We study a class of nonlinear kinetic Fokker-Planck type equations modeling quantum particles which obey the Bose-Einstein and Fermi-Dirac statistics, respectively. We establish the existence of classical solutions in the perturbative regime and prove exponential convergence towards the equilibrium.

The solutions of the nonlinear Schrödinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest being the local norm square conservation law. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we first give the conservation laws that can be calculated by means of Lie theory and then critically compare the quality of different finite difference methods that have been proposed in geometric integration with respect to conservation laws. We find that finite difference schemes derived by writing the Schrödinger equation as an (artificial) Hamiltonian system do not necessarily conserve important physical quantities better than other methods.

This article presents and evaluates a surface hopping algorithm for time-dependent two-level Schrödinger systems with conically intersecting eigenvalues. The algorithm implements an asymptotic semigroup for approximating the solution’s Wigner function, which was rigorously defined and derived from the Schrödinger equation by two of the authors in previous work. It is applied to two-dimensional isotropic systems, which include linear Jahn-Teller Hamiltonians and Gaussian initial data. It reproduces energy level populations and expectation values with an accuracy of two to three percent.

We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this paper we give a constructive proof for the existence of solutions in the case of weak shocks.

We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity.

Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive, more straightforward, proof that suits the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corre- sponding stable, unstable and center manifolds) for singular points of general dynamical systems of the same type as in the shock wave problem for DVMs.

The same approach can also be applied for DVMs for mixtures.

The instability of streamwise varying shear flow that is marginally stable to long Rossby waves is examined. Both Hamiltonian and non-Hamiltonian flows are considered within the framework of coupled wave instability (CWI). The CWI is shown to be mediated by a ‘physical’ wave and a ‘virtual’ wave. The physical/virtual wave model concisely describes the differences between the instabilities that develop on locally supercritical flow and those that develop on globally subcritical flow. Globally subcritical Hamiltonian flows are proven to be stable. In contrast, non- Hamiltonian flows may be unstable in both locally supercritical and globally subcritical regimes. In locally supercritical flow, the CWI grows via wave-resonance between the physical/virtual wave pair and is highly localized to the supercritical region. In contrast, in globally subcritical flow, the CWI grows via pseudomomentum extraction from the background flow and either radiates away or remains attached to the streamwise variation in the flow.

The non-local peridynamic theory describes the displacement field of a continuous body by the initial-value problem for an integro-differential equation that does not include any spatial derivative. The non-locality is determined by the so-called peridynamic horizon $\delta$ which is the radius of interaction between material points taken into account. Well-posedness and structural properties of the peridynamic equation of motion are established for the linear case corresponding to small relative displacements. Moreover the limit behavior as $\delta \rightarrow 0$ is studied.

We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is non-strictly hyperbolic and does not admit a fully conservative form, and we establish the existence of two-parameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit, and leads to formulas that can be implemented numerically for the approximation of the general initial-value problem.

In this paper, we consider two recently derived models: the Quantum Hydrodynamic model (QHD) and the Quantum Energy Transport model (QET). We propose different equivalent formulations of these models and we use a commutator formula for stating new properties of the models. A gauge invariance lemma permits to simplify the QHD model for irrotational flows. We finish by considering the special case of a slowly varying temperature and we discuss possible approximations which will be helpful for future numerical discretizations.

This short note presents the derivation of a new a priori estimate for the Oldroyd-B model. Such an estimate may provide useful information when investigating the long-time behaviour of macro-macro models, and the stability of numerical schemes. We show how this estimate can be used as a guideline to derive new estimates for other macroscopic models, like the FENE-P model.

We study the steady state phase behavior of homogeneous, rigid, extended (polar) nematic polymers or nematic dispersions under imposed magnetic (or electric fields), in which the intermolecular dipole-dipole and excluded volume interaction as well as external field contribution are accounted for. We completely characterize the phase diagram for polar nematics with and without permanent magnetic moments (or dipoles). For nematics without magnetic moments, the steady state is either purely nematic or polar depending on the strength of the excluded volume and the dipole-dipole interaction, in which the nonzero polarity vector (the first moment vector) is coaxial with the second moment tensor; thereby the steady state pdf is determined essentially by up to three scalar order parameters and a rotational group of SO(2) transverse to the imposed field direction. For nematics with permanent magnetic moments (or dipoles), the steady states are polar and the polarity vector is parallel to the external field direction when a necessary condition for parameters is met. When the condition is violated, only stable steady states have their polarity vector parallel to the external field direction and there are thermodynamically unstable nonparallel states. The stability of the steady states is inferred from the minimum of the free energy density.

We present an analytic study of an optimal boundary control problem for the diffusive $SP_{1}$-system modeling radiative heat transfer. The cost functional is of tracking-type and the control problem is considered as a constrained optimization problem, where the constraint is given by the nonlinear parabolic/elliptic $SP_{1}$-system. We prove the existence, uniqueness and regularity of bounded states, which allows for the introduction of the reduced cost functional. Further, we show the existence of an optimal control, derive the first-order optimality system and analyze the adjoint system, for which we prove existence, uniqueness and regularity of adjoint states.

This paper is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev (LSI) and the Poincaré inequalities. Assuming that a given probability measure gives rise to a LSI, we derive generalized Poincaré inequalities, improving upon the known constants from the literature. We also analyze the special case when these inequalities are restricted to functions with zero components for the first eigenspaces of the corresponding evolution operator.

In this paper we empirically evaluate a recently proposed Fast Approximate Discrete Fourier Transform (FADFT) algorithm, FADFT-2, for the first time. FADFT-2 returns approximate Fourier representations for frequency-sparse signals and works by random sampling. Its implemen- tation is benchmarked against two competing methods. The first is the popular exact FFT imple- mentation FFTW Version 3.1. The second is an implementation of FADFT-2’s ancestor, FADFT-1. Experiments verify the theoretical runtimes of both FADFT-1 and FADFT-2. In doing so it is shown that FADFT-2 not only generally outperforms FADFT-1 on all but the sparsest signals, but is also significantly faster than FFTW 3.1 on large sparse signals. Furthermore, it is demonstrated that FADFT-2 is indistinguishable from FADFT-1 in terms of noise tolerance despite FADFT-2’s better execution time.

We introduce a class of sub-linear scaling algorithms for analyzing the electronic structure of crystalline solids with isolated defects. We divide the localized orbitals of the electrons into two sets: one set associated with the atoms in the region where the deformation of the material is smooth (smooth region), and the other set associated with the atoms around the defects (non- smooth region). The orbitals associated with atoms in the smooth region can be approximated accurately using asymptotic analysis. The results can then be used in the original formulation to find the orbitals in the non-smooth region. For orbital-free density functional theory, one can simply partition the electron density into a sum of the density in the smooth region and a density in the non-smooth region. This kind of partition is not used for Kohn-Sham density functional theory and one uses instead the partition of the set of orbitals. As a byproduct, we develop the necessary real space formulations and we present a formulation of the electronic structure problem for a subsystem, when the electronic structure for the remaining part is known.

We present a seamless multiscale model and an efficient coupling scheme for the study of complex fluids. The multiscale model consists of macroscale conservation laws for mass and momentum, molecular dynamics on fiber bundles, as well as the Irving-Kirkwood formula which links the macroscale stress tensor with the microscopic variables. The macroscale and microscale models are solved with a macro time step and a micro time step respectively. The two models are synchronized at each time step by exchanging the velocity gradient (from macro to micro) and the stress tensor (from micro to macro). The multiscale method is applied to study the dynamics of polymer fluids in a channel driven by external forces.