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ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES
Yanqing WANG, Wei WEI, Gang WU, Yulin YE

Acta mathematica scientia,Series B. 2022, 42 (2 ):
671-689.
DOI: 10.1007/s10473-022-0216-7
In this paper, we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant $\varepsilon$ such that the solution $(\rho,u,\theta)$ to the full compressible Navier-Stokes equations can be extended beyond $t=T$ provided that one of the following two conditions holds: (1) $\rho \in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$, $u\in L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))$ and \begin{equation}\label{L1}\| u\|_{L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))}\leq \varepsilon, ~~\text{with}~~ {2/p}+ {3/q}=1,\ \ q > 3;\end{equation} (2) $\lambda < 3\mu,$ $\rho \in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$, $\theta\in L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))$ and \begin{equation}\label{L12}\|\theta\|_{L^{p,\infty}(0,T; L^{q,\infty}(\mathbb{R}^{3}))}\leq \varepsilon, ~~\text{with}~~ {2/p}+ {3/q}=2,\ \ q > 3/2.\end{equation} To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time $T^{\ast}$: (1) assuming that the pair $(p,\overrightarrow{q})$ satisfies $ {2}/{p }+{1}/{q_{1} }+{1}/{q_{2} }+{1}/{q_{3} }=1$ $(1 < q_{i} < \infty)$ and (1.17), then \begin{equation}\label{AL1}\limsup_{t\rightarrow T^*}( \|\rho \|_{L^{\infty}(0,t;L^{\infty}(\mathbb{R}^{3}))}+ \| u \|_{L^{p }(0,t; L_{1}^{ q_{1} }L_{2}^{ q_{2} } L_{3}^{q_{3} }(\mathbb{R}^{3}) )} )= \infty; \end{equation} (2) letting the pair $(p,\overrightarrow{q})$ satisfy ${2}/{p }+{1}/{q_{1} }+{1}/{q_{2} }+{1}/{q_{3} }=2$ $(1 < q_{i} < \infty)$ and (1.17), then \begin{equation}\label{AL2}\limsup_{t\rightarrow T^*}( \|\rho \|_{L^{\infty}(0,t;L^{\infty}(\mathbb{R}^{3}))}+ \| \theta \|_{L^{p }(0,t; L_{1}^{ q_{1} }L_{2}^{ q_{2} } L_{3}^{q_{3} }(\mathbb{R}^{3}) )} )= \infty, (\lambda < 3\mu). \end{equation} Third, without the condition on $\rho$ in (0.1) and (0.3), the results also hold for the 3D nonhomogeneous incompressible Navier-Stokes equations. The appearance of a vacuum in these systems could be allowed.

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