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Logarithmically improved extension criteria involving the pressure for the Navier-Stokes equations in \(\mathbb{R}^n\). (English) Zbl 1532.35346

The authors consider the Cauchý problem of the Navier-Stokes (N-S) system \(\partial u/\partial t -\triangle u+u \cdot \nabla u + \nabla\pi = 0\) in \(\mathbb{R}^n\times (0,T)\), \( \operatorname{div}{\, u}=0\) in \(\mathbb{R}^n\times (0,T)\), and \(u|_{t=0}=a\) in \(\mathbb{R}^n\), where \(u = u(x,t) = (u_1(x,t),\dots, u_n(x,t))\) and \(\pi =\pi (x,t)\) are the unknown velocity vector and the unknown scalar pressure of the fluid at the point \(x = (x_1,\dots, x_n)\in \mathbb{R}^n\) and the time \(t\in (0, T)\), respectively, while \(a = a(x) = (a_1 (x),\dots, a_n (x))\) is the given initial velocity vector, and \(n \geq 3\). Here the authors establish several new extension criteria involving the pressure and its gradient. They improve some known results by means of the homogeneous Besov space with negative differential orders and \(n = 3\). The main method is based on the interpolation inequality and the trilinear estimate due to P. Gerard et al. [Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math. Laurent Schwartz, Palaiseau 1996–1997, Exp. No. IV, 11 p. (1997; Zbl 1066.46501)]. The main statement concerns \(n=3\), \(s>1/2\) case for \(a\in H_{\sigma }^{s}\) provided that \(u\) is a strong solution of N-S in the class \(CL_s(0, T)\) with the associated pressure \(\pi \). In this frame it is proved that if one integral condition among five ones hold, as the first one is \[ \int\limits_{0}^{T}\frac {\|\pi (\tau)\|_{\dot{B}_{q,\infty}^{-3(1/p-1/q)}}^r} {\log(e+\|u(\tau)\|_{H^s})}d\tau < \infty, \] where \(2/r+3/p=2\), \(3/2<p<3\), \(p\leq q<3p/(3-p)\) and the other four conditions are defined similarly, then \(u\) can be continued to the strong solution in the class \(CL_s(0,T^{\prime})\) for some \(T^{\prime}>T\).
The second result concerns the case \(n\geq 4\) and \(s>n/2-1\), \(a\in H_{\sigma}^{s}\), \(u \) is a strong solution of N-S in the class \(CL_s(0,T)\) with an associated pressure \(\pi \). Besides, it is assumed that one among three special integral conditions hold, as the first one has the form \[ \int\limits_{0}^{T}\frac {\|\pi (\tau)\|_{L^p}^{r}} {\log(e+\|u(\tau)\|_{H^s})}d\tau < \infty, \] where \(2/r+n/p=2\), \(n/2<p<\infty \), and the rest two conditions have similar form. Under these conditions it turns out that \(u\) can be continued to the strong solution in the class \(CL_s(0,T^{\prime })\) \((T^{\prime } > T)\).

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D35 Strong solutions to PDEs

Citations:

Zbl 1066.46501
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References:

