# Forthcoming Papers

• L.F. Stokols and Alexis F. Vasseur
De Giorgi Techniques Applied to Hamilton-Jacobi Equations with Unbounded Right-Hand Side

In this article we obtain Holder estimates for solutions to second-order Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the diffusion and the smoothness of the Hamiltonion. Our work is in the spirit of a result by P. Cardaliaguet and L. Silvestre.

• Donghyun Lee
Initial value problem for the free-boundary Magnetohydrodynamics with zero magnetic boundary condition

We show local well-posedness of fluid-vacuum free-boundary magnetohydrodynamic(MHD) with both kinematic viscosity and magnetic diffusivity under the gravity force. We consider three-dimensional problem with finite depth and impose zero magnetic field condition on the free boundary and in vacuum. Sobolev-Slobodetskii space (Fractional Sobolev space) is used to perform energy estimates. Main difficulty is to control strong nonlinear couplings between velocity and magnetic fields. In [Lee, D., SIAM J. Math. Anal. 49, no.4, 2710-2789 (2017)], we send both kinematic viscosity and magnetic diffusivity to zero with same speed to get ideal (inviscid) free-boundary magnetohydrodynamics using the result of this paper.

• Wanrong Yang, Quansen Jiu, Jiahong Wu
The 3D incompressible Boussinesq equations with fractional partial dissipation

The 3D Boussinesq equations are one of the most important models for geophysical fluids. The fundamental problem of whether or not reasonably smooth solutions to the 3D Boussinesq equations with the standard Laplacian dissipation can blow up in a finite time is an outstanding open problem. The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations, but also are physically relevant and mathematically important. This paper focuses on a system of the 3D Boussinesq equations with fractional partial dissipation and proves that any $H^1$-initial data always leads to a unique and global-in-time solution. The result of this paper is part of our efforts devoted to the global well-posedness problem on the Boussinesq equations with minimal dissipation.

• Bo Tang
Global classical solutions to reaction-diffusion systems in one and two dimensions

The global existence of classical solutions to reaction-diffusion systems in dimensions one and two is proved. The considered systems are assumed to satisfy an {\it entropy inequality} and have nonlinearities with at most cubic growth in 1D or at most quadratic growth in 2D. This global existence was already proved in [T. Goudon and A. Vasseur, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 43 (2010), no. 1, 117--142] by a De Giorgi method. In this paper, we give a simplified proof by using a modified Gagliardo-Nirenberg inequality and the regularity of the heat operator. Moreover, the classical solution is proved to have $L^{\infty}$-norm growing at most polynomially in time. As an application, the solutions to chemical reaction-diffusion systems satisfying the so-called complex balance condition are proved to converge exponentially to equilibrium in $L^{\infty}$-norm.

• Martin Hutzenthaler, Arnulf Jentzen, Diyora Salimova
Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations

This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly non-globally monotone nonlinearities such as stochastic Kuramoto-Sivashinsky equations. The main result of this article proves that the proposed approximation scheme converges strongly and numerically weakly to the solution process of such an SPDE. Key ingredients in the proof of our convergence result are a suitable generalized coercivity-type condition, the specific design of the accelerated exponential Euler-type approximation scheme, and an application of Fernique's theorem.

• Ling Lin and Xiang Zhou
Asymptotic Expansion with Boundary Layer Analysis for Strongly Anisotropic Elliptic Equations

In this article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the Neumann boundary condition and the Dirichlet boundary condition on specific parts of the domain boundary, respectively. The ill-posedness arising from the Neumann boundary condition in the strongly anisotropic diffusion limit is handled by the decomposition of the solution into a mean part and a fluctuation part. The boundary layer analysis due to the Dirichlet boundary condition is conducted for each order in the expansion for the fluc tuation part. Our results suggest that the leading order is the combination of the mean part and the composite approximation of the fluctuation part for the general Dirichlet boundary condition. We also apply this method to derive the results for the state-dependent diffusion problems.

• A. Crestetto, N. Crouseilles and M. Lemou
A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step \Delta t in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in [A. Crestetto, N. Crouseilles, M. Lemou, Kin. Rel. Models 5, pp. 787-816 (2012)] where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit.

• Wanyang Dai
A Unified System of FB-SDEs with Levy Jumps and Double Completely-S Skew Reflections

We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs) with Levy jumps and double completely-S skew reflections. Owing to the reflections, the solution to an embedded Skorohod problem may be not unique, i.e., bifurcations may occur at reflection boundaries, the well-known contraction mapping approach can not be extended directly to solve our problem. Thus, we develop a weak convergence method to prove the well-posedness of an adapted 6-tuple weak solution in the sense of distribution to the unified system. The proof heavily depends on newly established Malliavin calculus for vector-valued Levy processes together with a generalized linear growth and Lipschitz condition that guarantees the well-posedness of the unified system even under a random environment. Nevertheless, if a more strict boundary condition is imposed, i.e., the spectral radii in certain sense for the reflections are strictly less than the unity, a unique adapted 6-tuple strong solution in the sense of sample pathwise is concerned. In addition, as applications and economical studies of our unified system, we also develop new techniques including deriving a generalized mutual information formula for signal processing over possible non-Gaussian channels with multi-input multi-output (MIMO) antennas and dynamics driven by Levy processes.

