The long time dynamics of the nonlinear Schrödinger equation, on a bounded domain, is very rich. Even for small amplitude initial data there can be quasi-periodic solutions, or solutions whose energy cascades between characteristically different length scales. Our aim in this talk is to explain how the long-time dynamics of the equation begin{equation*} left{ begin{array}{l} - i partial_t u + frac{1}{2pi} Delta u = epsilon^{2p} |u|^{2p} u qquad mbox{set on $(t,x) in mathbb{R} times mathbb{T}^n_L$} u(t=0) =epsilon u_0 end{array} right. end{equation*} can be described when $epsilon$ is small and $L$ is large. We will show how to derive an equation that describe the dynamics beyond the nonlinear time scale which is of order $mathcal{O}(frac1{epsilon^2})$.

• Thematic program: Classical and Quantum Transport (2016/2017)

The question whether given reduced density operators (marginals) for subsystems of a multipartite quantum system are compatible to a common total state is called quantum marginal problem (QMP). We present the solution found by A. Klyachko just a few years ago as well as the main steps for its derivation. Applying those concepts to fermionic systems reveals further constraints on fermionic occupation numbers beyond Pauli's famous exclusion principle. We introduce and discuss these so-called generalized Pauli constraints in great detail and comment on their potential physical relevance.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Generalized Pauli Constraints and Fermion Correlation" (2016)

It is now known that fermionic natural occupation numbers (NONs) do not only obey Pauli’s exclusion principle but are even stronger restricted by the so-called generalized Pauli constraints (GPC). Whenever given NONs lie on or close to the boundary of the allowed region the corresponding N-fermion quantum state has a significantly simpler structure. We explore this phenomenon in the context of quantum chemistry.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Generalized Pauli Constraints and Fermion Correlation" (2016)

We call the Hilbert space for three fermions in six orbitals the Borland-Dennis setting. It is isomorphic to the alternating tensor product of three copies of the standard 6-dimensional Hilbert space C^6. Slater determinant states in the Borland-Dennis setting correspond to "decomposable" trivectors, i.e., simple wedge products of three vectors from C^6. Generic wave functions in the Borland-Dennis setting can be written as a sum of just two decomposable trivectors. The wave functions that cannot be written as a sum of fewer than three decomposables constitute the "W-type entanglement class." I will discuss the geometry of the W-type class within the ambient Borland-Dennis space.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Generalized Pauli Constraints and Fermion Correlation" (2016)

Some of the electrons in a molecule are tightly bound to the nuclei. The closely bound "core electrons" can be relatively uncorrelated with the rest of the electrons in the molecule, and may even form what we call a "quasi-separated" pair. [Let F be the electronic wave function of a molecule with N+2 electrons. We say that F features a "quasi-separated pair" if it is approximately equal to the wedge product G ^ H of a geminal G that describes the state of the separated pair and an N-electron wave function H that is strongly orthogonal to G.] We have computational evidence of such quasi-separated electron pairs in the ground states of very small molecules (like LiH or the Be atom) whose correlated electronic structure can be very accurately approximated with full CI calculations.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Generalized Pauli Constraints and Fermion Correlation" (2016)

In this talk we will give an overview over our recent progress in solving time-dependent many-body problems using the two-particle reduced density matrix (2RDM) as the fundamental variable. The wavefunction is completely avoided and with this all problems arising from the exponentially increasing complexity with particle number. Key is the reconstruction of the 3RDM which couples to the dynamics of the 2RDM. At this point the approximation to the full solution of the Schrödinger equation enters: while two-particle correlations are fully incorporated, three-particle correlations are only approximated. We will discuss the reconstruction of the 3RDM, how we overcome the N-representability problem, and demonstrate the accuracy of our theory on two-examples: multi-electron atoms in strong fields, and ultra-cold atoms in optical lattices.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Generalized Pauli Constraints and Fermion Correlation" (2016)

