Space Of Divergence-free Vector Fields Research Articles

Chaos in the incompressible Euler equation on manifolds of high dimension

We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of any dimension. The main theorem of the paper, from which these families of solutions are obtained, states that for any given vector field X on a closed manifold N, there is a Riemannian manifold M on which the following holds: N is diffeomorphic to a finite dimensional manifold in the phase space of fluid velocities (the space of divergence-free vector fields on M) that is invariant under the Euler evolution, and on which the Euler equation reduces to a finite dimensional ODE that is given by an arbitrarily small smooth perturbation of the vector field X on N.

Turing Universality of the Incompressible Euler Equations and a Conjecture of Moore

Abstract In this article, we construct a compact Riemannian manifold of high dimension on which the time-dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao’s programme to study the blow-up problem for the Euler and Navier–Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete.

A variational interpretation of the Biot–Savart operator and the helicity of a bounded domain

In this note, we show that the projection of the Biot–Savart operator over the space of divergence-free vector fields that are tangential to the boundary is the solution of a suitable saddle-point variational problem. Since this projected Biot–Savart operator is shown to be compact, its spectrum can be completely characterized. In particular, through a suitable finite element discretization, it becomes possible to compute the helicity of a bounded domain of a general topological shape, via the determination of the eigenvalue of the projected Biot–Savart operator that has a maximum absolute value.

BIORTHOGONAL MULTIWAVELETS RELATED BY DIFFERENTIATION

We provide a procedure for constructing biorthogonal multiwavelets from a family of biorthogonal multiscaling functions compactly supported on [-1,1]. The scaling vectors and the associated multiwavelets are piecewise continuously differentiable, symmetrical and possess approximation order three. The construction of scaling vectors is accomplished using quadratic fractal interpolation functions. The filters corresponding to scaling vectors possess certain properties which enable us to construct a new pair of biorthogonal scaling vectors and associated multiwavelets with different regularity and approximation order, related to the old ones by differentiation. The old and new biorthogonal multiwavelet systems give rise to compactly supported biorthogonal multiwavelet basis for the space of divergence-free vector fields on the upper half plane with the Navier boundary condition.

AN AUGMENTED PROJECTION METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN ARBITRARY DOMAINS

A Cartesian grid method for computing flows with complex immersed, stationary and moving boundaries is presented in this paper. We introduce an augmented projection method for the numerical solution of the incompressible Navier-Stokes equations in arbitrary domains. In a projection method an intermediate velocity field is calculated from the momentum equations, which is then projected onto the space of divergence-free vector fields. In the proposed augmented projection method, we add one more step, which effectively eliminates spurious velocity field caused by complex immersed moving boundaries. The methodology is validated by comparing it with analytic, previous numerical and experimental results.

Hardy space of exact forms on $\mathbb {R}^N$

We show that the Hardy space of divergence-free vector fields on R 3 has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of BMO. Using the duality result we prove a div-curl type theorem: for b in L 2 loc (R 3 , R 3 ), sup f b.(⊇u × ⊇v) dx is equivalent to a BMO-type norm of b, where the supremum is taken over all u, u ∈ W 1,2 (R 3 ) with ∥⊇u∥ L 2, ∥⊇v∥ L 2 ≤ 1. This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on R N , study their atomic decompositions and dual spaces, and establish div-curl type theorems on R N .

Parallel computation of viscous incompressible flows using Godunov‐projection method on overlapping grids

AbstractThe Godunov‐projection method is implemented on a system of overlapping structured grids for solving the time‐dependent incompressible Navier–Stokes equations. This projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The Godunov procedure is applied to estimate the non‐linear convective term in order to provide a robust discretization of this terms at high Reynolds number. In order to obtain the pressure field, a separate procedure is applied in this modified Godunov‐projection method, where the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain, as they offer the flexibility of simplifying the grid generation around complex geometrical domains. This combination of projection method and overlapping grid is also parallelized and reasonable parallel efficiency is achieved. Numerical results are presented to demonstrate the performance of this combination of the Godunov‐projection method and the overlapping grid. Copyright © 2002 John Wiley & Sons, Ltd.

Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems

Vortex line and magnetic line representations are introduced for a description of flows in ideal hydrodynamics and magnetohydrodynamics (MHD), respectively. For incompressible fluids, it is shown with the help of this transformation that the equations of motion for vorticity Omega and magnetic field follow from a variational principle. By means of this representation, it is possible to integrate the hydrodynamic type system with the Hamiltonian H=integral|Omega|dr and some other systems. It is also demonstrated that these representations allow one to remove from the noncanonical Poisson brackets, defined in the space of divergence-free vector fields, the degeneracy connected with the vorticity frozenness for the Euler equation and with magnetic field frozenness for ideal MHD. For MHD, a new Weber-type transformation is found. It is shown how this transformation can be obtained from the two-fluid model when electrons and ions can be considered as two independent fluids. The Weber-type transformation for ideal MHD gives the whole Lagrangian vector invariant. When this invariant is absent, this transformation coincides with the Clebsch representation analog introduced by V.E. Zakharov and E. A. Kuznetsov [Dokl. Ajad. Nauk 194, 1288 (1970) [Sov. Phys. Dokl. 15, 913 (1971)]].

Trajectory attractors for the 2D Navier-Stokes system and some generalizations

(1) ∂tu+ νLu+B(u) = g(x, t), (∇, u) = 0, u|∂Ω = 0, x ∈ Ω b R, t ≥ 0, u = u(x, t) = (u, u) ≡ u(t), g = g(x, t) = (g, g) ≡ g(t). Here Lu = −P∆u is the Stokes operator, ν > 0, B(u) = P ∑2 i=1 ui∂xiu; P is the orthogonal projector onto the space of divergence-free vector fields (see Section 1). Consider the autonomous case: g(x, t) ≡ g(x), g ∈ H, to begin with. Suppose for t = 0 we are given the initial condition

A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection

In this method we present a fractional step discretization of the time-dependent incompressible Navier–Stokes equations. The method is based on a projection formulation in which we first solve diffusion–cnvection equations to predict intermediate velocities, which are then projected onto the space of divergence-free vector fields. Our treatment of the diffusion-convection step uses a specialized second-order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high Reynolds number. In contrast to conventional projection-type discretization that impose a discrete form of the divergence-free constraint, we only approximately impose the constraint; i.e., the velocity field we compute is not exactly divergence-free. The approximate projection is computed using a conventional discretization of the Laplacian and the resulting linear system is solved using conventional multigrid methods. Numerical examples are presented to validate the second-order convergence of the method for Euler, finite Reynolds number, and Stokes flow. A second example illustrating the behavior of the algorithm on an unstable shear layer is also presented.

A Second-Order Projection Method for the Incompressible Navier-Stokes Equations in Arbitrary Domains

We present a projection method for the numerical solution of the incompressible Navier-Stokes equations in an arbitrary domain that is second-order accurate in both space and time. The original projection method was developed by Chorin, in which an intermediate velocity field is calculated from the momentum equations which is then projected onto the space of divergence-free vector fields. Our method is based on the projection method developed by Bell and co-workers which is designed for problems in regular domains. We use the continuity equation to derive a pressure equation to compute the gradient part of the vector field. An integral form of the continuity equation is used to give us a natural way to define the discrete divergence operator for cells near the boundary which ensures the diagonal dominance of the resulting pressure equation. We then use the restarted version of the GMRES method to solve the pressure equation.

A second-order projection method for the incompressible navier-stokes equations

In this paper we describe a second-order projection method for the time-dependent, incompressible Navier-Stokes equations. As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. By introducing more coupling between the diffusion-convection step and the projection step we obtain a temporal discretization that is second-order accurate. Our treatment of the diffusion-convection step uses a specialized higher order Godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high Reynolds number. The Godunov procedure is second-order accurate for smooth flow and remains stable for discontinuous initial data, even in the zero-viscosity limit. We approximate the projection directly using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented validating the convergence properties of the method. We also apply the method to doubly periodic shear-layers to assess the performance of the method on more difficult applications.

On a magnetohydrodynamic problem of Euler type

The magnetohydrodynamic (or MHD) equations of an incompressible and homogeneous plasma are considered in the limit of vanishing viscosity and resistivity, the plasma spreading over a smoothly bounded domain G of R N ( N⩾2). The problem is posed and solved as an equation of evolution in a space of divergence-free vector fields parallel to the boundary of G. Main results are existence, uniqueness, and regularity of maximal solutions, and continuous dependence on forcing terms and initial data. The proof of existence is based on Galerkin's method and a priori estimates; the continuity result is obtained by use of a regularization procedure—The approach is inspired by the work of R. Temam and T. Kato and C. Y. Lai on the Euler equations of ideal fluid flow.