# Forthcoming Papers

• Ankita Jain, Iyla Timofeyev and Eric Vandern-Eijnden
Stochastic Mode-Reduction in Models with Conservative Fast Sub-Systems

A stochastic mode reduction strategy is applied to multiscale models with a deterministic energy-conserving fast sub-system. Specically, we consider situations where the slow variables are driven stochastically and interact with the fast sub-system in an energy-conserving fashion. Since the stochastic terms only affect the slow variables, the fast-subsystem evolves deter- ministically on a sphere of constant energy. However, in the full model the radius of the sphere slowly changes due to the coupling between the slow and fast dynamics. Therefore, the energy of the fast sub-system becomes an additional hidden slow variable that must be accounted for in order to apply the stochastic mode reduction technique to systems of this type.

• Jiao Li and Dexuan Xie
A New Linear Poisson- Boltzmann Equation and Finite Element Solver by Solution Decomposition Approach

The linear Poisson-Boltzmann equation (LPBE) is one well-known implicit solvent continuum model for computing the electrostatic potential of biomolecules in ionic solvent. To overcome its singular difficulty caused by Dirac delta distributions of point charges and to further improve its solution accuracy, we developed a new scheme for solving the current LPBE model, a new LPBE model, and a new LPBE finite element program package based on our previously proposed PBE solution decomposition. Numerical tests on biomolecules and a nonlinear Born ball model with an analytical solution validate the new LPBE solution decomposition schemes, demonstrate the effectiveness and efficiency of the new program package, and confirm that the new LPBE model can signicantly improve the solution accuracy of the current LPBE model.

• Zhilei Liang and Xiaoding Shi
Classical solution to the Cauchy problem for the 2D viscous polytropic fluids with vacuum and zero heat-conduction

This paper is concerned with the viscous polytropic uids in the two-dimensional (2D) space with vacuum as far field density. By means of weighted initial density, we obtain the local existence of classical solution to the Cauchy problem, in the case that the initial data satisfy a natural compatibility condition and the heat conduction coefficient is zero. Remember the blowup result of Xin [Z. Xin, Comm Pure Appl Math 51, 229-240, 1998], one should not expect the global smooth solution because the compactly supported initial density is included in our case.

• Ryo Takei, Richard Tsai and Yanina Landa
An efficient algorithm for a visibility-based surveillance-evasion game

We present an algorithm which computes the value function and optimal paths for a two-player static game, where the goal of one player is to maintain visibility of an adversarial player for as long as p ossible, and that of the adversarial player is to minimize this time. In a static game both players cho ose their controls at initial time and run in open-loop for t>0 until the end-game condition is met. Closed-loop (feedback strategy) games typically require solving PDEs in high dimensions and thus pose unsurmountable computational challenges. We demonstrate that, at the expense of a simpler information pattern that is more conservative towards one player, more memory and computationally efficient static games can be solved iteratively in the state space by the proposed PDE-based technique. In addition, we describe how this algorithm can be easily generalized to games with multiple evaders. Applications to target tracking and an extension to a feedback control game are also presented.

• Jianfeng Lu and Jeremy Marzuola
Strang splitting methods for a quasilinear Schrodinger equation -- convergence, instability and dynamics

We study the Strang splitting scheme for quasilinear Schrodinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to analytical breakdown of regularity of solutions to quasilinear Schrodinger equations. Numerical tests are performed for a modified version of the superfluid thin film equation.

• Christophe Pallard
A refined existence criterion for the relativistic Vlasov-Maxwell system

We show that a smooth compactly supported solution to the relativistic Vlasov-Maxwell system exists as long as the L^6 norm of the macroscopic density of particles remains bounded.

• Chuangchuang Liang and Kaijun Zhang
Asymptotic stability and quenching behavior of a hyperbolic nonlocal MEMS equation

We investigate a nonlocal wave equation with damping term and singular nonlinearity, which models an electrostatic micro-electro-mechanical system (MEMS) device. In the case of the relative strength parameter \lambda being small, the existence and uniqueness of the global solution are established. Moreover,the asymptotic result that the solution exponentially converges to the steady state solution is also proved. For large \lambda, quenching results of the solution are obtained.

