# Forthcoming Papers

• Xiuqing Chen, Xiaolong Li and Jian-Guo Liu
Existence and Uniqueness of Global Weak Solution to a Kinetic Model for the Sedimentation of Rod-like Particles

We investigate a kinetic model for the sedimentation of dilute suspensions of rod-like particles under gravity, deduced by Helzel, Otto and Tzavaras (2011), which couples the impressible (Navier-)Stokes equation with the Fokker-Planck equation. With no-flux boundary condition for distribution function, we establish the existence and uniqueness of global weak solution to the two dimensional model involving Stokes equation.

• Jishan Fan and Fucai Li
Uniform local well-posedness and regularity criterion for the density-dependent incompressible flow of liquid crystals

In this paper we rst prove the uniform local well-posedness for the density-dependent incompressible flow of liquid crystals in the whole space R^3. Next, we provide a regularity criterion to the strong solutions when the initial density may contain vacuum.

• Stephane Brull and Jacques Schneider
Derivation of a BGK model for reacting gas mixtures

In this paper we derive a new relaxation model for reacting gas mixtures. We prove that this model satisfies the fundamental properties (equilibrium states, conservation laws, H-theorem, ...). We also consider the slow reaction regime. In this case a rigorous Chapman-Enskog procedure is performed and Navier-Stokes equations are derived.

• P. Noundjeu and C. Chendjou
On the static solutions of the spherically symmetric Vlasov-Einstein-Maxwell system for low charge and ainsotropic pessure

We consider the Vlasov-Einstein-Maxwell (VEM) system in the spherical symmetry setting and we try to establish a global static solutions with isotropic or anisotropic pressure that approachs Minkowski spacetime at the spacial infinity and have a regular center. This work extends the previous one recently done by the first author, in which only the isotropic case is concerned .

• S. Moutari and M. Herty
A lagrangian approach for modeling road collisions using second order models for traffic flow

In [M. Herty, A. Klein, S. Moutari, IMA J. Appl. Math. 2012] and [M. Herty, and V. Schleper, ZAMM J. Appl. Math Mech. 2011], a macroscopic approach, derived from fluid-dynamics models, has been introduced to infer traffic conditions prone to road traffic collisions along highways' sections. In these studies, the governing equations are coupled within an eulerian framework, which assumes fixed interfaces between the models. A coupling in lagrangian coordinates would enable to get rid of this (not very realistic) assumption. In this paper, we investigate the well-posedness and the suitability of the coupling of the governing equations within the lagrangian framework. Further, we illustrate some features of the proposed approach through some numerical simulations.

• Zhigang Wu and Weike Wang
Decay of the solution for the bipolar Euler-Poisson system with damping in dimension three

The global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in dimension three are constructed when the initial data in H^3 norm is small. And what's more, by using a refined energy estimate together with the interpolation trick, we improve the decay estimate in [Y.P. Li, X.F. Yang, J. Diff. Eqn. 252, 2012], besides, we need not the smallness assumption of the initial data in L^1 space in the paper of Li and Yang.

• Peixin Zhang and Junning Zhao
The existence of local solutions for the compressible Navier-Stokes equations with the density-dependent viscosities

In this paper, we consider the isentropic compressible Navier-Stokes equations with density-dependent viscosities. We prove the local existence of the classical solutions, where the initial density is allowed to vanish.

• Boualem Khouider
A coarse grained stochastic multi-type particle interacting model for tropical convection: nearest neighbour interactions