The Pauli exclusion principle has a strong impact on the properties and the behavior of most fermionic quantum systems. Remarkably, even stronger restrictions on fermionic natural occupation numbers follow from the fermionic exchange symmetry. We develop an operationally meaningful measure which allows one to quantify the potential physical relevance of those generalized Pauli constraints beyond the well-established relevance of Pauli's exclusion principle. It is based on a geometric hierarchy induced by Pauli exclusion principle constraints. The significance of that measure is illustrated for a few-fermion model which also confirms such nontrivial relevance of the generalized Pauli constraints.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Generalized Pauli Constraints and Fermion Correlation" (2016)

By employing the simpler structure arising from pinning and quasipinnig a variational optimization method for few fermion ground states is elaborated. We quantitatively confirm its high accuracy for systems whose vector of NON is close to the boundary of the polytope. In particular, we derive an upper bound on the error of the correlation energy given by the ratio of the distance to the boundary of the polytope and the distance of the vector of NON to the Hartree-Fock point. These geometric insights shed some light on the concept of active spaces, correlation energy, frozen electrons and virtual orbitals.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Generalized Pauli Constraints and Fermion Correlation" (2016)

We describe recent results concerning the orbital and asymptotic stability of dark solitons and multi- solitons for the Landau-Lifshitz equation with an easy-plane anisotropy. This is joint work with André de Laire (University of Lille Nord de France), and by Yakine Bahri (Nice Sophia Antipolis University).

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

I will discuss a joint work with Bellazzini, Boussaid, Jeanjean about the existence and orbital stability of standing waves for NLS with a partial confinement in a supercritical regime. The main point is to show the existence of local minimizers of the constraint energy.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We consider the defocusing nonlinear Schrödinger equations on a two-dimensional compact Riemannian manifold without boundary or a bounded domain in dimension two. In particular, we discuss the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In form of a case study for the mKdV and the KdV2 equation we discuss a novel approach of representing frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include wellposedness results in spaces of low regularity as well as properties of the actions to frequencies map. This is joint work with Jan Molnar.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The onset of a dispersive shock in solutions to the Kadomtsev-Petviashvili (KP) equations is studied numerically. First we study the shock formation in the dispersionless KP equation by using a map inspired by the characteristic coordinates for the one-dimensional Hopf equation. This allows to numerically identify the shock and to unfold the singularity. A conjecture for the KP solution near this critical point in the small dispersion limit is presented. It is shown that dispersive shocks for KPI solutions can have a second breaking where modulated lump solutions appear.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The Green-Naghdi system is an asymptotic model for the water-waves system, describing the propagation of surface waves above a layer of ideal, homogeneous, incompressible and irrotational fluid, when the depth of the layer is assumed to be small with respect to wavelength of the flow. It can be seen as a perturbation of the standard quasilinear (dispersionless) Saint-Venant system, with additional nonlinear higher-order terms. Because of the latter, the well-posedness theory concerning the GN system is not satisfactory, in particular outside of the one-dimensional framework. We will discuss novel results, obtained with Samer Israwi, that emphasize the role of the irrotationality assumption.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In this lecture, we investigate the behavior of the solutions to the nonlinear Schrodinger equation: (1) ( i@tu +
u = f(u); ujt=0 = u0 2 H1 rad(R2); where the nonlinearity f : C ! C is dened by (2) f(u) = p( p 4 juj) u with p > 1 and p(s) = es2 􀀀 pX􀀀1 k=0 s2k k!
Recall that the solutions of the Cauchy problem (1)-(2) formally satisfy the conservation laws: (3) M(u; t) = Z R2 ju(t; x)j2dx = M(u0) and (4) H(u; t) = Z R2  jru(t; x)j2 + Fp(u(t; x))  dx = H(u0) ; where Fp(u) = 1 4 p+1 􀀀p 4 juj

