Abstract
AbstractThis paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor $$\mathcal {D}(u)$$
D
(
u
)
to the 3D Navier–Stokes equations in Lorentz spaces. It is shown that a Leray–Hopf weak solution is regular on (0, T] provided that the norm $$\Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} $$
‖
λ
2
+
‖
L
p
,
∞
(
0
,
T
;
L
q
,
∞
(
R
3
)
)
with $$ {2}/{p}+{3}/{q}=2$$
2
/
p
+
3
/
q
=
2
$$( {3}/{2}<q\le \infty )$$
(
3
/
2
<
q
≤
∞
)
is small. This generalizes the corresponding works of Neustupa–Penel and Miller.
Funder
National Natural Science Foundation of China
Publisher
Springer Science and Business Media LLC
Reference29 articles.
1. Agarwal, R.P., Gala, S., Ragusa, M.A.: A regularity criterion in weak spaces to Boussinesq equations. J. Math. 8, 920 (2020)
2. Beirao da Veiga, H.: A new regularity class for the Navier–Stokes equations in $$\mathbb{R}^{n}$$. Chin. Ann. Math. Ser. B 16, 407–412 (1995)
3. Beirao da Veiga, H., Yang, J.: On mixed pressure–velocity regularity criteria to the Navier–Stokes equations in Lorentz spaces. (2020). arXiv:2007.02089
4. Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)
5. Bosia, S., Pata, V., Robinson, J.: A weak-$$L^p$$ Prodi–Serrin type regularity criterion for the Navier–Stokes equations. J. Math. Fluid Mech. 16, 721–725 (2014)