Particle interacting systems on a lattice are widely used to model complex physical processes that occur on much smaller scales than the observed phenomenon one wishes to model. However, their full applicability is hindered by the curse of dimensionality so that in most practical applications a mean field equation is derived and used. Unfortunately, the mean field limit does not retain the inherent variability of the microscopic model. Recently, a systematic methodology is developed and used to derive stochastic coarse-grained birth-death processes which are intermediate between the microscopic model and the mean field limit, for the case of the one-type particle-Ising system. Here we consider a stochastic multicloud model for organized tropical convection introduced recently to improve the variability in climate models. Each lattice is either clear sky of occupied by one of three cloud types. In earlier work, local interaction between lattice sites were ignored in order to simplify the coarse graining procedure that leads to a multi-dimensional birth-death process; Changes in probability transitions depend only on changes in the large-scale atmospheric variables. Here the coarse-graining methodology is extended to the case of multi-type particle systems with nearest neighbour interactions and the multi-dimensional birth-death process is derived for this general case. The derivation is carried under the assumption of uniform redistribution of particles within each coarse grained cell given the coarse grained values. Numerical tests show that despite the coarse graining the birth-death process preserves the variability of the microscopic model. Moreover, while the local interactions do not increase considerably the overall variability of the system, they induce a signicant shift in the climatology and at the same time boost its intermittency from the build up of coherent patches of cloud clusters that induce long time excursions from the equilibrium state.

• Lyle Noakes
Approximating Near-Geodesic Natural Cubic Splines

A method is given for calculating approximations to natural Riemannian cubic splines in symmetric spaces with computational eort comparable to what is needed for the classical case of a natural cubic spline in Euclidean space. Interpolation of n+1 points in the unit sphere S^m requires the solution of a sparse linear system of 4mn linear equations. For n+1 points in bi-invariant SO(p) we have a sparse linear system of 2np(p-1) equations. Examples are given for the Euclidean sphere S^2 and for bi-invariant SO(3) showing signicant improvements over standard chart-based interpolants.

• Lili Du and Yongfu Wang
Blowup criterion for 3-dimensional compressible Navier-Stokes equations involving velocity divergence

In this paper, we provided a sufficient condition, in terms of only velocity divergence for global regularity of strong solutions to the there-dimensional Navier-Stokes equations with vacuum in the whole space, as well as for the case of bounded domain with Dirichlet boundary conditions. More precisely, we showed that the weak solutions of the Cauchy problem or the Dirichlet initial-boundary-value problem of the 3D compressible Navier-Stokes equations is indeed regular provided that the L^2(0,T; L^\infty)-norm of the divergence of the velocity is bounded. Additionally, initial vacuum states are allowed and the viscosity coe1cients are only restricted by the physical conditions.

• Stephane Brull
An Ellipsoidal Statistical Model for gas mixtures

In this paper, we propose a construction of a new BGK model generalizing the Ellipsoidal Statistical Model to the context of gas mixtures. The derivation of the model is based on the introduction of relaxation coefficients associated to some moments and the resolution of a minimization problem. We obtain in this work, an ESBGK model for gas mixtures satisfying the fundamental properties of the Boltzmann collision operator (conservation laws, H theorem, equilibrium states, ...) and that is able to give a range of Prandtl numbers including the indifferentiability situation.

• Changhong Guo, Shaomei Fang and Boling Guo
Global smooth solutions of the generalized KS-CGL equations for flames governed by a sequential reaction

In this paper, we investigate the periodic initial value problem and Cauchy problem of the generalized Kuramoto-Sivashinsky-complex Ginzburg-Landau (GKS-CGL) equations for flames governed by a sequential reaction. We prove the global existence and uniqueness of solutions to these two problems in various spatial dimensions via delicate a priori estimates, the Galerkin method and so-called continuity method.

• Harald Garcke, Kei Fong Lam and Bjorn Stinner
Diffuse interface modelling of soluble surfactants in two-phase flow

Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are derived from balance equations and an energy inequality so that thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they are related to sharp interface models. Both cases of dynamic as well as instantaneous adsorption are covered. Flexibility with respect to the choice of bulk and surface free energies allows to realise various isotherms and relations of state between surface tension and surfactant. Some numerical simulations display the effectiveness of the presented approach.

• David Cohen and Guillaume Dujardin
Energy-preserving integrators for stochastic Poisson systems

A new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix (in the Stratonovich sense) is proposed. These numerical integrators are of mean-square order one and also preserve quadratic Casimir functions. In the deterministic setting, our schemes reduce to methods proposed in [E. Hairer, J. Numer. Anal. Ind. Appl. Math., 5(1-2):73-84, 2011] and [D. Cohen and E. Hairer, BIT, 51(1):91-101, 2011.].

