lagrangian coordinate for the class of the BV functions and point wise $L^\infty$ around a constant state; and the underline pointwise structure of the solutions is constructed. It is also shown that for the class of BV solution the solution is at most piecewise $C^2$-solution even though the initial data is piecewise $C^\infty$.
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