arXiv subjects
Publications and source records attributed to James Glimm.
Smooth solutions of the Navier-Stokes equation with smooth but otherwise unconstrained initial conditions are constructed, to solve the Millennium fluids problem in the positive. The smooth solutions are the mean values of general weak solutions and are alternately characterized as the entropy production minimizing solutions. The construction occurs in a finite periodic cube.
A principle of maximum entropy is proposed in the context of viscous incompressible flow in Eulerian coordinates. The relative entropy functional, defined over the space of $L^2$ divergence-free velocity fields, is maximized relative to alternate measures supported over the energy--enstrophy surface. Since thermodynamic equilibrium distributions are characterized by maximum entropy, connections are drawn with stationary statistical solutions of the incompressible Navier-Stokes equations. Special emphasis is on the correspondence with the final statistics described by Kolmogorov's theory of fully developed turbulence.
The mean of Young measure solutions for the Navier-Stokes equations with general initial conditions are PDE solutions of the Navier-Stokes equation of the class considered by Leray and Hopf.
We construct several quantum gauge theories in 4 dimensional space time, including both Abelian and non Abelian gauge groups, with the Abelian gauge fields coupled to zero mass matter fields. The construction occurs in a fixed finite Euclidean spatial domain. The construction begins with the doubly cutoff bare field theory (with a finite space time volume and a mesh based ultraviolet cutoff) constructed in a separate paper. We use a limited range of renormalized perturbation theory, just sufficient to cancel all the divergences. We demonstrate convergence of the ultraviolet limit and of the renormalized perturbation theory, when summed to all orders. We construct the fully renormalized Lagrangian and Schwinger functions.
We present an extension to Kolmogorov's refined similarity hypothesis for universal fully developed turbulence. The extension is applied within Z. She and E. Leveque's multifractal model of inertial range scaling and its generalizations. Our modification rectifies an apparent gap between the implicit continuum of length scales in Obukhov's conception of a turbulent energy cascade, and scaling law models derived from Kolmogorov's refined similarity hypothesis that lack infinite divisibility. The development has relevance to universal fully developed turbulence, a state we describe explicitly in terms of the coupling between velocity fluctuations and averaged energy dissipation at all orders. This description is unique and leads to a reparametrization of the She-Leveque model that preserves its original forecasts and is infinitely divisible.
We formulate multifractal models for velocity differences and gradients which describe the full range of length scales in turbulent flow, namely: laminar, dissipation, inertial, and stirring ranges. The models subsume existing models of inertial range turbulence. In the localized ranges of length scales in which the turbulence is only partially developed, we propose multifractal scaling laws with scaling exponents modified from their inertial range values. In local regions, even within a fully developed turbulent flow, the turbulence is not isotropic nor scale invariant due to the influence of larger turbulent structures (or their absence). For this reason, turbulence that is not fully developed is an important issue which inertial range study can not address. In the ranges of partially developed turbulence, the flow can be far from universal, so that standard inertial range turbulence scaling models become inapplicable. The model proposed here serves as a replacement.Details of the fitting of the parameters for the $τ_p$ and $ζ_p$ models in the dissipation range are discussed. Some of the behavior of $ζ_p$ for larger $p$ is unexplained. The theories are verified by comparing to high resolution simulation data.
In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. This distribution is labeled as the physical solution in the remainder of the article. The dynamics is governed by an assumption of a Lagrangian formulation, with the velocity time derivatives as the momenta conjugate to the velocity configurations. From these definitions and assumptions, we show mathematically that a maximum entropy production principle selects the physical measure from among alternate solutions of the Navier-Stokes and Euler equations, but its transformation to an Eulerian frame is not established here, a topic that will be considered separately.
We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for compressible fluids in $\mathbb{R}^3$. Motivated by the Kolmogorov hypothesis (1941) for incompressible flow, we introduce a Kolmogorov-type hypothesis for barotropic flows, in which the density and the sonic speed normally vary significantly. We then observe that the compressible Kolmogorov-type hypothesis implies the uniform boundedness of some fractional derivatives of the weighted velocity and sonic speed in the space variables in $L^2$, which is independent of the viscosity coefficient $μ>0$. It is shown that this key observation yields the equicontinuity in both space and time of the density in $L^γ$ and the momentum in $L^2$, as well as the uniform bound of the density in $L^{q_1}$ and the velocity in $L^{q_2}$ independent of $μ>0$, for some fixed $q_1 >γ$ and $q_2 >2$, where $γ>1$ is the adiabatic exponent. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations for barotropic fluids in $\mathbb{R}^3$. Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of $μ>0$, that is in the high Reynolds number limit.
Three very different algorithms have been proposed for solution of the Rayleigh-Taylor turbulent mixing problem. They are based upon three different physical principles governing the Euler equations for fluid flow, which serve to complete these underspecified equations by selection of the physically relevant solution from among the many otherwise nonunique solutions of these equations. The disputed physical principle is the admissibility condition which selects the physically meaningful solution from among the myriad of nonphysical solutions. The three different algorithms, expressing the three physical admissibility principles, are formulated alternately in terms of the energy dissipation rate or the entropy production rate. The three alternatives are zero, minimal or maximal rates. The solutions are markedly different. We find strong validation evidence that supports the solution with the maximum rate of dissipated energy, based on a review of prior results and new results presented here. Our verification reasoning, consisting of mathematical analysis based on physics assumptions, also supports the maximum energy dissipation rate and reasons against the other two. The zero dissipation solution is based on claims of direct numerical simulation. We dispute these claims and introduce analysis indicating that such simulations are far from direct numerical simulations. Recommendations for the numerical modeling of the deflagration to detonation transition in type Ia supernova are discussed.