• Taoufik Hmidi and Mohamed Zerguine
Vortex patch problem for stratified Euler equations

We study in this paper the vortex patch problem for the stratified Euler equations in space dimension two. We generalize Chemin's result [J.Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, 1998] concerning the global persistence of the Holderian regularity of the vortex patches. Roughly speaking, we prove that if the initial density is smooth and the initial vorticity takes the form \omega_0=1_\Omega, with \Omega a C^{1+\epsilon}-bounded domain, then the velocity of the stratified Euler equations remains Lipschitz globally in time and the vorticity is split into two parts \omega = 1_{\Omega_t} + \tilda{\rho}(t), where \Omega_t denotes the image of \Omega by the flow and has the same regularity of the domain \Omega. The function \tilda{\rho} is a smooth function.

• Michael Herty, Alexander Kurganov and Dmitry Kurochkin
Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs

We develop a numerical method for the solution to linear adjoint equations arising, for example, in optimization problems governed by hyperbolic systems of nonlinear conservation and balance laws in one space dimension. Formally, the solution requires to numerically solve the hyperbolic system forward in time and a corresponding linear adjoint system backward in time. Numerical results for the control problem constrained by either the Euler equations of gas dynamics or isothermal gas dynamics equations are presented. Both smooth and discontinuous prescribed terminal states are considered.

• Mihaela Ignatova, Gautam Iyer, James P Kelliher, Robert L Pego, and Arghir Dani Zarnescu
Global existence for two extended Navier-Stokes systems

We prove global existence of weak solutions to two systems of equations which extend the dynamics of the Navier-Stokes equations for incompressible viscous flow with no-slip boundary condition. The systems of equations we consider arise as formal limits of time discrete pressure-Poisson schemes introduced by Johnston & Liu (J. Comp. Phys. 199 (2004) 221{259) and by Shiroko & Rosales (J. Comp. Phys. 230 (2011) 8619{8646) when the initial data does not satisfy the required compatibility condition. Unlike the results of Iyer et al. (J. Math. Phys. 53 (2012) 115605), our approach proves existence of weak solutions in domains with less than C^1-regularity. Our approach also addresses uniqueness in 2D and higher regularity.

• Chunxiao Guo and Bolin Guo
H^1-random attractors of stochastic monopolar non-Newtonian fluids with multiplicative noise

In this paper, the authors study the asymptotic dynamical behavior for stochastic monopolar non-Newtonian fluids with multiplicative noise defined on a two-dimensional bounded domain, and prove the existence of $H^1$-random attractor for the corresponding random dynamical system. Random attractor is a random compact set absorbing any bounded subset of the phase space $V$.

• Hayden Schaeffer, Noirin Duggan, Carole le Guyader and Liminita Vese
Topology preserving active contours

Active contours models are variational methods for segmenting complex scenes using edge or regional information. Many of these models employ the level set method to numerically minimize a given energy, which provides a simple representation for the resulting curve evolution problem. During the evolution, the curve can merge or break, thus these methods tend to have steady state solution which are not homeomorphic to the initial condition. In many applications, the topology of the edge set is known, and thus can be enforced. In this work, we combine a topology preserving variational term with the region based active contours models in order to segment images with known structure. The advantage of this method over current topology preserving methods is that our model locates boundaries of objects and not only edges. This is particularly useful for highly textured or noisy data.

• Xu Zhang and Zhong Tan
Decay estimates of the non-isentropic compressible fluid models of Korteweg type in $R^3$

The existence and optimal convergence rates of global-in-time classical solution to the Cauchy problem for compressible non-isotropic Navier-Stokes-Korteweg system for small initial perturbation is obtained. The global solution are obtained by combining the local existence and the priori estimates provided the initial perturbation around a constant state is small enough. The optimal convergence rates are obtained by energy estimates and interpolation inequalities among them without linear decay analysis.