It is known (see [4], [6] and [2]) that global well-posedness for the Cauchy problem (1)-(2) holds in both subcritical and critical regimes in the functional space C(R;H1(R2)) L4(R;W1;4(R2)). Here the notion of criticity is related to the size of the initial Hamiltonian H(u0) with respect to 1. More precisely, the concerned Cauchy problem is said to be subcritical if H(u0) < 1, critical if H(u0) = 1 and supercritical if H(u0) > 1. Structures theorems originates in the elliptic framework in the studies by H. Brezis and J.- M. Coron in [3] and M. Struwe in [8]. The approach that we shall adopt in this article consists in comparing the evolution of oscillations and concentration eects displayed by sequences of solutions of the nonlinear Schrodinger equation (1)-(2) and solutions of the linear Schrodinger equation associated to the same sequence of Cauchy data. Our source of inspiration here is the pioneering works [1] and [7] whose aims were to describe the structure of bounded sequences of solutions to semilinear defocusing wave and Schrodinger equations, up to small remainder terms in Strichartz norms. The analysis we conducted in this work emphasizes that the nonlinear eect in this framework only stems from the 1-oscillating component of the sequence of the Cauchy data, using the terminology introduced in [5]. This phenomenon is strikingly dierent from those obtained for critical semi linear dispersive equations, such as for instance in [1, 7] where all the oscillating components induce the same nonlinear eect, up to a change of scale. To carry out our analysis, we have been led to develop a prole decomposition of bounded sequences of solutions to the linear Schrodinger equation both in the framework of Strichartz and Orlicz norms. The linear structure theorem we have obtained in this work highlights the distinguished role of the 1-oscillating component of the sequence of the Cauchy data. It turns out that there is a form of orthogonality between the Orlicz and the Strichartz norms for the evolution under the ow of the free Schrodinger equation of the unrelated component to the scale 1 of the Cauchy data (according to the vocabulary of [5]), while this is not the case for the 1-oscillating component.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The incompressible Euler equation with free surface dictates the dynamics of the interface separating the air from a perfect incompressible fluid. This talk is about the controllability and the stabilization of this equation. The goal is to understand the generation and the absorption of water waves in a wave tank. These two problems are studied by two different methods: microlocal analysis for the controllability (this is a joint work with Pietro Baldi and Daniel Han-Kwan), and study of global quantities for the stabilization (multiplier method, Pohozaev identity, hamiltonian formulation, Luke’s variational principle, conservation laws…).

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In the 1960’s G. B. Whitham suggested a non-local version of the KdV equation as a model for water waves. Unlike the KdV equation it is not integrable, but it has certain other advantages. In particular, it has the same dispersion relation as the full water wave problem and it allows for wave breaking. The equation has a family of periodic, travelling wave solutions for any given wavelength. Whitham conjectured that this family contains a highest wave which has a cusp at the crest. I will outline a proof of this conjecture using global bifurcation theory and precise information about an integral operator which appears in the equation. Joint work with M. Ehrnström.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical quadratic shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of $2pi$-periodic traveling wave solutions. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like $|x|^{1/2}$ near its peak, the cusped waves obtained here behave like $|xlog|x||$ at their peak and are smooth away from their highest points. This is joint work with Mathew A. Johnson and Kyle M. Claassen at University of Kansas.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

It is conjectured that bounded solutions of the focusing energy-critical wave equation decouple asymptotically as a sum of a radiation term and a finite number of solitons . In this talk, I will review recent works on the subject, including the proof of a weak form of this conjecture (joint work with Hao Jia, Carlos Kenig and Frank Merle)

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We consider the linear wave equation and the linear Schr dingier equation outside a compact, strictly convex obstacle in R^d with smooth boundary. In dimension d = 3 we show that for both equations, the linear flow satises the (corresponding) dispersive estimates as in R^3. For d>3, if the obstacle is a ball, we show that there exists at least one point (the Poisson spot) where the dispersive estimates fail. This is joint work with Gilles Lebeau.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In this talk we will present some results on the dynamics of vortex filaments according to a model introduced by Klein, Majda and Damodaran, focusing on the issue of collisions. This is a joint work with Valeria Banica and Erwan Faou.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The aim of this talk is to present some qualitative properties of a coupled Maxwell-Schrödinger system. First, I will describe conditions for the existence of minimizers with prescribed charge in terms of a coupling constant e. Secondly, I will study the existence of ground states for the stationary problem, the uniqueness of ground states for small e and finish with the orbital stability for the quadratic nonlinearity. This is a joint work with Tatsuya Watanabe.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The Hartree equation can be derived from the N-body Heisenberg equation by the mean-field limit assuming that the particle number N tends to infinity. The first rigorous result in this direction is due to Spohn (1980) (see also [Bardos-Golse-Mauser, Meth. Applic. Anal. 7:275-294, (2000)] for more details), and is based on analyzing the Dyson series representing the solution of the BBGKY hierarchy in the case of bounded interaction potentials.This talk will (1) provide an explicit convergence rate for the Spohn method, and (2) interpolate the resulting convergence rate with the vanishing h bound obtained in [Golse-Mouhot-Paul, Commun. Math. Phys. 343:165-205 (2016)] by a quantum variant of optimal transportation modulo O(h) terms. The final result is a bound for a Monge-Kantorovich-type distance between the Husimi transforms of the Hartree solution and of the first marginal of the N-body Heisenberg solution which is independent of h and vanishes as N tends to infinity. (Work in collaboration with T. Paul and M. Pulvirenti).