• Lijun Pan, Xinli Han, Tong Li and Lihui Guo
The generalized Riemann problem and instability of delta shock to the chromatography equations

The generalized Riemann problem for the nonlinear chromatography equations in a neighborhood of the origin (t>0) on the (x,t) plane is considered. The problem is quite different from the previous generalized Riemann problems which have no delta shock wave in the corresponding Riemann solutions. With the method of characteristic analysis and the local existence and uniquness theorem proposed by Li Ta-tsien and Yu Wenci, we constructively solve the generalized Riemann problem and prove the existence and uniqueness of the solutions. It is proved that the generalized Riemann solutions possess a structure similar to the solution of the corresponding Riemann problem for most cases. In case that a delta shock wave in the corresponding Riemann solution, we discover that the generalized Riemann solution may turn into a combination of a shock wave and a contact discontinuity, which shows the instability and the internal mechanisms of a delta shock wave.

• Enrique Domingo Fernandez Nieto, Martin Parisot, Yohan Penel, Jacques Sainte-Marie
A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

In geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model.
In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to model the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.

• Marina Chugunova, Roman Taranets
Thin film flow dynamics on fiber nets

We analyze existence and qualitative behavior of non-negative solutions for fourth order degenerate parabolic equations on graph domains with Kirchhoff's boundary conditions at the inner nodes and Neumann boundary conditions at the boundary nodes. The problem is originated from industrial constructions of spray coated meshes which are used in water collection and in oil-water separation processes. For a certain range of parameter values we prove convergence toward a constant steady state that corresponds to the uniform distribution of coating on a fiber net.

• Alexei Novikov, Karim Shikh Khalil
Resetting of a particle system for the Navier-Stokes equations

This work is based on a formulation of the incompressible Navier-Stokes equations developed by P. Constantin and G.Iyer, where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. If we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with the empirical mean, then it was shown that the particle system for the Navier-Stokes equations does not dissipate all its energy as $t \to \infty$. In contrast to the true (unforced) Navier-Stokes equations, which dissipate all of its energy as $t \to \infty$. The objective of this short note is to describe a resetting procedure that removes this deficiency. We prove that if we repeat this resetting procedure often enough, then the new particle system for the Navier-Stokes equations dissipates all its energy.

• Yuanyuan Feng, Lei Li, Jian-Guo Liu
A note on semi-groups of stochastic gradient descent and online principal component analysis

We study the Markov semigroups for two important algorithms from machine learning: stochastic gradient descent (SGD) and online principal component analysis (PCA). We investigate the effects of small jumps on the properties of the semi-groups. Properties including regularity preserving, $L^{\infty}$ contraction are discussed. These semigroups are the dual of the semigroups for evolution of probability, while the latter are $L^{1}$ contracting and positivity preserving. Using these properties, we show that stochastic differential equations (SDEs) in $\bbR^d$ (on the sphere $\bbS^{d-1}$) can be used to approximate SGD (online PCA) weakly. These SDEs may be used to provide some insights of the behaviors of these algorithms.

• Frederic Charve
Global well-posedness and asymptotics for a penalized Boussinesq-type system without dispersion

J.-Y. Chemin proved the convergence (as the Rossby number \epsilon goes to zero) of the solutions of the Primitive Equations to the solution of the 3D quasi-geostrophic system when the Froude number F = 1 that is when no dispersive property is available. The result was proved in the particular case where the kinematic viscosity \nu and the thermal diffusivity \nu' are close. In this article we generalize this result for any choice of the viscosities, the key idea is to rely on a special feature of the quasi-geostrophic structure.

• Daniel G. Alfago Vigo and Gladys Calle Cardena
A family of asymptotic models for internal wave propagating in intermidiate/deep water

In this paper, we obtain a family of approximate systems of two partial differential equations for the modeling of weakly nonlinear long internal waves propagating at the interface between two immiscible and irrotational fluids in a channel of intermediate/infinite depth. These systems are approximations of the system of Euler equations that share the same asymptotic order. The analysis of the corresponding linearized systems, leads to the identification of several subfamilies (associated with different subsets in the space of parameters), for which the solutions of the linearized models are physically compatible with the solutions of the linearized system of Euler equations. Finally, for the class of weakly dispersive nonlinear systems which is formed by some of those subfamilies, we establish the existence and uniqueness of local in time solutions.

• Sergio Frigeri, Kei Fong Lam, Elisabetta Rocca, Giulio Schimperna
On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials

We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters p, d (proliferating and dead cells, respectively), u (cell velocity) and n (nutrient concentration). The variables p, d satisfy a Cahn-Hilliard type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas u obeys a form of the Darcy law and n satis fies a quasistatic diff usion equation. The main novelty of the present work stands in the fact that we are able to consider a configuration potential of singular type implying that the concentration vector (p,d) is constrained to remain in the range of physically admissible values. On the other hand, in view of the presence of nonzero forcing terms, this choice gives rise to a number of mathematical difficulties, especially related to the control of the mean values of p and d. For the resulting mathematical problem, by imposing suitable initial-boundary conditions, our main result concerns the existence of weak solutions in a proper regularity class.

• Jun Guo, Hao Wu, Lei Xiao
The direct and inverse elastic scattering problems for two scatterers in contact

This paper is concerned with the elastic scattering problem of a combined scatterer, which consists of a penetrable obstacle and a hard crack touching with each other. By using the boundary integral equation method, the direct scattering problem is formulated as a boundary integral system, then we obtain the existence and uniqueness of a weak solution according to Fredholm theory. The inverse scattering problem we are dealing with is the shape reconstruction of the combined scatterer from the knowledge of far field patterns due to the incident plane compressional and shear waves. Based on an analysis of a particular transmission eigenvalue problem, the linear sampling method is established to reconstruct the combined scatterer. The numerical experiments show the feasibility and validity of the proposed method.

