Space-time Cylinder Research Articles

A global Holmgren theorem for multidimensional hyperbolic partial differential equations

Holmgren's theorem guarantees unique continuation across non-characteristic surfaces of class C 1 for solutions to homogeneous linear partial differential equations with analytic coefficients. Based on this result a global uniqueness theorem for solutions to hyperbolic differential equations with analytic coefficients in a space-time cylinder Q = (0, T) × Ω is established. Zero Cauchy data on a part of the lateral C 1-boundary will force every solution to a homogeneous hyperbolic PDE to vanish at half time T/2 provided that T > T 0. The minimal time T 0 depends on the geometry of the slowness surface of the hyperbolic operator, and can be determined explicitly as demonstrated by several examples including the system of transversely isotropic elasticity and Maxwell's equations. Furthermore, it is shown that this global uniqueness result holds for hyperbolic operators with C 1-coefficients as long as they satisfy the conclusion of Holmgren's Theorem.

Estimates for deviations from exact solutions of the Cauchy problem for Maxwell's equations

We consider an initial boundary value problem for Maxwell's equations in the space-time cylinder generated by the time interval (0;T). For this hyperbolic type system, we derive guaranteed and computable upper bounds of the dierence between the exact solution and any pair of vector fields that belongs to the natural admissible energy class. Our analysis is based upon transformations of the canonical integral relation and Gronwall's inequality and generalizes the method suggested in (22) for the wave equation to the case of the Maxwell's equation.

Refinement of flexible space–time finite element meshes and discontinuous Galerkin methods

In this paper we present an algorithm to refine space–time finite element meshes as needed for the numerical solution of parabolic initial boundary value problems. The approach is based on a decomposition of the space–time cylinder into finite elements, which also allows a rather general and flexible discretization in time. This also includes adaptive finite element meshes which move in time. For the handling of three-dimensional spatial domains, and therefore of a four-dimensional space–time cylinder, we describe a refinement strategy to decompose pentatopes into smaller ones. For the discretization of the initial boundary value problem we use an interior penalty Galerkin approach in space, and an upwind technique in time. A numerical example for the transient heat equation confirms the order of convergence as expected from the theory. First numerical results for the transient Navier–Stokes equations and for an adaptive mesh moving in time underline the applicability and flexibility of the presented approach.

On the maximal L p -regularity of parabolic mixed-order systems

We study maximal L p -regularity for a class of pseudodifferential mixed-order systems on a space–time cylinder \({\mathbb{R}^n \times \mathbb{R}}\) or \({X \times \mathbb{R}}\) , where X is a closed smooth manifold. To this end, we construct a calculus of Volterra pseudodifferential operators and characterize the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms between suitable L p -Sobolev spaces of Bessel potential or Besov type. If the cross section of the space–time cylinder is compact, the inverse of a parabolic system belongs to the calculus again. As applications, we discuss time-dependent Douglis–Nirenberg systems and a linear system arising in the study of the Stefan problem with Gibbs–Thomson correction.

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space–time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole–Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.

A posteriori error estimates for optimal distributed control governed by the evolution equations

We describe a technique for a posteriori error estimates suitable to the optimal control problem governed by the evolution equations solved by the method of lines. It is applied to the control problem governed by the parabolic equation, convection-diffusion equation and hyperbolic equation. The error is measured with the aid of the L 2 -norm in the space–time cylinder combined with a special time weighted energy norm.

Finite volume spectrum of 2D field theories from Hirota dynamics

We propose, using the example of the O(4) sigma model, a general method for solving integrable two dimensional relativistic sigma models in a finite size periodic box. Our starting point is the so-called Y-system, which is equivalent to the thermodynamic Bethe ansatz equations of Yang and Yang. It is derived from the Zamolodchikov scattering theory in the cross channel, for virtual particles along the non-compact direction of the space-time cylinder. The method is based on the integrable Hirota dynamics that follows from the Y-system. The outcome is a nonlinear integral equation for a single complex function, valid for an arbitrary quantum state and accompanied by the finite size analogue of Bethe equations. It is close in spirit to the Destri-deVega (DdV) equation. We present the numerical data for the energy of various states as a function of the size, and derive the general Luscher-type formulas for the finite size corrections. We also re-derive by our method the DdV equation for the SU(2) chiral Gross-Neveu model.

