# Forthcoming Papers

• Yanyan Yu, Weihua Deng and Yujiang Wu
Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations

The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has space fractional derivative, which characterizes L\'{e}vy flights. Sometimes the infinite variance of L\'{e}vy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high order quasi-compact discretizations for space fractional derivative and tempered space fractional derivative. The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically.

• Daniel Han-Kwan and Mikaela Iacobelli
The quasineutral limit of the Vlasov-Poisson equation in Wasserstein metric

In this work, we study the quasineutral limit of the one-dimensional Vlasov-Poisson equation for ions with massless thermalized electrons. We prove new weak-strong stability estimates in the Wasserstein metric that allow us to extend and improve previously known convergence results. In particular, we show that given a possibly unstable analytic initial profile, the formal limit holds for sequences of measure initial data converging sufficiently fast in the Wasserstein metric to this profile. This is achieved without assuming uniform analytic regularity.

• Jishan Fan, Fucai Li, and Gen Nakamura
Non-relativistic and low Mach number limits of two P1 approximation model arising in radiation hydrodynamics

In this paper we study the non-relativistic and low Mach num-ber limits of two P1 approximation model arising in radiation hydrodynamics in T^3, i.e. the barotropic model and the Navier-Stokes-Fourier model. For the barotropic model, we consider the case that the initial data is a small perturbation of stable equilbria while for the Navier-Stokes-Fourier model, we consider the case that the initial data is large. For both models, we prove the convergence to the solution of the incompressible Navier-Stokes equations with/without stationary transport equations.

• Dong Li and Zhonghua Qiao
On the stabilization size of semi-implicit Fourier-spectral methods for 3D Cahn-Hilliard equations

The stabilized semi-implicit time-stepping method is an efficient algorithm to simulate phased field problems with fourth order dissipation. We consider the 3D Cahn-Hilliard equation and prove unconditional energy stability of the corresponding stabilized semi-implicit Fourier spectral scheme independent of the time step. We do not impose any Lipschitz-type assumption on the nonlinearity. It is shown that the size of the stabilization term depends only on the initial data and the diffusion coefficient. Unconditional Sobolev bounds of the numerical solution are obtained and the corresponding error analysis under nearly optimal regularity assumptions is established.

• Xiang Zhou, Ling Lin and Shuting Gu
Sensitivity Analysis and Optimization of Reaction Rate

The chemical reaction rate from reactant to product depends on the geometry of potential energy surface (PES) as well as the temperature. We consider a design problem of how to choose the best PES from a given family of smooth potential functions in order to maximize (or minimize) the reaction rate for a given chemical reaction. By utilizing the transition-path theory, we relate reaction rate to committor functions which solves boundary-value elliptic problems, and perform the sensitivity analysis of the underlying elliptic equations via adjoint approach. We derive the derivative of the reaction rate with respect to the potential function. The shape derivative with respect to the domains defining reactant and product is also investigated. The numerical optimization method based on the gradient is applied for two simple numerical examples to demonstrate the feasibility of our approach.

• Ansgar Jungel, Polina Shpartko and Nicola Zamponi
Energy-transport models for spin transport in ferromagnetic semiconductors

Explicit energy-transport equations for the spinorial carrier transport in ferromagnetic semiconductors are calculated from a general spin energy-transport system that was derived by Ben Abdallah and El Hajj from a spinorial Boltzmann equation. The novelty of our approach are the simplifying assumptions leading to explicit models which extend both spin drift-diffusion and semiclassical energy-transport equations. The explicit models allow us to examine the interplay between the spin and charge degrees of freedom. In particular, the dissipation of the entropy (or free energy) is quantified, and the existence of weak solutions to a time-discrete version of one of the models is proved, using novel truncation arguments. Numerical experiments in one-dimensional multilayer structures using a finite-volume discretization illustrate the effect of the temperature and the polarization parameter.

• Gangjoon Yoon, Jea-Hyun Park and Chohong Min
Convergence analysis on the Gibou-Min method for the Hodge projection

The Hodge projection of a vector field is the divergence-free component of its Helmholtz decomposition. In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. In the decomposition by the Gibou-Min method, an important $L^{2}$-orthogonality holds between the gradient field and the solenoidal field, which is similar to the continuous Hodge decomposition. Using the orthogonality, we present a novel analysis which shows that the convergence order is 1.5 in the $L^2$-norm for approximating the divergence-free vector field. Numerical results are presented to validate our analyses.

• Caidi Zhao and Ling Yang
Pullback attractors and invariant measures for the non-autonomous globally modified Navier-Stokes equations

This paper studies the non-autonomous globally modified Navier-Stokes equations. The authors first prove that the associated process possesses a pullback attractor. Then they establish that there exists a unique family of Borel invariant probability measures on the pullback attractor.