• Rinaldo M. Colombo and Elena Rossi
Hyperbolic Predators vs. Parabolic Preys

We present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for preys. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of predators can be directed towards region with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions.

• Ronghua Pan and Weizhe Zhang
Compressible Navier-Stokes equations with temperature dependent heat conductivity

We prove the existence and uniqueness of global strong solutions to the one dimensional, compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when heat conductivity depends on temperature in power law of Chapman-Enskog. The results reported in this article is valid for initial boundary value problem with non-slip and heat insulated boundary along with smooth initial data with positive temperature and density without smallness assumption.

• Gerard Awanou
Smooth approximations of the Aleksandrov solution of the Monge-Ampere equation

We prove the existence of piecewise polynomials strictly convex smooth functions which converge uniformly on compact subsets to the Aleksandrov solution of the Monge-Ampe quation. We extend the Aleksandrov theory to right hand side only locally integrable and on convex bounded domains not necessarily strictly convex. The result suggests that for the numerical resolution of the equation, it is enough to assume that the solution is convex and piecewise smooth.

• Anne C. Bronzi, Milton C. Lopes Filho and Helena J. Nussenzveig Lopes
Wild solutions for 2D incompressible ideal flow with passive tracer

In [C. De Lellis and L. Szekelyhidi, Ann. of Math. 170, 1417-1436m 2009] C. De Lellis and L. Szekelyhidi Jr. constructed wild solutions of the incompressible Euler equations using a reformulation of the Euler equations as a differential inclusion together with convex integration. In this article we adapt their construction to the system consisting of adding the transport of a passive scalar to the two-dimensional incompressible Euler equations.

• Remi Carles and Jorge Drumond Silva
Large time behavior in nonlinear Schrodinger equation with time dependent potential

We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space, uniformly with respect to the time variable. We show a general exponential control of first order derivatives and momenta, which yields a double exponential bound for higher Sobolev norms and momenta. On the other hand, we show that if the potential is an isotropic harmonic potential with a time dependent frequency which decays sufficiently fast, then Sobolev norms are bounded, and momenta grow at most polynomially in time, because the potential becomes negligible for large time: there is scattering, even though the potential is unbounded in space for fixed time.

• Kazuo Yamazaki
Regularity criteria of porous media equation in terms of one partial derivative or pressure scalar field

We obtain new regularity criteria and smallness condition for the global regularity of the N-dimensional supercritical porous media equation. In particular, it is shown that in order to obtain global regularity result, one only needs to bound a partial derivative in one direction or the pressure scalar eld. Our smallness condition is also in terms of one direction, dropping conditions on (N-1) other directions completely, or the pressure scalar eld. The proof relies on key observations concerning the incompressibility of the velocity vector eld and the special identity derived from Darcy's law.

• Teng Wang
Vanishing viscosity limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations with density-dependent viscosity

The vanishing viscosity limit of the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity c(\rho)=\epsilon \rho^\alpha (\alpha >0) is considered in the present paper. It is proved that given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to the compressible Navier-Stokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the convergence rate depending on \alpha is obtained for all \alpha >0. The main difficulty in our proof lies in the degeneracies of the density and the density-dependent viscosity at the vacuum region in the vanishing viscosity limit.

• Meili Li, Reinhard Illner, Rod Edwards and Junling Ma
Marketing New Products: Bass Models on Random Graphs

We consider the problem of marketing a new product in a population modelled as a random graph, in which each individual (node) has a random number of connections to other individuals. Marketing can occur via word of mouth along edges, or via advertising. Our main result is adaptation of the Miller model, describing the spread of an infectious disease, to this setting, leading to a generalized Bass marketing model. The Miller model can be directly applied to word- of-mouth marketing. The main challenge lies in revising the Miller model to incorporate advertisement, which we solve by introducing a marketing node that is connected to every individual in the popula- tion. We tested this model for Poisson and scale free random networks, and found excellent agreement with microscopic simulations. In the homogeneous limit where the number of individuals goes to \infty and the network is completely connected our model becomes the classical Bass model. We further present the generalization of this model to two competing products. For a completely connected network this model is again consistent with the known continuum limit. Numerical simulations show excellent agreement with microscopic simulations ob- tained via an adaptation of the Gillespie algorithm. Our model shows that, if the two products have the same word-of-mouth marketing rate on the network, then the ratio of their market shares is exactly the ratio of their advertisement rates.

