David G. Ebin

The initial-value problem for elastodynamics of incompressible bodies

AbstractWe prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials.Our method consists in:a)Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.b)Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:1) Using energy estimates to bound the growth of various Sobolev norms of solutions.2) Finding the solution as the limit of a sequence of solutions of linearized problems. Our equations differ from hyperbolic systems, however, in that the pressure gradient is a spatially non-local function of the position and velocity variables.

On the Limit of Large Surface Tension for a Fluid Motion with Free Boundary

We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary has constant curvature, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.

The free boundary Euler equations with large surface tension

Abstract We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.

MOTION OF SLIGHTLY COMPRESSIBLE FLUIDS IN A BOUNDED DOMAIN. II

We study the problem of inviscid slightly compressible fluids in a bounded domain. We find a unique solution to the initial-boundary value problem and show that it is near the analogous solution for an incompressible fluid provided the initial conditions for the two problems are close. In particular, the divergence of the initial velocity of the compressible flow at time zero is assumed to be small. Furthermore we find that solutions to the compressible motion problem in Lagrangian coordinates depend differentiably on their initial data, an unexpected result for this type of non-linear equations.

Groups of Diffeomorphisms and Fluid Motion: Reprise

Following Ebin and Marsden (Ann Math 92(1):102–163, 1970) we provide a concise proof of the well-posedness of the equations of perfect fluid motion. We use a construction which casts the equations as an ordinary differential equation on a non-linear function space.