In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs. The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erd\H{o}s-R{\' e}nyi, small-world, and power law graphs.

We consider a layer of an incompressible inviscid fluid, bounded below by a fixed general bottom and above by a free moving boundary, in a horizontally periodic setting. The fluid dynamics is governed by the gravity-driven incompressible Euler equations with damping, and the effect of surface tension is included on the free surface. We prove that the problem is globally well-posed for the small initial data; moreover, the solution decays exponentially to the equilibrium.

This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effctive quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.

By analyzing the geometric characteristic of related algebra equations, we first find the sharp energy criteria of singular solutions and global solutions for the Davey-Stewartson systems. Then, we study the limiting behavior of singular solutions, and obtain the rate of singular solutions, rate of concentration of singular solutions for Davey-Stewartson systems with the $L^2$ super-critical nonlinearity.

The Kreiss symmetrizer technique gives sharp estimates of the solutions of the first order hyperbolic initial-boundary value problems both in the interior and at the boundary of the domain. Such estimates imply both robustness and strong well-posedness in the generalized sense, and the corresponding problems are called strongly boundary stable. There are however problems that are not strongly boundary stable and yet are well-posed and robust. For such problems sharp estimates of the solution can be obtained only in the interior and not at the boundary. Examples include boundary-type wave phenomena, such as surface waves and glancing waves, that often occur in elastodynamics and electromagnetism, described by elastic wave equations and Maxwell's equations, respectively. We consider such class of hyperbolic problems, chracterized by generalized boundary-type eigenmodes, and refer to them as well-posed in the generalized sense. Using the Laplace-Fourier technique and through the construction of smooth symmetrizers, we show that the Kreiss theory can indeed be extended to this class of problems. In particular, we obtain sufficient algebraic conditions for well-posedness in the generalized sense.

We consider a non-stationary Stokes-Nernst-Planck-Poisson system posed in perforated domains. Our aim is to justify rigorously the homogenization limit for the upscaled system derived by means of two-scale convergence in \cite{RMK12}. In other words, we wish to obtain the so-called corrector homogenization estimates that specify the error obtained when upscaling the microscopic equations. Essentially, we control in terms of suitable norms differences between the micro- and macro-concentrations and between the corresponding micro- and macro-concentration gradients. The major challenges that we face are the coupled flux structure of the system, the nonlinear drift terms and the presence of the microstructures. Employing various energy-like estimates, we discuss several scalings choices and boundary conditions.

We consider hydrodynamic scaling limits for a class of reversible interacting particle systems, which includes the symmetric simple exclusion process and certain zero-range processes. We study a (non-quadratic) microscopic action functional for these systems. We analyse the behaviour of this functional in the hydrodynamic limit and we establish conditions under which it converges to the (quadratic) action functional of Macroscopic Fluctuation Theory. We discuss the implications of these results for rigorous analysis of hydrodynamic limits.

Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Lévy density of a Lévy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical results in challenging simulation examples. On the theoretical side, we establish that our nonparametric Bayesian procedure is consistent: in the low frequency data setting, with equispaced in time observations and intervals between successive observations remaining fixed, the posterior asymptotically, as the sample size n \to \infty, concentrates around the Lévy density under which the data have been generated. Finally, we test our method on a classical insurance dataset.

Nonlinear electromagneto-elasticity with moving boundary is presented in this work. We introduce a diffeomorphism operator that transforms the moving boundary system into an equivalent one with fixed boundary and we apply the Galerkin's method and results of compactness for obtain the existence and uniqueness of solution.

n this paper we study a crystal surface model first proposed by H.~Al Hajj Shehadeh, R.V.~Kohn and J.~Weare (2011 Physica D, 240,1771-1784). By seeking a solution of a particular function form, we are led to a boundary value problem for a fourth-order nonlinear elliptic equation. The mathematical challenge of the problem is due to the fact that the degeneracy in the equation is directly imposed by one of the two boundary conditions. An existence theorem is established in which a meaningful mathematical interpretation of the other boundary condition can not be found. Our investigations seems to suggest this to be an inherent property of the problem.

