Next Wednesday January 19, our Timothée Crin-Barat Postdoctoral Researcher at the Chair of Computational Mathematics at University of Deusto/Deusto Foundation will be talking on “Partially dissipative hyperbolic systems, results and perspectives” as part of the CCMSeminar series.
Abstract. First, we present a method for investigating global strong solutions of partially dissipative hyperbolic systems in a critical regularity setting. Using hybrid Besov norms, with different regularity exponents in low and high frequency, we are able to pinpoint optimal smallness conditions for global well-posedness and to get more accurate information on the qualitative properties of the constructed solutions.To handle the high frequencies, 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.
Then, we discuss the case where the damping term is only active outside of a ball. Our goal is to recover the classical decay properties due to Shizuta and Kawashima, with a time-delay depending on the time that each component of the system spends in the ball. This is an on-going work with Nicola De Nitti and Enrique Zuazua.
Wednesday January 17, 2022 at 11:00H