
ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIERSTOKES EQUATIONS IN LORENTZ SPACES
Yanqing WANG, Wei WEI, Gang WU, Yulin YE
Acta mathematica scientia,Series B. 2022, 42 (2):
671689.
DOI: 10.1007/s1047302202167
In this paper, we derive several new sufficient conditions of the nonbreakdown of strong solutions for both the 3D heatconducting compressible NavierStokes system and nonhomogeneous incompressible NavierStokes equations. First, it is shown that there exists a positive constant $\varepsilon$ such that the solution $(\rho,u,\theta)$ to the full compressible NavierStokes 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 blowup criteria in anisotropic Lebesgue spaces for the finite blowup 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 NavierStokes equations. The appearance of a vacuum in these systems could be allowed.
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