Liouville-type Equations Research Articles

Efficiency analysis of discontinuous Galerkin approaches for the application onto quantum Liouville-type equations

The simulation of nanodevices is computationally inefficient with current algorithms. The discontinuous Galerkin approach has been demonstrated in the field of computational fluid dynamics to deliver high order accuracy and efficiency due to its reliance on matrix–vector multiplications. Previously, the discontinuous Galerkin approach was successfully used in conjunction with the finite volume technique to solve the Liouville–von Neumann equation in center-mass coordinates and thus simulate nanodevices. To exploit its full potential regarding high-performance computing, this work aims to substitute the aforementioned finite volume technique with the discontinuous Galerkin method. To arrive at the said formalism, a finite element method is implemented as an intermediate step.

Application of the tight-binding method onto the Von Neumann equation

This paper presents a numerical framework for the analysis of quantum devices based on the Von Neumann (VN) equation, which involves the concept of the Tight-Binding Method (TBM). The model is based on the application of the Tight-Binding Hamiltonian within Quantum Liouville Type Equations and has the advantage that the atomic structure of the materials used is taken into account. Furthermore, the influence of a Complex Absorbing Potential (CAP) as a complementary boundary condition and its essential contribution to the system stability with respect to the eigenvalue spectrum is discussed.

On Liouville-type theorems for 𝑘-Hessian equations with gradient terms

In this paper, we investigate several Liouville-type theorems related to k k -Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S k [ D 2 u ] ≥ h ( u , | D u | ) S_k[D^2u]\ge h(u,|Du|) in R n \mathbb {R}^n . The results depend on some qualified growth conditions of h h at infinity. A Liouville-type result to subsolutions of a prototype equation S k [ D 2 u ] = f ( u ) + g ( u ) ϖ ( | D u | ) S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S k [ D 2 u ] = f ( u ) + g ( u ) | D u | q S_k[D^2u]=f(u)+g(u)|Du|^q for 0 ≤ q > k + 1 0\le q>k+1 is also given.

Blow-up Analysis of Stationary Solutions to a Liouville-Type Equation in 3-D

Blow-up Analysis of Stationary Solutions to a Liouville-Type Equation in 3-D

Positive solutions for the Riemann–Liouville-type fractional differential equation system with infinite-point boundary conditions on infinite intervals

In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda >0$\\end{document} by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results.

Time-resolved analysis of dual-gate FETs with non-parabolic energy dispersion for THz applications

The investigation of charge carrier transport in state-of-the-art nanoelectronic devices based on III/V semiconductors proves to be challenging, even more so when the highly non-parabolic energy dispersion exhibited by these materials is taken into account. Unlike the common approach of neglecting this behavior by the use of the parabolic band approximation, a novel combination of a tight-binding approach with a quantum Liouville-type equation is introduced here, where any arbitrary energy dispersion can effectively be included. This leads to a discretization based on the atomic structure without the need for finite difference approximations of the Hamiltonian. Because this allows for the stationary as well as the transient simulation of quantum charge carrier transport, it is well suited for the analysis of ultrathin FETs such as dual-gate FETs when it is combined with a mode-space approach. We demonstrate that the parabolic approximation not only vastly underestimates the current densities when compared to the non-parabolic case but also fails to capture transient effects such as gain compression when amplifier operation is considered.

Investigating the impact of quantum confinement on the THz behavior of Nanoscale FinFETs

The stationary and transient behavior of a 3 nm wide on-insulator type FinFET is investigated. The focus is laid on the evaluation and characterization of the effect of quantum confinement on the amplifier behavior. The high computational burden related to the time-resolved analysis of such three-dimensional simulations is dealt with by applying a mode-space approach onto a Quantum Liouville-type Equation. It is shown that the diagonal coupling terms related to the mode-space approach should be taken into consideration for the analysis of FinFETs. An increase in gain compression and higher order harmonic distortion is observed, when the fin height is decreased and, therefore, stronger quantum confinement occurs. The 1 dB compression point is shown to decrease by as much as 8 dB when reducing the fin height from 7 to 3 nm.

On some weighted fourth-order equations

This paper deals with the following radial Caffarelli-Kohn-Nirenberg-type inequality,∫RN|x|α|Δu|2dx≥Srad(N,α)(∫RN|x|l|u|pα⁎dx)2pα⁎,u∈Cc∞(RN), where N≥3, 2<α<N, l=4(α−2)(N−2)N−α−α and pα⁎=2(N+l)N−4+α. Then we consider the related Euler-Lagrange equation:Δ(|x|αΔu)=|x|lupα⁎−1,u>0inRN. For α≠0 or l≠0, it is known the solutions of above equation are invariant for dilations λN−4+α2u(λx) but not for translations. However we show that if α is an even integer, there exist new solutions to the linearized problem, related to the radial solution that “replace” the ones due to the translations invariance. As an application, we turn back to investigate the remainder terms of the above inequality. As well as some fourth-order Liouville-type equations with singular data.

The $$C^0$$-convergence at the Neumann boundary for Liouville equations

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li (Commun Math Phys 200(2):421–444, 1999) gave a uniform asymptotic estimate. Later, Zhang (Commun Math Phys 268(1):105–133, 2006) and Gluck (Nonlinear Anal 75(15):5787–5796, 2012) improved Li’s estimate in the sense of $$C^0$$ -convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao–Wang–Zhou (J Math Anal Appl 418:142–162, 2014) proved a similar asymptotic estimate of Li (1999). In this paper, we will prove a $$C^0$$ -convergence result in this boundary blow-up process. Our method is different from Gluck (2013), Zhang (2006).

