On weighted regularity criteria for the axisymmetric Navier–Stokes equations☆
Introduction
The three-dimensional Navier–Stokes equations governing the incompressible fluid read as where is the fluid velocity field, π is a scalar pressure, and u0 is the prescribed initial data satisfying the compatibility condition in the sense of distributions.
For initial data of finite energy, a class of weak solutions (the so-called Leray–Hopf weak solutions) satisfying was constructed by Leray [12] and Hopf [6]. However, the issue of regularity and uniqueness of u is an outstanding open problem in mathematical fluid dynamics. Thus, many researchers are devoted to finding sufficient conditions (called the regularity criterion) to ensure the smoothness of the solution after the works of [17], [18].
In this paper, we study the axisymmetric solutions to (1.1) of the form where and ur, uθ and uz are called the radial, swirl and axial components of u respectively. Thus, (1.1) can be reformulated as where denotes the convection derivative (or material derivative). We can also compute the vorticity as with The governing equations of ω can be easily deduced from (1.3) as In case we have known the global regularity of (1.3), see [10], [13], [19]. However, when uθ ≠ 0, the global strong solutions to (1.3) is still unknown. There are many sufficient conditions to ensure the regularity of the solution, see [2], [3], [4], [5], [7], [8], [9], [11], [15], [16], [20], [21] for example. In particular, we have the following regularity criterion
- (1)
with provided in [15] and the extension to in [21];
- (2)
with 0 ≤ d < 1 treated in [3, Theorem 1.1] and (under the assumption ) covered by interpolation method in [3, Remark 1.3];
- (3)
established in [3, Theorem 1.4];
- (4)
with obtained in [3, Theorem 1.3] and the improvement to in [21];
- (5)
with found in [2, Theorem 1] and the refinement to 0 ≤ d < 2 in [21].
In this paper, we would like to make a further contribution in this direction. We shall show that (1.12) with could also ensure the regularity of the solution. Precisely,
Theorem 1 Let be axially symmetric and be divergence-free, and be the unique axisymmetric solution of (1.3). If
then the solution can be smoothly extended beyond T. Remark 2 Although Theorem 1 tells us that we can add conditions on rdωθ with to ensure the extension of local strong solutions, it is not satisfactory, since the range of β is not of full range . We shall investigate this issue in the future.
Section snippets
Proof of Theorem 1
In this section, we shall prove Theorem 1. As stated in [3, Lemma 2.5] (see also [7], [15], [20] and [21, Proposition 12] for a simplified proof), we only need to obtain a bound of under the assumption (1.13). To this aim, we multiply (1.3)2 by integrate over to get To bound I, we invoke the magic identity of Miao–Zheng [14, Proposition 2.5] where
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2018, Journal of Mathematical Analysis and ApplicationsGlobal weighted regularity for the 3D axisymmetric MHD equations
2022, Zeitschrift fur Angewandte Mathematik und Physik
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This work is supported by the Natural Science Foundation of Jiangxi (grant no. 20151BAB201010) and the National Natural Science Foundation of China (grant no. 11501125).