We present two main results. The first is a plausible validation argument for the principle of a maximal rate of entropy production for Euler equation turbulence. This principle can be seen as an extension of the second law of thermodynamics. In our second main result, we examine competing models for large eddy simulations of Euler equation (fully developed) turbulence. We compare schemes with no subgrid modeling, implicit large eddy simulation (ILES) with limited subgrid modeling and those using dynamic subgrid scale models. Our analysis is based upon three fundamental physical principles: conservation of energy, the maximum entropy production rate and the principle of universality for multifractal clustering of intermittency. We draw the conclusion that the absence of subgrid modeling, or its partial inclusion in ILES solution violates the maximum entropy dissipation rate admissibility criteria. We identify circumstances in which the resulting errors have a minor effect on specific observable quantities and situations where the effect is major. Application to numerical modeling of the deflagration to detonation transition in type Ia supernova is discussed.
Fractal mesh refinement enables mesh refinement regimes to the Gibson scale and beyond. Conventional multifractal subgrid models have cost considerations which also lead to local mesh refinement, but more importantly, the application of these models to compressible turbulent deflagration fronts is a topic for future research, while subgrid models tuned to this complex physics lack a multifractal search focus, and for cost reasons do not allow the large search volumes required to find the sought for rare event to trigger a DDT. Here we propose methods to resolve fine scales and locate rare possible DDT trigger events within large volumes while addressing multifractal issues at a feasible computational cost. We are motivated by the goal of confirming or assessing the hypothesis of a delayed detonation in type Ia supernova and to assessing the delay in this event if it is found to occur.
Shattering of a brittle material such as glass occurs dynamically through a propagating failure wave, which however, can not be assigned to any of the classical waves of the elasto-plastic theories of materials. Such failure waves have been a topic of research for decades. In this paper, we build a thermodynamically consistent theory based on the idea that a failure wave is analogous to a deflagration wave. Our theory admits, as special cases, the classical models of Feng and Clifton. Two fundamental thermodynamic functions (the free energy and the entropy production rate) form the basis of our theory. Such a two-function approach allows for the construction of a new variational principle and a new Lagrangian formulation that produce the equations of motion. Finally, a linearization of this theory is examined to gain insight into the coupling between the diffusive and elastic wave phenomena.
We study the spatial statistics of velocity gradient volatility (i,e., the energy dissipation rate) in turbulent flow. We extend the Kolmogorov-Obukhov theory but also narrow its scope. The models are log normal, with verification from finely resolved large eddy and direct numerical simulations. They are parameterized by a mean and a covariance operator. Addressing applications to large eddy simulations, the mean and the covariance depend on the resolved scale solution. Removing this resolved scale dependence by a locally defined rescaling of the turbulent statistics yields a universal theory for the subgrid statistics, specifically for the mean and variance of turbulent fluctuations, i.e., the log of the energy dissipation rate ε. The variance of the velocity gradient statistics is found to be log normal, in accordance with Kolmogorov 1962. Rescaling by the resolved scale mean and variance removes the influence of the resolved scales on the subgrid scales, justifying conceptually the universality we observe. The universality is the basis of new power scaling laws. The new power laws, in turn, allow a simple parameterization of the mean and covariance of the subgrid rescaled dissipation rate, i.e., velocity gradient volatility. We treat the coarse grid resolved space- time location as a random variable. A restriction of the theory to a small range of unresolved but stochastically modelled scales is proposed, with renormalization group ideas offered to overcome this restriction.
We develop an embedded boundary method (EBM) to solve the two-phase incompressible flow with piecewise constant density. The front tracking method is used to track the interface. The fractional step methods are used to solve the incompressible Navier-Stokes equations while the EBM is used in the projection step to solve an elliptic interface problem for the pressure with a jump equal to the surface tension force across the interface. Several examples are used to verify the accuracy of the method.
In this paper, we perform a comparison study of two methods (the embedded boundary method and several versions of the mixed finite element method) to solve an elliptic boundary value problem.
We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in $\R^3$. We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the $α^{th}$-order fractional derivative of the velocity for some $α>0$ in the space variables in $L^2$, which is independent of the viscosity $μ>0$. Then it is shown that this key observation yields the $L^2$-equicontinuity in the time and the uniform bound in $L^q$, for some $q>2$, of the velocity independent of $μ>0$. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations in $\R^3$. We also consider passive scalars coupled to the incompressible Navier-Stokes equations and, in this case, find the weak-star convergence for the passive scalars with a limit in the form of a Young measure (pdf depending on space and time). Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of $μ>0$, that is in the high Reynolds number limit.
We have developed a Scalable Linear Augmented Slater-Type Orbital (LASTO) method for electronic-structure calculations on free-standing atomic clusters. As with other linear methods we solve the Schrödinger equation using a mixed basis set consisting of numerical functions inside atom-centered spheres and matched onto tail functions outside. The tail functions are Slater-type orbitals, which are localized, exponentially decaying functions. To solve the Poisson equation between spheres, we use a finite difference method replacing the rapidly varying charge density inside the spheres with a smoothed density with the same multipole moments. We use multigrid techniques on the mesh, which yield the Coulomb potential on the spheres and in turn defines the potential inside via a Dirichlet problem. To solve the linear eigen-problem, we use ScaLAPACK, a well-developed package to solve large eigensystems with dense matrices. We have tested the method on small clusters of palladium.
We develop and validate a computational approach to nanomagnets. It is built on the spin wave approximation to a Heisenberg ferromagnet whose parameters can be calculated from a first principles theory (e.g. density functional theory). The method can be used for high throughput analysis of a variety of nanomagnetic materials. We compute the dependence of the magnetization of an iron nanomagnet on temperature, size and shape. The approach is applied to nanomagnets in the range of 432 atoms to 59 million atoms, a size which is several orders of magnitude beyond the scalability of density functional theory.