Estimates of deviations from exact solutions of initial boundary value problems for the wave equation

We derive computable upper bounds of the difference between an exact solution of the initial boundary value problem for a linear hyperbolic equation and any function in a space-time cylinder that belongs to the respective energy class. We prove that the bounds vanish if and only if the approximate solution coincides with the exact one. Bibliography: 13 titles.

Localization and Geometric Depletion of Vortex-Stretching in the 3D NSE

Vortex-stretching and, consequently, the evolution of the vorticity is localized on an arbitrarily small space-time cylinder. This yields a complete localization of the geometric condition(s) for the regularity involving coherence of the vorticity direction. In particular, it implies the regularity of any geometrically constrained Leray solution independently of the type of the spatial domain or the boundary conditions.

Register allocation for software pipelined multidimensional loops

This article investigates register allocation for software pipelined multidimensional loops where the execution of successive iterations from an n -dimensional loop is overlapped. For single loop software pipelining, the lifetimes of a loop variable in successive iterations of the loop form a repetitive pattern. An effective register allocation method is to represent the pattern as a vector of lifetimes (or a vector lifetime using Rau's terminology [Rau 1992]) and map it to rotating registers. Unfortunately, the software pipelined schedule of a multidimensional loop is considerably more complex and so are the vector lifetimes in it. In this article, we develop a way to normalize and represent the vector lifetimes, which captures their complexity, while exposing their regularity that enables a simple solution. The problem is formulated as bin-packing of the multidimensional vector lifetimes on the surface of a space-time cylinder. A metric, called distance, is calculated either conservatively or aggressively to guide the bin-packing process, so that there is no overlapping between any two vector lifetimes, and the register requirement is minimized. This approach subsumes the classical register allocation for software pipelined single loops as a special case. The method has been implemented in the ORC compiler and produced code for the IA-64 architecture. Experimental results show the effectiveness. Several strategies for register allocation are compared and analyzed.

Nonstationary ideal incompressible fluid flows: Conditions of existence and uniqueness of solutions

The problem of formulating minimal conditions on input data that can guarantee the existence and uniqueness of solutions of the boundary value problems describing non-one-dimensional ideal incompressible fluid flow is considered using as an example the initial boundary value problem in a space-time cylinder constructed on a bounded flow domain with the nonpenetration condition on its boundary (which corresponds to fluid flow in a closed vessel). The existence problems are considered only for plane flows, and the uniqueness issues for three-dimensional flows as well. The required conditions are obtained in the form of conditions specifying that the vorticity belongs to definite functional Orlicz spaces. The results are compared with well-known results. Examples are given of admissible types of singularities for which the obtained results are valid, which is a physical interpretation of these results.

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.

Partial Boundary Regularity for the Navier-Stokes Equations

We prove two conditions of local Holder continuity for suitable weak solutions to the Navier-Stokes equations near a smooth curved part of the boundary of the domain. One of these conditions has the form of the Caffarelli-Kohn-Nirenberg condition for local boundedness of suitable weak solutions at interior points of the space-time cylinder. The corresponding results near a planar part of the boundary have been established earlier by Seregin. Bibliography: 21 titles.

Adjoint QCD1 + 1 in light-cone gauge, quantized at equal time

SU(2) gauge theory coupled to massless fermions in the adjoint representation is quantized in light-cone gauge by imposing the equal-time canonical algebra. The theory is defined on a space-time cylinder with “twisted” boundary conditions, periodic for one color component (the diagonal 3-component) and antiperiodic for the other two. The focus of the study is on the non-trivial vacuum structure and the fermion condensate. It is shown that the indefinite-metric quantization of free gauge bosons is not compatible with the residual gauge symmetry of the interacting theory. A suitable quantization of the unphysical modes of the gauge field is necessary in order to guarantee the consistency of the subsidiary condition and allow the quantum representation of the residual gauge symmetry of the classical Lagrangian: the 3-color component of the gauge field must be quantized in a space with an indefinite metric while the other two components require a positive-definite metric. The contribution of the latter to the free Hamiltonian becomes highly pathological in this representation, but a larger portion of the interacting Hamiltonian can be diagonalized, thus allowing perturbative calculations to be performed. The vacuum is evaluated through second order in perturbation theory and this result is used for an approximate determination of the fermion condensate.

