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The optimal decay estimates on the framework of Besov spaces for generally dissipative systems. (English) Zbl 1323.35141
Summary: 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 $${L^{p}(\mathbb{R}^{n})}$$ embedding and the improved Gagliardo-Nirenberg inequality, the optimal $${L^{p}(\mathbb{R}^{n})-L^{2}(\mathbb{R}^{n})(1\leqq p < 2)}$$ decay rates and $${L^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})(1\leqq p < 2\leqq q\leqq \infty)}$$ decay rates are further shown. Finally, as a direct application, the optimal decay rates for three dimensional damped compressible Euler equations are also obtained.

##### MSC:
 35Q31 Euler equations 42B25 Maximal functions, Littlewood-Paley theory 76N15 Gas dynamics, general
##### Keywords:
compressible Euler equations
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##### References:
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