Optimal \(L^{p}\) - \(L^{q}\) convergence rates for the compressible Navier-Stokes equations with potential force.

*(English)*Zbl 1121.35096In recent years the 3-dimensional Navier-Stokes flow of a compressible fluid received much attention. See for example authors’ references [T. Kobayashi, Y. Shibata, Commun. Math. Phys. 200, 621-659 (1999; Zbl 0921.35092); T. Kobayashi, J. Differ. Equations 184, 587–619 (2002; Zbl 1069.35051); S. Ukai, T. Yang and H. J. Zhao, J. Hyperbolic Differ. Equ. 3, 561–574 (2006; Zbl 1184.35251); R. J. Duan, T. Seiji, and H. J. Zhao, Optimal Convergence for the compressible Navier-Stokes equation with potential forces, Math. Models Methods Appl. Sci. 17, 1–22 (2007; Zbl 1122.35093)]. The compressible Navier-Stokes equations are of the form:

\[ \rho_t+\nabla \cdot(\rho u)=0, \]

\[ u_t+(u\cdot\nabla)u+(\nabla P(\rho))/ \rho=(\mu/\rho)\Delta u+ [(\mu+\mu')\nabla(nabla\cdot u)]/\rho-\nabla\varphi (x), \]

\[ (\rho(x), u(x))\to (\rho_\infty,0)\text{ as } x\to\infty. \] Here \(\rho\) is the density, \(u\) is the velocity, \(P\) the pressure, \(\varphi(x)\) is a time independent potential, \(\mu >0\) and \(\mu'\geq-^2/3\mu\) are viscosity coefficients. The domain \(\Omega\) is either all of \(\mathbb R^3\), the half-space, or an exterior domain of a bounded region in \(\mathbb R^3\), with smooth boundary. The authors use Sobolev inequalities (which may be found in [R. Adams, Sobolev spaces. New York: Academic Press (1975; Zbl 0314.46030)], to establish optimal convergence rates to a steady state of the solution in the \(L^q\) norm, \(2\leq q\leq 6\), and of the derivatives in the \(L^2\) norm, while a perturbation is bounded in the \(L^p\) norm. The existence of a global solution is proved by using the of Lyapunov energy inequality on derivatives of order one, two, or three. The authors conclude that higher-order derivatives decay at the same rate as the first derivative. The final estimates are also based on the properties of the semigroup generated by the linearized operator.

The reviewer observed in the authors’ list of references the absence of some familiar names of American, English and other European experts in Navier-Stokes equations.

\[ \rho_t+\nabla \cdot(\rho u)=0, \]

\[ u_t+(u\cdot\nabla)u+(\nabla P(\rho))/ \rho=(\mu/\rho)\Delta u+ [(\mu+\mu')\nabla(nabla\cdot u)]/\rho-\nabla\varphi (x), \]

\[ (\rho(x), u(x))\to (\rho_\infty,0)\text{ as } x\to\infty. \] Here \(\rho\) is the density, \(u\) is the velocity, \(P\) the pressure, \(\varphi(x)\) is a time independent potential, \(\mu >0\) and \(\mu'\geq-^2/3\mu\) are viscosity coefficients. The domain \(\Omega\) is either all of \(\mathbb R^3\), the half-space, or an exterior domain of a bounded region in \(\mathbb R^3\), with smooth boundary. The authors use Sobolev inequalities (which may be found in [R. Adams, Sobolev spaces. New York: Academic Press (1975; Zbl 0314.46030)], to establish optimal convergence rates to a steady state of the solution in the \(L^q\) norm, \(2\leq q\leq 6\), and of the derivatives in the \(L^2\) norm, while a perturbation is bounded in the \(L^p\) norm. The existence of a global solution is proved by using the of Lyapunov energy inequality on derivatives of order one, two, or three. The authors conclude that higher-order derivatives decay at the same rate as the first derivative. The final estimates are also based on the properties of the semigroup generated by the linearized operator.

The reviewer observed in the authors’ list of references the absence of some familiar names of American, English and other European experts in Navier-Stokes equations.

Reviewer: Vadim Komkov (Florida)

##### MSC:

35Q30 | Navier-Stokes equations |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

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\textit{R. Duan} et al., J. Differ. Equations 238, No. 1, 220--233 (2007; Zbl 1121.35096)

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##### References:

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