Blowing-up solutions for a nonlocal Liouville type equation

Blowing-up solutions for a nonlocal Liouville type equation

Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations

This paper is concerned with the invariant measures of non-autonomous stochastic reaction-diffusion equations on unbounded domains. We firstly investigate the existence of invariant measures for random dynamical systems over two parametric spaces as well as periodic invariant measures, and then discuss the convergence of (periodic) invariant measures with respect to some system parameters. Random Liouville type equation is also derived for invariant measures associated with non-autonomous random differential equations. Based on such abstract results, we prove the existence of (periodic) invariant measures for non-autonomous stochastic reaction-diffusion equation on unbounded domains, which are supported on the pullback attractors. And we further show every limit of a sequence of (periodic) invariant measures must be the (periodic) invariant measure of the corresponding limiting equation when the noise intensity varies in finite interval. Moreover, we prove that the invariant measures of the underlying equation satisfy a stochastic Liouville type equation.

Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results.

A blow-up phenomenon for a non-local Liouville-type equation

We consider the non-local Liouville equation $${\left( { - \Delta } \right)^{{1 \over 2}}}u = {h_\varepsilon }{e^u} - 1\,\,\,\,\,{\rm{in}}\,\,{\mathbb{S}^1},$$ corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up, when hε approaches a function h as ε → 0, at a critical point of the harmonic extension of h provided some generic assumptions are satisfied.

Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

On the blow-up analysis at collapsing poles for solutions of singular Liouville-type equations

We analyze a blow-up sequence of solutions for Liouville-type equations involving Dirac measures with “collapsing” poles. We consider the case where blow-up occurs exactly at a point where the poles coalesce. After proving that a” quantization” property still holds for the” blow-up mass,” we obtain precise pointwise estimates when blow-up occurs with the least blow-up mass. Interestingly, such estimates express the exact analogue of those previously obtained for solutions of “regular” Liouville equations where the “collapsing” Dirac measures are neglected. Such information will be used in a forthcoming paper to describe the asymptotic behavior of minimizers of the Donaldson functional introduced by Goncalves and Uhlenbeck in 2007, yielding to mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds.

THz gain compression in nanoscale FinFETs

The gain compression and excitation of higher order harmonics in 3 nm wide on-insulator type FinFETs is investigated. The high computational burden related to the time-resolved analysis of such three-dimensional devices is dealt with by applying a mode-space approach onto a Quantum Liouville-type Equation. With this approach an increase in gain compression and higher order harmonic distortion is observed when decreasing the fin height from 5 to 3 nm.

Multiple solutions for nonlinear Sturm–Liouville differential equations with possibly negative variable coefficients

The aim of this paper is to study a parameter-dependent nonlinear differential problem with Sturm–Liouville type equation and Dirichlet boundary conditions. In particular, by requiring a suitable behavior on the nonlinearity, we determine an interval of parameter λ for which the problem admits three or infinitely many solutions.

Quantized point vortex equilibria in a one-way interaction model with a Liouville-type background vorticity on a curved torus

We construct point vortex equilibria with strengths quantized by multiples of 2π in a fixed background vorticity field on the surface of a curved torus. The background vorticity consists of two terms: first, a term exponentially related to the stream function and a second term arising from the curvature of the torus, which leads to a Liouville-type equation for the stream function. By using a stereographic projection of the torus onto an annulus in a complex plane, the Liouville-type equation admits a class of exact solutions, given in terms of a loxodromic function on the annulus. We show that appropriate choices of the loxodromic function in the solution lead to stationary vortex patterns with 4n̂ point vortices of identical strengths, n̂∈N. The quantized point vortices are stationary in the sense that they are equilibria of a “one-way interaction” model where the evolution of point vortices is subject to the continuous background vorticity, while the background vorticity distribution is not affected by the velocity field induced by the point vortices. By choosing loxodromic functions continuously dependent on a parameter and taking appropriate limits with respect to this parameter, we show that there are solutions with inhomogeneous point vortex strengths, in which the exponential part of the background vorticity disappears. The point vortices are always located at the innermost and outermost rings of the torus owing to the curvature effects. The topological features of the streamlines are found to change as the modulus of the torus changes.

Liouville links and chains on the plane and associated stationary point vortex equilibria

&lt;p style='text-indent:20px;'&gt;Liouville links and chains are exact steady solutions of the Euler equation for two-dimensional, incompressible, homogeneous and planar fluid flow, uncovered recently in [&lt;xref ref-type="bibr" rid="b11"&gt;11&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b12"&gt;12&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b13"&gt;13&lt;/xref&gt;]. These solutions consist of a set of stationary point vortices embedded in a smooth non-zero and non-uniform background vorticity described by a Liouville-type partial differential equation. The solutions contain several arbitrary parameters and possess a rich structure. The background vorticity can be varied with one of the parameters, resulting in two limiting cases where it concentrates into some point vortex equilibrium configuration in one limit and another distinct point vortex equilibrium in the other limit. By a simple scaling of the point vortex strengths at a limit, a new steady solution can be constructed, and the procedure iterated indefinitely in some cases. The resulting sequence of solutions has been called a Liouville chain [&lt;xref ref-type="bibr" rid="b13"&gt;13&lt;/xref&gt;]. A transformation exists that can produce the limiting point vortex equilibria from a given seed equilibrium. In this paper, we collect together all these results in a review and present selected new examples corresponding to special sequences of 'collapse configurations.' The final section discusses possible applications to different geophysical flow scenarios.&lt;/p&gt;

Kinetic equations for processes on co-evolving networks

&lt;p style='text-indent:20px;'&gt;The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.&lt;/p&gt;