Quantisation without Witten Anomalies

It is argued that the gauge anomalies are only the artefacts of quantum field theory when certain subtleties are not taken into account. With the Berry's phase needed to satisfy certain boundary conditions of the generating path integral, the gauge anomalies associated with homotopically nontrivial gauge transformations are shown explicitly to be eliminated, without any extra quantum fields introduced. This is in contra-distinction to other quantisations of `anomalous' gauge theory where extra, new fields are introduced to explicitly cancel the anomalies.Comment: 9 pages, latex, no figure

Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids

The generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions on the degree of the Forchheimer polynomial are imposed. We derive, for all time, the interior $L^\infty$-estimates for the pressure and its partial derivatives, and the interior $L^2$-estimates for its Hessian. The De Giorgi and Ladyzhenskaya-Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity

Simulations with Complex Measures

Towards a solution to the sign problem in the simulations of systems having indefinite or complex-valued measures, we propose a new approach which yields statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. The 1D complex-coupling Ising model is employed as an illustration.Comment: 3 pages, postcript (95K), contribution to LAT93, UM-P-93/10

Fermionic Field Theory and Gauge Interactions on Random Lattices

Random-lattice fermions have been shown to be free of the doubling problem if there are no interactions or interactions of a non-gauge nature. However, gauge interactions impose stringent constraints as expressed by the Ward-Takahashi identities which could revive the free-field suppressed doubler modes in loop diagrams. After introducing a formulation for fermions on a new kind of random lattice, we compare random, naive and Wilson fermions in two dimensional Abelian background gauge theory. We show that the doublers are revived for random lattices in the continuum limit, while demonstrating that gauge invariance plays the critical role in this revival. Some implications of the persistent doubling phenomenon on random lattices are also discussed.Comment: 16 A4 pages, UM-P-93/0

A family of steady two-phase generalized Forchheimer flows and their linear stability analysis

We model multi-dimensional two-phase flows of incompressible fluids in porous media using generalized Forchheimer equations and the capillary pressure. Firstly, we find a family of steady state solutions whose saturation and pressure are radially symmetric and velocities are rotation-invariant. Their properties are investigated based on relations between the capillary pressure, each phase's relative permeability and Forchheimer polynomial. Secondly, we analyze the linear stability of those steady states. The linearized system is derived and reduced to a parabolic equation for the saturation. This equation has a special structure depending on the steady states which we exploit to prove two new forms of the lemma of growth of Landis-type in both bounded and unbounded domains. Using these lemmas, qualitative properties of the solution of the linearized equation are studied in details. In bounded domains, we show that the solution decays exponentially in time. In unbounded domains, in addition to their stability, the solution decays to zero as the spatial variables tend to infinity. The BernsteinComment: 33 page

Properties of Generalized Forchheimer Flows in Porous Media

The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids and study the initial boundary value problem for the resulting degenerate parabolic equation for pressure with the time-dependent flux boundary condition. We estimate $L^\infty$-norm for pressure and its time derivative, as well as other Lebesgue norms for its gradient and second spatial derivatives. The asymptotic estimates as time tends to infinity are emphasized. We then show that the solution (in interior $L^\infty$-norms) and its gradient (in interior $L^{2-\delta}$-norms) depend continuously on the initial and boundary data, and coefficients of the Forchheimer polynomials. These are proved for both finite time intervals and time infinity. The De Giorgi and Ladyzhenskaya-Uraltseva iteration techniques are combined with uniform Gronwall-type estimates, specific monotonicity properties, suitable parabolic Sobolev embeddings and a new fast geometric convergence result.Comment: 63 page