Location: Baker Board Room
Speaker: Steve Skoller
Title: Well-posedness of the 3-D compressible Euler equations moving vacuum boundary
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.