## Advances in Differential Equations

- Adv. Differential Equations
- Volume 18, Number 1/2 (2013), 101-146.

### Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit

#### Abstract

We consider the Navier--Stokes equations for viscous incompressible flows in the half plane under the no-slip boundary condition. In this paper we first establish a solution formula for the vorticity equations through the appropriate vorticity formulation. The formula is then applied to establish the asymptotic expansion of vorticity fields at $\nu\rightarrow 0$ that holds at least up to the time $c\nu^{1/3}$, where $\nu$ is the viscosity coefficient and $c$ is a constant. As a consequence, we get a natural sufficient condition on the initial data for the vorticity to blow up at the inviscid limit, together with explicit estimates.

#### Article information

**Source**

Adv. Differential Equations Volume 18, Number 1/2 (2013), 101-146.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867483

**Mathematical Reviews number (MathSciNet)**

MR3052712

**Zentralblatt MATH identifier**

1261.35111

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 76D05: Navier-Stokes equations [See also 35Q30] 76D10: Boundary-layer theory, separation and reattachment, higher-order effects

#### Citation

Maekawa, Yasunori. Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit. Adv. Differential Equations 18 (2013), no. 1/2, 101--146. https://projecteuclid.org/euclid.ade/1355867483