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We will discuss special regularity properties of solutions to the IVP associated to the k-generalized KdV equations. We show that for data u0 2 H3=4+(R) whose restriction belongs to Hk((b;1)) for some k 2 Z+ and b 2 R, the restriction of the corresponding solution u(
; t) belongs to Hk((;1)) for any  2 R and any t 2 (0; T). Thus, this type of regularity propagates with innite speed to its left as time evolves. This kind of regularity can be extended to a general class of nonlinear dispersive equations. Recently, we proved that the solution ow of the k-generalized KdV equation does not preserve other kind of regularities exhibited by the initial data u0.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In order to study the transverse (in) stability of a line soliton, we consider the 2-d Zakharov-Rubenchik/Benney-Roskes system with initial data localized by a line soliton. The new terms in perturbed system lead to some diculties, for example, the lack of mass conservation. In this talk, I will present our recent work on this problem. This is a joint work with Norbert Mauser and Jean-Claude Saut. 1

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In this talk, I will review recent research and progress using the multiconfigurational time-dependent Hartree for indistinguishable particles method to obtain highly accurate solutions of the time-dependent many-body Schr"odinger equation for interacting, indistinguishable particles. As an example, I will focus on ultracold bosonic particles with internal degrees of freedom described by the multiconfigurational time-dependent Hartree for bosons method. For the groundstate of N=100 parabolically confined bosons with two internal states, fragmentation emerges as a function of the separation between the state-dependent minima of the two parabolic potentials: for small separations, the bosons occupy only one single-particle state while for larger separations, two single-particle states contribute macroscopically. The coherence of the system is maintained within each internal state of the atoms. Between the different internal states, however, correlations are built up and the coherence is lost for larger separations. This is a hallmark of a new kind of fragmentation -- "composite fragmentation" -- which is absent in bosons without internal structure.

• Thematic program: Classical and Quantum Transport (2016/2017)

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• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Working group "Efficient numerical methods for quantum systems" (2017)

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• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Working group "Efficient numerical methods for quantum systems" (2017)

Convolution-type potential are common and important in many science and engineering fields. Efficient and accurate evaluation of such nonlocal potentials are essential in practical simulations.In this talk, I will focus on those arising from quantum physics/chemistry and lightning-shield protection, including Coulomb, dipolar and Yukawa potentials that are generated by isotropic and anisotropic smooth and fast-decaying density. The convolution kernel is usually singular or discontinuous at the origin and/or at the far field, and density might be anisotropic, which together present great challenges for numerics in both accuracy and efficiency. The state-of-art fast algorithms include Wavelet based Method(WavM), kernel truncation method(KTM), NonUniform-FFT based method(NUFFT) and Gaussian-Sumbased method(GSM). Gaussian-sum/exponential-sum approximation and kernel truncation technique, combined with finite Fourier series and Taylor expansion, finally lead to a O(NlogN) fast algorithm achieving spectral accuracy. Applications to NLSE are reviewed.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Working group "Efficient numerical methods for quantum systems" (2017)

