# Forthcoming Papers

• Dietmar Oelz
Convergence of the penalty method applied to a constrained curve straightening flow

We apply the penalty method to the curve straightening flow of inextensible planar open curves generated by the Kirchho bending energy. Thus we consider the curve straightening flow of extensible planar open curves generated by a combination of the Kirchho bending energy and a functional penalising deviations from unit arc-length. We start with the governing equations of the explicit parametrisation of the curve and derive an equivalent system for the two quantities indicatrix and arc-length. We prove existence and regularity of solutions and use the indicatrix/arc-length representation to compute the energy dissipation. We prove its coercivity and conclude exponential decay of the energy. Finally, by an application of the Lions-Aubin Lemma, we prove convergence of solutions to a limit curve which is the solution of an analogous gradient flow on the manifold of inextensible open curves. This procedure also allows to characterise the Lagrange multiplier in the limit model as a weak limit of force terms present in the relaxed model.

• Ciprian G Gal
Global attractor for a nonlocal model for biological aggregation

We investigate the long term behavior in terms of global attractors, as time goes to infinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. We consider the aggregation equation with both degenerate and non-degenerate diffusion in a bounded domain subject to various boundary conditions. In the degenerate case, we prove the existence of the global attractor and derive some optimal regularity results. Furthermore, in the non-degenerate case we give a complete structural characterization of the global attractor, and also discuss the convergence of any bounded solutions to steady states. In particular, under suitable assumptions on the parameters of the problem, we establish the convergence of the bounded solution u(t) to a single steady state and the rate of convergence. Finally, the existence of an exponential attractor is also demonstrated for suciently smooth kernels in the case of non-degenerate diffusion.

• Changsheng Dou and Qiangchang Ju
Low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain for all time

The low Mach number limit of global smooth solutions to the compressible magnetohydrodynamic equations in a bounded smooth domain in R^2 with perfectly conducting boundary is verified for all time, provided that the initial data are well-prepared.

• Igor Kukavica, Mihaela Ignatova and Lenya Ryzhik
The Harnack inequality for second-order elliptic equations with divergence-free drifts

We consider an elliptic equation with a divergence-free drift b. We prove that an inequality of Harnack type holds under the assumption b \in L^{n/2+\delta} where \delta>0. As an application we provide a one-sided Liouville's theorem provided that b\in L^{n/2+\delta}(R^n).

• Hui Zhang and Lizhi Cheng
Asymptotically optimal Johnson-Lindenstrauss Lemma for random circulant magtrices

This paper analyzes circulant Johnson-Lindenstrauss (JL) embeddings which, as an important class of structured random JL embeddings, are formed by randomizing the column signs of a circulant matrix generated by a random vector. With the help of recent decoupling techniques and matrix-valued Bernstein inequalities, we obtain a new bound $k=O(\epsilon^{-2}\log^(1+\delta)(n))$ for Gaussian circulant JL embeddings. Moreover, by using the Laplace transform technique (also called Bernstein's trick), we extend the result to subgaussian case. The bounds in this paper offer a small improvement over the current best bounds for Gaussian circulant JL embeddings for certain parameter regimes and are derived using more direct methods.

• Dongfeng Bian and Bolin Guo
Global existeance of smooth solutions to the k-epsilon model equations for turbulent flows

In this paper we are concerned with the global existence of smooth solutions to the k-epsilon model equations for turbulent flows in R^3. The global well-posedness is proved under the condition that the initial data are close to the standard equilibrium state in H^3-framework. The proof relies on energy estimates about velocity, temperature, turbulent kinetic energy and rate of viscous dissipation. We use several new techniques to overcome the difficulties from the product of two functions and higher order norms. This is the first result concerning k-epsilon model equations.