• Goro Akagi, Giulio Schimperna, Antonio Segatti, Laura Spinolo
Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i) regularity results for solutions of fractional Poisson problems in Besov spaces; (ii) quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii) quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.

• Andrea Tosin, Mattia Zanella
Boltzmann-type models with uncertain binary interactions

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods.

• Xue Jiang, Peijun Li, Junliang Lv and Weiying Zheng
Convergence of the PML solution for elastic wave scattering by biperiodic structures

This paper is concerned with the analysis of elastic wave scattering of a time-harmonic plane wave by a biperiodic rigid surface, where the wave propagation is governed by the threedimensional Navier equation. An exact transparent boundary condition is developed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing a PML equivalent transparent boundary condition. The proofs rely on a careful study of the error between the two transparent boundary operators. The work significantly extends the results from one-dimensional periodic structures to two-dimensional biperiodic structures. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.

• Joonas Ilmavirta and Alden Waters
Recovery of the sound speed for the acoustic wave equation from phaseless measurements

We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the determination of covector fields for the wave equation. Under some geometric assumptions, we reduce this to a question of boundary rigidity, which allows recovery of the sound speed for the acoustic wave equation. Previous techniques do not measure the full amplitude of the outgoing scattered wave, which is the main novelty in our approach.

• Christophe Gomez anf Kunt Solna
Wave Propagation in Random Waveguides with Long-Range Correlations.

The paper presents an analysis of acoustic wave propagation in a waveguide with random fluctuations of its sound speed profile. These random perturbations are assumed to have long-range correlation properties. In the waveguide a monochromatic wave can be decomposed in propagating modes and evanescent modes, and the random perturbation couples all these modes. The paper presents an asymptotic analysis of the mode-coupling mechanism and uses this to characterize transmitted wave. The paper presents the first fully rigorous characterization of wave propagation in long range non-layered media.

• Wenlong Sun and Yeping Li
Pullback dynamical behaviors of the non-autonomous micropolar fluid flows with minimally regular force and moment

In this paper, we investigate the pullback asymptotic behaviors of solutions for the non-autonomous micropolar fluid flows in 2D bounded domains. Firstly, when the force and the moment have a little additional regularity, we make use of the semigroup method and \epsiloni-regularity method to obtain the existence of a compact pullback absorbing family in \hat{H} and \hat{V}, respectively. Then, applying the global well-posedness and the estimates of the solutions, we verify the flattening property (also known as the "Condition (C)") of the generated evolution process for the universe of fixed bounded sets and for another universe with a tempered condition in spaces \hat{H} and \hat{V}, respectively. Further, we show the existence and regularity of the pullback attractors of the evolution process. Compared with the regularity of the force and the moment of [C. Zhao, W. Sun and C. Hsu, Dynamics of PDE, 12(2015), 265-288], here we only need the minimal regularity of the force and the moment.

• Hayden Schaeffer
A Penalty Method for Some Nonlinear Variational Obstacle Problems

We formulate a penalty method for the obstacle problem associated with a nonlinear variational principle. It is proven that the solution to the relaxed variational problem (in both the continuous and discrete settings) is exact for finite parameter values above some calculable quantity. To solve the relaxed variational problem, an accelerated forward-backward method is used, which ensures convergence of the iterates, even when the Euler-Lagrange equation is degenerate and nondifferentiable. Several nonlinear examples are presented, including quasi-linear equations, degenerate and singular elliptic operators, discontinuous obstacles, and a nonlinear two-phase membrane problem.

• Lihui Chai, Carlos Garcia-Cervera, Xu Yang
Diffusion limit of the Boltzmann-Landau-Lifshitz-Gilbert system in ferromagnetic materials

In this paper, we continue the study initiated in our previous work on the semiclassical limit for the Schr0dinger-Poisson-Landau-Lifshitz-Gilbert system in [L. Chai, C. J. Garcia-Cervera and X. Yang, to appear in Arch. Rational Mech. Anal.]. Specifically, we consider the s-wave form spin dynamics coupled with the magnetization dynamics governed by the Landau-Lifshitz-Gilbert system, and rigorously obtain the diffusion limit of the coupled system.

Anisotropic Challenges in Pedestrian Flow Modeling

Macroscopic models of crowd flow incorporating the individual pedestrian choices present many analytic and computational challenges. Anisotropic interactions are particularly subtle, both in terms of describing the correct "optimal" direction field for the pedestrians and ensuring that this field is uniquely defined. We develop sufficient conditions, which establish a range of "safe" densities and parameter values for each model. We illustrate our approach by analyzing several established intra-crowd and inter-crowd models. For the two-crowd case, we also develop sufficient conditions for the uniqueness of Nash Equilibria in the resulting non-zero-sum game.

