Seminar

Location:  Baker Board Room

Speaker: Steve Skoller
Title: Well-posedness of the 3-D compressible Euler equations moving vacuum boundary

Abstract:
In this lecture, I will dicuss the analysis and geometry of the existence, 
uniqueness, and regularity theory for the 3-D compressible Euler equations with a moving 
hypersurface of discontinuity, comprised of the so-called "physical" vacuum boundary, 
with an equation of state given by p(ρ) = ρ^γ for γ>1.

A vacuum state is called "physical" when it permits the gas-vacuum boundary to
 accelerate, and induces a singularity in the pressure gradient, requiring the sound 
speed to vanish at a rate of the square-root of the distance to the vacuum.

The vanishing of the sound speed, and hence the density of the gas, ensures that 
the Euler equations are a degenerate and characteristic hyperbolic free-boundary 
system of conservation laws, to which standard methods of symmetrizable multi-D 
conservation laws cannot be applied.

I will describe a fairly general methodology to treat such degenerate hyperbolic 
free-boundary problems. The method relies on the Lagrangian formulation of the
Euler equations, special weighted energy estimates for time-derivatives and tangential
derivatives, combined with elliptic-type estimates for normal derivatives.
Solutions are constructed by a unique parabolic-type regularization of the Euler
equations which preserves much of the transport structures of Euler, together with a 
new higher-order Hardy-type inequality.
This is joint work with D. Coutand.