Advances in Differential Equations

On uniqueness of symmetric Navier-Stokes flows around a body in the plane

Tomoyuki Nakatsuka

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Abstract

We investigate the uniqueness of symmetric weak solutions to the stationary Navier-Stokes equation in a two-dimensional exterior domain $\Omega$. It is known that, under suitable symmetry condition on the domain and the data, the problem admits at least one symmetric weak solution tending to zero at infinity. Given two symmetric weak solutions $u$ and $v$, we show that if $u$ satisfies the energy inequality $\| \nabla u \|_{L^2 (\Omega)}^2 \le (f,u)$ and $\sup_{x \in \Omega} (|x|+1)|v(x)|$ is sufficiently small, then $u=v$. The proof relies upon a density property for the solenoidal vector field and the Hardy inequality for symmetric functions.

Article information

Source
Adv. Differential Equations Volume 20, Number 3/4 (2015), 193-212.

Dates
First available in Project Euclid: 4 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1423055199

Mathematical Reviews number (MathSciNet)
MR3311432

Zentralblatt MATH identifier
1311.35187

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 76D05: Navier-Stokes equations [See also 35Q30]

Citation

Nakatsuka, Tomoyuki. On uniqueness of symmetric Navier-Stokes flows around a body in the plane. Adv. Differential Equations 20 (2015), no. 3/4, 193--212. https://projecteuclid.org/euclid.ade/1423055199.


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