• Andrew Majda and Tong Xin
Rigorous accuracy and robustness analysis for two-scale reduced random Kalman filters in high dimensions

Contemporary data assimilation often involves millions of prediction variables. The classical Kalman filter is no longer computationally feasible in such a high dimensional context. This problem can often be resolved by exploiting the underlying multiscale structure, applying the full Kalman filtering procedures only to the large scale variables, and estimating the small scale variables with proper statistical strategies, including multiplicative inflation, representation model error in the observations, and crude localization. The resulting two-scale reduced filters can have close to optimal numerical filtering skill based on previous numerical evidence. Yet, no rigorous explanation exists for this success, because these modifications create unavoidable bias and model error. This paper contributes to this issue by establishing a new error analysis framework for two different reduced random Kalman filters, valid independent of the large dimension. The first part of our results examines the fidelity of the covariance estimators, which is essential for accurate uncertainty quantification. In a simplified setting, this is demonstrated by showing the true error covariance is dominated by its estimators. In general settings, the Mahalanobis error and its intrinsic dissipation can be used as simplified quantification of the same property. The second part develops upper bounds for the covariance estimators by comparing with proper Kalman filters. Combining both results, the classical tools for Kalman filters can be used as a-priori performance criteria for the reduced filters. In applications, these criteria guarantee the reduced filters are robust, and accurate for small noise systems. They also shed light on how to tune the reduced filters for stochastic turbulence.

• Yanxiang Zhao, Yanping Ma, Hui Sun, Bo Li, Qiang Du
A New Phase-field Approach to Variational Implicit Solvation of Charged Molecules with the Coulomb-field Approximation

We construct a new phase-field model for the solvation of charged molecules with a variational implicit solvent. Our phase-field free-energy functional includes the surface energy, solute-solvent van der Waals dispersion energy, and electrostatic interaction energy that is described by the Coulomb-field approximation, all coupled together self-consistently through a phase field. By introducing a new phase-field term in the description of the solute-solvent van der Waals and electrostatic interactions, we can keep the phase-field values closer to those describing the solute and solvent regions, respectively, making it more accurate in the free-energy estimate. We first prove that our phase-field functionals Gamma-converge to the corresponding sharp-interface limit. We then develop and implement an efficient and stable numerical method to solve the resulting gradient-flow equation to obtain equilibrium conformations and their associated free energies of the underlying charged molecular system. Our numerical method combines a linear splitting scheme, spectral discretization, and exponential time differencing Runge-Kutta approximations. Applications to the solvation of single ions and a two-plate system demonstrate that our new phase-field implementation improves the previous ones by achieving the localization of the system forces near the solute-solvent interface and maintaining more robustly the desirable hyperbolic tangent profile for even larger interfacial width. This work provides a scheme to resolve the possible unphysical feature of negative values in the phase-field function found in the previous phase-field modeling (cf. H. Sun et al. J. Chem. Phys., 2015) of charged molecules with the Poisson--Boltzmann equation for the electrostatic interaction.

• Sean Lawley
Blowup from randomly switching between stable boundary conditions for the heat equation

We find a pair of boundary conditions for the heat equation such that the solution goes to zero for either boundary condition, but if the boundary condition randomly switches, then the solution becomes unbounded in time. To our knowledge, this is the first PDE example showing that randomly switching between two globally asymptotically stable systems can produce a blowup. We devise several methods to analyze this random PDE. First, we use the method of lines to approximate the switching PDE by a large number of switching ODEs and then apply recent results to determine if they grow or decay in the limit of fast switching. We then use perturbation theory to obtain more detailed information on the switching PDE in this fast switching limit. To understand the case of finite switching rates, we characterize the parameter regimes in which the first and second moments of the random PDE grow or decay. This moment analysis reveals rich dynamical behavior, including a region of parameter space in which the mean of the random PDE oscillates with ever increasing amplitude for slow switching rates, grows exponentially for fast switching rates, but decays to zero for intermediate switching rates. We also highlight cases in which the second moment is necessary to understand the switching system's qualitative behavior, rather than just the mean. Finally, we give a PDE example in which randomly switching between two unstable systems produces a stable system. All of our analysis is accompanied by numerical simulation.

• Thi-Thao-Phuong Hoang, Lili Ju, Zhu Wang
Overlapping Localized Exponential Time Differencing Methods for Diffusion Problems

The localized exponential time differencing (ETD) based on overlapping domain decomposition has been recently introduced for extreme-scale phase field simulations of coarsening dynamics, which displays excellent parallel scalability in supercomputers. This paper serves as the first step toward building a solid mathematical foundation for this approach. We study the overlapping localized ETD schemes for a model time-dependent diffusion equation discretized in space by the standard central difference. Two methods are proposed and analyzed for solving the fully discrete localized ETD systems: the first one is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second one is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergences of the associated iterative solutions to the corresponding fully discrete localized ETD solution and to the exact semidiscrete solution are rigorously proved. Numerical experiments are also carried out to confirm theoretical results and to compare the performance of the two methods.

• Ekaterina Merkurjev, Andrea L. Bertozzi and Fan Chung
A semi-supervised heat kernel pagerank MBO algorithm for data classification

We present an accurate and efficient graph-based algorithm for semi-supervised classification that is motivated by recent successful threshold dynamics approaches and derived using heat kernel pagerank. Two different techniques are proposed to compute the pagerank, one of which proceeds by simulating random walks of bounded length. The algorithm produces accurate results even when the number of labeled nodes is very small, and avoids computationally expensive linear algebra routines. Moreover, the accuracy of the procedure is comparable with or better than that of state-of-the-art methods and is demonstrated on benchmark data sets. In addition to the main algorithm, a simple novel procedure that uses heat kernel pagerank directly as a classifier is outlined. We also provide detailed analysis, including information about the time complexity, of all proposed methods.

