Non-vanishing Gradient Research Articles

Influence of thermal flux on Marangoni convection in weakly viscoelastic thin films on uniformly heated substrate

This study examines the hydrodynamic stability of weakly viscoelastic thin film of Walters liquid B″ type on a uniformly heated rigid inclined plane. The current model incorporates a heat flux (HF) boundary condition, which is a realistic representation. This approach accounts for heat flux and heat loss at the wall–liquid and wall–air interfaces, allowing for an examination of the effects of a nonvanishing temperature gradient at the top of the rigid wall. This consideration is essential, as maintaining a constant temperature throughout the wall in laboratory settings poses significant challenges. A Benney-type evolution equation is derived using the long-wave expansion of the flow variables, describing the film thickness as a function of x, t. A linear stability analysis in normal mode is conducted to determine the onset of instability to the critical Reynolds number. The linear study indicates the dual role of the wall film Biot number (Bw). For small values of (Bw), a destabilizing effect is observed; however, upon reaching a critical value, a stabilizing effect is produced. The linear study confirms that the presence of Bw consistently results in a destabilizing effect on the viscoelastic parameter. The nonlinear free surface equation is numerically simulated using the Fourier Spectral method. The temporal evolution of hmax and hmin in the presence of (Bw) proves the viscoelastic parameter's destabilizing effect.

Phonon-photon conversion as mechanism for cooling and coherence transfer

The dynamical Casimir effect is the physical phenomenon where the mechanical energy of a movable wall of a cavity confining a quantum field can be converted into quanta of the field itself. This effect has been recognized as one of the most astonishing predictions of quantum field theory. At the quantum scale, the energy conversion can also occur incoherently, namely, without a physical motion of the wall. By means of quantum thermodynamics, we show that this phenomenon can be employed as a tool to cool down the wall when there is a nonvanishing temperature gradient between the wall and the cavity. At the same time, the mechanism responsible for the heat transfer enables sharing the coherence from one cavity mode, driven by a laser, to the wall, thereby forcing its coherent oscillation. Finally, we show how to employ one laser drive to cool the entire system, including the case when it is composed of other subsystems. Published by the American Physical Society 2024

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green’s first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

Nonequilibrium currents in stochastic field theories: A geometric insight.

We introduce a formalism to study nonequilibrium steady-state probability currents in stochastic field theories. We show that generalizing the exterior derivative to functional spaces allows identification of the subspaces in which the system undergoes local rotations. In turn, this allows prediction of the counterparts in the real, physical space of these abstract probability currents. The results are presented for the case of the Active Model B undergoing motility-induced phase separation, which is known to be out of equilibrium but whose steady-state currents have not yet been observed, as well as for the Kardar-Parisi-Zhang equation. We locate and measure these currents and show that they manifest in real space as propagating modes localized in regions with nonvanishing gradients of the fields.

Hamiltonian Deep Neural Networks Guaranteeing Nonvanishing Gradients by Design

Deep Neural Networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretization of continuous-time Hamiltonian systems and include several existing DNN architectures based on ordinary differential equations. Our main result is that a broad set of H-DNNs ensures non-vanishing gradients by design for an arbitrary network depth. This is obtained by proving that, using a semi-implicit Euler discretization scheme, the backward sensitivity matrices involved in gradient computations are symplectic. We also provide an upper-bound to the magnitude of sensitivity matrices and show that exploding gradients can be controlled through regularization. The good performance of H-DNNs is demonstrated on benchmark classification problems, including image classification with the MNIST dataset.

Conductivity reconstruction from power density data in limited view

In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.

The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal{A}$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval $(0; 1/4]$, then the boundary curve is an ellipse. In the language of first integrals one can assert that if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $\gamma$ admits a $C^1$-smooth first integral with non-vanishing gradient on $\mathcal{A}$, then the curve $\gamma$ is an ellipse. The main ingredients of the proof are (1) the non-standard generating function for convex billiards; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach for rigidity results that was invented by the first named author for circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.

Global Existence Analysis of Energy-Reaction-Diffusion Systems

We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.

Thermoelectric transport coefficients of hot and dense QCD matter

The presence of a nonvanishing thermal gradient and/or a chemical potential gradient in a conducting medium can lead to an electric field—an effect known as thermoelectric effect or Seebeck effect. We discuss here the thermoelectric effects for hot and dense strongly interacting matter within the framework of relativistic Boltzmann equation in the relaxation time approximation. In the context of heavy-ion collisions, the Seebeck coefficients for the quark matter as well as for the hadronic matter are estimated within this approach. The quark matter is modeled by the two flavor Nambu–Jona–Lassinio (NJL) model and the hadronic medium is modeled by the hadron resonance gas (HRG) model with hadrons and their resonances up to a mass cutoff $${\tilde{\varLambda }}\sim 2.6$$ GeV. For the estimation of thermoelectric transport coefficients, for the quark matter, we calculate the relaxation times for the quarks and antiquarks from the quark–quark and quark–antiquark scattering through meson exchange within the NJL model. On the other hand, for the hadronic medium, the relaxation times of hadrons and their resonances are estimated within a hard sphere scattering approximation. We also discuss the formalism of the thermoelectric effect in the presence of a nonvanishing external magnetic field. We give an estimation of the associated magneto-Seebeck coefficient and the Nernst coefficient for the hot and dense QCD matter.

