Steve Shkoller

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order

Discrete Euler-Poincaré and Lie-Poisson equations

In this paper, discrete analogues of Euler–Poincare and Lie–Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L : TG → R that are G-invariant. These discrete equations provide ‘reduced’ numerical algorithms which manifestly preserve the symplectic structure. The manifold G x G is used as an approximation of TG, and a discrete Langragian L : G x G → R is constructed in such a way that the G-invariance property is preserved. Reduction by G results in a new ‘variational’ principle for the reduced Lagrangian l : G → R, and provides the discrete Euler–Poincare (DEP) equations. Reconstruction of these equations recovers the discrete Euler–Lagrange equations developed by Marsden et al (Marsden J E, Patrick G and Shkoller S 1998 Commun. Math. Phys. 199 351–395) and Wendlandt and Marsden (Wendlandt J M and Marsden J E 1997 Physica D 106 223–246) which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie–Poisson (DLP) algorithm. It is shown that when G = SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian L are equivalent to the Moser–Veselov scheme for the generalized rigid body.

Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence

Capabilities for turbulence calculations of the Lagrangian averaged Navier-Stokes (LANS-alpha) equations are investigated in decaying and statistically stationary three-dimensional homogeneous and isotropic turbulence. Results of the LANS-alpha computations are analyzed by comparison with direct numerical simulation (DNS) data and large eddy simulations. Two different decaying turbulence cases at moderate and high Reynolds numbers are studied. In statistically stationary turbulence two different forcing techniques are implemented to model the energetics of the energy-containing scales. The resolved flows are examined by comparison of the energy spectra of the LANS-alpha with the DNS computations. The energy transfer and the capability of the LANS-alpha equations in representing the backscatter of energy is analyzed by comparison with the DNS data. Furthermore, the correlation between the vorticity and the eigenvectors of the rate of the resolved strain tensor is studied. We find that the LANS-alpha equations capture the gross features of the flow, while the wave activity below the scale alpha is filtered by a nonlinear redistribution of energy.

Variational Methods, Multisymplectic Geometry and Continuum Mechanics

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper.

Motion of an Elastic Solid inside an Incompressible Viscous Fluid

Abstract.The motion of an elastic solid inside an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of regularity which has left the basic question of existence open until now.In this paper, we prove the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. Our functional framework is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.

Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains

We prove the global well–posedness and regularity of the (isotropic) Lagrangian averaged Navier–Stokes (LANS–α) equations on a three–dimensional bounded domain with a smooth boundary with no–slip boundary conditions for initial data in the set {u ∈ Hs ∩ H10| Au = 0 on ∂Ω, div u = 0}, s ∈ [3, 5), where A is the Stokes operator. As with the Navier–Stokes equations, one has parabolic–type regularity; that is, the solutions instantaneously become space–time smooth when the forcing is smooth (or zero). The equations are an ensemble average of the Navier–Stokes equations over initial data in an α–radius phase–space ball, and converge to the Navier–Stokes equations as α → 0. We also show that classical solutions of the LANS–α equations converge almost all in Hs for s ∈ 2.5, 3), to solutions of the inviscid equations (ν = 0), called the Lagrangian averaged Euler (LAE–α) equations, even on domains with boundary, for time–intervals governed by the time of existence of solutions of the LAE–α equations.

The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.

The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

Abstract. This paper develops the geometric analysis of geodesic ow of a new right invariant metric

The vortex blob method as a second-grade non-Newtonian fluid

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Well-posedness of the full Ericksen–Leslie model of nematic liquid crystals

on two subgroups of the volume preserving diffeomorphism group of a smooth n-dimensional compact subset