• Kresimir Burazin, Ivana Crnjac, Marko Vrdoljak
Variant of optimality criteria method for multiple state optimal design problems

We consider multiple state optimal design problems, aiming to find the best arrangement of two given isotropic materials, such that the obtained body has some optimal properties regarding $m$ different right-hand sides. Using the homogenization method as the relaxation tool, the standard variational techniques lead to necessary conditions of optimality. These conditions are the basis for the optimality criteria method, a commonly used numerical (iterative) method for optimal design problems. In Vrdoljak (2010) one variant of this method is presented, which is suitable for the energy maximization problems. We study another variant of the method, which works well for energy minimization problems. The explicit calculation of the design update is presented, which makes the implementation simple and similar to the case of single state equation. The method is tested on examples, showing that exact solutions are well approximated with the obtained numerical solutions.

• Yinghui Wang, Lei Yao
Time periodic solutions for a three-dimensional non-conservative compressible two-fluid model

In this paper, we consider the existence of time periodic solution to a non-conservative compressible two-fluid model with constant viscosity coefficients and unequal pressure functions $P^+\neq P^-$ in periodic domains of $\mathbb{R}^3.$ Based on the topological degree theory, we obtain the existence of the time periodic solution under some smallness assumptions.

• Qing Chen, Guochun Wu
The 3D compressible viscoelastic fluid in a bounded domain

In this paper, we prove the global existence and uniqueness of strong solution for the 3D compressible viscoelastic fluid in a bounded domain under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Based on the standard energy estimate, the estimation of the exponential convergence rates of the strong solution is also obtained.

• Yongyong Cai, Wenfan Yi
Error estimates of finite difference time domain methods for the Klein-Gordon-Dirac system in the nonrelativistic limit regime

In this paper, we establish error estimates of finite difference time domain (FDTD) methods for the Klein-Gordon-Dirac (KGD) system in the nonrelativistic limit regime, involving a small dimensionless parameter 0< \varepsilon\ll 1 inversely proportional to the speed of light. In this limit regime, the solution of the KGD system propagates waves with O(\varepsilon^2) and O(1)-wavelength in time and space respectively. The high oscillation and the nonlinear coupling between the real scalar Klein-Gordon field and the complex Dirac vector field bring great challenges to the analysis of the numerical methods for the KGD system in the nonrelativistic limit regime. Four implicit/semi-implicit/explicit FDTD methods are rigorously analyzed. By applying the energy method and cut-off technique, we obtain the error bounds for the FDTD methods at O(\tau^2/\varepsilon^6+h^2/\varepsilon) with time step \tau and mesh size h. Thus, in order to compute `correct' solutions when 0 < \varepsilon \ll 1, the estimates suggest that the meshing strategy requirement of the FDTD methods is \tau = O(\varepsilon^3) and h=O(\sqrt{\varepsilon}). In addition, numerical results are reported to support our conclusions. Our approach is valid in one, two and three dimensions.

• Pei Liu, Simo Wu, Chun Liu
Non-Isothermal Electrokinetics: Energetic Variational Approach

Fluid dynamics accompanies with the entropy production thus increases the local temperature, which plays an important role in charged systems such as the ion channel in biological environment and electrodiffusion in capacitors/batteries. In this article, we propose a general framework to derive the transport equations with heat flow through the Energetic Variational Approach. According to thermodynamic first law, the total energy is conserved and we can then use the Least Action Principle to derive the conservative forces. From thermodynamic second law, the entropy increases and the dissipative forces can be computed with the Maximum Dissipation Principle. Combining these two laws, we then conclude with the force balance equations and a temperature equation. To emphasis, our method provide a self consistent procedure to obtain dynamical equations satisfying proper energy laws and it not only works for the charge systems but also for general systems.

• Hongzhi Tong, Michael Ng
Regularized Semi-supervised Least Squares Regression with Dependent Samples

In this paper, we study regularized semi-supervised least squares regression with dependent samples. We analyze the regularized algorithm based on reproducing kernel Hilbert spaces, and show with the use of unlabelled data that the regularized least squares algorithm can achieve the minimax learning rate with a logarithmic term for dependent samples. Our new results are better than existing results in the literature.

• Yue Pu, Robert Pego, Denys Dutykh, Didier Clamond
Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations

We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh (Commun. Nonl. Sci. Numer. Simulat. 55 (2018) 237--247). This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.

• Shouming Zhou, Chunlai Mu, Li Yang
Global dissipative solutions of the Novikov equation

This paper is concerned with the continuation of solutions to the Novikov equation beyond wave breaking. By constructing a ''good'' set of new variables based on the characteristic, we transform the Novikov equation to a closed semilinear system on these new variables, which resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global dissipative solutions, which depends continuously on the initial data. Note that the characteristic of Novikov equation is $u^2$ whereas the characteristic of the CH equation is only $u$, otherwise, the cubic nonlinearity term $u_x^3$ in the Novikov equation can't be controlled by the $H^1$ energy, so, we need to chose the high-order energy density $(1+u_x^2)^2$ and another conservative law $\int_\mathbb{R}\left(u^4+2u^2u_x^2-\frac{1}{3}u_x^4\right)dx$.

• Xin Yang Lu
Analytical validation of a 2+1 dimensional continuum model for epitaxial growth with elastic substrate

We consider the evolution equation $$\label{abs1} h_t=\ddt[ \F^{-1}(-aE \F(h)) - r/h^2 -\ddt h ],$$ introduced in {\cite{TS}} by Tekalign and Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, $h$ denotes the surface height of the film, $\F$ is the Fourier transform, and $a$, $E$, $r$ are positive material constants. For simplicity, we set $aE=r=1$. As this equation does not have any particular structure, its analysis is quite challenging. Therefore, we introduce the auxiliary equation (with $c$ being a given constant) $$\label{abs2} u_t=\gr[ - \div u - (\div u+c)^{-2} -\ddt \div u ],$$ which has a variational structure. Equivalency between \eqref{abs1} and \eqref{abs2} will hold under sufficient regularity on the solution. The main aim of this paper is to provide an analytical validation to \eqref{abs2}, by proving existence and regularity properties for weak solutions, under suitable assumptions on the initial datum.