• Louis-Philippe Saumier, Boualem Khouider and Martial Agueh
Optimal transport for particle image velocimetry

We present a new method for particle image velocimetry, a technique using successive laser images of particles immersed in a uid to measure the velocity eld of the uid ow. The main idea is to recover this velocity eld via the solution of the L^2-optimal transport problem associated with each pair of successive distributions of tracers. We model the tracers by a network of Gaussian-like distributions and derive rigorous bounds on the approximation error in terms of the model's parameters. To obtain the numerical solution, we employ Newton's method combined with an efficient spectral method, to solve the Monge-Ampere equation associated with the transport problem. We present numerical experiments based on two synthetic flow fields, a plane shear and an array of vortices. Although the theoretical results are derived for the case of a single particle in dimensions one and two, the results are valid in R^d, d\ge 1. Moreover, the numerical experiments demonstrate that these results hold for the case of multiple particles, provided the Monge-Ampere equation is solved on a fine enough grid.

• Juan Pabo Borgna, Mariano de Leo, Constanza Sanchez de la Vega and Diego Rial
General splitting methods for abstract semilinear evolution equations

In this paper we present a unified picture concerning general splitting methods for solving a large class of semilinear problems: nonlinear Schrodinger, Schrodinger-Poisson, Gross-Pitaevskii equations, etc. This picture includes as particular instances known schemes such as Lie-Trotter, Strang and Ruth-Yoshida. The convergence result is presented in suitable Hilbert spaces related with the time regularity of the solution and is based on Lipschitz estimates for the nonlinearity. In addition, with extra requirements both on the regularity of the initial datum and on the nonlinearity, we show the linear convergence of these methods. We finally mention that in some special cases in which the linear convergence result is known the assumptions we made are less restrictive.

• Hui Zhang, LiZhi Cheng and Wotao Yin
A dual algorithm for a class of augmented convex signal recovery models

Convex optimization models find interesting applications, especially in signal/image processing and compressive sensing. We study some augmented convex models, which are perturbed by strongly convex functions, and propose a dual gradient algorithm. The proposed algorithm includes the linearized Bregman algorithm and the singular value thresholding algorithm as special cases. Based on fundamental properties of proximal operators, we present a concise approach to establish the convergence of both primal and dual sequences, improving the results in the existing literature. Extensions to models with gauge functions are provided.

• Sean D. Lawley, Jonathan C. Mattingly and Michael C. Reed
Sensitivity to switching rates in stochastically switched ODEs

We consider a stochastic process driven by a linear ordinary differential equation whose right-hand side switches at exponential times between a collection of different matrices. We construct planar examples that switch between two matrices where the individual matrices and the average of the two matrices are all Hurwitz (all eigenvalues have strictly negative real part), but nonetheless the process goes to infinity at large time for certain values of the switching rate. We further construct examples in higher dimensions where again the two individual matrices and their averages are all Hurwitz, but the process has arbitrarily many transitions between going to zero and going to infinity at large time as the switching rate varies. In order to construct these examples, we first prove in general that if each of the individual matrices is Hurwitz, then the process goes to zero at large time for sufficiently slow switching rate and if the average matrix is Hurwitz, then the process goes to zero at large time for sufficiently fast switching rate. We also give simple conditions that ensure the process goes to zero at large time for all switching rates.

• Liqun Qi
Hankel Tensors: Associated Hankel matrices and Vandermonde decomposition

Hankel tensors arise from applications such as signal processing. In this paper, we make an initial study on Hankel tensors. For each Hankel tensor, we associate it with a Hankel matrix and a higher order two-dimensional symmetric tensor, which we call the associated plane tensor. If the associated Hankel matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel tensor. We show that an mth-order n-dimensional tensor is a Hankel tensor if and only if it has a Vandermonde decomposition. We call a Hankel tensor a complete Hankel tensor if it has a Vandermonde decomposition with positive coefficients. We prove that if a Hankel tensor is copositive or an even order Hankel tensor is positive semi-definite, then the associated plane tensor is copositive or positive semi-definite, respectively. We show that even order strong and complete Hankel tensors are positive semi-definite, the Hadamard product of two strong Hankel tensors is a strong Hankel tensor, and the Hadamard product of two complete Hankel tensors is a complete Hankel tensor. We show that all the H-eigenvalue of a complete Hankel tensors (maybe of odd order) are nonnegative. We give some upper bounds and lower bounds for the smallest and the largest Z-eigenvalues of a Hankel tensor, respectively. Further questions on Hankel tensors are raised.

