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David G. Ebin

Publications and source records attributed to David G. Ebin.

6 recordsLinked to original sources
arXiv2016

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.

math.AP↗
arXiv2016

The free boundary Euler equations with large surface tension

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.

math.AP↗
arXiv2014

Riemannian geometry of the contactomorphism group

We define a right-invariant Riemannian metric on the group of contactomorphisms and study its Euler-Arnold equation. If the metric is associated to the contact form, the Euler-Arnold equation reduces to $m_t + u(m) + (n+2) mE(f) = 0$, in terms of the Reeb field $E$, a stream function $f$, the contact vector field $u$ defined by $f$, and the momentum $m = f - Δf$. Here the equation is considered on a compact manifold $M$ of dimension $2n+1$. When $n=0$ this reduces to the Camassa-Holm equation, and we emphasize the analogy with the higher-order equation. We use the usual momentum conservation law for Euler-Arnold equations to rewrite the geodesic equation as a smooth first-order equation on the contactomorphism group of Sobolev class $H^s$, and thus obtain local existence in time of solutions which depend smoothly on initial data. In addition we prove a global existence criterion analogous to the Beale-Kato-Majda criterion in fluid mechanics, and show how this criterion is automatically satisfied on the totally geodesic subgroup of quantomorphisms. Finally we briefly discuss singular solutions and conservation laws of the Euler-Arnold equation.

math.AP↗
arXiv2013

Riemannian geometry on the quantomorphism group

We are interested in the geometry of the group $\mathcal{D}_q(M)$ of diffeomorphisms preserving a contact form $θ$ on a manifold $M$. We define a Riemannian metric on $\mathcal{D}_q(M)$, compute the corresponding geodesic equation, and show that solutions exist for all time and depend smoothly on initial conditions. In certain special cases (such as on the 3-sphere), the geodesic equation is a simplified version of the quasigeostrophic equation, so we obtain a new geodesic interpretation of this geophysical system. We also show that the genuine quasigeostrophic equation on $S^2$ can be obtained as an Euler-Arnold equation on a one-dimensional central extension of $T_{\id}\mathcal{D}_q(M)$, and that our global existence result extends to this case. If $E$ is the Reeb field of $θ$ and $μ$ is the volume form, assumed compatible in the sense that $\text{div} E=0$, we show that $\mathcal{D}_q(M)$ is a smooth submanifold of $\mathcal{D}_{E,μ}(M)$, the space of diffeomorphisms preserving the vector field $E$ and the volume form $μ$, in the sense of $H^s$ Sobolev completions. The latter manifold is related to symmetric motion of ideal fluids. We further prove that the corresponding geodesic equations and projections are $C^{\infty}$ objects in the Sobolev topology.

math.DG↗
arXiv1993

Global Solutions of the Equations of Elastodynamics of Incompressible Neo-Hookean Materials

We prove that the initial-value problem for the motion of a certain type of elastic body has a solution for all time if the initial data are sufficiently small. The body must fill all of three space, obey a ``neo-Hookean'' stress-strain law, and be incompressible. The proof takes advantage of the delayed singularity formation which occurs for solutions of quasi-linear hyperbolic equations in more than one space dimension. It turns out that the curl of the displacement of the body obeys such an equation. Thus using Klainerman's inequality, one derives the necessary estimates to gaurantee that solutions persist for all time.

chao-dyn↗