Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

Abstract

We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach a constant equilibrium state in the L p -norm at a rate O(t −(m/2)(1−1/ p) ) as t → ∞ for p ∈ [min{m, 2}, ∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m ≥ 2, and by a parabolic equation, in the spirit of Chapman-Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem.

Decay property of regularity-loss type for the Euler-Maxwell system

“…Consequently, our system (2.2) is not included in a class of systems considered in [1,6,10,11,12,20,22,24].…”

“…We can check that our system verifies the stability condition formulated above; for the details, we refer the reader to [23]. However, the dissipation matrix L is not real symmetric and therefore we can not apply the general theory on the dissipative structure developed in [20,22,10,6,24,11,1,12] to our system. This situation is quite similar to that for the dissipative Timoshenko system.…”

“…We refer the reader to [6,24,1,12]. All these works were, however, based on the standard dissipative structure formulated in [20,22,10] and hence are different from the present paper.…”

Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method

“…[3,11,15,24,25,38]). Since three decades the Lyapunov direct method has found extensions in the studies of feedback stabilization and feedback stabilizability of distributed parameter systems governed by partial differential equations (PDE) (see [1][2][3][4]8,10,13,17,18,32]). Up to today numerous interesting results have been obtained using the elaborated techniques adapted to the classical PDEs such as the wave equation, the beam equation and the Petrovsky system.…”

“…The paper [4] has studied the asymptotic behaviour of global smooth solutions for general entropy dissipative hyperbolic systems of balance law. The systems are supposed to satisfy the strictly entropy dissipative condition and the Shizuta-Kawashima condition.…”