• Harbir Antil, Johannes Pfefferer, Sergejs Rogovs
Fractional Operators with Inhomogeneous Boundary Conditions: Analysis, Control, and Discretization

In this paper we introduce new characterizations of the spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We apply our definition to fractional elliptic equations of order $s \in (0,1)$ with nonzero Dirichlet and Neumann boundary condition. Here the domain $\Omega$ is assumed to be a bounded, quasi-convex Lipschitz domain. To impose the nonzero boundary conditions, we construct fractional harmonic extensions of the boundary data. It is shown that solving for the fractional harmonic extension is equivalent to solving for the standard harmonic extension in the very-weak form. The latter result is of independent interest as well. The remaining fractional elliptic problem (with homogeneous boundary data) can be realized using the existing techniques. We introduce finite element discretizations and derive discretization error estimates in natural norms, which are confirmed by numerical experiments. We also apply our characterizations to Dirichlet and Neumann boundary optimal control problems with fractional elliptic equation as constraints.

• Xiaoping Zhai, Zhimin Chen
Global well-posedness for $n$-dimensional Boussinesq system with viscosity depending on temperature

In this paper, we mainly study the global well-posedness issue for the Boussinesq system with the temperature-dependent viscosity in $\mathbb{R}^n(n\ge 2)$. With a temperature damping term, we first get a global solution in $\R^2$ provided the initial temperature is exponential small compared with the initial velocity field. Then using a weighted Chemin-Lerner type norm, we can also give a global large solution in $\mathbb{R}^n$ if the initial data satisfying a nonlinear smallness condition. In particular, our results imply the global large solutions without any smallness conditions imposed on the initial velocity.

• Yue Zhao, Guanghui Hu, Xiaokai Yuan
Direct and Inverse Elastic Scattering From a Locally Perturbed Half-plane

This paper is concerned with time-harmonic elastic scattering from a locally perturbed rough surface in two dimensions. We consider a rigid scattering interface given by the graph of a one-dimensional Lipschitz function which coincides with the real axis in the complement of some compact set. Given the incident field and the scattering interface, the direct problem is to determine the field distribution, whereas the inverse problem is to determine the shape of the interface from the measurement of the field on an artificial boundary in the upper half plane. We propose a symmetric coupling method between finite element and boundary integral equations to show uniqueness and existence of weak solutions. The synthetic data is computed via the finite element method with the Perfectly Matched Layer (PML) technique. To investigate the inverse problem, we derive the domain derivatives of the field with respect to the scattering interface. An iterative continuation method with multi-frequency data is used for recovering the unknown scattering interface.

• Wancheng Sheng, Qinglong Zhang
Interaction of the elementary waves of isentropic flow in a variable cross-section duct

The equations of fluid in a variable cross-section duct is nonconservative because of the source term. Recently, the Riemann solutions of the equations for the compressible duct flow have been obtained. The authors also obtained the elementary waves including rarefaction waves, shock waves and stationary wave. In this paper, we mainly discuss the interactions of rarefaction wave and shock wave with the stationary wave, in which the cross-section area is either decrease or increase. The large time behaviour is shown in each case.

• Rangrang Zhang, Guoli Zhou, Boling Guo
The asymptotic behavior of primitive equations with multiplicative noise

This paper is concerned with the existence of random attractor and the existence of the invariant measure for 3D stochastic primitive equations driven by linear multiplicative noise under $non$-$periodic$ $boundary$ $conditions$. The common method is to apply Sobolev imbedding theorem to obtain a compact absorbing ball in the solution space. However, this method fails because of the highly non-linearity and non-periodic boundary conditions of the stochastic primitive equations. To overcome the difficulties, we firstly use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to establish compactness of the solution operators. Then taking advantages of the continuous dependence on the initial data of the strong solution in an $good$ regular space we construct a compact absorbing ball. Finally the existence of the invariant measure under $non$-$periodic$ $boundary$ $conditions$ follows from the asymptotic compactness of the stochastic flow.

• Klaus Widmayer
Convergence to Stratified Flow for an Inviscid 3D Boussinesq system

We study the stability of special, stratified solutions of a 3d Boussinesq system describing an incompressible, inviscid 3d fluid with variable density (or temperature, depending on the context) under the effect of a uni-directional gravitational force. The behavior is shown to depend on the properties of an anisotropic dispersive operator with weak decay in time. However, the dispersive decay also depends on the strength of the gravity in the system and on the profile of the stratified solution, whose stability we study. We show that as the strength of the dispersion in the system tends to infinity, the 3d system of equations tends to a stratified system of 2d Euler equations with stratified density.

• Marc Josien
Some mathematical properties of the Weertman equation

We derive here some mathematical properties of the Weertman equation and show it is the limit of an evolution equation. The Weertman equation is a semilinear integrodifferential equation involving a fractional Laplacian. In addition to this purely theoretical interest, the results proven here give a solid ground to a numerical approach that we have implemented elsewhere.

Quantum Kac's Chaos

We study the notion of quantum Kac’s chaos which was implicitly introduced by Spohn and explicitly formulated by Gottlieb. We prove the analogue of a result of Sznitman which gives the equivalence of Kac’s chaos to 2-chaoticity and to convergence of empirical measures. Finally we give a simple, diﬀerent proof of a result of Spohn which states that chaos propagates with respect to certain Hamiltonians that deﬁne the evolution of the mean ﬁeld limit for interacting quantum systems.

