2012/07/09 by Marta Strani, Strani, Marta
Mathematics · Engineering · #Navier-Stokes equation solutions #Computational Fluid Dynamics and Aerodynamics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1207.2024
This paper considers the slow motion of the shock layer exhibited by the\nsolution to the initial-boundary value problem for a scalar hyperbolic system\nwith relaxation. Such behavior, known as metastable dynamics, is related to the\npresence of a first small eigenvalue for the linearized operator around an\nequilibrium state; as a consequence, the time-dependent solution approaches its\nsteady state in an asymptotically exponentially long time interval. In this\ncontest, both rigorous and asymptotic approaches are used to analyze such slow\nmotion for the Jin-Xin system. To describe this dynamics we derive an ODE for\nthe the position of the internal transition layer, proving how it drifts\ntowards the equilibrium location with a speed rate that is exponentially slow.\nThese analytical results are also validated by numerical computations.\n