Optimal decay estimates of a regularity-loss type system with constraint condition

In the paper [14], the authors formulated a new structural condition which includes the Kawashima–Shizuta condition, and analyzed the weak dissipative structure called the regularity-loss type for general systems which contain the Timoshenko system and the Euler–Maxwell system. However, this new structural condition can not cover all of dissipative systems. Indeed we introduce a dissipative system which does not satisfy the new condition and analyze the weaker dissipative structure in this paper. Precisely we first derive the $L^2$ decay estimate of solutions and discuss the type of the corresponding regularity-loss structure. Moreover, in order to show the optimality of the decay estimate, we analyze the expansion for the corresponding eigenvalue of our problem and derive that the solution approaches the diffusion wave as time tends to infinity.     

Decay property of regularity-loss type for solutions in elastic solids with voids

In this paper, we consider the Cauchy problem for a system of elastic solids with voids. First, we show that a linear porous dissipation leads to decay rates of regularity-loss type of the solution. We show some decay estimates for initial data in H^s(R)∩L¹(R)$H^s\left(\mathbb{R}\right)\cap L^1\left(\mathbb{R}\right)$. Furthermore, we prove that by restricting the initial data to be in H^s(R)∩L^1,γ(R)$H^s\left(\mathbb{R}\right)\cap L^{1,\gamma }\left(\mathbb{R}\right)$ and γ∈[0,1]$\gamma \in [0,1]$, we can derive faster decay estimates of the solution. Second, we show that by adding a viscoelastic damping term, then we gain the regularity of the solution and obtain the optimal decay rate.