Liouville-type Equations Research Articles

Subdomain-based exponential integrators for quantum Liouville-type equations

In order to describe quantum mechanical effects, the use of the von-Neumann equation is apparent. In this work, we present a unified numerical framework so that the von-Neumann equation in center-of-mass coordinates leads to a Quantum Liouville-type equation when choosing a suitable basis. In particular, the proposed approach can be related to the conventional Wigner equation when a plane wave basis is used. The drawback of the numerical methods is the high computational cost. Our presented approach is extended to allow reducing the dimension of the basis, which leads to a computationally efficient and accurate subdomain approach. Not only the steady-state behavior is of interest, but also the dynamic behavior. In order to solve the time-dependent case, suitable approximation methods for the time-dependent exponential integrator are necessary. For this purpose, we also investigate approximations of the exponential integrator based on Faber polynomials and Krylov methods. In order to evaluate and justify our approach, various test cases, including a resonant tunnel diode as well as a double-gate field-effect transistor, are investigated and validated for the stationary and the dynamic device behavior.

Concentration solutions to the singularly prescribed Gaussian and geodesic curvatures problem

We consider the following Liouville-type equation with exponential Neumann boundary condition:{−Δu˜=ε2K(x)e2u˜,x∈D,∂u˜∂n+1=εκ(x)eu˜,x∈∂D, where D⊂R2 is the unit disk, ε2K(x)>0 and εκ(x)>0 stand for the prescribed Gaussian curvature and geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if κ(x)+K(x)+κ(x)2 (x∈∂D) has a local extremum point, which is a new result for exponential Neumann boundary problems.

Boundary value problems for local and nonlocal Liouville type equations with several exponential type nonlinearities. Radial and nonradial solutions

This paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc B_{1} subset {mathbf{R}}^{2} and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at partial B_{1} and under appropriate conditions is constructed. It develops logarithmic singularities for r = 0 , r = infty . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.

Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices

We first verify the global well-posedness of the impulsive reaction-diffusion equations on infinite lattices. Then we establish that the generated process by the solution operators has a pullback attractor and a family of Borel invariant probability measures. Furthermore, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results show that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, and the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point.

Subdynamics of fluctuations in an equilibrium classical many-particle system and generalized linear Boltzmann and Landau equations.

Exact completely closed homogeneous generalized master equations (GMEs) governing the evolution in time of equilibrium two-time correlation functions for dynamic variables of a subsystem of s particles (s<N) selected from N≫1 particles of a classical many-body system are obtained. These time-convolution and time-convolutionless GMEs differ from the known GMEs (e.g., Nakajima-Zwanzig GME) by the absence of inhomogeneous terms containing correlations between all N particles at the initial moment of time and preventing the closed description of s-particle subsystem evolution. Closed homogeneous GMEs describing the subdynamics of fluctuations are obtained by applying a special projection operator to the Liouville-type equation governing the dynamics of the correlation function with the related to the Gibbs distribution initial state, which is more natural than the conventional factorized initial state. No common approximation, like the "molecular chaos," is needed. In the linear approximation in the particles' density, the linear generalized Boltzmann equation accounting for initial correlations and valid at all timescales is obtained. This equation for a weak interparticle interaction converts into the generalized linear Landau equation in which the initial correlations are also accounted for. The connection of these equations to the nonlinear Boltzmann and Landau equations is discussed. The same approach is applicable to studying the kinetics of the conventional reduced s-particle distribution functions for a classical N-particle system driven from an equilibrium state by an external force.

Liouville-type equations for the n-particle distribution functions of an open system

In this work, we derive a mathematical model for an open system that exchanges particles and momentum with a reservoir from their joint Hamiltonian dynamics. The complexity of this many-particle problem is addressed by introducing a countable set of n-particle phase space distribution functions just for the open subsystem, while accounting for the reservoir only in terms of statistical expectations. From the Liouville equation for the full system, we derive a set of coupled Liouville-type equations for the n-particle distributions by marginalization with respect to reservoir states. The resulting equation hierarchy describes the external momentum forcing of the open system by the reservoir across its boundaries, and it covers the effects of particle exchanges, which induce probability transfers between the n- and (n + 1)-particle distributions. Similarities and differences with the Bergmann–Lebowitz model of open systems [P. G. Bergmann and J. L. Lebowitz, Phys. Rev. 99, 578–587 (1955)] are discussed in the context of the implementation of these guiding principles in a computational scheme for molecular simulations.

The HK-Sobolev space and applications to one-dimensional boundary value problems

In this paper, we introduce the HK-Sobolev space and establish a fundamental theorem of calculus and an integration by parts formula, then we give sufficient conditions for the existence and uniqueness of a solution to a variational problem associated with a Sturm–Liouville type equation involving Henstock-Kurzweil integrable functions as source terms.

Remarks on Solutions of the Generalized Jackiw-Pi Model with Self-Dual Potential

We derive the self-dual equations of the generalized Jackiw-Pi model with the self-dual potential (1.7). We also show that the finite energy solution of the generalized Jackiw-Pi model with special self-dual potential (1.9) satisfies the self-dual system which can be transformed into the Liouville type equation.

Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications

In this article, we first prove, from the viewpoint of infinite dynamical system, sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. Then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a Liouville type equation. Moreover, we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the Liouville type equation. Finally, we study the equations of three-dimensional incompressible magneto-micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations.

