Daniel Coutand

Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit

We prove that the three-dimensional compressible Euler equations with surface tension along the moving free-boundary are well-posed; we then establish the limit as surface tension tends to zero. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs [Supersonic Flow and Shock Waves, Appl. Math. Sci. 21, Springer-Verlag, New York, 1976] as

Global existence and decay for solutions of the Hele–Shaw flow with injection

p(\rho) = \alpha \rho^ \gamma - \beta

On the Limit as the Density Ratio Tends to Zero for Two Perfect Incompressible Fluids Separated by a Surface of Discontinuity

for consants

Well-Posedness in Smooth Function Spaces for Moving-Boundary 1-D Compressible Euler Equations in Physical Vacuum

\gamma >1

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

and

On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations

\alpha , \beta > 0

A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

. The analysis is made difficult by two competing nonlinearities associated with the potential energy: compression in the bulk and surface area dynamics on the free-boundary. Unlike the analysis of the incompressible Euler equations, wherein boundary regularity controls regularity in the interior, the compressible Euler equation requires the additional analysis of nonlinear wave equations generating sound waves. An existence theory is developed by a specially chosen parabolic regularization together with th...

A SIMPLE PROOF OF WELL-POSEDNESS FOR THE FREE-SURFACE INCOMPRESSIBLE EULER EQUATIONS

Author(s): Cheng, C. H. Arthur; Coutand, Daniel; Shkoller, Steve | Abstract: We study the global existence and decay to spherical equilibrium of Hele-Shaw flows with surface tension. We prove that without injection of fluid, perturbations of the sphere decay to zero exponentially fast. On the other hand, with a time-dependent rate of fluid injection into the Hele-Shaw cell, the distance from the moving boundary to an expanding sphere (with time-dependent radius) also decays to zero but with an algebraic rate, which depends on the injection rate of the fluid.

On the motion of vortex sheets with surface tension in three‐dimensional Euler equations with vorticity

We study the asymptotic limit as the density ratio ρ−/ρ+ → 0, where ρ+ and ρ− are the densities of two perfect incompressible 2-D/3-D fluids, separated by a surface of discontinuity along which the pressure jump is proportional to the mean curvature of the moving surface. Mathematically, the fluid motion is governed by the two-phase incompressible Euler equations with vortex sheet data. By rescaling, we assume the density ρ+ of the inner fluid is fixed, while the density ρ− of the outer fluid is set to ε. We prove that solutions of the free-boundary Euler equations in vacuum are obtained in the limit as ε → 0.