• DOI:10.1512/IUMJ.2019.68.7650
• Corpus ID: 20158522

Stationary solution to the Navier-Stokes equations in the scaling invariant Besov space and its regularity

@article{Kaneko2019StationaryST,
  title={Stationary solution to the Navier-Stokes equations in the scaling invariant Besov space and its regularity},
  author={Kenta Kaneko and Hideo Kozono and Senjo Shimizu},
  journal={Indiana University Mathematics Journal},
  year={2019}
}

• K. Kaneko, H. Kozono, S. Shimizu
• Published 2019
• Mathematics
• Indiana University Mathematics Journal

We consider the stationary problem of the Navier-Stokes equations in R for n ≥ 3. We show existence, uniqueness and regularity of solutions in the homogeneous Besov space Ḃ −1+np p,q which is the scaling invariant one. As a corollary of our results, a self-similar solution is obtained. For the proof, several bilinear estimates are established. The essential tool is based on the paraproduct formula and the imbedding theorem in homogeneous Besov spaces. Introduction. Let us consider the stationary Navier-Stokes equation in Rn for n ≥ 3; { −∆u+ u · ∇u+∇π = f, div u = 0, (NS) where u = u(x) = (u1(x), · · · , un(x)) and π = π(x) denote the unknown velocity vector and the unknown pressure at the point x = (x1, · · · , xn) ∈ Rn, respectively, while f = f(x) = (f1(x), · · · , fn(x)) denotes the given external force. In the literature, there are a number of results on weak and strong solutions to the Navier-Stokes equations, and modern techniques arising in functional and harmonic analysis have been applied to solve fundamental problems on both stationary and non-stationary cases such as existence, uniqueness, regularity and stability. The existence of the solution of the stationary Navier-Stokes equations (NS) had been founded by Leray [19], Ladyzhenskaya [18] and Fujita [9]. Then Finn [8] established a fundamental framework of the stationary problem in exterior domains. It was shown by Heywood [10] that the solution of (NS) can be obtained as a limit of the solutions of the non-stationary equation. Their methods can be also applied to the whole space Rn, and these results have been developed