• Daniel Sanz-Alonso and Andrew M. Stuart
Gaussian Approximations of Small Noise Diffusions in Kullback-Leibler Divergence

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.

• Michael Winkler and Christina Surulescu
Global weak solutions to a strongly degenerate haptotaxis model

We consider a one-dimensional version of a model obtained in [Engwer, C., Hunt, A., Surulescu, IMA J. Math. Med. Bio, 2015] and describing the anisotropic spread of tumor cells in a tissue network. The model consists of a reaction-diffusion-taxis equation for the density of tumor cells coupled with an ODE for the density of tissue fibers and allows for strong degeneracy both in the diffusion and the haptotaxis terms. In this setting we prove the global existence of weak solutions to an associated no-flux initial-boundary value problem. Numerical simulations are performed in order to illustrate the model behavior.

• Renhui Wan
Global well-posedness of strong solutions to the 2D damped Boussinesq and MHD equations with large velocity

In this paper, we obtain global well-posedness for the 2D damped Boussinesq equations. Based on the estimate of the damped Euler equations leading to the uniform corresponding bound (do not grow in time), we can achieve this goal by using a new decomposition technic. Comparing with the previous works, we do not need any small assumptions of the initial velocity. As an application of our method, we obtain a similar result for the 2D damped MHD equations.

• M.-O. Bristeau, C. Guichard, B. Di Martino and J. Sainte-Marie
Layer-averaged Euler and Navier-Stokes equations

In this paper we propose a strategy to approximate incompressible hydrostatic free surface Euler and Navier-Stokes models. The main advantage of the proposed models is that the water depth is a dynamical variable of the system and hence the model is formulated over a fixed domain. The proposed strategy extends previous works approximating the Euler and Navier-Stokes systems using a multilayer description. Here, the needed closure relations are obtained using an energy-based optimality criterion instead of an asymptotic expansion. Moreover, the layer-averaged description is successfully applied to the Navier-Stokes system with a general form of the Cauchy stress tensor.

• Leif Arkeryd and Anne Nouri
On the Cauchy problem with large data for a space-dependent Boltzmann-Nordheim boson equation

This paper studies a Boltzmann Nordheim equation in a slab with two-dimensional velocity space and pseudo-Maxwellian forces. Strong solutions are obtained for the Cauchy problem with large initial data in an L^1\cap L^\infty setting. The main results are existence, uniqueness and stability of solutions conserving mass, momentum and energy that explode in L^\infty if they are only local in time. The solutions are obtained as limits of solutions to corresponding anyon equations.

• John W. Barrett, Yong Lu, and Endre Suli
Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model

A compressible Oldroyd-B type model with stress diffusion is derived from a compressible Navier-Stokes-Fokker-Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent, are idealized as pairs of massless beads connected with Hookean springs. We develop a priori bounds for the model, including a logarithmic bound, which guarantee the nonnegativity of the elastic extra stress tensor, and we prove the existence of large data global-in-time finite-energy weak solutions in two space dimensions.

On the global regularity of the 2D critical Boussinesq system with $\alpha>2/3$

This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation (\alpha, ]beta): \alpha+\beta=1. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data, as long as \alpha>2/3. The main new idea is the introduction of a new, second generation Hmidi-Keraani-Rousset type, change of variables, which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by new set of commutator estimates (in both negative and positive index Sobolev spaces!), which may be of independent interest.

• Simon Axmann, Piotr Boguslaw Mucha and Milan Pokorny
Steady solutions to viscous shallow water equations. The case of heavy water

In this note, we show the existence of regular solutions to the stationary version of the Navier--Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods for weak solutions to the compressible Navier--Stokes system.

• Pierre Degond, Michael Herty and Jian-Guo Liu
Meanfield games and model predictive control

Mean-Field Games are games with a continuum of players that incorporate the time dimension through a control-theoretic approach. Recently, simpler approaches relying on the Best Reply Strategy have been proposed. They assume that the agents navigate their strategies towards their goal by taking the direction of steepest descent of their cost function (i.e. the opposite of the utility function). In this paper, we explore the link between Mean-Field Games and the Best Reply Strategy approach. This is done by introducing a Model Predictive Control framework, which consists of setting the Mean-Field Game over a short time interval which recedes as time moves on. We show that the Model Predictive Control offers a compromise between a possibly unrealistic Mean-Field Game approach and the sub-optimal Best Reply Strategy.

