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.

We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter $\eps$ and of the frequency $\lambda$ is in general not continuous at a point $(\eps,\lambda)=(0,\lambda^0)$ where trapped modes exist. In particular, we prove that for a given $\eps\ne0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $\lambda^0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

For the non cutoff radially symmetric homogeneous Boltzmann equation with Maxwellian molecules, we give the numerical solutions using symbolic manipulations and spectral decomposition of Hermit functions. The initial data can belong to some measure space.

In this paper, we study the mean field limit of weakly interacting particles with memory that are governed by a system of non-Markovian Langevin equations. Under the assumption of quasi-Markovianity (i.e. that the memory in the system can be described using a finite number of auxiliary processes), we pass to the mean field limit to obtain the corresponding McKean-Vlasov equation in an extended phase space. For the case of a quadratic confining potential and a quadratic (Curie-Weiss) interaction, we obtain the fundamental solution (Green's function). For nonconvex conﬁning potentials, we characterize the stationary state(s) of the McKean-Vlasov equation, and we show that the bifurcation diagram of the stationary problem is independent of the memory in the system. In addition, we show that the McKean-Vlasov equation for the non-Markovian dynamics can be written in the GENERIC formalism and we study convergence to equilibrium and the Markovian asymptotic limit.

We study a stochastic perturbation of the Nos\'{e}-Hoover equation (called the Nos\'{e}-Hoover equation under Brownian heating) and show that the dynamics converges at a geometric rate to the augmented Gibbs measure in a weighted total variation distance. The joint marginal distribution of the position and momentum of the particles in turn converges exponentially fast in a similar sense to the canonical Boltzmann-Gibbs distribution. The result applies to a general number of particles interacting through wide class of potential functions, including the usual polynomial type as well as the singular Lennard-Jones variety.

We study the Landau equation for a mixture of two species in the whole space, with initial condition of one species near a vacuum and the other near a Maxwellian equilibrium state. For the linearized level, without any smoothness assumption on the initial data, it is shown that the solution becomes smooth instantaneously in both the space and momentum variables. Moreover, the large time behavior of the solution is also obtained.

The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvature tensor is introduced from the minimization of the associated quadratic form. The M-eigenvalues of the Riemann curvature tensor always exist and are real. They are invariants of the Riemann curvature tensor. The associated quadratic form of the Riemann curvature tensor is always positive at a point if and only if the M-eigenvalues of the Riemann curvature tensor are all positive at that point. We investigate the M-eigenvalues for the simple cases, such as the 2D case, the 3D case, the constant curvature and the Schwarzschild solution, and all the calculated M-eigenvalues are related to the curvature invariants.

The phase retrieval problem has garnered significant attention since the development of the PhaseLift algorithm, which is a convex program that operates in a lifted space of matrices. Because of the substantial computational cost due to lifting, many approaches to phase retrieval have been developed, including non-convex optimization algorithms which operate in the natural parameter space, such as Wirtinger Flow. Recently, a convex formulation called PhaseMax has been discovered, and it has been proven to achieve phase retrieval via linear programming in the natural parameter space under optimal sample complexity. The original proofs of PhaseMax rely on statistical learning theory or geometric probability theory. Here, we present a short and elementary proof that PhaseMax exactly recovers real-valued vectors from random measurements under optimal sample complexity. Our proof only relies on standard probabilistic concentration and covering arguments, yielding a simpler and more direct proof than those that require statistical learning theory, geometric probability or the highly technical arguments for Wirtinger Flow-like approaches.

The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. These small scales influence the large scale behavior, hence their effects should be accurately modelled in a numerical simulation. A direct numerical simulation is prohibitively expensive since a minimum of two points per wavelength are needed to resolve the small scales. A multiscale method, under the equation free methodology, is proposed to approximate the coarse scale behaviour of the exact solution at a cost independent of the small scales in the problem. We prove convergence rates for the upscaled quantities in one as well as in multi-dimensional periodic settings. Moreover, numerical results in one and two dimensions are provided to support the theory.

In this paper, we prove the global existence of small solutions to partially damped Euler-Poisson ``two fluid " system in three dimensional periodic domain. Different from the previous ``two fluid" Euler-Poisson systems, our model describes two fluids obey different dynamical evolutions, one is compressible Euler and another is compressible Euler with damping. We use a carefully designed time-weighted energy frame to prove our theorem.

This paper deals with the analysis of an instationary drift-diffusion model for organic semiconductor devices including Gauss--Fermi statistics and application-specific mobility functions. The charge transport in organic materials is realized by hopping of carriers between adjacent energetic sites and is described by complicated mobility laws with a strong nonlinear dependence on temperature, carrier densities and the electric field strength.