• Songting Li, Douglas Zhou and David Cai
Analysis of the dendritic integration of excitatory and inhibitory inputs using cable models

We address the question of how a neuron integrates excitatory (E) and inhibitory (I ) synaptic inputsolutions. Using these asymptotic solutions, in the presence of E and I inputs, we can successfully reveal the underlying mechanisms of a dendritic integration rule, which was discovered in a recent experiment. Our analysis can be extended to the multi-branch case to characterize the E-I dendritic integration on any branches. The novel characterization is confirmed by the numerical simulation of a biologically realistic neuron.

• Guanghui Hu, Xiaodong Liu, Fenglong Qu and Bo Zhang
Variational Approach to Scattering by Unbounded Rough Surfaces with Neumann and Generalized Impedance Boundary Conditions'

This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an infinite penetrable medium with a finite height modeled by the Helmholtz equation. On the lower boundary of the rough layer the Neumann or generalized impedance boundary condition is imposed. The scattered field in the unbounded homogeneous medium is required to satisfy the upward angular-spectrum representation. Using the variational approach, we prove uniqueness and existence of solutions in the standard space of finite energy for inhomogeneous source terms, and in appropriate weighted Sobolev spaces for incident point source waves in R^m (m=2,3) and incident plane waves in R^2. To avoid guided waves, we assume that the penetrable medium satisfies certain non-trapping and geometric conditions.

• Donglong Li, Yanfeng Guo and Bolin Guo
Random Attractor and Stationary Measure for Stochastic Long-short Wave Equations

Asymptotic behaviors of stochastic long-short equations driven by random force, which is smooth enough in space and white noise in time, are mainly considered. The existence and uniqueness of solutions for stochastic long-short equations are obtained via Galerkin approximation by the stopping time and Borel-Cantelli Lemma on the basis of a priori estimates in the sense of expectation. A global random attractor and the existence of a stationary measure are investigated by Birkhoff ergodic theorem and Chebyshev inequality.

• Jacques Schneider
On a well-posed simulation model for multicomponent reacting gases

We aim to present a relaxation model that can be used in real simulations of dilute multicomponent reacting gases. The kinetic framework is the semi-classical approach with only one variable for the internal energy modes. The relaxation times for the internal energy modes are assumed to be smaller than the chemistry characteristic times. The strategy is the same as in [S. Brull, J. Schneider, Comm. Math. Sci. to appear]. That is a sum of operators for respectively the mechanical and chemical processes. The mechanical operator(s) is the "natural" extension to polyatomic gases of the method of moment relaxations presented in [S. Brull, J. Schneider, Cont. Mech. Thermodyn. 20, 63-74, 2008] and [S. Brull, V. Pavan and J. Schnieder, Eur. J. Mech. (B-Fluids) 33, 74-86, 2012]. The derivation of the chemical model lies on the chemical processes at thermal equilibria. It is shown that this BGK approach features the same properties as the Boltzmann equation: conservations and entropy production. Moreover null entropy production states are characterized by vanishing chemical production rates. We also study the hydrodynamic limit in the slow chemistry regime. Finally we show that the whole set of parameters that are used in the derivation of the model can be calculated by softwares such as EGlib or STANJAN.

• Zachary Bradshaw and Z. Grujic
A note on the surface quasi-geostrophic temperature variance cascade

In this note we examine the dynamical role played by inertial forces on the sur face temperature (or buoyancy) variance in strongly rotating, stratified flows with uniform potential vorticity fields and fractional dissipation. In particu lar, using a dynamic, multi-scale averaging process, we identify a sufficient c ondition for the existence of a direct temperature variance cascade across an i nertial range. While the result is consistent with the physical and numerical t heories of SQG turbulence, the condition triggering the cascade is more exotic, a fact reflecting the non-locality introduced by fractional dissipation. A comment regarding the scale-locality of the temperature variance flux is also included.