Idealized models of reduced complexity are important tools to understand key processes underlying a complex system. In climate science in particular, they are important for helping the community improve our ability to predict the eect of climate change on the earth system. Climate models are large computer codes based on the discretization of the uid dynamics equations on grids of horizontal resolution in the order of 100 km, whereas unresolved processes are handled by subgrid models. For instance, simple models are routinely used to help understand the interactions between small-scale processes due to atmospheric moist convection and large-scale circulation patterns. Here, a zonally symmetric model for the monsoon circulation is presented and solved numerically. The model is based on the Galerkin projection of the primitive equations of atmospheric synoptic dynamics onto the rst modes of vertical structure to represent free tropospheric circulation and is coupled to a bulk atmospheric boundary layer (ABL) model. The model carries bulk equations for water vapor in both the free troposphere and the ABL, while the processes of convection and precipitation are represented through a stochastic model for clouds. The model equations are coupled through advective nonlinearities, and the resulting system is not conservative and not necessarily hyperbolic. This makes the design of a numerical method for the solution of this system particularly dicult. We develop a numerical scheme based on the operator time-splitting strategy, which decomposes the system into three pieces: a conservative part and two purely advective parts, each of which is solved iteratively using an appropriate method. The conservative system is solved via a central scheme, which does not require hyperbolicity since it avoids the Riemann problem by design. One of the advective parts is a hyperbolic diagonal matrix, which is easily handled by classical methods for hyperbolic equations, while the other advective part is a nilpotent matrix, which is solved via the method of lines. Validation tests using a synthetic exact solution are presented, and formal second-order convergence under grid renement is demonstrated. Moreover, the model is tested under realistic monsoon conditions, and the ability of the model to simulate key features of the monsoon circulation is illustrated in two distinct parameter regimes. This is joint work with Michale De La Chevrotiare.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

The dynamics of atmospheric vortices such as tropical storms, hurricanes and mid-latitude cyclones is driven by a variety of interacting scales. [1] developed an asymptotic theory for the dynamics of strongly tilted atmospheric vortices in the gradient-wind regime, embedded into a synoptic-scale geostrophic background eld. One central outcome of the theory is the evolution equation for the nearly axisymmetric primary circulation. It predicts that Fourier-mode 1 of asymmetric diabatic heating/ cooling patterns can spin up or spin down a vortex depending on the relative arrangement of the heating dipole relative to the vortex tilt. Based on this methodology further investigations led to the conclusion that this theory is generalizable to Rossby numbers of order 1 and higher, i.e. cyclostrophic balance. Accompaning the asymptotics numerical experiments are conducted to test the theory within an anelastic model [2]. In this talk we present the latest results showing consistency of numerical simulations and theoretical predictions. [1] E. Paschke, P. Marschalik, A. Z. Owinoh and R. Klein, Motion and structure of at- mospheric mesoscale baroclinic vortices: dry air and weak environmental shear, J. Fluid Mech. 701: 137{170, (2012) [2] J. M. Prusa, P. K. Smolarkiewicz and A. A. Wyszogrodzki, EULAG, a computational model for multiscale ows, Comput. Fluids 37: 1193{1207 (2008)

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

Paeschke et al (2012) showed analytically how non-axisymmetric external diabatic forcing of a tilted vortex in dry air can amplify or attenuated the ow depending on the relative orientation of vortex tilt and the "heating dipole". Here we include a bulk moist microphysics closure and describe how boundary layer processes and multiscale deep moist convection can interact to produce this eect self-consistently.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

This work extends to moist-precipitating dynamics a recently documented high-performance nite-volume integrators for simulating global all-scale atmospheric ows (doi:10.1016/j.jcp. 2016.03.015). A key objective of the current development is a seamless coupling of the conservation laws for moist variables engendered by cloud physics with the semi-implicit, non-oscillatory forward-in-time integrators already proven for dry dynamics. The representation of the water substance and the associated processes in weather and climate models can vary widely in formulation details and complexity levels. The adopted representation assumes a canonical warm-rain" bulk microphysics parametrisation, recognised for its minimal physical intricacy while accounting for the essential mathematical complexity of cloud-resolving models. A key feature of the presented numerical approach is global conservation of the water substance to machine precision | implied by the local conservativeness and positivity preservation of the numerics | for all water species including water vapour, cloud water, and precipitation. The moist formulation assumes the compressible Euler equations as default, but includes reduced anelastic equations as an option. The theoretical considerations are illustrated with a benchmark simulation of a tornadic thunderstorm on a reduced size planet, supported with a series of numerical experiments addressing the accuracy of the associated water budget.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