• Yekaterina Epshteyn
Algorithms Composition Approach based on Difference Potentials Method for Parabolic Problems

In this work we develop an efficient and flexible Algorithms Composition Approach based on the idea of the difference potentials method (DPM) for parabolic problems in composite and complex domains. Here, the parabolic equation serves both as the simplied model, and as the first step towards future development of the proposed framework for more realistic systems of materials, fluids, or chemicals with different properties in the different domains. Some examples of such models include the ocean-atmosphere models, chemotaxis models in biology, and blood flow models. Very often, such models are heterogeneous systems - described by different types of partial differential equations (PDEs) in different domains, and they have to take into consideration the complex structure of the computational subdomains. The major challenge here is to design an efficient and flexible numerical method that can capture certain properties of analytical solutions in different domains, while handling the arbitrary geometries and complex structures of the subdomains. The Algorithms Compositions principle, as well as the Domain Decomposition idea, is one way to overcome these difficulties while developing very efficient and accurate numerical schemes for the problems. The Algorithms Composition Approach proposed here can handle the complex geometries of the domains without the use of unstructured meshes, and can be employed with fast Poisson solvers. Our method combines the simplicity of the finite difference methods on Cartesian meshes with the flexibility of the Difference Potentials method. The developed method is very well suited for parallel computations as well, since most of the computations in each domain are performed independently of the others.

• Gil Ariel, Bjorn Engquist, Seong Jun Kim and Richard Tsai
Iterated averaging of three-scale oscillatory systems

A theory of iterated averaging is developed for a class of highly oscillatory ordinary differential equations (ODEs) with three well separated time scales. The solutions of these equations are assumed to be (almost) periodic in the fastest time scales. It is proved that the dynamics on the slowest time scale can be approximated by an effective ODE obtained by averaging out oscillations. In particular, the effective dynamics of the considered class of ODEs is always deterministic and does not show any stochastic effects. This is in contrast to systems in which the dynamics on the fastest time scale is mixing. The systems are studied from three perspectives: first, using the tools of averaging theory; second, by formal asymptotic expansions; and third, by averaging with respect to fast oscillations using nested convolutions with averaging kernels. The latter motivates a hierarchical numerical algorithm consisting of nested integrators.

• Argus A. Dunca and Roger Lewandowski
Modeling error in approximate deconvolution models

We investigate the asymptotic behavior of the modeling error in 3D periodic Approximate Deconvolution Models, when the order N of deconvolution goes to \infty. We consider generalized Helmholtz lters of order p, then the Gaussian filter. For Helmholtz filters, we estimate the rate of convergence to zero thanks to energy budgets, Gronwall's Lemma and sharp inequalities applied to the Fourier coefficients of the residual stress. We next explain why the same analysis does not imply convergence to zero of the modeling error in the case of the Gaussian filter, leaving open issues.

• Ninghao Zhang and Guillaume Bal
Convergence of SPDE of the Schrodinger equation with large, random potential

We study the asymptotic behavior of solutions to the Schrodinger equation with large-amplitude, highly oscillatory, random potential. In dimension d < m, where m is the order of the leading operator in the Schrodinger equation, we construct the heterogeneous solution by using a Duhamel expansion and prove that it converges in distribution, as the correlation length $\epsilon$ goes to 0, to the solution of a stochastic differential equation, whose solution is represented as a sum of iterated Stratonovich integral, over the space $C([0, \infty), S')$. The uniqueness of the limiting solution in a dense space of $L^2(\Omega \times R^d)$ is shown by verifying the property of conservation of mass for the Schrodinger equation. In dimension d > m, the solution to the Schrodinger equation is shown to converge in $L^2(\Omega \times R^d)$ to a deterministic Schrodinger solution in [N. Zhang and G. Bal, Homogenization of the Schroedinger equation with large, random potential, published in Stochastics and Dynamics, 2012].