To prove the existence of global weak solutions, we consider a problem with (for small densities) regularized state equations on any arbitrarily chosen finite time interval. We ensure its solvability by time discretization and passage to the time-continuous limit. Positive lower a priori estimates for the densities of its solutions that are independent of the regularization level ensure the existence of solutions to the original problem. Furthermore, we derive for these solutions global positive lower and upper bounds strictly below the density of transport states for the densities. The estimates rely on Moser iteration techniques.

We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. Well-posedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.

This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently sample the invariant distribution of the diffusion process, thanks to the approximation of the associated free energy function (relative to a reaction coordinate). The bias which is introduced in the dynamics is computed adaptively; it depends on the past of the trajectory of the process through some time-averages.

We give a detailed and general construction of such methods. We prove the consistency of the approach (almost sure convergence of well-chosen weighted empirical probability distribution). We justify the efficiency thanks to several qualitative and quantitative additional arguments. To prove these results , we revisit and extend tools from stochastic approximation applied to self-interacting diffusions, in an original context.

We consider a multi component gas mixture with translational and internal energy degrees of freedom assuming that the number of particles of each species remains constant. We will illustrate the derived model in the case of two species, but the model can be easily generalized to multiple species. The two species are allowed to have different degrees of freedom in internal energy and are modelled by a system of kinetic ES-BGK equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the temperature, H-theorem and convergence to a global equilibrium in the form of a global Maxwell distribution. Thus, we are able to derive the usual macroscopic conservation laws. For numerical purposes we apply the Chu reduction to the developed model for polyatomic gases and give an application for a gas consisting of a mono atomic and a diatomic species.

In this article, we study the dynamic transition of the Rayleigh-Benard Convection with internal heating and varying gravity. We show that this problem can only undergo a continuous or catastrophic transition, and the specific type is completely determined by the sign of a parameter -- referred to as the transition number -- that depends on the aspect ratio. Through numerical simulations we compute, for six qualitatively different heating sources, the corresponding value of the transition number, and find that the transition is always continuous. In particular, after transition, the system bifurcates from a basic steady state to a family of stable steady-states, homeomorphic to S^{1}, that describe the heating convection. Furthermore, upon varying the aspect ratio immediately after the first transition has occurred, we find the existence of a second transition, which is always catastrophic. More precisely, there exists a family of discrete values of the aspect ratio, which are the discontinuity points of the transition number, at which the transition is catastrophic and the number of convection rolls changes.

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.

This paper proposes a quantitative description of the low energy edge states at the interface between two-dimensional topological insulators. They are modeled by continuous Hamiltonians as systems of Dirac equations that are amenable to a large class of random perturbations. We consider general as well as fermionic time reversal symmetric models. In the former case, Hamiltonians are classified using the index of a Fredholm operator. In the latter case, the classification involves a Z2 index. These indices dictate the number of topologically protected edge states. A remarkable feature of topological insulators is the asymmetry (chirality) of the edge states, with more modes propagating, say, up than down. In some cases, backscattering off imperfections is prevented when no mode can carry signals backwards. This is a desirable feature from an engineering perspective, which raises the question of how back-scattering is protected topologically. A major motivation for the derivation of continuous models is to answer such a question. We quantify how backscattering is affected but not suppressed by the non- trivial topology by introducing a scattering problem along the edge and describ- ing the effects of topology and randomness on the scattering matrix. Explicit macroscopic models are then obtained within the diffusion approximation of field propagation to show the following: the combination of topology and randomness results in un-hindered transport of the protected modes while all other modes (Anderson) localize.

We present a method for determining optimal walking paths in steep terrain using the level set method and an optimal control formulation. By viewing the walking direction as a control variable, we can determine the optimal control by solving a Hamilton-Jacobi-Bellman equation. We then calculate the optimal walking path by solving an ordinary differential equation. We demonstrate the effectiveness of our method by computing optimal paths which travel throughout mountainous regions of Yosemite National Park. We include details regarding the numerical implementation of our model and address a specific application of a law enforcement agency patrolling a nationally protected area.

We adapt the improved duality estimates for bounded coefficients derived by Canizo et al. to the framework of cross diffusion. Since the estimates can not be directly applied we need to derive a time discrete version of their results and applied to an implicit semi-discretization in time of the cross diffusion systems. This leads to new global existence results for cross diffusion systems with bounded cross diffusion pressures and potentially superquadratic reaction.

It is well known that, for fast rotating fluids with the axis of rotation being perpendicular to the boundary, the boundary layer is of Ekman-type, described by a linear ODE system. In this paper we consider fast rotating fluids, with the axis of rotation being parallel to the boundary. We show that, for certain initial data with special asymptotic expansion, the corresponding boundary layer is described by a nonlinear, degenerated PDE system which is similar to the $2D$ Prandtl system. Finally, we prove the well-posedness of the governing system of the boundary layer in the space of analytic functions with respect to tangential variable.