Condensation of water vapor to form and grow cloud droplets is the most fundamental process of cloud and precipitation formation. It drives cloud dynamics through the release of latent heat and determines the strength of convective updrafts. Cloud-scale models simulate condensation by applying two drastically dierent methods. The rst one is the bulk condensation where condensation/evaporation is assumed to always maintain saturated conditions. The second approach involves prediction of the in-cloud super- or sub-saturation and can be used in models that predict not only condensate mass but also relevant features of the droplet size distribution (e.g., models with the 2-moment microphysics or with the bin microphysics). This presentation will address the question whether the dierence between the two approaches has a noticeable impact on convective dynamics. Model simulations with the bin microphysics for shallow non-precipitating convection and with the double-moment bulk microphysics for deep convection will be discussed to document the dierences in cloud eld simulations applying the two methodologies. For the shallow convection, the dierences in cloud eld simulated with bulk and bin schemes come not from small dierences in the condensation, but from more signicant dierences in the evaporation of cloud water near cloud edges as a result of entrainment and mixing. For the deep convection, results show a signicant dynamical impact of nite supersaturations and a strong microphysical eect associated with upper-tropospheric anvils. Implications of these results for modeling convective dynamics will be discussed and a possible intermediate modeling methodology will be suggested.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

In atmospheric ows, the Dynamic State Index (DSI) indicates local deviations from a steady wind solution. This steady wind solution is based on the primitive equations under adiabatic and inviscid conditions. Hence, from theoretical point of view, atmospheric dynamics is regarded relative to a solution derived from uid mechanic's rst principles. Thus, this parameter provides a tool to capture diabatic processes. The DSI can be designed for dierent uid mechanical models on distinguished scales, we will introduce a DSIQG for the quasi-geostrophic ow, a DSIRo for the Rossby model and DSImois that is based on the equations of motions including moisture processes.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

In this talk, we derive and discuss thermodynamically consistent models for heat-conduction fluids. Our approach is based on the entropy principle.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

Diusional growth is the most important growth mechanism for newly formed cloud droplets and ice crystals. Non-linear diusion equations control the transport of water vapor towards the cloud particles. Although the solution of these diusion equations is circumvented in numerical cloud models, it remains computationally expensive to include the details of diusional growth due to severe timestep restrictions. Moreover, as soon as ice crystals are present in a cloud consisting mostly of cloud droplets, the Wegener- Bergeron-Findeisen process becomes active and the ice crystals grow at the expense of the cloud droplets. In the rst part of the talk, we discuss the aspect of locality of the Wegener-Bergeron- Findeisen process, i.e. an ice crystal does only aect its immediate vicinity. Its presence decouples the diusional growth behavior of nearby droplets from environmental conditions. We show some simulation results and a possible way to include locality in the context of bulk-microphysics. The second part considers the case of a liquid cloud. In the context of numerical models, the microphysical details of the diusional growth and the timestep restrictions are eectively avoided through the technique of saturation adjustment. We will show some of these techniques and analyze an air parcel model containing activation of new droplets using asymptotics.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

