ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES

doi:10.1007/s10473-022-0216-7

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (2): 671-689.doi: 10.1007/s10473-022-0216-7

ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES

王艳青¹, 魏巍², 吴刚³, 叶嵎林⁴

1. Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China;
2. Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi'an 710127, China;
3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
4. School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

收稿日期:2020-09-24 修回日期:2020-12-15 出版日期:2022-04-25 发布日期:2022-04-22
通讯作者: Wei WEI,E-mail:ww5998198@126.com E-mail:ww5998198@126.com
作者简介:Yanqing WANG,E-mail:wangyanqing20056@gmail.com;Gang WU,E-mail:wugang2011@ucas.ac.cn;Yulin YE,E-mail:ylye@vip.henu.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (11971446, 12071113, 11601423, 11771352, 11871057, 11771423, 11671378, and 11701145) and Project funded by China Postdoctoral Science Foundation (2020M672196).

ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES

Yanqing WANG¹, Wei WEI², Gang WU³, Yulin YE⁴

1. Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China;
2. Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi'an 710127, China;
3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
4. School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

Received:2020-09-24 Revised:2020-12-15 Online:2022-04-25 Published:2022-04-22
Supported by:
This work was supported by the National Natural Science Foundation of China (11971446, 12071113, 11601423, 11771352, 11871057, 11771423, 11671378, and 11701145) and Project funded by China Postdoctoral Science Foundation (2020M672196).

摘要/Abstract

摘要： 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.

关键词: Navier-Stokes equations, strong solutions, regularity

Abstract: 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.

Key words: Navier-Stokes equations, strong solutions, regularity

中图分类号:

76D03

王艳青, 魏巍, 吴刚, 叶嵎林. ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES[J]. 数学物理学报(英文版), 2022, 42(2): 671-689.

Yanqing WANG, Wei WEI, Gang WU, Yulin YE. ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES[J]. Acta mathematica scientia,Series B, 2022, 42(2): 671-689.

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ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES

ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES

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