Symmetric ideal magnetofluidostatic equilibria with nonvanishing pressure gradients in asymmetric confinement vessels

We study the possibility of constructing steady magnetic fields satisfying the force balance equation of ideal magnetohydrodynamics with tangential boundary conditions in asymmetric confinement vessels, i.e., bounded regions that are not invariant under continuous Euclidean isometries (translations, rotations, or their combination). This problem is often encountered in the design of next-generation fusion reactors. We show that such configurations are possible if one relaxes the standard assumption that the vessel boundary corresponds to a pressure isosurface. We exhibit a smooth solution that possesses a Euclidean symmetry and yet solves the boundary value problem in an asymmetric ellipsoidal domain while sustaining a nonvanishing pressure gradient. This result provides a definitive answer to the problem of existence of regular ideal magnetofluidostatic equilibria in asymmetric bounded domains. The question remains open whether regular asymmetric solutions of the boundary value problem exist.

Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad–Shafranov equation, which is a single elliptic partial differential equation in two space dimensions. By revealing a hidden symmetry, we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad–Shafranov equation. In contrast to solutions of the classical Grad–Shafranov equation, solutions of the generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad–Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

A Comparison Theorem for Nonsmooth Nonlinear Operators

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity f is Lp function with p > 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.

Compatibility conditions from a Gram–Schmidt decomposition of deformation gradient in two dimensions

Planar deformations are analyzed in the context of a Gram-Schmidt decomposition of deformation gradient into a rotation and upper triangular stretch, where the latter consists of up to three possibly nonzero terms. Necessary—and sufficient under suitable restrictions—conditions for integrability of deformation gradient and stretch (geometrically, vanishing torsion and curvature of a certain affine connection) reduce to a system of three partial differential equations (PDEs). This system appears more convenient than corresponding compatibility conditions from polar decompositions, especially for motions involving simple shear. Admissible families of triangular stretch corresponding to simple shear, pure shear, dilation, and uniaxial strain are analyzed. Also considered are simultaneous simple shear and axial stretch, for which a non-trivial solution with non-vanishing rotation gradient is constructed.

Liquid crystal distortions revealed by an octupolar tensor.

The classical theory of liquid crystal elasticity as formulated by Oseen and Frank describes the (orientable) optic axis of these soft materials by a director n. The ground state is attained when n is uniform in space; all other states, which have a nonvanishing gradient ∇n, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from n and ∇n a third-rank, symmetric, and traceless tensor A (the octupolar tensor). The (nonlinear) eigenvectors of A associated with the local maxima of its cubic form Φ on the unit sphere (its octupolar potential) designate the directions of distortion concentration. The octupolar potential is illustrated geometrically and its symmetries are charted in the space of distortion characteristics, so as to educate the eye to capture the dominating elastic modes. Special distortions are studied, which have everywhere either the same octupolar potential or one with the same shape but differently inflated.

Mathematical Model for Isothermal Wire-Coating From a Bath of Giesekus Viscoelastic Fluid

A mathematical model of wire-coating based on Giesekus constitutive equation is analyzed under isothermal conditions. It is desired to see the functional dependence of Giesekus model parameters on the important operating variables in the wire-coating process, which include volumetric flow rate (later referred to as flow rate), shear stress at the wire surface (later referred to as shear stress), force required for pulling the wire (later referred to as force), and radius of the coated wire (later referred to as coated wire thickness). To this end, the equation governing the laminar, incompressible, and rectilinear flow is first derived and then solved analytically for the case of vanishing axial pressure gradient. A numerical procedure is described to obtain the solution for the case of nonvanishing pressure gradient. Our results indicate that the magnitude of shear stress and force follow a decreasing trend with increasing Giesekus model parameters in both cases. The flow rate and coated wire thickness decrease on increasing the Giesekus model parameters when there is no imposed pressure gradient. However, in the presence of pressure gradient these variables first decrease with increasing Giesekus model parameters and then follow an increasing trend.

Effects of fluid velocity gradients on heavy quark energy loss

We use holographic duality to analyze the drag force on, and consequent energy loss of, a heavy quark moving through a strongly coupled conformal fluid with non-vanishing gradients in its velocity and temperature. We derive the general expression for the drag force to first order in the fluid gradients. Using this general expression, we show that a quark that is instantaneously at rest, relative to the fluid, in a fluid whose velocity is changing with time feels a nonzero force. And, we show that for a quark that is moving ultra-relativistically, the first order gradient "corrections" become larger than the zeroth order drag force, suggesting that the gradient expansion may be unreliable in this regime. We illustrate the importance of the fluid gradients for heavy quark energy loss by considering a fluid with one-dimensional boost invariant Bjorken expansion as well as the strongly coupled plasma created by colliding sheets of energy.