In this talk, we will present some recent mathematical results, mainly the global wellposedness and convergence of the relaxation limit, on two kinds of dynamical models for the atmosphere with moisture. In the rst part of this talk, which is a joint work with Edriss S. Titi [1], we will consider a tropical atmosphere model introduced by Frierson, Majda, and Pauluis (Commum. Math. Sci. 2004); for this model, we will present the global well-posedness of strong solutions and the strong convergence of the relaxation limit, as the relaxation time " tends to zero. It will be shown that, for both the nite-time and instantaneous-relaxation systems, the H1 regularities on the initial data are sucient for both the global existence and uniqueness of strong solutions, but slightly more regularities than H1 are required for both the continuous dependence and strong convergence of the relaxation limit. In the second part of this talk, which is a joint work with Sabine Hittmeir, Rupert Klein, and Edriss S. Titi [2], we will consider a moisture model for warm clouds used by Klein and Majda (Theor. Comput. Fluid Dyn. 2006), where the phase changes are allowed, and we will present the global well-posedness of this system. [1] Jinkai Li; Edriss S. Titi: A tropical atmosphere model with moisture: global well- posedness and relaxation limit, Nonlinearity, 29 (2016), 2674{2714. [2] Sabine Hittmeir; Rupert Klein; Jinkai Li; Edriss S. Titi: Global well-posedness for passively transported nonlinear moisture dynamics with phase changes, arXiv:1610.00060

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Mathematics of Moist Atmospheric Dynamics: Modeling, Analysis and Computations" (2017)

We consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase and/or initial amplitude. To estimate quantities of interest (QoI) related to the solution $u^\varepsilon$ and their statistics, we combine a high-frequency method based on Gaussian beams with sparse stochastic collocation. In the talk we will discuss how the rate of convergence for the stochastic collocation and the complexity of evaluating the QoI depend on the short wavelength $\varepsilon$. We find in particular that QoIs based on local averages of $\vert u^\varepsilon\vert ^2$ can give fast convergence rates, despite the fact that $u^\varepsilon$ is highly oscillatory in both physical and stochastic space.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Quantum dynamics and uncertainty Quantification" (2017)

A brief description of averaging theory for highly-oscillatory problems will be first presented with an emphasis on the so-called classical and stroboscopic averaging methods. Then I will present two general strategies to construct efficient numerical schemes for a class of highly oscillatory PDEs: the so-obtained numerical schemes have a uniform accuracy with respect to the frequency. Two applications will be considered: the Vlasov kinetic equation with strong magnetic field and the Klein-Gordon equation in the non-relativistic regime.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Quantum dynamics and uncertainty Quantification" (2017)

Band-crossing is a quantum dynamical behavior that contributes to important physics and chemistry phenomena such as quantum tunneling, Berry connection, charge transfer, chemical reaction etc. In this talk, we will discuss some recent works in developing semiclassical methods for band-crossing in surface hopping. For such systems we will also introduce an nonlinear geometric optics method based "asymptotic-preserving" method that is accurate uniformly for all wave numbers, including the problem with random uncertain band gaps.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Quantum dynamics and uncertainty Quantification" (2017)

We will review some results concerning uncertainties in the derivation of kinetic equations from wave propagation in random media, that is modeled by a wave or a Schroedinger equation. Kinetic equations usually describe quadratic quantities in the wavefield such as the energy or wave-wave correlations, and can be used to solve some imaging problems in complex media.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Quantum dynamics and uncertainty Quantification" (2017)

We propose an explicit bound for the convergence rate in the semiclassical limit for the Schrödinger equation which holds for potentials with Lipschitz continuous gradient. This bound is based on an analogue of the Wasserstein metric used in optimal transportation, adapted to measuring the distance between a quantum and a classical density.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Quantum dynamics and uncertainty Quantification" (2017)

We present some mathematical methods occurring in the modeling and simulation of Nonlinear Schrödinger equations and nonlocal potentials. We focus on Gross-Pitaevskii equations describing Bose Einstein Condensates and stray field calculations in micro-magnetism.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop on "Quantum dynamics and uncertainty Quantification" (2017)

Recent experiments with Bose-Einstein condensates of dysprosium [1] and erbium [2] atoms have observed the formation of droplets that can preserve their form, even in the absence of any external confinement [3]. These droplets occur when the long-ranged dipole-dipole interaction between the atoms dominates over the short-ranged contact interaction. In this regime meanfield theory predicts that the condensate is unstable to collapse, however the Lee-Huang-Yang corrections to the meanfield energy [3] can stabilize the system as one or many finite sized droplets. I will discuss our current understanding of these droplets, and introduce a new type of nonlinear Schrodinger equation used to describe their equilibrium and dynamical properties.

• Thematic program: Classical and Quantum Transport (2016/2017)