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.

Markov jump processes are widely used to model natural and engineered processes. In the context of biological or chemical applications one typically refers to the chemical master equation (CME), which models the evolution of the probability mass of any copy-number combination of the interacting particles. When many interacting particles ("species") are considered, the complexity of the CME quickly increases, making direct numerical simulations impossible. This is even more problematic when one aims at controlling the Markov jump processes defined by the CME. In this work, we study both open loop and feedback optimal control problems of the Markov jump processes in the case that the controls can only be switched at fixed control stages. Based on Kurtz's limit theorems, we prove the convergence of the respective control value functions of the underlying Markov decision problem as the copy numbers of the species go to infinity. In the case of the optimal control problem on a finite time-horizon, we propose a hybrid control policy algorithm to overcome the difficulties due to the curse of dimensionality when the copy number of the involved species is large. Two numerical examples demonstrate the suitability of both the analysis and the proposed algorithms.

We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the tendency to concentrate dominates the strength of diffusion and solutions emanating from sufficiently localised initial data may explode in finite time. The main purpose of this paper is to show that under suitable spectral conditions on the flow, which guarantee good mixing properties, for any regular initial datum the solution to the corresponding advection-aggregation-diffusion equation is global if the prescribed flow is sufficiently fast. This paper can be seen as a partial extension of Kiselev and Xu (Arch. Rat. Mech. Anal. 222(2), 2016), and our arguments show in particular that the suppression mechanism for the classical 2D parabolic-elliptic Keller-Segel model devised by Kiselev and Xu also applies to the fractional Keller-Segel model (where $\triangle$ is replaced by $-(-\triangle)^\frac{\gamma}{2}$) requiring only that $\gamma >1$. In addition, we remove the restriction to dimension $d< 4$. As a by-product, a characterisation of the class of relaxation enhancing flows on the $d$-torus is extended to the case of fractional dissipation.

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.

In this paper, we study the quasi-neutral limit of the full quantum Navier-Stokes-Maxwell equation as the Debye length tends to zero. We justify rigorously the quasi-neutral limit by establishing rigorous uniform estimates on the error functions with respect to the Debye length and by using the formal asymptotic expansion and singular perturbation methods combined with curl-div decomposition of the gradient. The key difficulty is to deal with the quantum effect, which do play important roles in establishing a priori estimates.

We address the far field regularity for solutions of the surface quasi-geostrophic equation \begin{align*} &\theta_t+u\cdot\nabla\theta+\Lambda^{2\alpha}\theta=0 \\ &u=\RR^{\perp}\theta=(-\RR_2\theta,\RR_1\theta), \end{align*} in the supercritical range $0< \alpha< 1/2$ with $\alpha$ sufficiently close to $1/2$. We prove that if the datum is sufficiently regular, then the set of space-time singularities is compact in ${\mathbb R}^2\times{\mathbb R}$. The proof depends on a new spatial decay result on solutions in the supercritical range.

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.

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.

In this paper, we will introduce a mathematical model of jerk equation to simulate the unstable oscillations of the motion of a falling sphere in the wormlike micellar solution.This differential/algebraic equation (DAE) is established only by learning the experimental data of time vs velocity with the sparse optimization method. To simulate the solutions of the DAE, four discretization schemes are proposed and compared. Periodic and damped harmonic motion,and nonuniform transient and sustaining oscillations can be observed for the sedimentation of a sphere through the non-Newtonian fluid in the numerical experiments.

In this paper, we provide a complete description of the selected spreading speed of systems of reaction- diffusion equations with unilateral coupling and prove the existence of anomalous spreading speeds for systems with monostable nonlinearities. Our work extends known results for systems with linear and quadratic couplings, and Fisher-KPP type nonlinearities. Our proofs rely on the construction of appropriate sub- and super-solutions.

We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.

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.

Dislocations are the main carriers of the permanent deformation of crystals. For simulations of engineering applications, continuum models where material microstructures are represented by continuous density distributions of dislocations are preferred. It is challenging to capture in the continuum model the short-range dislocation interactions, which vanish after the standard averaging procedure from discrete dislocation models. In this study, we consider systems of parallel straight dislocation walls and develop continuum descriptions for the short-range interactions of dislocations by using asymptotic analysis. The obtained continuum short-range interaction formulas are incorporated in the continuum model for dislocation dynamics based on a pair of dislocation density potential functions that represent continuous distributions of dislocations. This derived continuum model is able to describe the anisotropic dislocation interaction and motion. Mathematically, these short-range interaction terms ensure strong stability property of the continuum model that is possessed by the discrete dislocation dynamics model. The derived continuum model is validated by comparisons with the discrete dislocation dynamical simulation results.

In this work we prove local and global well-posedness of the Cauchy problem of the regularized intermediate long-wave (rILW) equation in periodic and nonperiodic Sobolev spaces.

We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker--Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached.

We consider the Helmholtz equation with a complex attenuation coefficient on a bounded, strictly convex domain in R^d. We prove a Holder conditional stability estimate for identifying attenuation coefficients from phaseless boundary value measurements, when the initial excitation state is in the form of a Gaussian bump. We use the Gaussian beam Ansatz and stability results for the X-ray transform on strictly convex domains to establish these estimates.

We study the global in time classical solutions to the two-fluid incompressible Navier-Stokes-Maxwell system with (solenoidal) Ohm's law with small initial data. This system is a coupling of the incompressible Navier-Stokes equations with the Maxwell equations through the Lorenz force and Ohm's law for the current. Comparing to the previous results, we employ the decay properties of both the electric field $E$ and the wave equation with linear damping of the divergence free magnetic field $B$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 satisfies a quasistatic diffusion 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.