Last week, April 25th, 2022 Timothée Crin-Barat Postdoctoral researcher at DeustoCCM Chair of Computational Mathematics at University of Deusto talked on “Seminar in Partial differential equations” at the School of Mathematical Sciences in Tongji University in Shanghai.
Abstract.- We present a method to study global strong solutions of partially dissipative hyperbolic systems in a critical regularity setting. Introducing hybrid Besov norms, with different regularity exponents in low and high frequency, we pinpoint optimal smallness conditions for global well-posedness and get more accurate information on the qualitative properties of the constructed solutions.To handle the high frequencies of the solution, our analysis relies on the construction of a Lyapunov functional in the spirit of the one constructed by Beauchard and Zuazua (ARMA 2011). And concerning the low frequencies, exhibiting a damped mode with faster time decay than the whole solution plays a key role. Our analysis allows us to justify the relaxation limit of the compressible Euler with damping to the porous media equations and to derive an explicit rate of convergence of the process in the multidimensional setting. This is a work in collaboration with Raphaël Danchin.