A note on a sinh-Poisson type equation with variable intensities on pierced domains

We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain Δ u + ρ ( V 1 ( x ) e u − V 2 ( x ) e − τ u ) = 0 in Ω ϵ : = Ω ∖ ⋃ i = 1 m B ( ξ i , ϵ i ) ‾ u = 0 on ∂ Ω ϵ , where ρ &gt; 0, V 1 , V 2 &gt; 0 are smooth potentials in Ω, τ &gt; 0, Ω is a smooth bounded domain in R 2 and B ( ξ i , ϵ i ) is a ball centered at ξ i ∈ Ω with radius ϵ i &gt; 0, i = 1 , … , m. When ρ &gt; 0 is small enough and m 1 ∈ { 1 , … , m − 1 }, there exist radii ϵ = ( ϵ 1 , … , ϵ m ) small enough such that the problem has a solution which blows-up positively at the points ξ 1 , … , ξ m 1 and negatively at the points ξ m 1 + 1 , … , ξ m as ρ → 0. The result remains true in cases m 1 = 0 with V 1 ≡ 0 and m 1 = m with V 2 ≡ 0, which are Liouville type equations.

Strong trajectory statistical solutions and Liouville type equation for dissipative Euler equations

The main aim of this letter is to use the strong compact strong trajectory attractor to construct the strong trajectory statistical solutions for two-dimensional dissipative Euler equations. Further, it is established that the constructed trajectory statistical solutions possess an invariant property and satisfy a Liouville type equation.

Domain Wall Solitons Arising in Classical Gauge Field Theories

Domain wall solitons are basic constructs realizing phase transitions in various field-theoretical models and are solutions to some nonlinear ordinary differential equations descending from the corresponding full sets of governing equations in higher dimensions. In this paper, we present a series of domain wall solitons arising in several classical gauge field theory models. In the context of the Abelian gauge field theory, we unveil the surprising result that the solutions may explicitly be constructed, which enriches our knowledge on integrability of the planar Liouville type equations in their one-dimensional limits. In the context of the non-Abelian gauge field theory, we obtain some existence theorems for domain wall solutions arising in the electroweak type theories by developing some methods of calculus of variations formulated as direct and constrained minimization problems over a weighted Sobolev space.

On a singular Liouville-type equation and the Alexandrov isoperimetric inequlity

We obtain a generalized version of an inequality, first derived by C. Bandle in the analytic setting, for weak subsolutions of a singular Liouville-type equation. As an application we obtain a new proof of the Alexandrov isoperimetric inequality on singular abstract surfaces. Interestingly enough, motivated by this geometric problem, we obtain a seemingly new characterization of local metrics on Alexandrov's surfaces of bounded curvature. At least to our knowledge, the characterization of the equality case in the isoperimetric inequality in such a weak framework is new as well.

Special Class of Liouville-Type Hyperbolic Equations

In this paper, we obtain differential substitutions for a special class of Liouville-type equations that simplify these equations. We present necessary and sufficient conditions of the existence of y-integrals of the second order. We examine the principal case of necessary and sufficient conditions of the existence of y-integrals.

Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.

Liouville type equation with exponential Neumann boundary condition and with singular data

In this paper we will analyze the blow-up behaviors of solutions to the singular Liouville type equation with exponential Neumann boundary condition. We generalize the Brezis–Merle type concentration-compactness theorem to this Neumann problem. Then along the line of the Li–Shafrir type quantization property we show that the blow-up value $$m(0) \in 2\pi \mathbb N\cup \{ 2\pi (1+\alpha )+2\pi (\mathbb N\cup \{0\})\}$$ if the singular point 0 is a blow-up point. In the end, when the boundary value of solutions has an additional condition, we can obtain the precise blow-up value $$m(0)=2\pi (1+\alpha )$$ .

Asymptotic properties of solutions to difference equations of Sturm–Liouville type

We consider the discrete Sturm–Liouville type equation of the formΔ(rnΔxn)=anf(xσ(n))+bn.Assume s is a given nonpositive real number. We present sufficient conditions for the existence of solution x with the asymptotic behaviorxn=c(r1−1+⋯+rn−1−1)+d+o(ns)where c, d are given real numbers. Moreover, we establish conditions under which for a given solution x there exist real numbers c, d such that x has the above asymptotic behavior.

Weighted Trudinger-Moser inequalities and associated Liouville type equations

We discuss some Trudinger-Moser inequalities with weighted Sobolev norms. Suitable logarithmic weights in these norms allow an improvement in the maximal growth for integrability when one restricts to radial functions. The main results concern the application of these inequalities to the existence of solutions for certain mean-field equations of Liouville type. Sharp critical thresholds are found such that for parameters below these thresholds the corresponding functionals are coercive, and hence solutions are obtained as global minima of these functionals. In the critical cases the functionals are no longer coercive and solutions may not exist. We also discuss a limiting case, in which the allowed growth is of double exponential type. Surprisingly, we are able to show that in this case a local minimum persists to exist for critical and also for slightly supercritical parameters. This allows us to obtain the existence of a second (mountain-pass) solution for almost all slightly supercritical parameters using the Struwe monotonicity trick. This result is in contrast to the non-weighted case, where positive solutions do not exist (in star-shaped domains) in the critical and supercritical cases.

Prescribing Gaussian and Geodesic Curvature on Disks

Abstract In this paper, we consider the problem of prescribing the Gaussian and geodesic curvature on a disk and its boundary, respectively, via a conformal change of the metric. This leads us to a Liouville-type equation with a non-linear Neumann boundary condition. We address the question of existence by setting the problem in a variational framework which seems to be completely new in the literature. We are able to find minimizers under symmetry assumptions.