• Yongkai Liao and Huijiang Zhao
Global Solutions to One-dimensional Equations for a Self-gravitating Viscous Radiative and Reactive Gas with Density-dependent Viscosity

In this paper we are concerned with the global existence of smooth solutions to two types of initial-boundary value problems to a system of equations describing one-dimensional motion of selfgravitating, radiative and chemically reactive gas whose viscosity coefficient depends on density. The main ingredient of the analysis is to derive the positive lower and upper bounds on both the specific volume and the absolute temperature.

• Changhong Guo and Shaomei Fang
Global existence and pointwise estimates of solutions for the generalized sixth-order Boussinesq equation

This paper studied the Cauchy problem for the generalized sixth-order Boussinesq equation in multi-dimension (n \ge 3), which was derived in the shallow fluid layers and nonlinear atomic chains. Firstly the global classical solution for the problem are obtained by means of long wave-short wave decomposition, energy method and the Green’s function. Secondly and what’s more, the pointwise estimates of the solutions are derived by virtue of the Fourier analysis and Green’s function.

• Ke Ye and Shenglong Hu
Inverse eigenvalue problem for tensors

• Julien Deschamps, Erwan Hingant and Romain Yvinec
Quasi steady state approximation of the small clusters in Becker-DOrings equations leads to boundary conditions in the Lifschitz-Slyozov limit

The following paper addresses the connection between two classical models of phase transition phenomena describing different stages of clusters growth. The first one, the Becker-Doring model (BD) that describes discrete-sized clusters through an infinite set of ordinary differential equations. The second one, the Lifshitz-Slyozov equation (LS) that is a transport partial differential equation on the continuous half-line $x\in (0,+\infty)$. We introduce a scaling parameter $\veps>0$, which accounts for the grid size of the state space in the BD model, and recover the LS model in the limit $\veps\to 0$. The connection has been already proven in the context of outgoing characteristic at the boundary $x=0$ for the LS model when small clusters tend to shrink. The main novelty of this work resides in a new estimate on the growth of small clusters, which behave at a fast time scale. Through a rigorous quasi steady state approximation, we derive boundary conditions for the incoming characteristic case, when small clusters tend to grow.

• Liqun Qi, Guofeng Zhang, Daniel Braun, Fabian Bohnet-Waldraff and Olivier Giraud
Regularly decomposable tensors and classical spin states

A spin-jj state can be represented by a symmetric tensor of order N=2j and dimension 4. Here, j can be a positive integer, which corresponds to a boson; j can also be a positive half-integer, which corresponds to a fermion. In this paper, we introduce regularly decomposable tensors and show that a spin-j state is classical if and only if its representing tensor is a regularly decomposable tensor. In the even-order case, a regularly decomposable tensor is a completely decomposable tensor but not vice versa; a completely decomposable tensors is a sum-of-squares (SOS) tensor but not vice versa; an SOS tensor is a positive semi-definite (PSD) tensor but not vice versa. In the odd-order case, the first row tensor of a regularly decomposable tensor is regularly decomposable and its other row tensors are induced by the regular decomposition of its first row tensor. We also show that complete decomposability and regular decomposability are invariant under orthogonal transformations, and that the completely decomposable tensor cone and the regularly decomposable tensor cone are closed convex cones. Furthermore, in the even-order case, the completely decomposable tensor cone and the PSD tensor cone are dual to each other. The Hadamard product of two completely decomposable tensors is still a completely decomposable tensor. Since one may apply the positive semi-definite programming algorithm to detect whether a symmetric tensor is an SOS tensor or not, this gives a checkable necessary condition for classicality of a spin-j state. Further research issues on regularly decomposable tensors are also raised.

• Deqin Zhou, Chunlai Mu and Ke Lin
On the global attractor of the dampled Rosenau equation on the real line

We consider the asymptotic behaviour of the solution for the damped Rosenau equation on R^1. By applying the I-method and a variant form of Riesz-Rellich criteria, we prove that this damped Rosenau equation possesses a global attractor in H^s(R) for any s \in (1/2, 2). Moreover, the global attractor As is contained in H^2(R) for any s\in (1/2, 2). Our results establish the lower regularity of the global attractor for the damped Rosenau equation in fractional order Sobolev space and give a partial answer to the open problem in [D. Zhou, C. Mu, Appl. Anal. (2016), 1-10].

• Ryan W. Murray and Robert L. Pego
Cutoff estimates for the linearized Becker-Doring equations

This paper continues the authors’ previous study (SIAM J. Math. Anal. 48 (2016) 2819-2842) of the trend toward equilibrium of the Becker-D¨oring equations with subcritical mass, by characterizing certain fine properties of solutions to the linearized equation. In particular, we partially characterize the spectrum of the linearized operator, showing that it contains the entire imaginary axis in polynomially weighted spaces. Moreover, we prove detailed cutoff estimates that establish upper and lower bounds on the lifetime of a class of perturbations to equilibrium.