This article is focused on the bound estimate and convergence analysis of an unconditionally energy stable scheme for the MMC-TDGL equation, a Cahn-Hilliard equation with a Flory-Huggins-deGennes energy. The numerical scheme, a finite difference algorithm based on a convex splitting technique of the energy functional, was proposed in [Sci. China Math. 59(2016),1815]. We provide a theoretical justification of the unique solvability for the proposed numerical scheme, in which a well-known difficulty associated with the singular nature of the logarithmic energy potential has to be handled. Meanwhile, a careful analysis reveals that, such a singular nature prevents the numerical solution of the phase variable reaching the limit singular values, so that the positivity preserving property could be proved at a theoretical level. In particular, the natural structure of the deGennes diffusive coefficient also ensures the desired positivity-preserving property. In turn, the unconditional energy stability becomes an outcome of the unique solvability and the convex-concave decomposition for the energy functional. Moreover, an optimal rate convergence analysis is presented in the $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm, in which the the convexity of nonlinear energy potential has played an essential role. In addition, a rewritten form of the surface diffusion term has facilitated the convergence analysis, in which we have made use of the special structure of concentration-dependent deGennes type coefficient.

In this paper, we consider numerical approximations of a binary fluid-surfactant phase-field model coupled with the fluid flow, in which the system consists the incompressible Navier-Stokes equations and two Cahn-Hilliard type equations. We develop two, linear and second order time marching schemes for solving this system, by combining the ``Invariant Energy Quadratization" approach for the nonlinear potentials, the projection method for the Navier-Stokes equation, and a subtle implicit-explicit treatment for the stress and convective terms. We prove the well-posedness of the linear system and its unconditional energy stability rigorously. Various 2D and 3D numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.

In this paper, we give an integral approximation for the elliptic operators with anisotropic coefficients on smooth manifold. Using the integral approximation, the elliptic equation is transformed to an integral equation. The integral approximation preserves the symmetry and coercivity of the original elliptic operator. Based on these good properties, we prove the convergence between the solutions of the integral equation and the original elliptic equation.

We define the space of functions of bounded variation ($BV$) on the graph. Using the notion of divergence of flows on graphs, we show that the unit ball of the dual space to $BV$ in the graph setting can be described as the image of the unit ball of the space $\ell^{\infty}$ by the divergence operator. Based on this result, we propose a new algorithm to find the exact minimizer for the total variation (TV) denoising problem on the graph. The algorithm is provable convergent and its performance on image denoising test examples is illustrated. Furthermore, we review two well known methods for solving the TV denoising problem, Split Bregman and Primal-Dual splitting, and compare them with the proposed algorithm. We show highly competitive empirical convergence rates and visual quality.

We consider the quadratic Schrödinger system iu_t+\Delta_{\gamma_1}u+\bar{u}v=0\\ 2iv_t+\Delta_{\gamma_2}v-\beta v+\frac 12 u^2=0,& t\in\er,\,x\in \er^d\times \er, in dimensions $1\leq d\leq 4$ and for $\gamma_1,\gamma_2 >0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.

We study the existence and non-existence of a class of regional fractional equations arising in optics.

We introduce the use of conservation laws in multi-player consensus games. Indeed, a general non anticipative strategy is proposed within a rigorous analytic framework. By means of numerical integrations, we describe peculiar features of this strategy, such as its effectiveness, the automatic formation of coalitions and the effects of competitions.

We consider a 1-dimensional Brownian motion whose diffusion coefficient varies when it crosses the origin. We study the long time behavior and we establish different regimes, depending on the variations of the diffusion coefficient: emergence of a non-Gaussian multipeaked probability distribution and a dynamical transition to an absorbing static state. We compute the generator and we study the partial differential equation which involves its adjoint. We discuss global existence and blow-up of the solution to this latter equation.

We are interested in the derivation of optimality conditions for controlled interacting agent systems. We establish the relation between mean field optimality conditions and the optimality condition of the mean field control problem. This link is important for many recently published articles on control strategies for agent based systems since it establishes the precise relation between multipliers for the individual agents and the probability density distribution of the multipliers in the mean field limit. The relation to different notions of differentiability are also shown.

In this paper, we consider the Cauchy problem of the compressible magneto--micropolar fluids system in $\mathbb{R}^3$ with initial data close to some constant steady state. Based on the spectral analysis on the semigroup generated by the linearized equations and the nonlinear energy estimates, we show that the solution of the magneto--micropolar fluids system converges to its constant equilibrium state at the exact same $L^2$--decay rate as the linearized equations, which shows that the convergence rate is optimal.

This paper is devoted to the study of the construction of a viscous approximation of the nonconservative bitemperature Euler system. Starting from a BGK model coupled with Ampère and Poisson equations proposed in a previous article, we perform a Chapman-Enskog expansion up to order 1 leading to a Navier-Stokes system. Next, we prove that this system is compatible with the entropy of the bitemperature Euler system.

We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to pre- scribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle.