In this paper we study a limit connecting a scaled wave map with the heat flow into the unit sphere $\mathbb{S}^2$. We show quantitatively how that the two equations are connected by means of an initial layer correction. This limit is motivated as a first step into understanding the limit of zero inertia for the hyperbolic-parabolic Ericksen-Leslie's liquid crystal model.

We study smooth, global-in-time solutions of the relativistic Vlasov-Maxwell system that possess arbitrarily large charge densities and electric fields. In particular, we construct spherically symmetric solutions that describe a thin shell of equally charged particles concentrating arbitrarily close to the origin and which give rise to charge densities and electric fields as large as one desires at some finite time. We show that these solutions exist even for arbitrarily small initial data or any desired mass. In the latter case, the time at which solutions concentrate can also be made arbitrarily large.

Let $u$ and $v$ be two $n$-dimensional complex vectors with $v^T u = 0$, so that the elementary matrix $A = I - uv^T$ is not diagonalizable. We find all solutions of the quadratic matrix equation $AXA=XAX$. This is a continuation of the work ({\it Computers Math. Appl.} 72(6), 1541-1548, 2016) from the case of diagonalizable elementary matrices to non-diagonalizable ones.

In this paper, a full hydrodynamic semiconductor model with a time periodic external force is concerned. First, we regularize the system under consideration and prove the existence of time periodic solutions to the linearized approximate system by applying Tychonoff fixed point theorem combined with the energy method and the decay estimates. This idea is from the Massera-type criteria for linear periodic evolution equations. Then, the existence of a strong time periodic solution under some smallness assumptions is established by using the topological degree theory and an approximation scheme. The uniqueness of time periodic solutions is proved basing on the energy estimates. Also, the existence of the stationary solution is obtained.

In this paper we perform a formal asymptotic analysis on a kinetic model for reactive mixtures in order to derive a reaction-diffusion system of Maxwell-Stefan type. More specifically, we start from the kinetic model of simple reacting spheres for a quaternary mixture of monatomic ideal gases that undergoes a reversible chemical reaction of bimolecular type. Then, we consider a scaling describing a physical situation in which mechanical collisions play a dominant role in the evolution process, while chemical reactions are slow, and compute explicitly the production terms associated to the concentration and momentum balance equations for each species in the reactive mixture. Finally, we prove that, under isothermal assumptions, the limit equations for the scaled kinetic model is the reaction diffusion system of Maxwell-Stefan type.

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.

Energy transport equations are derived directly from full a many-particle description as a coarse-grained description. This effort is motivated by the observation that the conventional heat equation is unable to describe the heat conduction process at the nano-mechanical scale. With the local energy chosen as the coarse-grained variables, we apply the Mori-Zwanzig formalism to derive a reduced model, in the form of a generalized Langevin equation. A Markovian embedding technique is then employed to eliminate the history dependence, by introducing auxiliary variables. The auxiliary equations govern the dynamics of the energy flux. In sharp contrast to conventional energy transport models, this derivation yields stochastic dynamical models for the spatially averaged energy. The random force in the generalized Langevin equation is typically modeled by additive white noise. As an initial attempt, we consider multiplicative noises, to ensure the correct statistics of the solution.

This paper concerns the convergence of an iterative scheme for 2D stochastic primitive equations on a bounded domain. The stochastic system is split into two equations: a deterministic 2D primitive equations with random initial value and a linear stochastic parabolic equation, which are both simpler for numerical computations. An estimate of approximation error is given, which implies that the strong speed rate of the convergence in probability is almost 1/2.

We study asymptotically and numerically the fundamental gap -- the difference between the first two smallest (and distinct) eigenvalues -- of the fractional Schrodinger operator (FSO) and formulate a gap conjecture on the fundamental gap of the FSO. We begin with an introduction of the FSO on bounded domains with homogeneous Dirichlet boundary conditions, while the fractional Laplacian operator defined either via the local fractional Laplacian (i.e. via the eigenfunctions decomposition of the Laplacian operator) or via the classical fractional Laplacian (i.e. zero extension of the eigenfunctions outside the bounded domains and then via the Fourier transform). For the FSO on bounded domains with either the local fractional Laplacian or the classical fractional Laplacian, we obtain the fundamental gap of the FSO analytically on simple geometry without potential and numerically on complicated geometries and/or with different convex potentials. Based on the asymptotic and extensive numerical results, a gap conjecture on the fundamental gap of the FSO is formulated. Surprisingly, for two and higher dimensions, the lower bound of the fundamental gap depends not only on the diameter of the domain, but also the diameter of the largest inscribed ball of the domain, which is completely different from the case of the Schrodinger operator. Extensions of these results for the FSO in the whole space and on bounded domains with periodic boundary conditions are presented.

We provide an asymptotic analysis of a non-local Fisher-KPP type equation in periodic media and with a non-local stable operator of order\alpha \in (0,2). We perform a long time-long range scaling in order to prove that the stable state invades the unstable state with a speed which is exponential in time.

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.