23 research outputs found
L^2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method
We consider scalar nonviscous conservation laws with strictly convex flux in
one spatial dimension, and we investigate the behavior of bounded L^2
perturbations of shock wave solutions to the Riemann problem using the relative
entropy method. We show that up to a time-dependent translation of the shock,
the L^2 norm of a perturbed solution relative to the shock wave is bounded
above by the L^2 norm of the initial perturbation.Comment: 17 page
Hydrodynamic limit of the kinetic Cucker-Smale flocking model
The hydrodynamic limit of a kinetic Cucker-Smale model is investigated. In
addition to the free-transport of individuals and the Cucker-Smale alignment
operator, the model under consideration includes a strong local alignment term.
This term was recently derived as the singular limit of an alignment operator
due to Motsch and Tadmor. The model is enhanced with the addition of noise and
a confinement potential. The objective of this work is the rigorous
investigation of the singular limit corresponding to strong noise and strong
local alignment. The proof relies on the relative entropy method and entropy
inequalities which yield the appropriate convergence results. The resulting
limiting system is an Euler-type flocking system.Comment: 23 page
Relative entropy in diffusive relaxation
We establish convergence in the diffusive limit from entropy weak solutions of
the equations of compressible gas dynamics with friction to the porous media equation away from vacuum.
The result is based on a Lyapunov type of functional provided by a calculation of the relative entropy.
The relative entropy method is also employed to establish convergence from entropic weak solutions
of viscoelasticity with memory to the system of viscoelasticity of the rate-type
Classical solutions to a BGK-type model relaxing to the isentropic gas dynamics
In this paper, we consider a BGK-type kinetic model relaxing to the
isentropic gas dynamics in the hydrodynamic limit. We introduce a linearization
of the equation around the global equilibrium. Then we prove the global
existence of classical solutions with an exponential convergence rate toward
the equilibrium state in the periodic domain when the initial data is a small
perturbation of the global equilibrium