A Liouville theorem for the axially-symmetric Navier-Stokes equations

作者：Lei Zhen^*; Zhang Qi S

来源：Journal of Functional Analysis, 2011, 261(8): 2323-2345.

DOI：10.1016/j.jfa.2011.06.016

• Axially-symmetric Navier Stokes equations
• Liouville theorem
• Ancient solutions
• Nash method

Let v(x, t) = v(r)e(r) + v(theta)e(theta) + v(z)e(z) be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by b = v(r)e(r) + v(z)e(z) the radial-axial vector field. Under a general scaling invariant condition on b, we prove that the quantity Gamma = rv(theta) is Holder continuous at r = 0, t = 0. As an application, we prove that the ancient weak solutions of axi-symmetric Navier-Stokes equations must be zero (which was raised by Koch, Nadirashvili, Seregin and Sverak (2009) in [15] and Seregin and Sverak (2009) in [26] as a conjecture) under the condition that b is an element of L(infinity)([0, T], BMO(-1)). As another application, we prove that if b is an element of L(infinity)([0, T], BMO(-1)), then v is regular.

• 出版日期2011-10-15
• 单位复旦大学