In this study, we aim to describe the first dynamic transitions of the MHD equations in a thin spherical shell. It is well known that the MHD equations admit a motionless steady state solution with constant vertically aligned magnetic field and linearly conducted temperature. This basic solution is stable for small Rayleigh numbers R and loses its stability at a critical threshold R$_c$. There are two possible sources for this instability. Either a set of real eigenvalues or a set of non-real eigenvalues cross the imaginary axis at R$_c$. We restrict ourselves to the study of the first case. In this case, by the center manifold reduction, we reduce the full PDE to a system of $2l_c+1$ ODE's where $l_c$ is a positive integer. We exhibit the most general reduction equation regardless of $l_c$. Then, it is shown that for $l_c=1,2$, the system either exhibits a continuous transition accompanied by an attractor homeomorphic to $2l_c$ dimensional sphere or a drastic transition accompanied by a repeller bifurcated on R$< $R$_c$. We show that there are parameter regimes where both types of transitions are realized. Besides, several identities involving the triple products of gradients of spherical harmonics are derived, which are useful for the study of related problems.

We consider the asymptotic limit of a diffuse interface model for tumor-growth when a parameter $\varepsilon$ proportional to the thickness of the diffuse interface goes to zero. An approximate solution which shows explicitly the behavior of the true solution for small $\varepsilon$ will be constructed by using matched expansion method. Based on the energy method, a spectral condition in particular, we establish a smallness estimate of the difference between the approximate solution and the true solution.

This paper focuses on the general $d$-dimensional ($d\ge 2$) Boussinesq equations with the fractional dissipation $(-\Delta)^\alpha u$ and without thermal diffusion. Our primary goal here is the uniqueness of weak solutions to this partially dissipated system in a weakest possible setting. The issue on the uniqueness of weak solutions is very important and can be quite difficult as in the case of the Leray-Hopf weak solutions to the 3D Navier-Stokes equations. We present two main results. The first is the global existence and uniqueness of weak solutions which assesses the global existence of $L^2$-weak solutions for any $\alpha >0$ and the uniqueness of the weak solutions when $\alpha\ge \frac12 + \frac{d}{4}$ for $d\ge 2$. Especially the 2D Boussinesq equations without thermal diffusion have a unique and global $L^2$ weak solutions. The second result establishes the zero thermal diffusion limit with an explicit convergence rate for the aforementioned weak solutions. This convergence result appears to be the very first one on weak solutions of partially dissipated Boussinesq systems.

Consideration herein is the inviscid limit of the 3-D incompressible axisymmetric Navier-Stokes-Boussinesq system with partial viscosity. We obtain uniform estimates of the solutions of this system with respect to the viscosity. We then provide a strong convergence result in the $H^{s-2}$ norm of the viscous solutions of this Navier-Stokes Boussinesq system to the one of Euler-Boussinesq equations.

This article studies the aggregation diffusion equation \[ \partial_t\rho = \Delta^\frac{\alpha}{2} \rho + \lambda\,\mathrm{div}((K*\rho)\rho), \] where $\Delta^\frac{\alpha}{2}$ denotes the fractional Laplacian and $K = \frac{x}{|x|^a}$ is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modelling of the motion of cells. In the \textit{diffusion dominated} case $a < \alpha$, we prove global well-posedness for an $L^1_k$ initial condition, and in the \textit{fair competition} case $a = \alpha$ for an $L^1_k\cap L\ln L$ initial condition. In the \textit{aggregation dominated} case $a > \alpha$, we prove global or local well-posedness for an $L^p$ initial condition, depending on some smallness condition on the $L^p$ norm of the initial condition. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria.

The structure of the electric double layer has long been described by the classical Poisson--Boltzmann (PB) theory, in which a uniform dielectric coefficient is often assumed. Experimental data and molecular simulations evidence that the effective dielectric coefficient decreases as local ionic concentrations. In this work, we develop a modified PB theory with concentration-dependent dielectrics that are described by the Bruggeman equation, which defines an implicit function of the effective dielectric coefficient on both the size and dielectric coefficients of hydrated ions. The Bruggeman equation, which takes into account contributions from both counterions and coions systematically, provides a closure to the modified PB theory, making the theory self-consistent. In addition to ionic size and valence, our theory introduces another source of ion-specificity, i.e., the dielectric coefficient of hydrated ions, to the continuum modeling of electrostatics. Asymptotic analysis reveals the connection between the modified PB theory and previous linear decrement models, and derives a criterion for the presence of counterion saturation. Robust numerical methods with efficient acceleration techniques are proposed to solve the resulting coupled equations. Dielectric coefficients predicted by our theory show good agreement with the experimental data for certain electrolytes. The dielectric decrement effect on the ionic structure of electric double layers is assessed in extensive numerical simulations. With ion-specific parameters, our theory predicts asymmetric camel-shape profiles of differential capacitance against applied potentials for electrolytes with low salinity, and asymmetric bell-shape profiles for electrolytes with high salinity. The impact of counterion saturation, arising from the steric effects and dielectric decrement, on the shape of differential capacitance profiles is demonstrated through analysis and numerical investigations. To further understand the effect of concentration-dependent dielectrics, the modified PB theory is also applied to study the distribution of counterions around charged cylinders with various dielectric coefficients.

