The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems
By Jiang Xu, Shuichi Kawashima. 2014
79 citations recorded by the source corpus.
We give a new decay framework for the general dissipative hyperbolic system and the hyperbolic–parabolic composite system, which allows us to pay less attention to the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood–Paley pointwise energy estimates and new time-weighted energy functionals to establish optimal decay estimates on the framework of spatially critical Besov spaces for the degenerately dissipative hyperbolic system of balance laws. Based on the $${L^{p}(\mathbb{R}^{n})}$$Lp(Rn) embedding and the improved Gagliardo–Nirenberg inequality, the optimal $${L^{p}(\mathbb{R}^{n})-L^{2}(\mathbb{R}^{n})(1\leqq p < 2)}$$Lp(Rn)-L2(Rn)(1≦p<2) decay rates and $${L^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})(1\leqq p < 2\leqq q\leqq \infty)}$$Lp(Rn)-Lq(Rn)(1≦p<2≦q≦∞) decay rates are further shown. Finally, as a direct application, the optimal decay rates for three dimensional damped compressible Euler equations are also obtained.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bd9aacd818d6a6300788559add9d33e907fd6867
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In Applied Mathematics
In Control and Systems Engineering
In Mathematical Physics
Citations in the corpus
- Optimal Time-decay Estimates for the Compressible Navier–Stokes Equations in the Critical Lp Framework (2016, background): “or [27]. One can get more Lq-Lr decay estimates, as a consequence of the following GagliardoNirenberg type inequalities which parallel the work of Sohinger and Strain [28] (see also [1], Chap. 2, and [29]): Proposition 4.1. The following interpolation inequality holds true: kΛℓfk Lr .kΛmfk1−θ Lq kΛ ”