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Steve Shkoller

Publications and source records attributed to Steve Shkoller.

67 records · Page 4Linked to original sources
arXiv2000

Enhanced inverse-cascade of energy in the averaged Euler equations

For a particular choice of the smoothing kernel, it is shown that the system of partial differential equations governing the vortex-blob method corresponds to the averaged Euler equations. These latter equations have recently been derived by averaging the Euler equations over Lagrangian fluctuations of length scale $\a$, and the same system is also encountered in the description of inviscid and incompressible flow of second-grade polymeric (non-Newtonian) fluids. While previous studies of this system have noted the suppression of nonlinear interaction between modes smaller than $\a$, we show that the modification of the nonlinear advection term also acts to enhance the inverse-cascade of energy in two-dimensional turbulence and thereby affects scales of motion larger than $\a$ as well. This latter effect is reminiscent of the drag-reduction that occurs in a turbulent flow when a dilute polymer is added.

math.NA↗
arXiv2000

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.

math.DG↗
arXiv2000

Symmetry reduction of discrete Lagrangian mechanics on Lie groups

For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group $G$ of the canonical discrete Lagrange 2-form $ω_\mathbb{L}$ on $G \times G$. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example.

math.NA↗
arXiv2000

On Incompressible Averaged Lagrangian Hydrodynamics

This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the analytical and geometrical properties of the Lagrangian flow map. We prove existence and uniqueness of smooth-in-time solutions for initial data in $H^s$, $s > n/2 +1$ by establishing the existence of smooth geodesics of a new weak right invariant metric on new subgroups of the volume-preserving diffeomorphism group. We establish smooth limits of zero viscosity for the second-grade fluids equations even on manifolds with boundary. We prove that the weak curvature operator of the weak invariant metric is continuous in the $H^s$ topology for $s> n/2+2$, thus proving existence and uniqueness for the Jacobi equation. We show that this new metric stabilizes the Lagrangian flow of the original Euler equations by changing the sign of the sectional curvature.

math.AP↗
arXiv1999

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

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$α$ equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in ${\mathcal M}({\mathbb R}^2)$ such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods.

math.AP↗
arXiv1999

Second-order multisymplectic field theory: A variational approach to second-order multisymplectic field theory

This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ as well as the Noether theorem arise from the first variation of the action function. We generalize the multisymplectic form formula derived for first order field theories in \cite{MPS}, to the case of second-order field theories, and we apply our theory to the Camassa-Holm (CH) equation in both the continuous and discrete settings. Our discretization produces a multisymplectic-momentum integrator, a generalization of the Moser-Veselov rigid body algorithm to the setting of nonlinear PDEs with second order Lagrangians.

math.DG↗
arXiv1999

Discrete Euler-Poincaré and Lie-Poisson Equations

In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups $G$ with Lagrangians $L:TG \to {\mathbb R}$ that are $G$-invariant. These discrete equations provide ``reduced'' numerical algorithms which manifestly preserve the symplectic structure. The manifold $G \times G$ is used as an approximation of $TG$, and a discrete Langragian ${\mathbb L}:G \times G \to {\mathbb R}$ is construced in such a way that the $G$-invariance property is preserved. Reduction by $G$ results in new ``variational'' principle for the reduced Lagrangian $\ell:G \to {\mathbb R}$, and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed in \cite{MPS,WM} 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 $G=\text{SO} (n)$, the DEP and DLP algorithms for a particular choice of the discrete Lagrangian ${\mathbb L}$ are equivalent to the Moser-Veselov scheme for the generalized rigid body. %As an application, a reduced symplectic integrator for two dimensional %hydrodynamics is constructed using the SU$(n)$ approximation to the volume %preserving diffeomorphism group of ${\mathbb T}^2$.

math.NA↗
arXiv1999

A Nonlinear Analysis of the Averaged Euler Equations

This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter $α$; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order $α$. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold's theorem), but with respect to a right invariant $H^1$ metric instead of the $L^2$ metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, {\it even in the presence of boundaries}.

chao-dyn↗
arXiv1998

Persistence of invariant manifolds for nonlinear PDEs

We prove that under certain stability and smoothing properties of the semi-groups generated by the partial differential equations that we consider, manifolds left invariant by these flows persist under $C^1$ perturbation. In particular, we extend well known finite-dimensional results to the setting of an infinite-dimensional Hilbert manifold with a semi-group that leaves a submanifold invariant. We then study the persistence of global unstable manifolds of hyperbolic fixed-points, and as an application consider the two-dimensional Navier-Stokes equation under a fully discrete approximation. Finally, we apply our theory to the persistence of inertial manifolds for those PDEs which possess them. te

math.AP↗
arXiv1998

Multisymplectic geometry, covariant Hamiltonians, and water waves

This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. In this theory, solutions of a PDE are sections of a fiber bundle $Y$ over a base manifold $X$ of dimension $n$$+$1, typically taken to be spacetime. Given a connection on $Y$, a covariant Hamiltonian density ${\mathcal H}$ is then intrinsically defined on the primary constraint manifold $P_{\mathcal L}$, the image of the multisymplectic version of the Legendre transformation. One views $P_{\mathcal L}$ as a subbundle of $J^1(Y)^\star$, the affine dual of $J^1(Y)$, the first jet bundle of $Y$. A canonical multisymplectic ($n$$+$2)-form $Ω_{\mathcal H}$ is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangian. We show that the $n$$+$1 2-forms $ω^{(μ)}$ defined by Bridges [1997] are a particular coordinate representation for a single multisymplectic ($n$$+$2)-form, and in the presence of symmetries, can be assembled into $Ω_{\mathcal H}$. A generalized Hamiltonian Noether theory is then constructed which recovers the vanishing of the divergence of the vector of $n$$+$1 distinct momentum mappings defined in Bridges [1997] and, when applied to water waves, recovers Whitham's conservation of wave action. We also show the utility of this theory in the study of periodic pattern formation and wave instability.

math.DG↗
arXiv1998

Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics

Recently, Holm, Marsden, and Ratiu [1998] have derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation $\dot{V}(t) + \nabla_{U(t)} V(t) - α^2 [\nabla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- α^2 \triangle)U$. In this model, the momentum $V$ is transported by the velocity $U$, with the effect that nonlinear interaction between modes corresponding to length scales smaller than $α$ is negligible. We generalize this equation to the setting of an $n$ dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincaré equation associated with the geodesic flow of the $H^1$ right invariant metric on ${\mathcal D}^s_μ$, the group of volume preserving Hilbert diffeomorphisms of class $H^s$. We prove that the geodesic spray is continuously differentiable from $T{\mathcal D}_μ^s(M)$ into $TT{\mathcal D}_μ^s(M)$ so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [1966]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant $H^1$ metric on ${\mathcal D}^s_μ$ is a bounded trilinear map in the $H^s$ topology, from which it follows that solutions to Jacobi's equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.

math.AP↗
arXiv1998

Multisymplectic geometry, variational integrators, and nonlinear PDEs

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes.

math.DG↗