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# Recent advances in the inviscid limit and Prandtl expansions for the incompressible Navier-Stokes under no-slip boundary conditions

### Early Career Math Colloquium

Recent advances in the inviscid limit and Prandtl expansions for the incompressible Navier-Stokes under no-slip boundary conditions
Series: Early Career Math Colloquium
Location: Online
Presenter: Trinh Nguyen, USC

Abstract:

In this talk, I will give an overview of recent results on the inviscid limit of the 2D incompressible Navier-Stokes equations. When the viscosity $\nu$ tends to zero (inviscid limit), the problem is very delicate, due to the mismatch in the boundary conditions of the Euler equations (\$\nu=0) and the Navier-Stokes equations. This leads to a large gradient of the velocity field and vorticity near the boundary. The problem becomes more delicate for domains with curved boundaries, due to a loss of two derivatives in the equation. We discuss recent key ideas and techniques to overcome these difficulties, for initial data that are only required to be analytic near the boundary. On the half-space, we fully justify the Prandtl asymptotic expansion. For curved-boundary domains, we give a precise control for the vorticity at the boundary.