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Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations. (English) Zbl 1060.35104
The following Cauchy problem is considered: \[ \begin{aligned} &\frac{\partial\rho }{\partial t}+\gamma\,\text{div}\,v=0 \quad \text{in} \;(0,\infty)\times \mathbb R^n\\ &\frac{\partial v}{\partial t}-\alpha\Delta v -\beta\nabla\text{div}\,v + \gamma\,\nabla \rho=0\quad \text{in}\;(0,\infty)\times \mathbb R^n\\ &v(x,0)=v_0(x),\quad \rho(x,0)=\rho_0(x)\quad \text{in}\;\mathbb R^n \end{aligned} \] Here the velocity \(v(x,t)\) and the density \(\rho(x,t)\) are unknown functions, \(n\geq 2\). The norms
\(\| D^j_tD^{\mu}_x\rho(\cdot,t)\|_{L_{\infty}(\mathbb R^n)}\), \(\| D^j_tD^{\mu}_x v(\cdot,t)\|_{L_{\infty}(\mathbb R^n)}\), \(\| D^j_tD^{\mu}_x \rho(\cdot,t)\|_{L_1(\mathbb R^n)}\),
\(\| D^j_tD^{\mu}_x v(\cdot,t)\|_{L_1(\mathbb R^n)}\) are estimated via power decreasing time dependent functions multiplied by \(L_1\) or \(L_{\infty}\) norms of initial data \(\rho_0\) and \(v_0\). Estimates are obtained by using the Fourier transform method. The estimates of the paper are better than in D. Hoff and K. Zumbrun [Z. Angew. Math. Phys. 48, 517–614 (1997; Zbl 0882.76074)] and Y. Shibata [Math. Meth. Appl. Sci. 23, 203–226 (2000; Zbl 0947.35020)] .

35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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