• Michal Branicki and Andrew Majda
Quantifying Bayesian Filter Performance for Turbulent Dynamical Systems through Information Theory

Incomplete knowledge of the true dynamics and its partial observations pose a notoriously diffcult problem in many scientic applications which require predictions of high-dimensional dynamical systems with physical instabilities and energy fluxes across a wide range of scales. In such cases assimilation of real data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of imperfect models. However, the practically implementable data assimilation/filtering strategies are also imperfect and not optimal due to the formidably complex nature of the underlying dynamics. Here, the connections between information theory and the filtering problem are exploited in order to establish bounds on the filter error statistics, and to systematically study the statistical accuracy of various Kalman filters with model error for estimating the dynamics of spatially extended, partially observed turbulent systems. The effects of model error on filter stability and accuracy in this high-dimensional setting are analyzed through appropriate information measures which naturally extend the common path-wise estimates of filter performance, like the mean-square error or pattern correlation, to the statistical superensemble setting that involves all possible initial conditions and all realizations of noisy observations of the truth signal. Particular emphasis is on the notion of practically achievable filter skill which requires trade-offs between different facets of filter performance; a new information criterion is introduced in this context. This information-theoretic framework for assessment of filter performance is an important complement to the path-wise approach, and it has natural generalizations to Kalman filtering with non-Gaussian statistically exactly solvable forecast models. Here, this approach is utilized to study the performance of imperfect, reduced-order filters involving Gaussian forecast models which use various spatio-temporal discretizations to approximate the dynamics of the stochastically forced advection-diffusion equation; important examples in this conguration include effects of biases due to model error in the filter estimates for the mean dynamics which are quantied through appropriate information measures.

• Rustum Choksi, Irene Fonseca and Barbara Zwicknagl
A few remarks on variational models for denoising

Variational models for image and signal denoising are based on the minimization of energy functionals consisting of a delity term together with higher-order regularization. In addition to the choices of function spaces to measure delity and impose regularization, different scaling exponents appear. In this note we present a few simple, yet novel, remarks on (i) the stability with respect to deterministic noise perturbations, captured via oscillatory sequences converging weakly to zero, and (ii) exact reconstruction.

• 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.

• Steffi Winkelmann, Christof Schutte and Max von Kleist
Markov control processes with rare state observation: Theory and application to treatment scheduling in in HIV-1

Markov Decision Processes (MDP) or Partially Observable MDPs (POMDP) are used for modelling situations in which the evolution of a process is partly random and partly controllable. These MDP theories allow for computing the optimal control policy for processes that can continuously or frequently be observed, even if only partially. However, they cannot be applied if state observation is very costly and therefore rare (in time). We present a novel MDP theory for rare, costly observations and derive the corresponding Bellman equation. In the new theory, state information can be derived for a particular cost after certain, rather long time intervals. The resulting, information costs enter into the total cost and thus into the optimization criterion. This approach applies to many real world problems, particularly in the medical context, where the medical condition is examined rather rarely because examination costs are high. At the same time, the approach allows for efficient numerical realization. We demonstrate the usefulness of the novel theory by determining, from the national economic perspective, optimal therapeutic policies for the treatment of the human immunodeficiency virus (HIV) in resource-rich and resource-poor settings. Based on the developed theory and models, we discover that available drugs may not be utilized efficiently in resource-poor settings due to exorbitant diagnostic costs.

• Guoliang Xu and Chong Chen
Blended Finite Element Method and its Convergence of 3D Image Reconstruction Using $L^2$-Gradient Flow

In an earlier paper, we presented an iterative algorithm for reconstructing a three dimensional density function from a set of two dimensional electron microscopy images. By minimizing an energy functional consisting of a fidelity term and a regularization term, an L2-gradient flow was derived. The flow was integrated by an explicit finite element method. Numerical experiments have shown that the L2-gradient flow method yields very desirable results. However, the convergence result of the numerical algorithm has not been established. In this paper, we present a blended finite element method, which includes the explicit finite element method as a special case, for solving the same flow. Furthermore, theoretical analysis for the convergence of the blended finite element method is presented. Numerical results are also presented which show that the blended nfiite element method is more effcient than explicit one.

