Differential Equations

In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the free boundary isentropic compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time.

We show that if the adiabatic exponent gamma belongs to the interval (1, 5/3] then these affine motions are nonlinearly stable; small perturbations lead to global-in-time solutions that remain "close" to the affine solutions, are smooth in the interior of their support, and no shocks are formed in the process.

Our strategy relies on two key ingredients: a new interpretation of the affine motions using an (almost) invariant action of GL(3) on the compressible Euler system and the use of Lagrangian coordinates. The former suggests a particular rescaling of time and a change of variables that elucidates a stabilisation mechanism, while the latter requires introducing new ideas with respect to the existing well-posedness theory for vacuum free boundary fluid equations. This is joint work with Juhi Jang (USC). Speaker(s): Mahir Hadzic (King`s College London)