Local Regularity of Suitable Weak Solutions to the Navier--Stokes Equations Near the Boundary

We prove a condition of local Holder continuity for suitable weak solutions to the Navier—Stokes equations near the plane boundary. This condition has the form of the Caffarelli—Kohn—Nirenberg condition for local boundedness of suitable weak solutions at the interior points of the space-time cylinder.

Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications

We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.

COHERENT STATES AND THE QUANTIZATION OF (1+1)-DIMENSIONAL YANG–MILLS THEORY

This paper discusses the canonical quantization of (1+1)-dimensional Yang–Mills theory on a spacetime cylinder from the point of view of coherent states, or equivalently, the Segal–Bargmann transform. Before gauge symmetry is imposed, the coherent states are simply ordinary coherent states labeled by points in an infinite-dimensional linear phase space. Gauge symmetry is imposed by projecting the original coherent states onto the gauge-invariant subspace, using a suitable regularization procedure. We obtain in this way a new family of "reduced" coherent states labeled by points in the reduced phase space, which in this case is simply the cotangent bundle of the structure group K. The main result explained here, obtained originally in a joint work of the author with B. Driver, is this: The reduced coherent states are precisely those associated to the generalized Segal–Bargmann transform for K, as introduced by the author from a different point of view. This result agrees with that of K. Wren, who uses a different method of implementing the gauge symmetry. The coherent states also provide a rigorous way of making sense out of the quantum Hamiltonian for the unreduced system. Various related issues are discussed, including the complex structure on the reduced phase space and the question of whether quantization commutes with reduction.

On one approach to a posteriori error estimates for evolution problems solved by the method of lines

In this paper, we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of our goals is to apply known estimates derived for elliptic problems to evolution equations. We apply the new technique to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic operator, a linear nonstationary convection-diffusion problem, and a linear second order hyperbolic problem. The error is measured with the aid of the $L^2$-norm in the space-time cylinder combined with a special time-weighted energy norm. Theory as well as computational results are presented.

On the Existence Theorem for Weak Solutions of Increased Smoothness of the Multidimensional Navier7ndash;Stokes Equations

Hopf [1] defined the notion of a weak solution of the 3D Navier–Stokes equations and proved the existence of such solutions. Later, it was proved that, under certain conditions, the weak solutions (the 3D viscous fluid flow velocities) have second derivatives 5/4th-power integrable over the space-time cylinder [2–4]. The present paper deals with some development of these results. We consider viscous fluid flows in an n-dimensional space with velocities 2π-periodic in each of the space variables and show that if the initial velocity is square integrable over the period cube, its derivatives are integrable, and the mass forces satisfy similar integrability conditions over the obvious space-time parallelepiped, then there exist weak solutions whose second derivatives are (4/3 − e)th-power integrable over the space-time parallelepiped for any e > 0.

Yang-Mills Theory and the Segal-Bargmann Transform

We use a variant of the Segal–Bargmann transform to study canonically quantized Yang–Mills theory on a space-time cylinder with a compact structure group K. The non-existent Lebesgue measure on the space of connections is “approximated” by a Gaussian measure with large variance. The Segal–Bargmann transform is then a unitary map from the L 2 space over the space of connections to a holomorphic L 2 space over the space of complexified connections with a certain Gaussian measure. This transform is given roughly by followed by analytic continuation. Here is the Laplacian on the space of connections and is the Hamiltonian for the quantized theory. On the gauge-trivial subspace, consisting of functions of the holonomy around the spatial circle, the Segal–Bargmann transform becomes followed by analytic continuation, where Δ K is the Laplacian for the structure group K. This result gives a rigorous meaning to the idea that reduces to Δ K on functions of the holonomy. By letting the variance of the Gaussian measure tend to infinity we recover the standard realization of the quantized Yang–Mills theory on a space-time cylinder, namely, −½Δ K is the Hamiltonian and L 2(K) is the Hilbert space. As a byproduct of these considerations, we find a new one-parameter family of unitary transforms from L 2(K) to certain holomorphic L 2-spaces over the complexification of K. This family of transformations interpolates between the two previously known unitary transformations. Our work is motivated by results of Landsman and Wren and uses probabilistic techniques similar to those of Gross and Malliavin.