arXiv subjects

Ansgar Jüngel

Publications and source records attributed to Ansgar Jüngel.

At least 127 records · Page 7Linked to original sources

arXiv2015

Qualitative behavior of solutions to cross-diffusion systems from population dynamics

A general class of cross-diffusion systems for two population species in a bounded domain with no-flux boundary conditions and Lotka-Volterra-type source terms is analyzed. Although the diffusion coefficients are assumed to depend linearly on the population densities, the equations are strongly coupled. Generally, the diffusion matrix is neither symmetric nor positive definite. Three main results are proved: the existence of global uniformly bounded weak solutions, their convergence to the constant steady state in the weak competition case, and the uniqueness of weak solutions. The results hold under appropriate conditions on the diffusion parameters which are made explicit and which contain simplified Shigesada-Kawasaki-Teramoto population models as a special case. The proofs are based on entropy methods, which rely on convexity properties of suitable Lyapunov functionals.

math.AP↗

arXiv2015

Hypocoercivity for a linearized multi-species Boltzmann system

A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cut-off condition. Two proofs are given: a non-constructive one, based on the decomposition of the collision operator into a compact and a coercive part, and a constructive one, which exploits the "cross-effects" coming from collisions between different species and which yields explicit constants. Furthermore, the essential spectra of the linearized collision operator and the linearized Boltzmann operator are calculated. Based on the spectral-gap estimate, the exponential convergence towards global equilibrium with explicit rate is shown for solutions to the linearized multi-species Boltzmann system on the torus. The convergence is achieved by the interplay between the dissipative collision operator and the conservative transport operator and is proved by using the hypocoercivity method of Mouhot and Neumann.

math.AP↗

arXiv2015

A higher-order gradient flow scheme for a singular one-dimensional diffusion equation

A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow, is numerically solved in one space dimension using a higher-order minimizing movement scheme based on the BDF (backward differentiation formula) discretization. In each time step, the approximation is obtained as the solution of a constrained quadratic minimization problem on a finite-dimensional space consisting of piecewise quadratic basis functions. The numerical scheme conserves the mass and dissipates the $G$-norm of the two-step BDF time approximation. Numerically, also the discrete entropy and variance are decaying. The decay turns out to be exponential in all cases. The corresponding decay rates are computed numerically for various grid numbers.

math.NA↗

arXiv2015

Analysis of a coupled spin drift-diffusion Maxwell-Landau-Lifshitz system

The existence of global weak solutions to a coupled spin drift-diffusion and Maxwell-Landau-Lifshitz system is proved. The equations are considered in a two-dimensional magnetic layer structure and are supplemented with Dirichlet-Neumann boundary conditions. The spin drift-diffusion model for the charge density and spin density vector is the diffusion limit of a spinorial Boltzmann equation for a vanishing spin polarization constant. The Maxwell-Landau-Lifshitz system consists of the time-dependent Maxwell equations for the electric and magnetic fields and of the Landau-Lifshitz-Gilbert equation for the local magnetization, involving the interaction between magnetization and spin density vector. The existence proof is based on a regularization procedure, $L^2$-type estimates, and Moser-type iterations which yield the boundedness of the charge and spin densities. Furthermore, the free energy is shown to be nonincreasing in time if the magnetization-spin interaction constant in the Landau-Lifshitz equation is sufficiently small.

math.AP↗

arXiv2015

Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations

Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.

math.NA↗

arXiv2015

Analysis of degenerate cross-diffusion population models with volume filling

A class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions. The equations are formally derived from a random-walk lattice model in the diffusion limit. Compared to previous results in the literature, the novelty is the combination of general degenerate diffusion and volume-filling effects. Conditions on the nonlinear diffusion coefficients are identified, which yield a formal gradient-flow or entropy structure. This structure allows for the proof of global-in-time existence of bounded weak solutions and the exponential convergence of the solutions to the constant steady state. The existence proof is based on an approximation argument, the entropy inequality, and new nonlinear Aubin-Lions compactness lemmas. The proof of the large-time behavior employs the entropy estimate and convex Sobolev inequalities. Moreover, under simplifiying assumptions on the nonlinearities, the uniqueness of weak solutions is shown by using the $H^{-1}$ method, the $E$-monotonicity technique of Gajewski, and the subadditivity of the Fisher information.

math.AP↗

arXiv2015

A Finite-Volume Scheme for a Spinorial Matrix Drift-Diffusion Model for Semiconductors

An implicit Euler finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors is analyzed. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin-vector densities, coupled to the Poisson equation for the elec-tric potential. The equations are solved in a bounded domain with mixed Dirichlet-Neumann boundary conditions. The charge and spin-vector fluxes are approximated by a Scharfetter-Gummel discretization. The main features of the numerical scheme are the preservation of positivity and L $\infty$ bounds and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. For undoped semiconductor materials, the numerical scheme is uncon-ditionally stable. The fundamental ideas are reformulations using spin-up and spin-down densities and certain projections of the spin-vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic-layer field-effect transistor in two space dimensions are presented.

math.NA↗

arXiv2014

Performance Portability Study of Linear Algebra Kernels in OpenCL

The performance portability of OpenCL kernel implementations for common memory bandwidth limited linear algebra operations across different hardware generations of the same vendor as well as across vendors is studied. Certain combinations of kernel implementations and work sizes are found to exhibit good performance across compute kernels, hardware generations, and, to a lesser degree, vendors. As a consequence, it is demonstrated that the optimization of a single kernel is often sufficient to obtain good performance for a large class of more complicated operations.

cs.MS↗

arXiv2014

Boundedness of weak solutions to cross-diffusion systems from population dynamics