• Hai-Yang Jin, Zhi-an Wang and Linjie Xiong
Cauchy problem of the magnetohydrodynamic Burgers system

In this paper, the asymptotic nonlinear stability of solutions to the Cauchy problem of a strongly coupled Burgers system arising in magnetohydro-dynamic (MHD) turbulence is established. It is shown that, as time tends to infinity, the solutions of the Cauchy problem converge to constant states or rarefac- tion waves with large data, or viscous shock waves with arbitrarily large amplitude, where the precise asymptotic behavior depends on the relationship between the left and right end states of the initial value. Our results confirm the existence of shock waves (or turbulence) numerically found in [J. Fleischer, P.H. Diamond, Phys. Rev. E 61, 3912-3925, 2000; S. Yanase, Phys. Plasma 4, 1010-1027, 1997].

• Martin Burger, Luis Caffarelli, Peter A. Markowich and Marie-Therese Wolfram
On the asymptotic behavior of a Boltzmann-type price formation model

In this paper we study the asymptotic behavior of a Boltzmann type price formation model, which describes the trading dynamics in a financial market. In many of these markets trading happens at high frequencies and low transactions costs. This observation motivates the study of the limit as the number of transactions k tends to infinity, the transaction cost a to zero and ka=const. Furthermore we illustrate the price dynamics with numerical simulations.

• Dehua Wang, Xiang Xu and Cheng Yu
Global weak solution for a coupled compressible Navier-Stokes and Q-tensor system

In this paper, we study a coupled compressible Navier-Stokes/Q-tensor system modeling the nematic liquid crystal flow in a three-dimensional bounded spatial domain. The existence and long time dynamics of globally defined weak solutions for the coupled system are established, using weak convergence methods, compactness and interpolation arguments. The symmetry and traceless properties of the Q-tensor play key roles in this process.

• Paul Hand
Conditions for Existence of Dual Certificates in Rank-One Semidefinite Problems

We study the existence of dual certificates in convex minimization problems where a matrix X is to be recovered under semidefinite and linear constraints. Dual certificates exist if and only if the problem satisfies strong duality. In the case that X is rank-one, a dual certificate may fail to exist if there are measurement matrices that are positive and orthogonal to X. We present a completeness condition on the measurement matrices and prove dual certificate existence if this completeness condition holds. For the case where the condition fails, we present a completion process which produces an equivalent program for which dual certificates are guaranteed to exist. We also prove a weak form of necessity for the completeness condition. These results inform the search space for the analytical construction of dual certificates in rank-one matrix completion problems. As an illustration, we present a semidefinite relaxation for the task of finding the sparsest element in a subspace. One formulation of this program does not admit dual certificates. The completion process we describe produces an equivalent formulation which does admit dual certificates.

• Boling Guo, Yongqian Han and Guoli Zhou
Global well-posedness of stochastic Burgers system

In this paper stochastic Burgers system in Ito form is considered. The global well-posedness is proved. The proof relies on energy estimates about velocity. Maximum principle of deterministic parabolic equations is used to overcome the difficulties arising from higher order norms. The methods and results can be applied to other parabolic equations with additive white noise such as stochastic reaction diffusion equations.

• Christian Poschl and Otmar Scherzer
Exact solutions of one-dimensional Total generalized variation

Total generalized variation regularization has been introduced by Bredies, Kunisch, and Pock. This regularization method requires careful tuning of two regularization parameters. The focus of this paper is to derive analytical results, which allow for characterizing parameter settings, which make this method in fact dierent from total variation regularization (that is the Rudin-Osher-Fatmi model) and the second order variation model regularization, respectively. In this paper we also provide explicit solutions of total generalized variation denoising for particular one-dimensional function data.

• 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.

• Mingtao Chen, Xinying Xu and Jianwen Zhan
Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum

In this paper, we study the global existence of weak solutions to the Cauchy problem for three-dimensional equations of compressible micropolar fluids with discontinuous initial data. Here it is assumed that the initial energy is suitable small, and that the initial density is bounded in L^\infty and the gradients of initial velocity and microrotational velocity are bounded in L^2. Particularly, this implies that the initial data may contains vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial-energy.

• 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. Comombo 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.