• Bingkang Huang, Lan Zhang
A regularity criterion of strong solutions to the 2D cauchy problem of the kinetic-fluid model for flocking

In this paper, we consider the blow up criterion for the two dimensional kinetic-fluid model in the whole space. For particle and fluid dynamics, we employ the Cucker-Smale-Fokker- Planck model for the flocking particle part, and the isentropic compressible Navier-Stokes equations for the fluid part, and the separate systems are coupled through the drag force. We show that the strong solution exists globally if the L∞(0, T ; L∞) norm of the fluid density ρ(t, x) is bounded.

• Faker Ben Belgacem, Tarik Fahlaoui, Faten Jelassi, Maimouna Mint Brahim

The purpose is a finite element approximation of the diffusion problem in composite media, with non-linear contact resistance at the interfaces. As already explained in [Journal of Scientific Computing, {\bf 63}, 478-501 (2015)], hybrid dual formulations are well fitted to complicated composite geometries and provide tractable approaches to variationally express the jumps of the temperature. The finite element spaces are standard. Interface contributions are added to the variational problem to account for the contact resistance. This is an important advantage for computing codes developers. We undertake the analysis of the semi-linear diffusion problem for a large range of contact resistance and we investigate its discretization by hybrid dual finite element methods. Numerical experiments are presented to support the theoretical results.

• Gregory Eskin

In recent years a remarkable progress was made in the construction of spatial cloaks using the methods of transformation optics and metamaterials. The temporal cloaking, i.e. the cloaking of an event in spacetime, was also widely studied by using transformations on spacetime domains. We propose a simple and general method for the construction of temporal cloaking using the change of time variables only.

• Jinghua Yao

We rigorously show that a class of systems of partial differential equations (PDEs) modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics on the center manifold(s). This class of systems is related to the theory of hyberbolic conservation laws and supplies a new class of PDE examples for stationary $O(2)$-bifurcation. A direct consequence of our result is that the oscillations of the dynamics are \textit{not} due to rotation waves though the system exhibits Euclidean symmetries. The main difficulties of carrying out the program are: 1) the system under study contains multi bifurcation parameters and we do not know \textit{a priori} how they come into play in the bifurcation dynamics. 2) the representation of the linear operator on the center space is a $2\times 2$ zero matrix, which makes the characteristic condition in the well-known normal form theorem trivial. We overcome the first difficulty by using projection method. We managed to overcome the second subtle difficulty by using a conjugate pair coordinate for the center space and applying duality and projection arguments. Due to the specific complex pair parametrization, we could naturally obtain a form of the center manifold reduction function, which makes the study of the current dynamics on the center manifold possible. The symmetry of the system plays an essential role in excluding the possibility of bifurcating rotation waves.

• Seung-Yeal Ha, Jinyeong Park and Xiongtao Zhang
On the first-order reduction of the Cucker-Smale model and its clustering dynamics

We present a first-order reduction for the Cucker-Smale (C-S) model \textcolor{red}{on the real line}, and discuss its clustering dynamics in terms of spatial configurations an system parameters. In previous literature, flocking estimates for the C-S model were mainly focused on the relaxation dynamics of the particle's velocities toward the common velocity. In contrast, the relaxation dynamics of spatial configurations was treated as a secondary issue except for the uniform boundedness of the spatial diameter. In this paper, we first derive a first-order system for the spatial coordinate that can be rewritten as a gradient flow, and then use this first-order formulation to derive several sufficient conditions on the clustering dynamics based on the spatial positions depending on the natural velocities characterized by initial position-velocity configurations.

• Alex Lin, Yat Tin Chow, Stanley Osher
A Splitting Method For Overcoming the Curse of Dimensionality in Hamilton-Jacobi Equations Arising from Nonlinear Optimal Control and Differential Games with Applications to Trajectory Generation

Recent observations have been made that bridge splitting methods arising from optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with Hamiltonians H(p). This has produced extremely fast algorithms in computing solutions of these PDEs. More recent observations were made in generalizing the Hopf and Lax formulas to state-and-time-dependent cases H(x,p,t). In this article, we apply a new splitting method based on the Primal Dual Hybrid Gradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control and differential games problems, based on techniques from the derivation of the new Hopf and Lax formulas, which allow us to compute solutions at points (x,t) directly, i.e. without the use of grids in space. This algorithm also allows us to create trajectories directly. Thus we are able to lift the curse of dimensionality a bit, and therefore compute solutions in much higher dimensions than before. And in our numerical experiments, we actually observe that our computations scale polynomially in time. Furthermore, this new algorithm is embarrassingly parallelizable.

• Elena Beretta, Luca Ratti, Marco Verani
Detection of conductivity inclusions in a semilinear elliptic problem arising from cardiac electrophysiology

In this work we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic equation from the knowledge of the solution on the boundary of the planar bounded domain, an inverse problem motivated by an application in cardiac electrophysiology. We formulate a constraint minimization problem involving a quadratic mismatch functional enhanced with a regularization term which penalizes the perimeter of the inclusion to be identified. We introduce a phase-field relaxation of the problem, employing a Ginzburg-Landau-type energy and assessing the Γ-convergence of the relaxed functional to the original one. After computing the optimality conditions of the phase-field optimization problem and introducing a discrete Finite Element formulation, we propose an iterative algorithm and prove convergence properties. Several numerical results are reported, assessing the effectiveness and the robustness of the algorithm in identifying arbitrarily-shaped inclusions. Finally, we compare our approach to a shape derivative based technique, both from a theoretical point of view (computing the sharp interface limit of the optimality conditions) and from a numerical one.

