In 1752, Euler first formulated the system of equations describing the dynamics of a perfect fluid. This system was complemented by Clausius in the 19th century, by introducing the concept of entropy of thermodynamics. This self-contained system is called compressible Euler system (CE).
The most important feature of CE is the finite-time breakdown of smooth solutions, that is, the formation of shock as severe singularity due to irreversibility and discontinuity. Therefore, a fundamental question (since Riemann 1858) is on what happens after a shock occurs. This is the problem on well-posedness (that is, existence, uniqueness, stability) of weak solutions satisfying the 2nd law of thermodynamics, which is called entropy solution.
This issue has been conjectured as follows:
Well-posedness of entropy solutions for CE can be obtained in a class of vanishing viscosity limits of solutions to the Navier-Stokes system.
This conjecture for the fundamental issue remains wide open even for the one-dimensional CE.
My recent result (arXiv:1902.01792) provides a first answer to the conjecture in the case of the 1D isentropic CE starting from a shock.
The proof crucially uses our new methodology (arXiv:1712.07348) to get the contraction of any large perturbations from viscous shock to the Navier-Stokes. This will be a main part of my talk.
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