• Fabien Campillo, Nicolas Champagnat and Coralie Fritsch
On the variations of the principle eigenvalue and the probability of survival with respect to a parameter in growth-fragmentation-death models

We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model, using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we deduce the variations of the eigenvalue with respect to the parameter of the model.

• Etienne Bernard, Laurent Desvillettes, Francois Golse and Valeria Riccci
A derivation of the Vlasov-Navier-Stokes model for aerosol flows from kinetic theory

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the collision kernels, we prove that the sequences of solutions to the multiphase Boltzmann system converge to distributional solutions to the Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit. Specifically, we assume (a) that the mass ratio of the gas molecules to the dust particles/droplets is small, (b) that the thermal speed of the dust particles/droplets is much smaller than that of the gas molecules and (c) that the mass density of the gas and of the dispersed phase are of the same order of magnitude. The class of kernels modelling the interaction between the dispersed phase and the gas includes, among others, elastic collisions and inelastic collisions of the type introduced in [F. Charles: in “Proceedings of the 26th International Symposium on Rarefied Gas Dynamics”, AIP Conf. Proc. 1084, (2008), 409-414].

• Zuoqiang Shi
Enforce the Dirichlet boundary condition by volume constraint in Point Integral method

Poisson equation on point cloud with Dirichlet boundary condition plays important role in many problems. In this paper, we use the volume constraint proposed by Du et.al to handle the Dirichlet boundary condition in the point integral method for Poisson equation on point cloud. We prove that the solution given by volume constraint converges to the true solution as the point cloud converges to the underlying smooth manifold.

• Jonathan Siegel and Omer Tekin
Compact Support of L1 Penalized Variational Problems

We investigate the solutions to L1 constrained variational problems. In particular, we are interested in the case where the L1 term is weighted by some non-negative function. Extending previous results of Brezis et al., we prove that for a wide range of variational problems, the solutions have compact support. Additionally, we provide the results of some numerical experiments, where we computed the solutions to L1 constrained elliptic and parabolic problems using splitting and ADMM.

• Hui Huang and Jian-Guo Liu
Discrete-in-time random particle blob method for the Keller-Segel equation and convergence analysis

• Jiankai Xu, Huoxiong Wu and Zhong Tan
Integrability and asymptotic behaviors of positive solutions for some integral equations

In this paper, we investigate the integrability and asymptotic behaviors of positive solutions for a nonlinear integral system in a functional setting. Using the regularity lifting lemma and some delicate analysis techniques, we obtain the optimal integral intervals and the asymptotic estimates for such solutions around the origin and near infinity. Moreover, the index of regular solution is distinct from one of the previous several related systems.

• Nicolas van Goethem
Front migration for the dislocation strain in single crystals

In this paper we consider a tensor version of Cahn-Hilliard system in the context of incompatible linearized elasticity. We interpret the trace of the incompatible strain as the density of point defects and propose a model for point defects collapse into dislocation loops in single crystals as based on Cahn-Hilliard dynamics. By means of a asymptotic analysis, we determine the front dynamics and obtain as a result a tensor version of Mullins-Sekerka law which allows for flat interfaces, i.e., such as polygonization.

• Marco A. Iglesias, Kui Lin, Shuai Lu, Andrew M. Stuart
Filter Based Methods For Statistical Linear Inverse Problems

Ill-posed inverse problems are ubiquitous in applications. Understanding of algorithms for their solution has been greatly enhanced by a deep understanding of the linear inverse problem. In the applied communities ensemble-based filtering methods have recently been used to solve inverse problems by introducing an artificial dynamical system. This opens up the possibility of using a range of other filtering methods, such as 3DVAR and Kalman based methods, to solve inverse problems, again by introducing an artificial dynamical system. The aim of this paper is to analyze such methods in the context of the linear inverse problem. Statistical linear inverse problems are studied in the sense that the observational noise is assumed to be derived via realization of a Gaussian random variable. We investigate the asymptotic behavior of filter based methods for these inverse problems. Rigorous convergence rates are established for 3DVAR and for the Kalman filters, including minimax rates in some instances. Blowup of 3DVAR and a variant of its basic form is also presented, and optimality of the Kalman filter is discussed. These analyses reveal a close connection between (iterated) regularization schemes in deterministic inverse problems and filter based methods in data assimilation. Numerical experiments are presented to illustrate the theory.