• Guoli Zhou, Boling Guo
The global attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

Absorbing ball in $H^{1}(\mho)$ is obtained for the strong solution to the three dimensional viscous moist primitive equations under the natural assumption $Q_{1},Q_{2}\in L^{2}(\mho)$ which is weaker than the assumption $Q_{1},Q_{2}\in H^{1}(\mho)$ in previous work. In view of the structure of the manifold and the special geometry involved with vertical velocity, the continuity of the strong solution in $H^{1}(\mho)$ is established with respect to time and initial data. To obtain the existence of the global attractor for the moist primitive equations, the common method is to obtain the absorbing ball in $H^{2}(\mho)$ for the strong solution to the equations. But it is difficult due to the complex structure of the moist primitive equations. To overcome the difficulty, we try to use Aubin-Lions lemma and the continuous property of the strong solutions to the moist primitive equations to prove the the existence of the global attractor which improves the result, the existence of weak attractor obtained before.

• Cristina Pignotti, Emmanuel Trelat
Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays

We consider the celebrated Cucker-Smale model in finite dimension, modelling interacting collective dynamics and their possible evolution to consensus. The objective of this paper is to study the effect of time delays in the general model. By a Lyapunov functional approach, we provide convergence results to consensus for symmetric as well as nonsymmetric communication weights under some structural conditions.

• Jack Xin, Jiancheng Lyu, Yifeng Yu
Residual Diffusivity in Elephant Random Walk Models with Stops

We study the enhanced diffusivity in the so called elephant random walk model with stops (ERWS) by including symmetric random walk steps at small probability epsilon. At any epsilon> 0, the large time behavior transitions from sub-diffusive at epsilon = 0 to diffusive in a wedge shaped parameter regime where the diffsivity is strictly above that in the un-perturbed ERWS model in the epsilon - > 0 limit. The perturbed ERWS model is shown to be solvable with the first two moments and their asymptotics calculated exactly in both one and two space dimensions. The model provides a discrete analytical setting of the residual diffusion phenomenon known for the passive scalar transport in chaotic flows (e.g. generated by time periodic cellular flows and statistically sub-diffusive) as molecular diffusivity tends to zero.

• Duanmei Zhou, Guoliang Chen, Jiu Ding, Noah Rhee
A Quadratic Spline Least Squares Method for Computing Absolutely Continuous Invariant Measures

We develop a quadratic spline approximation method for the computation of absolutely continuous invariant measures of one dimensional mappings, based on the orthogonal projection of $L^2$ spaces. We prove the norm convergence of the numerical scheme and present the numerical experiments.

• Monika Eisenmann, Raphael Kruse
wo quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity

In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in $W^{\sigma,p}(0,T)$, $\sigma \in (0,2)$, $p \in [2,\infty)$. We introduce two quadrature rules: The first is best suited for the parameter range $\sigma \in (0,1)$ and consists of a Riemann--Maruyama approximation on a randomly shifted grid. The second quadrature rule considered in this paper applies to the case of a deterministic integrand of fractional Sobolev regularity with $\sigma \in (1,2)$. In both cases the order of convergence is equal to $\sigma$ with respect to the $L^p$-norm. As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical results are accompanied by numerical experiments.

• Andreas Hiltebrand, Sandra May
Entropy stable spacetime discontinuous Galerkin methods for the two-dimensional compressible Navier-Stokes equations

In this paper, we present entropy stable schemes for solving the compressible Navier-Stokes equations in two space dimensions. Our schemes use entropy variables as degrees of freedom. They are extensions of an existing spacetime discontinuous Galerkin method for solving the compressible Euler equations. The physical diffusion terms are incorporated by means of the symmetric (SIPG) or nonsymmetric (NIPG) interior penalty method, resulting in the two versions ST-SDSC-SIPG and ST-SDSC-NIPG. The streamline diffusion (SD) and shock-capturing (SC) terms from the original scheme have been kept, but have been adjusted appropriately. This guarantees that the new schemes essentially reduce to the original scheme for the compressible Euler equations in regions with underresolved physical diffusion. We show entropy stability for both versions under suitable assumptions. We also present numerical results confirming the accuracy and robustness of our schemes.

• Monika Eisenmann, Raphael Kruse
wo quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity

In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in $W^{\sigma,p}(0,T)$, $\sigma \in (0,2)$, $p \in [2,\infty)$. We introduce two quadrature rules: The first is best suited for the parameter range $\sigma \in (0,1)$ and consists of a Riemann--Maruyama approximation on a randomly shifted grid. The second quadrature rule considered in this paper applies to the case of a deterministic integrand of fractional Sobolev regularity with $\sigma \in (1,2)$. In both cases the order of convergence is equal to $\sigma$ with respect to the $L^p$-norm. As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical results are accompanied by numerical experiments.

• Jitao Liu, Shu Wang
Initial-boundary value problem for 2D incompressible micropolar equations without angular viscosity

This paper concerns the initial-boundary value problem for 2D incompressible micropolar equations without angular viscosity in a smooth bounded domain. It is shown that such a system admits a unique and global strong solution. The main contribution of this paper is to fully exploit the structure of this system and establish high order estimates via introducing an auxiliary field which is at the energy level of one order lower than micro-rotation.