[1] H.Beirão da Veiga, A new regularity class for the Navier-Stokes equations in \({\bf R}^n\), Chin. Ann. Math. B16 (1995), no. 4, 407-412. · Zbl 0837.35111
[2] J.Bergh and J.Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer‐Verlag, Berlin, 1976. · Zbl 0344.46071
[3] L. C.Berselli and G. P.Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Am. Math. Soc.130 (2002), no. 12, 3585-3595. · Zbl 1075.35031
[4] Q.Chen and Z.Zhang, Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations, Proc. Am. Math. Soc.135 (2007), no. 6, 1829-1837. · Zbl 1126.35047
[5] J.Fan, S.Jiang, and G.Nakamura, On logarithmically improved regularity criteria for the Navier-Stokes equations in \(\mathbb{R}^n\), IMA J. Appl. Math.76 (2011), no. 2, 298-311. · Zbl 1219.35169
[6] J.Fan, S.Jiang, and G.Ni, On regularity criteria for the n‐dimensional Navier‐Stokes equations in terms of the pressure, J. Dif. Equ.244 (2008), no. 11, 2963-2979. · Zbl 1143.35081
[7] J.Fan and T.Ozawa, Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure, J. Inequal. Appl. (2008), 412678. · Zbl 1162.35060
[8] J. Fan S.Jiang, G.Nakamura, and Y.Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech.13 (2011), no. 4, 557-571. · Zbl 1270.35339
[9] H.Fujita and T.Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal.16 (1964), 269-315. · Zbl 0126.42301
[10] P.Gérard, Y.Meyer, and F.Oru, Inégalités de sobolev précisées, Séminaire Équations aux dérivées partielles1996-1997 (1996-1997), 1-8. · Zbl 1066.46501
[11] Y.Giga, Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier-Stokes system, J. Diff. Equ.62 (1986), no. 2, 186-212. · Zbl 0577.35058
[12] Z.Guo, P.Kučera, and Z.Skalák, Regularity criterion for solutions to the Navier-Stokes equations in the whole 3D space based on two vorticity components, J. Math. Anal. Appl.458 (2018), no. 1, 755-766. · Zbl 1378.35217
[13] H.Hajaiej, L.Molinet, T.Ozawa, and B.Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, Harmonic analysis and nonlinear partial differential equations, RIMS Kôkyûroku Bessatsu, B26, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 159-175. · Zbl 1270.42026
[14] X.He and S.Gala, Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class \(L^2(0,T;\dot{B}^{-1}_{\infty,\infty }(\mathbb{R}^3))\), Nonlinear Anal. Real World Appl.12 (2011), no. 6, 3602-3607. · Zbl 1231.35146
[15] T.Kato, Strong \(L^p\)‐solutions of the Navier-Stokes equation in \({\bf R}^m\), with applications to weak solutions, Math. Z.187 (1984), no. 4, 471-480. · Zbl 0545.35073
[16] H.Kozono, T.Ogawa, and Y.Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi‐linear evolution equations, Math. Z.242 (2002), no. 2, 251-278. · Zbl 1055.35087
[17] H.Kozono and Y.Taniuchi, Bilinear estimates in BMO and the Navier‐Stokes equations, Math. Z.235 (2000), no. 1, 173-194. · Zbl 0970.35099
[18] D.Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam.35 (2019), no. 1, 23-100. · Zbl 1412.35261
[19] S.Montgomery‐Smith, Conditions implying regularity of the three dimensional Navier-Stokes equation, Appl. Math.50 (2005), no. 5, 451-464. · Zbl 1099.35086
[20] Y.Sawano, Homogeneous Besov spaces, Kyoto J. Math.60 (2020), no. 1, 1-43. · Zbl 1437.42034
[21] H.Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces, J. Evol. Equ.1 (2001), no. 4, 441-467. · Zbl 1007.35051
[22] E. M.Stein, Harmonic analysis: real‐variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001
[23] M.Struwe, On a Serrin‐type regularity criterion for the Navier-Stokes equations in terms of the pressure, J. Math. Fluid Mech.9 (2007), no. 2, 235-242. · Zbl 1131.35060
[24] F.Wu, Navier-Stokes regularity criteria in Vishik spaces, Appl. Math. Optim.84 (2021), no. 1, S39-S53. · Zbl 1477.35138
[25] X.Zhang, Y.Jia, and B.‐Q.Dong, On the pressure regularity criterion of the 3D Navier-Stokes equations, J. Math. Anal. Appl.393 (2012), no. 2, 413-420. · Zbl 1248.35151
[26] Y.Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \(\mathbb{R}^N\), Z. Angew. Math. Phys.57 (2006), no. 3, 384-392. · Zbl 1099.35091
[27] Y.Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in \(\mathbb{R}^3\), Proc. Am. Math. Soc.134 (2006), no. 1, 149-156. · Zbl 1075.35044
[28] Y.Zhou and J.Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math.24 (2012), no. 4, 691-708. · Zbl 1247.35115
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