In this paper, an asymptotic traveling wave of a free boundary problem related to a pricing model for corporate bond with multiple credit rating migration risk is studied. The pricing model is captured by a free boundary problem and the existence, uniqueness and regularity of the solution are obtained such that the rationality of the model guaranteed. The existence of the unique traveling wave in the free boundary problem is established with some condition of the risk discount rate satisfied. The inductive method is applied to overcome the multiplicity of free boundaries. We prove that the solution of the pricing model for corporate bond is convergent to the traveling wave, which shows a clear dynamics of price change for the corporate bond.

Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and another is nonstandard diffusion including the use of fractional derivatives. A major obstacle for application of these methods is the high computational cost from the numerical implementation of the nonlocal operators. It is natural to consider fast methods of fast multipole or hierarchical matrix type to overcome this challenge. Unfortunately the relevant kernels do not satisfy the standard necessary conditions. In this work a new class of fast algorithms is developed and analyzed, which is some cases reduces the computational complexity of applying nonlocal operators to essentially the same order of magnitude as the complexity of standard local numerical methods.

Consider the scattering of an acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous medium. This paper concerns the transient analysis of such a coupled acoustic-elastic wave propagation problem. A compressed coordinate transformation is proposed to reduce equivalently the scattering problem into an initial-boundary value problem in a bounded domain over a finite time interval. The reduced problem is shown to have a unique weak solution by using the Galerkin method. The stability estimate and an a priori estimate with explicit time dependence are obtained for the weak solution. The proposed method and the reduced model problem are useful for numerical simulations.

In this paper, we prove the existence of weak solutions to the steady two-phase flow. The result holds in a bounded domain $\om\in \mathbb{R}^3$ on the condition that the adiabatic constants $\gamma,\theta >1$ and $\gamma >\frac{7}{3}$, $\theta= 1$. By constructing a special example, we show that the weak solutions are non-unique. It turns out that the uniform approximation scheme restricts the type of weak solutions, which leads to some open problems.

We are concerned with an inverse problem associated with the fractional Helmholtz system that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We are particularly interested in the case that both the medium parameter and the internal source of the wave equation are unknown. Moreover, we consider a general class of source functions which can be frequency-dependent. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.

We prove that the famous diffusive Brusselator model can support more complicated spatial-temporal wave structure than the usual temporal-oscillation from a standard Hopf bifurcation. In our current investigation, we discover that the diffusion term in the model is neither a usual parabolic stabilizer nor a destabilizer as in the Turing instability of uniform state, but rather plays the role of maintaining an equivariant Hopf bifurcation spectral mechanism. At the same time, we show that such a mechanism can occur around any nonzero wave number and this finding is also different from the former works where oscillations caused by diffusion can cause the growth of wave structure only at a particular wavelength. Our analysis also demonstrates that the complicated spatial-temporal oscillation is not solely driven by the inhomogeneity of the reactants.

Bending vibrations of thin beams and plates may be described by nonlinear Euler-Bernoulli beam equations with $x$-dependent coefficients. In this paper we investigate existence of families of time-periodic solutions to such a model using Lyapunov-Schmidt reduction and a differentiable Nash-Moser iteration scheme. The results hold for all parameters $(\epsilon,\omega)$ in a Cantor set with asymptotically full measure as $\epsilon\rightarrow 0$.

We propose an efficient algorithm to analyze $3D$ crystal structure at the individual particle level based on a fast $3D$ synchrosqueezed wave packet transform. The proposed algorithm can automatically extract microscopic information from $3D$ atomic/particle resolution crystal images, e.g., crystal orientation, defects, and deformation, which are important information for characterizing material properties as well as potentially understanding the underlying formation processes. The effectiveness of our algorithms is illustrated by experiments of synthetic datasets and real $3$D microscopic colloidal images.

In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, for a canonical ensemble, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.

We prove existence of solutions to an anisotropic Cahn-Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate auxiliary results which play an essential part in the present existence proof. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1.