Regularity of 3D axisymmetric Navier-Stokes equations

Abstract

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair ( ω r r , ω θ r ), we get several Prodi-Serrin type regularity criteria based on the angular velocity, u θ . Moreover, we obtain the global well-posedness result if the initial angular velocity u θ 0 is appropriate small in the critical space L 3 (R 3 ). Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say ω 3 or u 3 .

On regularity criterion to the 3D axisymmetric incompressible MHD equations

“…Therefore, this result shows that if an axisymmetric solution develops a singularity, it can only be a singularity of an other type. For recent generations to this result, see [15][16][17][18] and references therein.…”

Global regularity for a family of 3D models of the axi-symmetric Navier–Stokes equations

“…In [18], the global regularity was obtained if |Γ| ≤ C| ln r| −2 , and this result was later improved to |Γ| ≤ C| ln r| − 3 2 in [25]. The cancellation property in the equation of u 1,z is crucial for the results in [3,18,25]. However, for the family of models (2.4) that we study in this paper, this cancellation is destroyed due to the change of strength in the convection terms in (2.4).…”

“…Using a cancellation property in the equation for u 1,z , they proved the global regularity of the 1D model with or without viscosity. In [3], the cancellation property used in [14] was further exploited, and several critical regularity criteria concerning only the angular velocity are proved. In particular, the authors of [3] showed that if r d u θ ∈ L q (L p (R 3 ), (0, T )) with…”

Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations

“…As a complementary of their work, Pan [26] proved the regularity of solutions by assuming r|u r | ≤ Cr α or r|u z | ≤ Cr α , α > 0. Later, Lei-Zhang in [23] improved the result in [8] by assuming r|v θ | ≤ C| ln r| −2 for small r. Also Wei in [30] improved the log power from −2 to − 3 2 . When the initial data satisfies some integral conditions, Abidi-Zhang in [2] give the global smooth axially symmetric solutions of 3-D inhomogeneous incompressible Navier-Stokes equations.…”

“…Pan [25] proved the regularity of solutions under a slightly supercritical assumption on the drift term b. Recently, Chen-Fang-Zhang in [8] proved that if ru θ satisfies r|u θ | ≤ Cr α , α > 0, then u is regular without any other a prior assumptions. As a complementary of their work, Pan [26] proved the regularity of solutions by assuming r|u r | ≤ Cr α or r|u z | ≤ Cr α , α > 0.…”