• Benoit Perthame, Nicolas Seguin and Magali Tournus
A simple derivation of BV bounds for inhomogeneous relaxation systems

We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. They are the most standard and simplest way to prove compactness and convergence. We provide a novel and simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, but closely related, notion. We give two examples motivated by renal flows which consist of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general.

• Giuliano Lazzaroni, Mariapia Palombaro and Anja Schloemerkemper
A discrete to continuum analysis of dislocations in nanowire heterostructures

Epitaxially grown heterogeneous nanowires present dislocations at the interface between the phases if their radius is big. We consider a corresponding variational discrete model with quadratic pairwise atomic interaction energy. By employing the notion of Gamma-convergence and a geometric rigidity estimate, we perform a discrete to continuum limit and a dimension reduction to a one-dimensional system. Moreover, we compare a defect-free model and models with dislocations at the interface and show that the latter are energetically convenient if the thickness of the wire is sufficiently large.

• Stefan Possanner, Luigi Barletti, Florian Hehats and Claudia Negulescu
Numerical study of a quantum-diffusive spin model for two-dimensional electron gases

We investigate the time evolution of spin densities in a two-dimensional electron gas subjected to Rashba spin-orbit coupling on the basis of the quantum drift-diffusive model derived in [L. barletti and F. Mehats, J. Math. Phys. 51, 053304, 2010]. This model assumes the electrons to be in a quantum equilibrium state in the form of a Maxwellian operator. The resulting quantum drift-diffusion equations for spin-up and spin-down densities are coupled in a non-local manner via two spin chemical potentials (Lagrange multipliers) and via off-diagonal elements of the equilibrium spin density and spin current matrices, respectively. We present two space-time discretizations of the model, one semi-implicit and one explicit, which comprise also the Poisson equation in order to account for electron-electron interactions. In a first step pure time discretization is applied in order to prove the well-posedness of the two schemes, both of which are based on a functional formalism to treat the non-local relations between spin densities. We then use the fully space-time discrete schemes to simulate the time evolution of a Rashba electron gas confined in a bounded domain and subjected to spin-dependent external potentials. Finite difference approximations are first order in time and second order in space. The discrete functionals introduced are minimized with the help of a conjugate gradient-based algorithm, where the Newton method is applied in order to find the respective line minima. The numerical convergence in the long-time limit of a Gaussian initial condition towards the solution of the corresponding stationary Schr\"odinger-Poisson problem is demonstrated for different values of the parameters $\eps$ (semiclassical parameter), $\alpha$ (Rashba coupling parameter), $\Delta x$ (grid spacing) and $\Delta t$ (time step). Moreover, the performances of the semi-implicit and the explicit scheme are compared.

• Solene Ozere and Carole Le Guyader
Topology preservation for image-registration-related deformation fields

In this paper, we address the issue of designing a theoretically well- motivated and computationally efficient method ensuring topology preservation on image-registration-related deformation elds. The model is motivated by a mathematical characterization of topology preservation for a deformation eld mapping two subsets of Z^2, namely, positivity of the four approximations to the Jacobian determinant of the deformation on a square patch. The first step of the proposed algorithm thus consists in correcting the gradient vector field of the deformation (that does not comply with the topology preservation criteria) at the discrete level in order to fulfill this positivity condition. Once this step is achieved, it thus remains to reconstruct the deformation field, given its full set of discrete gradient vectors. We propose to decompose the reconstruction problem into independent problems of smaller dimensions, yielding a natural parallelization of the computations and enabling us to reduce drastically the computational time (up to 80 in some applications). For each subdomain, a functional minimization problem under Lagrange interpolation constraints is introduced and its well-posedness is studied: existence/uniqueness of the solution, characterization of the solution, convergence of the method when the number of data increases to infinity, discretization with the Finite Element Method and discussion on the properties of the matrix involved in the linear system. Numerical simulations based on OpenMP parallelization and MKL multi-threading demonstrating the ability of the model to handle large deformations (contrary to classical methods) and the interest of having decomposed the problem into smaller ones are provided.