• Alexey Miroshnikov and Konstantina Trivisa
Hyperbolic Relaxation for Balance Laws

We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies on the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras for systems of hyperbolic conservation laws.

• Liqun Qi
H^+ -eigenvalues of Laplacian and signless Laplacian tensors

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H^+ -eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H^{++}-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor and the adjacency tensor has at most one H^{++}-eigenvalue, but has several other H^+ -eigenvalues. We identify their largest and smallest H^+ -eigenvalues, and establish some maximum and minimum properties of these H^+ -eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.

• Luca Rossi and Lenya Ryzhik
Transition waves for a class of space-time dependent monostable equations

We investigate the existence of generalized transition waves for reaction-diffusion KPP equations depending explicitly on time and space. In the case of spatially periodic diffusion and drift, and general temporal dependence of the nonlinearity, we almost completely characterize the set of admissible speeds of the waves in terms of a suitable notion of mean introduced by Naldi and Rossi. A lower bound for the speeds is also derived for equations with non-periodic, spatially dependent coefficients, without assuming the KPP condition.

• Shixin Xu, Ping Sheng and Chun Liu
An enegertic variational approach for ion transport

The transport and distribution of charged particles are crucial in the study of many physical and biological problems. In this paper, we employ an Energy Variational Approach to derive the coupled Poisson-Nernst-Planck-Navier-Stokes system. All physics is included in the choices of corresponding energy law and kinematic transport of particles. The variational derivations give the coupled force balance equations in a unique deterministic fashion. We also discuss the situations with different types of boundary conditions. Finally, under axisymmetric cylinder coordinate, we show the Onsager relation near the initial time.

• Nikita Mandrik and Yuri Trakhinin
Influence of vacuum electric field on the stability of a plasma-vacuum interface

We study the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics. Unlike the classical statement, when the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. We show that a suciently large vacuum electric field can make the planar interface violently unstable. We find and analyze a sucient condition on the vacuum electric field that precludes violent instabilities. Under this condition satisfied at each point of the unperturbed nonplanar plasma-vacuum interface, we prove the well-posedness of the linearized problem in anisotropic weighted Sobolev spaces.

• David Angeli, Murad Banaji and Casian Pantea
Combinatorial approaches to Hopf bifurcations in systems in interacting elements

We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed "interaction networks", of which networks of chemical reactions are a special case.

• Mingjie Li and Teng Wang
Zero dissipation limit to rarefaction wave with vacuum for one-dimensional full compressible Navier-Stokes equations

We study the zero dissipation limit of the full compressible Navier-Stokes equations to a rarefaction wave which connects to vacuum at one side. It is shown that there exist a family of smooth solutions to the full compressible Navier-Stokes equations converging to the rarefaction wave with vacuum away from the initial layers at a uniform rate as the viscosity and the heat conductivity coe cients tend to zero. Our method of proof consists of a scaling argument and elementary energy analysis, based on the underlying wave structure.

• Carlos J. Garcia-Cervera and Sookyung Joo
An efficient numerical scheme for Chen-Lubensky energy

We study the Chen-Lubensky energy to investigate layer undulations in smectic liquid crystals in response to an applied magnetic eld. In earlier work, the authors obtained an asymptotic expression of the unstable modes and a sharp estimate of the critical field using the Landau-de Gennes model for smectic A liquid crystals. In this paper, we extend our theory to the Chen-Lubensky energy, which includes a second order smectic order parameter gradient. Analysis based on \Gamma-convergence theory and bifurcation theory provide the estimate of the critical field and frequency of the undulations. Furthermore, we present a new numerical formulation of fourth order partial dierential equations. With this formulation, the fourth order system reduces to a second order equation with a constraint, which resembles the incompressible Navier-Stokes equations from fluid dynamics. We use this method to illustrate the presence of layer undulations near the critical field and to confirm that the results from our analysis agree with this numerical simulations. We also use asymptotic analysis to determine the structure of the domain wall at high fields under the assumption that the layer density is constant.

• 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.