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.
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
.
DOI: 10.1007/s10473-022-0216-7
[1] Benedek A, Panzone R. The space Lp with mixed norm. Duke Math J, 1961, 28:301-324
[2] Bergh J, Löfström J. Interpolation Spaces. Berlin:Springer-Verlag, 1976
[3] Bosia S, Pata V, Robinson J. A Weak-Lp Prodi-Serrin Type Regularity Criterion for the Navier-Stokes Equations. J Math Fluid Mech, 2014, 16:721-725
[4] Carrillo J A, Ferreira L C F. Self-similar solutions and large time asymptotics for the dissipative quasigeostrophic equation. Monatsh Math, 2007, 151:111-142
[5] Chen Z, Price W G. Blow-up rate estimates for weak solutions of the Navier-Stokes equations. (English summary) R Soc Lond Proc Ser A Math Phys Eng Sci, 2001, 457:2625-2642
[6] Cho Y, Choe H J, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluid. J Math Pures Appl, 2004, 83:243-275
[7] Cho Y, Kim H. Existence results for viscous polytropic fluids with vacuum. J Differential Equations, 2006, 228:377-411
[8] Choe H J, Kim H. Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids. Comm PDE, 2003, 28:1183-1201
[9] Choe H, Yang M. Blow up criteria for the compressible Navier-Stokes equations. Mathematical analysis in fluid mechanics-selected recent results. Contemp Math, 710:65-84. Providence, RI:Amer Math Soc, 2018
[10] Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford:Oxford Univ Press, 2004
[11] Guo Z, Caggio M, Skalàk Z. Regularity criteria for the Navier-Stokes equations based on one component of velocity. Nonlinear Anal Real World Appl, 2017, 35:379-396
[12] Gustafson S, Kang K, Tsai T. Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations. Commun Math Phys, 2007, 273:161-176
[13] Grafakos L. Classical Fourier analysis. 2nd Edition, Springer, 2008
[14] Huang X, Li J. Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows. Comm Math Phys, 2013, 324:147-171
[15] Huang X, Li J, Wang Y. Serrin-type blowup criterion for full compressible Navier-Stokes system. Arch Ration Mech Anal, 2013, 207:303-316
[16] Huang X, Li J, Xin Z. Serrin type criterion for the three-dimensional compressible flows. SIAM J Math Anal, 2011, 43:1872-1886
[17] Huang X, Xin Z. A blow-up criterion for classical solutions to the compressible Navier-Stokes equations. Sci China Math, 2010, 53:671-686
[18] Jiu Q, Wang Y, Ye Y. Refined blow up criteria for the full compressible Navier-Stokes equations involving temperature. J Evol Equ, 2021, 21:1895-1916
[19] Ji X, Wang Y, Wei W. New regularity criteria based on pressure or gradient of velocity in Lorentz spaces for the 3D Navier-Stokes equations. J Math Fluid Mech, 2020, 22:13
[20] Kim H, Kozono H. Interior regularity criteria in weak spaces for the Navier-Stokes equations. Manuscripta Math, 2004, 115:85-100
[21] Kim H. A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J Math Anal, 2006, 37:1417-1434
[22] Lei Z, Xin Z. On scaling invariance and type-I singularities for the compressible Navier-Stokes equations. Sci China Math, 2019, 62:2271-2286
[23] Lions P L. Mathematical topics in fluid mechanics. Vol 2. Compressible models. New York:Oxford University Press, 1998
[24] Malý J. Advanced theory of differentiation-Lorentz spaces. March 2003. http://www.karlin.mff.cuni.cz/?maly/lorentz.pdf
[25] O'Neil R. Convolution operaters and Lp,q spaces. Duke Math J, 1963, 30:129-142
[26] Rubio de Francia J L, Ruiz F J, Torrea J L. Calderon-Zygmund theory for operator-valued kernels. Adv Math, 1986, 62:7-48
[27] Serrin J. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch Rational Mech Anal, 1962, 9:187-195
[28] Sun Y, Wang C, Zhang Z. A Beale-Kato-Majda blow-up criterion for the 3D compressible Navier-Stokes equations. J Math Pures Appl, 2011, 95:36-47
[29] Sun Y, Wang C, Zhang Z. A Beale-Kato-Majda criterion for three dimensional compressible viscous heatconductive flows. Arch Ration Mech Anal, 2011, 201:727-742
[30] Sun Y, Zhang Z. Blow-up criteria of strong solutions and conditional regularity of weak solutions for the compressible Navier-Stokes equations//Handbook of mathematical analysis in mechanics of viscous fluids. Cham:Springer, 2018:2263-2324
[31] Struwe M. On partial regularity results for the Navier-Stokes equations. Comm Pure Appl Math, 1988, 41:437-458
[32] Sohr H. A regularity class for the Navier-Stokes equations in Lorentz spaces. J Evol Equ, 2001, 1:441-467
[33] Tartar L. Imbedding theorems of Sobolev spaces into Lorentz spaces. Bollettino dell'Unione Matematica Italiana, 1998, 1:479-500
[34] Wang Y, Wu G, Zhou D. ε-regularity criteria in anisotropic Lebesgue spaces and Leray's self-similar solutions to the 3D Navier-Stokes equations. Z Angew Math Phys, 2020, 71:164
[35] Wang Y, Wei W, Yu H. ε-regularity criteria for the 3D Navier-Stokes equations in Lorentz spaces. J Evol Equ, 2021, 21:1627-1650
[36] Wang Y. Weak Serrin-type blowup criterion for three-dimensional nonhomogeneous viscous incompressible heat conducting flows. Phys D, 2020, 402:132203
[37] Wen H, Zhu C. Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum. Adv Math, 2013, 248:534-572
[38] Xu X, Zhang J. A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum. Math Models Method Appl Sci, 2012, 22:1150010
[39] Zheng X. A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component. J Differential Equations, 2014, 256:283-309
