Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution
Ansgar Jüngel Josipa-Pina Milišić
Navier-Stokes equations for compressible quantum fluids, including the energy equation, are derived from a collisional Wigner equation, using the quantum entropy maximization method of Degond and Ringhofer. The viscous corrections are obtained from a Chapman-Enskog expansion around the quantum equilibrium distribution and correspond to the classical viscous stress tensor with particular viscosity coefficients depending on the particle density and temperature. The energy and entropy dissipations are computed and discussed. Numerical simulations of a one-dimensional tunneling diode show the stabilizing effect of the viscous correction and the impact of the relaxation terms on the current-voltage charcteristics.
keywords: tunneling diode finite differences Chapman-Enskog expansion semiconductors. density-dependent viscosity Quantum Navier-Stokes equations
First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation
Ansgar Jüngel Ingrid Violet
A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is analyzed. Using a new semi-discrete approximation in time, a first-order entropy–entropy dissipation inequality is proved. Passing to the limit of vanishing time discretization parameter, some regularity results are deduced. Moreover, it is shown that the solution is strictly positive for large time if it does so initially.
keywords: Entropy--entropy dissipation inequality regularity of solutions existence of weak solutions entropy construction method implicit Euler scheme long-time behavior of solutions.
Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations
José A. Carrillo Jean Dolbeault Ivan Gentil Ansgar Jüngel
In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus $S^1$ and explore their implications for the long-time asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, and an asymptotically exponential convergence of positive solutions towards their average. The asymptotically exponential regime is valid for a larger range of parameters for all relevant cases of application: porous medium/fast diffusion, thin film and logarithmic fourth order nonlinear diffusion equations. The techniques are inspired by direct entropy-entropy production methods and based on appropriate Poincaré type inequalities.
keywords: Poincare inequality higher-order nonlinear PDEs entropy production entropy-entropy production method Sobolev estimates entropy fast diffusion equation parabolic equations Logarithmic Sobolev inequality long-time behavior thin film equation. porous media equation
Mixed entropy estimates for the porous-medium equation with convection
Ansgar Jüngel Ingrid Violet
In this paper, we answer the question under which conditions the porous-medium equation with convection and with periodic boundary conditions possesses gradient-type Lyapunov functionals (first-order entropies). It is shown that the weighted sum of first-order and zeroth-order entropies are Lyapunov functionals if the weight for the zeroth-order entropy is sufficiently large, depending on the strength of the convection. This provides new a priori estimates for the convective porous-medium equation. The proof is based on an extension of the algorithmic entropy construction method which is based on systematic integration by parts, formulated as a polynomial decision problem.
keywords: systematic integration by parts. a priori estimates Slow diffusion first-order entropies fast diffusion entropy methods
A discrete Bakry-Emery method and its application to the porous-medium equation
Ansgar Jüngel Stefan Schuchnigg

The exponential decay of the relative entropy associated to a fully discrete porous-medium equation in one space dimension is shown by means of a discrete Bakry-Emery approach. The first ingredient of the proof is an abstract discrete Bakry-Emery method, which states conditions on a sequence under which the exponential decay of the discrete entropy follows. The second ingredient is a new nonlinear summation-by-parts formula which is inspired by systematic integration by parts developed by Matthes and the first author. Numerical simulations illustrate the exponential decay of the entropy for various time and space step sizes.

keywords: Finite differences Bakry-Emery method large-time asymptotics systematic integration by parts
Positive entropic schemes for a nonlinear fourth-order parabolic equation
José A. Carrillo Ansgar Jüngel Shaoqiang Tang
A finite-difference scheme with positivity-preserving and entropy-decreasing properties for a nonlinear fourth-order parabolic equation arising in quantum systems and interface fluctuations is derived. Initial-boundary value problems, the Cauchy problem and a rescaled equation are discussed. Based on this scheme we elucidate properties of the long-time asymptotics for this equation.
keywords: long-time behavior of discrete solutions. Finite difference method discrete entropy estimates discrete positive solutions
Semiclassical limit in a simplified quantum energy-transport model for semiconductors
Li Chen Xiu-Qing Chen Ansgar Jüngel
The semiclassical limit in a quantum energy-transport model for semiconductors is proved. The system consists of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; a degenerate elliptic heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The asymptotic limit is based on a priori estimates independent of the scaled Planck constant, obtained from entropy functionals, on the use of Gagliardo-Nirenberg inequalities, and weak compactness methods.
keywords: semiconductors. Quantum energy transport degenerate elliptic equation semiclassical limit fourth-order equation
On the Lagrangian structure of quantum fluid models
Philipp Fuchs Ansgar Jüngel Max von Renesse
Some quantum fluid models are written as the Lagrangian flow of mass distributions and their geometric properties are explored. The first model includes magnetic effects and leads, via the Madelung transform, to the electromagnetic Schrödinger equation in the Madelung representation. It is shown that the Madelung transform is a symplectic map between Hamiltonian systems. The second model is obtained from the Euler-Lagrange equations with friction induced from a quadratic dissipative potential. This model corresponds to the quantum Navier-Stokes equations with density-dependent viscosity. The fact that this model possesses two different energy-dissipation identities is explained by the definition of the Noether currents.
keywords: magnetic Schrödinger equation symplectic form Noether theory quantum Navier-Stokes equations osmotic velocity. Geometric mechanics
Small velocity and finite temperature variations in kinetic relaxation models
Kazuo Aoki Ansgar Jüngel Peter A. Markowich
A small Knuden number analysis of a kinetic equation in the diffusive scaling is performed. The collision kernel is of BGK type with a general local Gibbs state. Assuming that the flow velocity is of the order of the Knudsen number, a Hilbert expansion yields a macroscopic model with finite temperature variations, whose complexity lies in between the hydrodynamic and the energy-transport equations. Its mathematical structure is explored and macroscopic models for specific examples of the global Gibbs state are presented.
keywords: hydrodynamic equations diffusive limit Kinetic equation energy-transport equations. Gibbs state
A kinetic equation for economic value estimation with irrationality and herding
Bertram Düring Ansgar Jüngel Lara Trussardi

A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.

keywords: Inhomogeneous Boltzmann equation public information herding Fokker-Planck equation existence of solutions sociophysics

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