• Jose Raul Quintero Henao
Stability of 2D solitons for a sixth order Boussinesq type model

We study orbital stability of solitary wave of the least energy for a nonlinear 2D Benney-Luke model of higher order related with long water waves with small amplitude in the presence of strong surface tension. We follow a variational approach which includes the characterization of the ground state solution set associated with solitary waves. We use the Hamiltonian structure of this model to establish the existence of an energy functional conserved in time for the modulated equation associated with this Benney-Luke type model. For wave speed near to zero or one, and in the regime of strong surface tension, we prove the orbital stability result by following a variational approach.

• Pierre Degond, Giacomo Dimarco, Thi Bich Ngoc Mac, Nan Wang
Macroscopic models of collective motion with repulsion

We study a system of self-propelled particles which interact with their neighbors via alignment and repulsion. The particle velocities result from self-propulsion and repulsion by close neighbors. The direction of self-propulsion is continuously aligned to that of the neighbors, up to some noise. A continuum model is derived starting from a mean-field kinetic description of the particle system. It leads to a set of non conservative hydrodynamic equations. We provide a numerical validation of the continuum model by comparison with the particle model. We also provide comparisons with other self-propelled particle models with alignment and repulsion.

• Giacomo Albi, Michael Herty, and Lorenzo Pareschi
Kinetic description of optimal control problems and applications to opinion consensus

In this paper an optimal control problem for a large system of interacting agents is considered using a kinetic perspective. As a prototype model we analyze a microscopic model of opinion formation under constraints. For this problem a Boltzmann type equation based on a model predictive control formulation is introduced and discussed. In particular, the receding horizon strategy permits to embed the minimization of suitable cost functional into binary particle interactions. The corresponding Fokker-Planck asymptotic limit is also derived and explicit expressions of stationary solutions are given. Several numerical results showing the robustness of the present approach are finally reported.

• G. Furioli, A. Pulvirenti, E. Terraner and G. Toscani
On Rosenau-type approximations to fractional diffusion equations

Owing to the Rosenau argument [Phys. Rev. A 46, 12-15, 1992], originally proposed to obtain a regularized version of the Chapman-Enskog expansion of hydrodynamics, we introduce a non-local linear kinetic equation which approximates a fractional diffusion equation. We then show that the solution to this approximation, apart of a rapidly vanishing in time perturbation, approaches the fundamental solution of the fractional diffusion (a Levy stable law) at large times.

• Emeric Bouin and Sepideh Mirrahimi
A Hamilton-Jacobi approach for a model of population structured by space and trait

We study a non-local parabolic Lotka-Volterra type equation describing a population struc- tured by a space variable x\in R^d and a phenotypical trait \theta \in {\Cal \Theta}. Considering diffusion, mu- tations and space-local competition between the individuals, we analyze the asymptotic (long time/longreal phase WKB ansatz, we prove that the propagation of the population in space can be de- scribed by a Hamilton-Jacobi equation with obstacle which is independent of \theta. The effective Hamiltonian is derived from an eigenvalue problem. The main difficulties are the lack of regularity estimates in the space variable, and the lack of comparison principle due to the non-local term.

• Tadele Mengesha and Qiang Du
Multiscale analysis of linearized peridynamics

In this paper, we study the asymptotic behavior of a state-based multiscale heterogeneous peridynamic model. The model involves nonlocal interaction forces with highly oscillatory perturbations representing the presence of heterogeneities on a finer spatial length scale. The two-scale convergence theory is established for a steady state variational problem associated with the multiscale linear model. We also examine the regularity of the limit nonlocal equation and present the strong approximation to the solution of the peridyanmic model via a suitably scaled two-scale limit.

• Nicola Bellomo and Abdelghani Bellouquid
On Multiscale Models of Pedestrian Crowds - From Mesoscopic to Macroscopic

This paper deals with the derivation of macroscopic equations from the underlying mesoscopic description that is suitable to capture the main features of pedestrian crowd dynamics. The interactions are modeled by means of theoretical tools of game theory, while the macroscopic equations are derived from asymptotic limits.