The global-in-time existence of nonnegative bounded weak solutions to a class of cross-diffusion systems for two population species is proved. The diffusivities are assumed to depend linearly on the population densities in such a way that a certain formal gradient-flow structure holds. The main feature of these systems is that the diffusion matrix may be neither symmetric nor positive definite. The key idea of the proof is to employ the boundedness-by-entropy principle which yields at the same time the existence of global weak solutions and their boundedness. In particular, the uniform boundedness of weak solutions to the population model of Shigesada, Kawasaki, and Teramoto in several space dimensions under certain conditions on the diffusivities is shown for the first time.

math.AP↗

arXiv2014

The boundedness-by-entropy principle for cross-diffusion systems

A novel principle is presented which allows for the proof of bounded weak solutions to a class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea of the principle is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles.

math.AP↗

arXiv2014

A degenerate fourth-order parabolic equation modeling Bose-Einstein condensation. Part II: Finite-time blow-up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted $L^1$ norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the $L^\infty$ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.

math.AP↗

arXiv2014

A degenerate fourth-order parabolic equation modeling Bose-Einstein condensation. Part I: Local existence of solutions

math.AP↗

arXiv2013

Bounded weak solutions to matrix drift-diffusion model for spin-coherent electron transport in semiconductors

The global-in-time existence and uniqueness of bounded weak solutions to a spinorial matrix drift-diffusion model for semiconductors is proved. Developing the electron density matrix in the Pauli basis, the coefficients (charge density and spin-vector density) satisfy a parabolic $4\times 4$ cross-diffusion system. The key idea of the existence proof is to work with different variables: the spin-up and spin-down densities as well as the parallel and perpendicular components of the spin-vector density with respect to the magnetization. In these variables, the diffusion matrix becomes diagonal. The proofs of the $L^\infty$ estimates are based on Stampacchia truncation as well as Moser- and Alikakos-type iteration arguments. The monotonicity of the entropy (or free energy) is also proved. Numerical experiments in one space dimension using a finite-volume discretization indicate that the entropy decays exponentially fast to the equilibrium state.

math.AP↗

arXiv2013

Perfectly Matched Layers versus discrete transparent boundary conditions in quantum device simulations

Discrete transparent boundary conditions (DTBC) and the Perfectly Matched Layers (PML) method for the realization of open boundary conditions in quantum device simulations are compared, based on the stationary and time-dependent Schrödinger equation. The comparison includes scattering state, wave packet, and transient scattering state simulations in one and two space dimensions. The Schrödinger equation is discretized by a second-order Crank-Nicolson method in case of DTBC. For the discretization with PML, symmetric second-, fourth, and sixth-order spatial approximations as well as Crank-Nicolson and classical Runge-Kutta time-integration methods are employed. In two space dimensions, a ring-shaped quantum waveguide device is simulated in the stationary and transient regime. As an application, a simulation of the Aharonov-Bohm effect in this device is performed, showing the excitation of bound states localized in the ring region. The numerical experiments show that the results obtained from PML are comparable to those obtained using DTBC, while keeping the high numerical efficiency and flexibility as well as the ease of implementation of the former method.

math.NA↗

arXiv2013

Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations

New one-leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the nonnegativity and the entropy-dissipation structure of the diffusive equations. The key ideas are to combine Dahlquist's G-stability theory with entropy-dissipation methods and to introduce a nonlinear transformation of variables which provides a quadratic structure in the equations. It is shown that G-stability of the one-leg scheme is sufficient to derive discrete entropy dissipation estimates. The general result is applied to a cross-diffusion system from population dynamics and a nonlinear fourth-order quantum diffusion model, for which the existence of semi-discrete weak solutions is proved. Under some assumptions on the operator of the evolution equation, the second-order convergence of solutions is shown. Moreover, some numerical experiments for the population model are presented, which underline the theoretical results.

math.NA↗

arXiv2013

Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system

The incompressible Navier-Stokes equations coupled to the Maxwell-Stefan relations for the molar fluxes are analyzed in bounded domains with no-flux boundary conditions. The system models the dynamics of a multicomponent gaseous mixture under isothermal conditions. The global-in-time existence of bounded weak solutions to the strongly coupled model and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to the singular Maxwell-Stefan diffusion matrix, the cross-diffusion terms, and the Navier-Stokes coupling. The key idea of the proof is the use of a new entropy functional and entropy variables, which allows for a proof of positive lower and upper bounds of the mass densities without the use of a maximum principle.

math.AP↗

arXiv2013

Global existence analysis for degenerate energy-transport models for semiconductors

A class of energy-transport equations without electric field under mixed Dirichlet-Neumann boundary conditions is analyzed. The system of degenerate and strongly coupled parabolic equations for the particle density and temperature arises in semiconductor device theory. The global-in-time existence of weak nonnegative solutions is shown. The proof consists of a variable transformation and a semi-discretization in time such that the discretized system becomes elliptic and semilinear. Positive approximate solutions are obtained by Stampacchia truncation arguments and a new cut-off test function. Nonlogarithmic entropy inequalities yield gradient estimates which allow for the limit of vanishing time step sizes. Exploiting the entropy inequality, the long-time convergence of the weak solutions to the constant steady state is proved. Because of the lack of appropriate convex Sobolev inequalities to estimate the entropy dissipation, only an algebraic decay rate is obtained. Numerical experiments indicate that the decay rate is typically exponential.

math.AP↗

arXiv2013

A note on Aubin-Lions-Dubinskii lemmas

Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinski\uı lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a result of Dubinski\uı (1965) for seminormed cones. The second theorem applies to piecewise constant functions in time and sharpens slightly the results of Dreher and Jüngel (2012) and Chen and Liu (2012). An application is given, which is useful in the study of porous-medium or fast-diffusion type equations.

math.FA↗