Cosmic ray transport in non-uniform magnetic fields: consequences of gradient and curvature drifts

AbstractLarge-scale spatial variations of the guide magnetic field of interplanetary and interstellar plasmas give rise to the mirror force −(p⊥2/2mγB)∇B). The parallel component of this mirror force causes adiabatic focusing of the cosmic ray guiding center whereas the perpendicular component of the mirror force gives rise to the gradient and curvature drifts of the cosmic ray guiding center. Adiabatic focusing and the gradient and curvature drift terms additionally enter the Fokker–Planck transport equation for the gyrotropic cosmic ray particle phase space density in partially turbulent non-uniform magnetic fields. For magnetohydrodynamic turbulence with dominating magnetic fluctuations, the diffusion approximation is justified, which results in a modification of the diffusion–convection transport equation for the isotropic part of the gyrotropic phase space density from the additional focusing and drift terms. For axisymmetric undamped slab Alfvenic turbulence we show that all perpendicular spatial diffusion coefficients are caused by the non-vanishing gradient and curvature drift terms. For a specific (symmetric in μ) choice of the pitch-angle Fokker–Planck coefficients we show that the ratio of the perpendicular to parallel spatial diffusion coefficients apart from a constant is determined by the spatial first derivatives of the non-constant cosmic ray Larmor radius in the non-uniform magnetic field.

Extremality conditions for the quasi-concavity function and applications

A maximum principle for the lower envelope of two strictly subharmonic functions is proved, and subsequently used to investigate the first- and second-order extremality conditions for the quasi-concavity function. An application is done to the Dirichlet problem associated to elliptic equations involving the Laplacian as well as the minimal surface operator, when the domain of the problem is a convex ring and two constant boundary values are prescribed. The right-hand side may depend on the solution and on any of its first derivatives, and must depend on the space variable. The solution is proved to have convex level sets and a non-vanishing gradient. Assumptions are translation-invariant. Poisson’s equation is considered explicitly.

Modeling and 3D object reconstruction by implicitly defined surfaces with sharp features

We propose a new method for describing sharp features (i.e., edges and vertices) of implicitly defined surfaces. We consider an initial implicitly defined surface, which is represented as the zero set of a C^1 smooth scalar field with non-vanishing gradients. In order to represent sharp edges and vertices, this surface is augmented by adding new types of implicit representations, called edge descriptors and vertex descriptors. They are defined with the help of the distance field of edge curves. In our implementation, we use circular splines to describe these edge curves, since they support a fast and non-iterative closest point computation. After adding the edge and vertex descriptors to the initial scalar field, the zero set of the augmented function contains the sharp features. We apply the new representation to surface modeling by implicitly defined surfaces with sharp features and to object reconstruction. In the latter case we describe an algorithm for detecting the sharp curves and vertices of a shape which is given by an unorganized point cloud, which are then approximated by circular splines, in order to define the edge and vertex descriptors.

Semi-global simulations of the magneto-rotational instability in core collapse supernovae

Context. Possible effects of magnetic fields in core collapse supernovae rely on an efficient amplification of the weak pre-collapse fields. It has been suggested that the magneto-rotational instability (MRI) leads to a rapid growth for these weak seed fields. Although plenty of MRI studies exist for accretion disks, the application of their results to core collapse supernovae is inhibited as the physics of supernova cores is substantially different from that of accretion discs. Aims. We address the problem of growth and saturation of the MRI in core collapse supernovae by studying its evolution by means of semi-global simulations, which combine elements of global and local simulations by taking the presence of global background gradients into account and using a local computational grid. We investigate, in particular, the termination of the growth of the MRI and the properties of the turbulence in the saturated state.Methods. We analyze the dispersion relation of the MRI to identify different regimes of the instability. This analysis is complemented by semi-global ideal MHD simulations, where we consider core matter in a local computational box (size ∼ km) rotating at sub-Keplerian velocity and where we allow for the presence of a radial entropy gradient, but neglect neutrino radiation. Results. We identify six regimes of the MRI depending on the ratio of the entropy and angular velocity gradient. Our numerical models confirm the instability criteria and growth rates for all regimes relevant to core-collapse supernovae. The MRI grows exponentially on time scales of milliseconds, the flow and magnetic field geometries being dominated by channel flows. We find MHD turbulence and efficient transport of angular momentum. The MRI growth ceases once the channels are disrupted by resistive instabilities (stemming from to the finite conductivity of the numerical code), and MHD turbulence sets in. From an analysis of the growth rates of the resistive instabilities, we deduce scaling laws for the termination amplitude of the MRI, which agree well with our numerical models. We determine the dependence of the development of large-scale coherent flow structures in the saturated state on the aspect ratio of the simulation boxes. Conclusions. The MRI can grow rapidly under the conditions considered here, i.e., a rapidly rotating core in hydrostatic equilibrium, possibly endowed with a nonvanishing entropy gradient, leading to fields exceeding . More investigations are required to cover the parameter space more comprehensively and to include more physical effects.