In this paper, we are concerned with a class of multi-dimensional balance laws with a non-local dissipative source which arise as simplified models for the hydrodynamics of radiating gases. At first we introduce the energy method in the setting of smooth perturbations and study the stability of constants states. Precisely, we use Fourier space analysis to quantify the energy dissipation rate and recover the optimal time-decay estimates for perturbed solutions via an interpolation inequality in Fourier space. As application, the developed energy method is used to prove stability of smooth planar waves in all dimensions n2, and also to show existence and stability of time-periodic solutions in the presence of the time-periodic source. Optimal rates of convergence of solutions towards the planar waves or time-periodic states are also shown provided initially L1-perturbations. © 2009 Elsevier Masson SAS.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: R.-J. Duan would like to thank Prof. Peter Markowich and Dr. Massimo Fornasier for their support during the postdoctoral studies of the year 2008-2009 in RICAM. K. Fellner's work has been supported by the KAUST Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). The research of C.-J. Zhu was supported by the National Natural Science Foundation of China #10625105 and The Key Laboratory of Mathematical Physics of Hubei Province.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.